The Motion of Point Particles in Curved Spacetime, Eric Poisson, published in LivingReviews in Relativity on 2004-05-27

In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.

The field's action on the particle is difficult to calculate because of its singular nature: The field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle – its only effect is to contribute to the particle's inertia. What remains after subtraction is a smooth field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free (radiative) field that interacts with the particle; it is this interaction that gives rise to the self-force.

The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch.

The review begins with a discussion of the basic theory of bitensors (Section 2 ). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line (Section 3 ). It continues with a thorough discussion of Green's functions in curved spacetime (Section 4 ).

The review concludes with a detailed derivation of each of the three equations of motion (Section 5 ).

1 Introduction and Summary

1.1 Invitation

The motion of a point electric charge in flat spacetime was the subject of active investigation since the early work of Lorentz, Abrahams, and Poincaré, until Dirac [25] produced a proper relativistic derivation of the equations of motion in 1938. (The field's early history is well related in [52] .) In 1960 DeWitt and Brehme [24] generalized Dirac's result to curved spacetimes, and their calculation was corrected by Hobbs [29] several years later. In 1997 the motion of a point mass in a curved background spacetime was investigated by Mino, Sasaki, and Tanaka [39] , who derived an expression for the particle's acceleration (which is not zero unless the particle is a test mass); the same equations of motion were later obtained by Quinn and Wald [49] using an axiomatic approach.

The case of a point scalar charge was finally considered by Quinn in 2000 [48] , and this led to the realization that the mass of a scalar particle is not necessarily a constant of the motion.

This article reviews the achievements described in the preceding paragraph; it is concerned with the motion of a point scalar charge

q

, a point electric charge

e

, and a point mass

m

in a specified background spacetime with metric

g_{α β}

. These particles carry with them fields that behave as outgoing radiation in the wave zone. The radiation removes energy and angular momentum from the particle, which then undergoes a radiation reaction – its world line cannot be simply a geodesic of the background spacetime. The particle's motion is affected by the near-zone field which acts directly on the particle and produces a self-force. In curved spacetime the self-force contains a radiation-reaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angular-momentum transport.

The self-force is proportional to

q^{2}

in the case of a scalar charge, proportional to

e^{2}

in the case of an electric charge, and proportional to

m^{2}

in the case of a point mass.

In this review I derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely self-contained, and all relevant materials are developed ab initio. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (Section 2 ), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle's world line (Section 3 ). One must next formulate a theory of Green's functions in curved spacetimes (Section 4 ), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Section 5 ). The review is very long, but the payoff, I hope, will be commensurate.

In this introductory section I set the stage and present an impressionistic survey of what the review contains. This should help the reader get oriented and acquainted with some of the ideas and some of the notation. Enjoy!

1.2 Radiation reaction in flat spacetime

Let us first consider the relatively simple and well-understood case of a point electric charge

e

moving in flat spacetime [52, 30, 56, 47] . The charge produces an electromagnetic vector potential

A^{α}

that satisfies the wave equation

\begin{matrix} □ A^{α} = - 4 π j^{α} \end{matrix}

(1)

together with the Lorenz gauge condition

\partial_{α} A^{α} = 0

. (On page 294 in [30] Jackson explains why the term “Lorenz gauge” is preferable to “Lorentz gauge”.) The vector

j^{α}

is the charge's current density, which is formally written in terms of a four-dimensional Dirac functional supported on the charge's world line: The density is zero everywhere, except at the particle's position where it is infinite. For concreteness we will imagine that the particle moves around a centre (perhaps another charge, which is taken to be fixed) and that it emits outgoing radiation. We expect that the charge will undergo a radiation reaction and that it will spiral down toward the centre. This effect must be accounted for by the equations of motion, and these must therefore include the action of the charge's own field, which is the only available agent that could be responsible for the radiation reaction. We seek to determine this self-force acting on the particle.

An immediate difficulty presents itself: The vector potential, and also the electromagnetic field tensor, diverge on the particle's world line, because the field of a point charge is necessarily infinite at the charge's position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.

Difficult but not impossible. To find a way around this problem I note first that the situation considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the time-reversal invariance of Maxwell's theory. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose

A_{r e t}^{α}

, the retarded solution, as the physically-relevant solution. Choosing instead the advanced solution

A_{a d v}^{α}

would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition

\begin{matrix} A_{S}^{α} = \frac{1}{2} (A_{r e t}^{α} + A_{a d v}^{α}) \end{matrix}

(2)

would restore time-reversal invariance: Outgoing and incoming radiation would be present in equal amounts, there would be no net loss nor gain of energy by the system, and the charge would not undergo any radiation reaction. In Equation ( 2 ) the subscript `S' stands for `symmetric', as the vector potential depends symmetrically upon future and past.

My second key observation is that while the potential of Equation ( 2 ) does not exert a force on the charged particle, it is just as singular as the retarded potential in the vicinity of the world line.

This follows from the fact that

A_{r e t}^{α}

A_{a d v}^{α}

, and

A_{S}^{α}

all satisfy Equation ( 1 ), whose source term is infinite on the world line. So while the wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line – all three electromagnetic fields are dominated by the particle's Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript `S': It stands for `singular' as well as `symmetric'.

Because

A_{S}^{α}

is just as singular as

A_{r e t}^{α}

, removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because

A_{S}^{α}

is known not to affect the motion of the charged particle, this new potential must be entirely responsible for the radiation reaction. We therefore introduce the new potential

\begin{matrix} A_{R}^{α} = A_{r e t}^{α} - A_{S}^{α} = \frac{1}{2} (A_{r e t}^{α} - A_{a d v}^{α}) \end{matrix}

(3)

and postulate that it, and it alone, exerts a force on the particle. The subscript `R' stands for `regular', because

A_{R}^{α}

is nonsingular on the world line. This property can be directly inferred from the fact that the regular potential satisfies the homogeneous version of Equation ( 1 ),

□ A_{R}^{α} = 0

; there is no singular source to produce a singular behaviour on the world line. Since

A_{R}^{α}

satisfies the homogeneous wave equation, it can be thought of as a free radiation field, and the subscript `R' could also stand for `radiative'.

The self-action of the charge's own field is now clarified: A singular potential

A_{S}^{α}

can be removed from the retarded potential and shown not to affect the motion of the particle. (Establishing this last statement requires a careful analysis that is presented in the bulk of the paper; what really happens is that the singular field contributes to the particle's inertia and renormalizes its mass.) What remains is a well-behaved potential

A_{R}^{α}

that must be solely responsible for the radiation reaction. From the radiative potential we form an electromagnetic field tensor

F_{α β}^{R} = \partial_{α} A_{β}^{R} - \partial_{β} A_{α}^{R}

, and we take the particle's equations of motion to be

\begin{matrix} m a_{μ} = f_{μ}^{e x t} + e F_{μ ν}^{R} u^{ν}, \end{matrix}

(4)

where

u^{μ} = d z^{μ} / d τ

is the charge's four-velocity (

z^{μ} (τ)

gives the description of the world line and

τ

is proper time),

a^{μ} = d u^{μ} / d τ

its acceleration,

m

its (renormalized) mass, and

f_{e x t}^{μ}

an external force also acting on the particle. Calculation of the radiative field yields the more concrete expression

\begin{matrix} m a^{μ} = f_{e x t}^{μ} + \frac{2 e^{2}}{3 m} (δ_{ν}^{μ} + u^{μ} u_{ν}) \frac{d f_{e x t}^{ν}}{d τ}, \end{matrix}

(5)

in which the second-term is the self-force that is responsible for the radiation reaction. We observe that the self-force is proportional to

e^{2}

, it is orthogonal to the four-velocity, and it depends on the rate of change of the external force. This is the result that was first derived by Dirac [25] ^$1$ .

^$1$ Dirac's original expression actually involved the rate of change of the acceleration vector on the right-hand side. The resulting equation gives rise to the well-known problem of runaway solutions. To avoid such unphysical behaviour I have submitted Dirac's equation to a reduction-of-order procedure whereby $d a^{ν} / d τ$ is replaced with $m^{- 1} d f_{e x t}^{ν} / d τ$ . This procedure is explained and justified, for example, in [47, 26] .

1.3 Green's functions in flat spacetime

To see how Equation ( 5 ) can eventually be generalized to curved spacetimes, I introduce a new layer of mathematical formalism and show that the decomposition of the retarded potential into symmetric-singular and regular-radiative pieces can be performed at the level of the Green's functions associated with Equation ( 1 ). The retarded solution to the wave equation can be expressed as

\begin{matrix} A_{r e t}^{α} (x) = \int G_{+ β^{'}}^{α} (x, x^{'}) j^{β^{'}} (x^{'}) d V^{'} \end{matrix}

(6)

in terms of the retarded Green's function

G_{+ β^{'}}^{α} (x, x^{'}) = δ_{β^{'}}^{α} δ (t - t^{'} - | x - x^{'} |) / | x - x^{'} |

. Here

x = (t, x)

is an arbitrary field point,

x^{'} = (t^{'}, x^{'})

is a source point, and

d V^{'} \equiv d^{4} x^{'}

; tensors at

x

are identified with unprimed indices, while primed indices refer to tensors at

x^{'}

. Similarly, the advanced solution can be expressed as

\begin{matrix} A_{a d v}^{α} (x) = \int G_{- β^{'}}^{α} (x, x^{'}) j^{β^{'}} (x^{'}) d V^{'} \end{matrix}

(7)

in terms of the advanced Green's function

G_{- β^{'}}^{α} (x, x^{'}) = δ_{β^{'}}^{α} δ (t - t^{'} + | x - x^{'} |) / | x - x^{'} |

. The retarded Green's function is zero whenever

x

lies outside of the future light cone of

x^{'}

, and

G_{+ β^{'}}^{α} (x, x^{'})

is infinite at these points. On the other hand, the advanced Green's function is zero whenever

x

lies outside of the past light cone of

x^{'}

, and

G_{- β^{'}}^{α} (x, x^{'})

is infinite at these points. The retarded and advanced Green's functions satisfy the reciprocity relation

\begin{matrix} G_{β^{'} α}^{-} (x^{'}, x) = G_{α β^{'}}^{+} (x, x^{'}); \end{matrix}

(8)

this states that the retarded Green's function becomes the advanced Green's function (and vice versa) when

x

and

x^{'}

are interchanged.

From the retarded and advanced Green's functions we can define a singular Green's function by

\begin{matrix} G_{S β^{'}}^{α} (x, x^{'}) = \frac{1}{2} [G_{+ β^{'}}^{α} (x, x^{'}) + G_{- β^{'}}^{α} (x, x^{'})] \end{matrix}

(9)

and a radiative Green's function by

\begin{matrix} G_{R β^{'}}^{α} (x, x^{'}) = G_{+ β^{'}}^{α} (x, x^{'}) - G_{S β^{'}}^{α} (x, x^{'}) = \frac{1}{2} [G_{+ β^{'}}^{α} (x, x^{'}) - G_{- β^{'}}^{α} (x, x^{'})] . \end{matrix}

(10)

By virtue of Equation ( 8 ) the singular Green's function is symmetric in its indices and arguments:

G_{β^{'} α}^{S} (x^{'}, x) = G_{α β^{'}}^{S} (x, x^{'})

. The radiative Green's function, on the other hand, is antisymmetric.

The potential

\begin{matrix} A_{S}^{α} (x) = \int G_{S β^{'}}^{α} (x, x^{'}) j^{β^{'}} (x^{'}) d V^{'} \end{matrix}

(11)

satisfies the wave equation of Equation ( 1 ) and is singular on the world line, while

\begin{matrix} A_{R}^{α} (x) = \int G_{R β^{'}}^{α} (x, x^{'}) j^{β^{'}} (x^{'}) d V^{'} \end{matrix}

(12)

satisfies the homogeneous equation

□ A^{α} = 0

and is well behaved on the world line.

Equation ( 6 ) implies that the retarded potential at

x

is generated by a single event in spacetime:

the intersection of the world line and the past light cone of

x

' (see Figure 1 ). I shall call this the retarded point associated with

x

and denote it

z (u)

;

u

is the retarded time, the value of the proper-time parameter at the retarded point. Similarly we find that the advanced potential of Equation ( 7 ) is generated by the intersection of the world line and the future light cone of the field point

x

. I shall call this the advanced point associated with

x

and denote it

z (v)

;

v

is the advanced time, the value of the proper-time parameter at the advanced point.

Figure 1 : In flat spacetime, the retarded potential at

x

depends on the particle's state of motion at the retarded point

z (u)

on the world line; the advanced potential depends on the state of motion at the advanced point

z (v)

1.4 Green's functions in curved spacetime

In a curved spacetime with metric

g_{α β}

the wave equation for the vector potential becomes

\begin{matrix} □ A^{α} - R_{β}^{α} A^{β} = - 4 π j^{α}, \end{matrix}

(13)

where

□ = g^{α β} \nabla_{α} \nabla_{β}

is the covariant wave operator and

R_{α β}

is the spacetime's Ricci tensor; the Lorenz gauge conditions becomes

\nabla_{α} A^{α} = 0

, and

\nabla_{α}

denotes covariant differentiation. Retarded and advanced Green's functions can be defined for this equation, and solutions to Equation ( 13 ) take the same form as in Equations ( 6 ) and ( 7 ), except that

d V^{'}

now stands for

\sqrt{- g (x^{'})} d^{4} x^{'}

The causal structure of the Green's functions is richer in curved spacetime: While in flat spacetime the retarded Green's function has support only on the future light cone of

x^{'}

, in curved spacetime its support extends inside the light cone as well;

G_{+ β^{'}}^{α} (x, x^{'})

is therefore nonzero when

x \in I^{+} (x^{'})

, which denotes the chronological future of

x^{'}

. This property reflects the fact that in curved spacetime, electromagnetic waves propagate not just at the speed of light, but at all speeds smaller than or equal to the speed of light ; the delay is caused by an interaction between the radiation and the spacetime curvature. A direct implication of this property is that the retarded potential at

x

is now generated by the point charge during its entire history prior to the retarded time

u

associated with

x

: The potential depends on the particle's state of motion for all times

τ \leq u

(see Figure 2 ).

Figure 2 : In curved spacetime, the retarded potential at

x

depends on the particle's history before the retarded time

u

; the advanced potential depends on the particle's history after the advanced time

v

Similar statements can be made about the advanced Green's function and the advanced solution to the wave equation. While in flat spacetime the advanced Green's function has support only on the past light cone of

x^{'}

, in curved spacetime its support extends inside the light cone, and

G_{- β^{'}}^{α} (x, x^{'})

is nonzero when

x \in I^{-} (x^{'})

, which denotes the chronological past of

x^{'}

. This implies that the advanced potential at

x

is generated by the point charge during its entire future history following the advanced time

v

associated with

x

: The potential depends on the particle's state of motion for all times

τ \geq v

The physically relevant solution to Equation ( 13 ) is obviously the retarded potential

A_{r e t}^{α} (x)

, and as in flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike nature of the source, and the presence of spacetime curvature does not change the fact that the potential diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall attempt to decompose the retarded solution into a singular part that exerts no force, and a smooth radiative part that produces the entire self-force.

To decompose the retarded Green's function into singular and radiative parts is not a straightforward task in curved spacetime. The flat-spacetime definition for the singular Green's function, Equation ( 9 ), cannot be adopted without modification: While the combination half-retarded plus half-advanced Green's functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Equation ( 13 ), this candidate for the singular Green's function would produce a self-force with an unacceptable dependence on the particle's future history. For suppose that we made this choice. Then the radiative Green's function would be given by the combination half-retarded minus half-advanced Green's functions, just as in flat spacetime. The resulting radiative potential would satisfy the homogeneous wave equation, and it would be smooth on the world line, but it would also depend on the particle's entire history, both past (through the retarded Green's function) and future (through the advanced Green's function). More precisely stated, we would find that the radiative potential at

x

depends on the particle's state of motion at all times

τ

outside the interval

u < τ < v

; in the limit where

x

approaches the world line, this interval shrinks to nothing, and we would find that the radiative potential is generated by the complete history of the particle. A self-force constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and radiative Green's functions.

The proper definitions were identified by Detweiler and Whiting [23] , who proposed the following generalization to Equation ( 9 ):

\begin{matrix} G_{S β^{'}}^{α} (x, x^{'}) = \frac{1}{2} [G_{+ β^{'}}^{α} (x, x^{'}) + G_{- β^{'}}^{α} (x, x^{'}) - H_{β^{'}}^{α} (x, x^{'})] . \end{matrix}

(14)

The two-point function

H_{β^{'}}^{α} (x, x^{'})

is introduced specifically to cure the pathology described in the preceding paragraph. It is symmetric in its indices and arguments, so that

G_{α β^{'}}^{S} (x, x^{'})

will be also (since the retarded and advanced Green's functions are still linked by a reciprocity relation); and it is a solution to the homogeneous wave equation,

□ H_{β^{'}}^{α} (x, x^{'}) - R_{γ}^{α} (x) H_{β^{'}}^{γ} (x, x^{'}) = 0

, so that the singular, retarded, and advanced Green's functions will all satisfy the same wave equation.

Furthermore, and this is its key property, the two-point function is defined to agree with the advanced Green's function when

x

is in the chronological past of

x^{'}

H_{β^{'}}^{α} (x, x^{'}) = G_{- β^{'}}^{α} (x, x^{'})

when

x \in I^{-} (x^{'})

. This ensures that

G_{S β^{'}}^{α} (x, x^{'})

vanishes when

x

is in the chronological past of

x^{'}

In fact, reciprocity implies that

H_{β^{'}}^{α} (x, x^{'})

will also agree with the retarded Green's function when

x

is in the chronological future of

x^{'}

, and it follows that the symmetric Green's function vanishes also when

x

is in the chronological future of

x^{'}

The potential

A_{S}^{α} (x)

constructed from the singular Green's function can now be seen to depend on the particle's state of motion at times

τ

restricted to the interval

u \leq τ \leq v

(see Figure 3 ).

Because this potential satisfies Equation ( 13 ), it is just as singular as the retarded potential in the vicinity of the world line. And because the singular Green's function is symmetric in its arguments, the singular potential can be shown to exert no force on the charged particle. (This requires a lengthy analysis that will be presented in the bulk of the paper.)

Figure 3 : In curved spacetime, the singular potential at

x

depends on the particle's history during the interval

u \leq τ \leq v

; for the radiative potential the relevant interval is

- \infty < τ \leq v

The Detweiler–Whiting [23] definition for the radiative Green's function is then

\begin{matrix} G_{R β^{'}}^{α} (x, x^{'}) = G_{+ β^{'}}^{α} (x, x^{'}) - G_{S β^{'}}^{α} (x, x^{'}) = \frac{1}{2} [G_{+ β^{'}}^{α} (x, x^{'}) - G_{- β^{'}}^{α} (x, x^{'}) + H_{β^{'}}^{α} (x, x^{'})] . \end{matrix}

(15)

The potential

A_{R}^{α} (x)

constructed from this depends on the particle's state of motion at all times

τ

prior to the advanced time

v

τ \leq v

. Because this potential satisfies the homogeneous wave equation, it is well behaved on the world line and its action on the point charge is well defined.

And because the singular potential

A_{S}^{α} (x)

can be shown to exert no force on the particle, we conclude that

A_{R}^{α} (x)

alone is responsible for the self-force.

From the radiative potential we form an electromagnetic field tensor

F_{α β}^{R} = \nabla_{α} A_{β}^{R} - \nabla_{β} A_{α}^{R}

, and the curved-spacetime generalization to Equation ( 4 ) is

\begin{matrix} m a_{μ} = f_{μ}^{e x t} + e F_{μ ν}^{R} u^{ν}, \end{matrix}

(16)

where

u^{μ} = d z^{μ} / d τ

is again the charge's four-velocity, but

a^{μ} = D u^{μ} / d τ

is now its covariant acceleration.

1.5 World line and retarded coordinates

To flesh out the ideas contained in the preceding Section 1.4 I add yet another layer of mathematical formalism and construct a convenient coordinate system to chart a neighbourhood of the particle's world line. In the next Section 1.6 I will display explicit expressions for the retarded, singular, and radiative fields of a point electric charge.

Let

γ

be the world line of a point particle in a curved spacetime. It is described by parametric relations

z^{μ} (τ)

in which

τ

is proper time. Its tangent vector is

u^{μ} = d z^{μ} / d τ

and its acceleration is

a^{μ} = D u^{μ} / d τ

; we shall also encounter

{\dot{a}}^{μ} \equiv D a^{μ} / d τ

γ

we erect an orthonormal basis that consists of the four-velocity

u^{μ}

and three spatial vectors

e^{μ} a

labelled by a frame index

a = (1, 2, 3)

. These vectors satisfy the relations

g_{μ ν} u^{μ} u^{ν} = - 1

g_{μ ν} u^{μ} e^{ν} a = 0

, and

g_{μ ν} e^{μ} a e^{ν} b = δ_{a b}

. We take the spatial vectors to be Fermi–Walker transported on the world line:

D e^{μ} a / d τ = a_{a} u^{μ}

, where

\begin{matrix} a_{a} (τ) = a_{μ} e^{μ} a \end{matrix}

(17)

are frame components of the acceleration vector; it is easy to show that Fermi–Walker transport preserves the orthonormality of the basis vectors. We shall use the tetrad to decompose various tensors evaluated on the world line. An example was already given in Equation ( 17 ) but we shall also encounter frame components of the Riemann tensor,

\begin{matrix} R_{a 0 b 0} (τ) = R_{μ λ ν ρ} e^{μ} a u^{λ} e^{ν} b u^{ρ}, R_{a 0 b c} (τ) = R_{μ λ ν ρ} e^{μ} a u^{λ} e^{ν} b e^{ρ} c, R_{a b c d} (τ) = R_{μ λ ν ρ} e^{μ} a e^{λ} b e^{ν} c e^{ρ} d, \end{matrix}

(18)

as well as frame components of the Ricci tensor,

\begin{matrix} R_{00} (τ) = R_{μ ν} u^{μ} u^{ν}, R_{a 0} (τ) = R_{μ ν} e^{μ} a u^{ν}, R_{a b} (τ) = R_{μ ν} e^{μ} a e^{ν} b . \end{matrix}

(19)

We shall use

δ_{a b} = diag (1, 1, 1)

and its inverse

δ^{a b} = diag (1, 1, 1)

to lower and raise frame indices, respectively.

Consider a point

x

in a neighbourhood of the world line

γ

. We assume that

x

is sufficiently close to the world line that a unique geodesic links

x

to any neighbouring point

z

γ

. The two-point function

σ (x, z)

, known as Synge's world function [55] , is numerically equal to half the squared geodesic distance between

z

and

x

; it is positive if

x

and

z

are spacelike related, negative if they are timelike related, and

σ (x, z)

is zero if

x

and

z

are linked by a null geodesic. We denote its gradient

\partial σ / \partial z^{μ}

σ_{μ} (x, z)

, and

- σ^{μ}

gives a meaningful notion of a separation vector (pointing from

z

x

To construct a coordinate system in this neighbourhood we locate the unique point

x^{'} \equiv z (u)

γ

which is linked to

x

by a future-directed null geodesic (this geodesic is directed from

x^{'}

x

); I shall refer to

x^{'}

as the retarded point associated with

x

, and

u

will be called the retarded time.

To tensors at

x^{'}

we assign indices

α^{'}

β^{'}

, . . . ; this will distinguish them from tensors at a generic point

z (τ)

on the world line, to which we have assigned indices

μ

ν

, . . . . We have

σ (x, x^{'}) = 0

, and

- σ^{α^{'}} (x, x^{'})

is a null vector that can be interpreted as the separation between

x^{'}

and

x

Figure 4 : Retarded coordinates of a point

x

relative to a world line

γ

. The retarded time

u

selects a particular null cone, the unit vector

Ω^{a} \equiv {\hat{x}}^{a} / r

selects a particular generator of this null cone, and the retarded distance

r

selects a particular point on this generator.

The retarded coordinates of the point

x

are

(u, {\hat{x}}^{a})

, where

{\hat{x}}^{a} = - e^{a} α^{'} σ^{α^{'}}

are the frame components of the separation vector. They come with a straightforward interpretation (see Figure 4 ). The invariant quantity

\begin{matrix} r \equiv \sqrt{δ_{a b} {\hat{x}}^{a} {\hat{x}}^{b}} = u_{α^{'}} σ^{α^{'}} \end{matrix}

(20)

is an affine parameter on the null geodesic that links

x

x^{'}

; it can be loosely interpreted as the time delay between

x

and

x^{'}

as measured by an observer moving with the particle. This therefore gives a meaningful notion of distance between

x

and the retarded point, and I shall call

r

the retarded distance between

x

and the world line. The unit vector

\begin{matrix} Ω^{a} = {\hat{x}}^{a} / r \end{matrix}

(21)

is constant on the null geodesic that links

x

x^{'}

. Because

Ω^{a}

is a different constant on each null geodesic that emanates from

x^{'}

, keeping

u

fixed and varying

Ω^{a}

produces a congruence of null geodesics that generate the future light cone of the point

x^{'}

(the congruence is hypersurface orthogonal). Each light cone can thus be labelled by its retarded time

u

, each generator on a given light cone can be labelled by its direction vector

Ω^{a}

, and each point on a given generator can be labelled by its retarded distance

r

. We therefore have a good coordinate system in a neighbourhood of

γ

To tensors at

x

we assign indices

α

β

, . . . . These tensors will be decomposed in a tetrad

(e^{α} 0, e^{α} a)

that is constructed as follows: Given

x

we locate its associated retarded point

x^{'}

on the world line, as well as the null geodesic that links these two points; we then take the tetrad

(u^{α^{'}}, e^{α^{'}} a)

x^{'}

and parallel transport it to

x

along the null geodesic to obtain

(e^{α} 0, e^{α} a)

1.6 Retarded, singular, and radiative electromagnetic fields of a point electric charge

The retarded solution to Equation ( 13 ) is

\begin{matrix} A^{α} (x) = e \int_{γ} G_{+ μ}^{α} (x, z) u^{μ} d τ, \end{matrix}

(22)

where the integration is over the world line of the point electric charge. Because the retarded solution is the physically relevant solution to the wave equation, it will not be necessary to put a label `ret' on the vector potential.

From the vector potential we form the electromagnetic field tensor

F_{α β}

, which we decompose in the tetrad

(e^{α} 0, e^{α} a)

introduced at the end of Section 1.5 . We then express the frame components of the field tensor in retarded coordinates, in the form of an expansion in powers of

r

. This gives

\begin{matrix} F_{a 0} (u, r, Ω^{a}) & \equiv & F_{α β} (x) e^{α} a (x) e^{β} 0 (x) \end{matrix}

\begin{matrix} = & \frac{e}{r^{2}} Ω_{a} - \frac{e}{r} (a_{a} - a_{b} Ω^{b} Ω_{a}) + \frac{1}{3} e R_{b 0 c 0} Ω^{b} Ω^{c} Ω_{a} - \frac{1}{6} e (5 R_{a 0 b 0} Ω^{b} + R_{a b 0 c} Ω^{b} Ω^{c}) \end{matrix}

\begin{matrix} + \frac{1}{12} e (5 R_{00} + R_{b c} Ω^{b} Ω^{c} + R) Ω_{a} + \frac{1}{3} e R_{a 0} - \frac{1}{6} e R_{a b} Ω^{b} + F_{a 0}^{t a i l} + O (r), \end{matrix}

(23)

\begin{matrix} F_{a b} (u, r, Ω^{a}) & \equiv & F_{α β} (x) e^{α} a (x) e^{β} b (x) \end{matrix}

\begin{matrix} = & \frac{e}{r} (a_{a} Ω_{b} - Ω_{a} a_{b}) + \frac{1}{2} e (R_{a 0 b c} - R_{b 0 a c} + R_{a 0 c 0} Ω_{b} - Ω_{a} R_{b 0 c 0}) Ω^{c} \end{matrix}

\begin{matrix} - \frac{1}{2} e (R_{a 0} Ω_{b} - Ω_{a} R_{b 0}) + F_{a b}^{t a i l} + O (r), \end{matrix}

(24)

where

\begin{matrix} F_{a 0}^{t a i l} = F_{α^{'} β^{'}}^{t a i l} (x^{'}) e^{α^{'}} a u^{β^{'}}, F_{a b}^{t a i l} = F_{α^{'} β^{'}}^{t a i l} (x^{'}) e^{α^{'}} a e^{β^{'}} b \end{matrix}

(25)

are the frame components of the “tail part” of the field, which is given by

\begin{matrix} F_{α^{'} β^{'}}^{t a i l} (x^{'}) = 2 e \int_{- \infty}^{u^{-}} \nabla_{[α^{'}} G_{+ β^{'}] μ} (x^{'}, z) u^{μ} d τ . \end{matrix}

(26)

In these expressions, all tensors (or their frame components) are evaluated at the retarded point

x^{'} \equiv z (u)

associated with

x

; for example,

a_{a} \equiv a_{a} (u) \equiv a_{α^{'}} e^{α^{'}} a

. The tail part of the electromagnetic field tensor is written as an integral over the portion of the world line that corresponds to the interval

- \infty < τ \leq u^{-} \equiv u - 0^{+}

; this represents the past history of the particle. The integral is cut short at

u^{-}

to avoid the singular behaviour of the retarded Green's function when

z (τ)

coincides with

x^{'}

; the portion of the Green's function involved in the tail integral is smooth, and the singularity at coincidence is completely accounted for by the other terms in Equations ( 23 ) and ( 24 ).

The expansion of

F_{α β} (x)

near the world line does indeed reveal many singular terms. We first recognize terms that diverge when

r \to 0

; for example the Coulomb field

F_{a 0}

diverges as

r^{- 2}

when we approach the world line. But there are also terms that, though they stay bounded in the limit, possess a directional ambiguity at

r = 0

; for example

F_{a b}

contains a term proportional to

R_{a 0 b c} Ω^{c}

whose limit depends on the direction of approach.

This singularity structure is perfectly reproduced by the singular field

F_{α β}^{S}

obtained from the potential

\begin{matrix} A_{S}^{α} (x) = e \int_{γ} G_{S μ}^{α} (x, z) u^{μ} d τ, \end{matrix}

(27)

where

G_{S μ}^{α} (x, z)

is the singular Green's function of Equation ( 14 ). Near the world line the singular field is given by

\begin{matrix} F_{a 0}^{S} (u, r, Ω^{a}) & \equiv & F_{α β}^{S} (x) e^{α} a (x) e^{β} 0 (x) \end{matrix}

\begin{matrix} = & \frac{e}{r^{2}} Ω_{a} - \frac{e}{r} (a_{a} - a_{b} Ω^{b} Ω_{a}) - \frac{2}{3} e {\dot{a}}_{a} + \frac{1}{3} e R_{b 0 c 0} Ω^{b} Ω^{c} Ω_{a} - \frac{1}{6} e (5 R_{a 0 b 0} Ω^{b} + R_{a b 0 c} Ω^{b} Ω^{c}) \end{matrix}

\begin{matrix} + \frac{1}{12} e (5 R_{00} + R_{b c} Ω^{b} Ω^{c} + R) Ω_{a} - \frac{1}{6} e R_{a b} Ω^{b} + O (r), \end{matrix}

(28)

\begin{matrix} F_{a b}^{S} (u, r, Ω^{a}) & \equiv & F_{α β}^{S} (x) e^{α} a (x) e^{β} b (x) \end{matrix}

\begin{matrix} = & \frac{e}{r} (a_{a} Ω_{b} - Ω_{a} a_{b}) + \frac{1}{2} e (R_{a 0 b c} - R_{b 0 a c} + R_{a 0 c 0} Ω_{b} - Ω_{a} R_{b 0 c 0}) Ω^{c} \end{matrix}

\begin{matrix} - \frac{1}{2} e (R_{a 0} Ω_{b} - Ω_{a} R_{b 0}) + O (r) . \end{matrix}

(29)

Comparison of these expressions with Equations ( 23 ) and ( 24 ) does indeed reveal that all singular terms are shared by both fields.

The difference between the retarded and singular fields defines the radiative field

F_{α β}^{R} (x)

. Its frame components are

\begin{matrix} F_{a 0}^{R} & = & \frac{2}{3} e {\dot{a}}_{a} + \frac{1}{3} e R_{a 0} + F_{a 0}^{t a i l} + O (r), \end{matrix}

(30)

\begin{matrix} F_{a b}^{R} & = & F_{a b}^{t a i l} + O (r), \end{matrix}

(31)

and at

x^{'}

the radiative field becomes

\begin{matrix} F_{α^{'} β^{'}}^{R} = 2 e u_{[α^{'}} (g_{β^{'}] γ^{'}} + u_{β^{'}]} u_{γ^{'}}) (\frac{2}{3} {\dot{a}}^{γ^{'}} + \frac{1}{3} R_{δ^{'}}^{γ^{'}} u^{δ^{'}}) + F_{α^{'} β^{'}}^{t a i l}, \end{matrix}

(32)

where

{\dot{a}}^{γ^{'}} = D a^{γ^{'}} / d τ

is the rate of change of the acceleration vector, and where the tail term was given by Equation ( 26 ). We see that

F_{α β}^{R} (x)

is a smooth tensor field, even on the world line.

1.7 Motion of an electric charge in curved spacetime

I have argued in Section 1.4 that the self-force acting on a point electric charge is produced by the radiative field, and that the charge's equations of motion should take the form of

m a_{μ} = f_{μ}^{e x t} + e F_{μ ν}^{R} u^{ν}

, where

f_{μ}^{e x t}

is an external force also acting on the particle. Substituting Equation ( 32 ) gives

\begin{matrix} m a^{μ} = f_{e x t}^{μ} + e^{2} (δ_{ν}^{μ} + u^{μ} u_{ν}) (\frac{2}{3 m} \frac{D f_{e x t}^{ν}}{d τ} + \frac{1}{3} R_{λ}^{ν} u^{λ}) + 2 e^{2} u_{ν} \int_{- \infty}^{τ^{-}} \nabla^{[μ} G_{+ λ^{'}}^{ν]} (z (τ), z (τ^{'})) u^{λ^{'}} d τ^{'}, \end{matrix}

(33)

in which all tensors are evaluated at

z (τ)

, the current position of the particle on the world line.

The primed indices in the tail integral refer to a point

z (τ^{'})

which represents a prior position; the integration is cut short at

τ^{'} = τ^{-} \equiv τ - 0^{+}

to avoid the singular behaviour of the retarded Green's function at coincidence. To get Equation ( 33 ) I have reduced the order of the differential equation by replacing

{\dot{a}}^{ν}

with

m^{- 1} {\dot{f}}_{e x t}^{ν}

on the right-hand side; this procedure was explained at the end of Section 1.2 .

Equation ( 33 ) is the result that was first derived by DeWitt and Brehme [24] and later corrected by Hobbs [29] . (The original equation did not include the Ricci-tensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green's function vanishes everywhere within the domain of integration), and Equation ( 33 ) reduces to Dirac's result of Equation ( 5 ). In curved spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an external force: It is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that, in general, the self-force is nonlocal in time: It depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to Equation ( 33 ) is simply an interaction between the charge and a free electromagnetic field

F_{α β}^{R}

; it is this field that carries the information about the charge's past.

1.8 Motion of a scalar charge in curved spacetime

The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the electromagnetic theory. The particle's charge

q

produces a scalar field

Φ (x)

, which satisfies a wave equation

\begin{matrix} (□ - ξ R) Φ = - 4 π μ \end{matrix}

(34)

that is very similar to Equation ( 13 ). Here,

R

is the spacetime's Ricci scalar, and

ξ

is an arbitrary coupling constant; the scalar charge density

μ (x)

is given by a four-dimensional Dirac functional supported on the particle's world line

γ

. The retarded solution to the wave equation is

\begin{matrix} Φ (x) = q \int_{γ} G_{+} (x, z) d τ, \end{matrix}

(35)

where

G_{+} (x, z)

is the retarded Green's function associated with Equation ( 34 ). The field exerts a force on the particle, whose equations of motion are

\begin{matrix} m a^{μ} = q (g^{μ ν} + u^{μ} u^{ν}) \nabla_{ν} Φ, \end{matrix}

(36)

where

m

is the particle's mass; this equation is very similar to the Lorentz-force law. But the dynamics of a scalar charge comes with a twist: If Equations ( 34 ) and ( 36 ) are to follow from a variational principle, the particle's mass should not be expected to be a constant of the motion. It is found instead to satisfy the differential equation

\begin{matrix} \frac{d m}{d τ} = - q u^{μ} \nabla_{μ} Φ, \end{matrix}

(37)

and in general

m

will vary with proper time. This phenomenon is linked to the fact that a scalar field has zero spin: The particle can radiate monopole waves and the radiated energy can come at the expense of the rest mass.

The scalar field of Equation ( 35 ) diverges on the world line, and its singular part

Φ_{S} (x)

must be removed before Equations ( 36 ) and ( 37 ) can be evaluated. This procedure produces the radiative field

Φ_{R} (x)

, and it is this field (which satisfies the homogeneous wave equation) that gives rise to a self-force. The gradient of the radiative field takes the form of

\begin{matrix} \nabla_{μ} Φ_{R} = - \frac{1}{12} (1 - 6 ξ) q R u_{μ} + q (g_{μ ν} + u_{μ} u_{ν}) (\frac{1}{3} {\dot{a}}^{ν} + \frac{1}{6} R_{λ}^{ν} u^{λ}) + Φ_{μ}^{t a i l} \end{matrix}

(38)

when it is evaluated of the world line. The last term is the tail integral

\begin{matrix} Φ_{μ}^{t a i l} = q \int_{- \infty}^{τ^{-}} \nabla_{μ} G_{+} (z (τ), z (τ^{'})) d τ^{'}, \end{matrix}

(39)

and this brings the dependence on the particle's past.

Substitution of Equation ( 38 ) into Equations ( 36 ) and ( 37 ) gives the equations of motion of a point scalar charge. (At this stage I introduce an external force

f_{e x t}^{μ}

and reduce the order of the differential equation.) The acceleration is given by

\begin{matrix} m a^{μ} = f_{e x t}^{μ} + q^{2} (δ_{ν}^{μ} + u^{μ} u_{ν}) [\frac{1}{3 m} \frac{D f_{e x t}^{ν}}{d τ} + \frac{1}{6} R_{λ}^{ν} u^{λ} + \int_{- \infty}^{τ^{-}} \nabla^{ν} G_{+} (z (τ), z (τ^{'})) d τ^{'}], \end{matrix}

(40)

and the mass changes according to

\begin{matrix} \frac{d m}{d τ} = - \frac{1}{12} (1 - 6 ξ) q^{2} R - q^{2} u^{μ} \int_{- \infty}^{τ^{-}} \nabla_{μ} G_{+} (z (τ), z (τ^{'})) d τ^{'} . \end{matrix}

(41)

These equations were first derived by Quinn [48] ^$2$ .

In flat spacetime the Ricci-tensor term and the tail integral disappear, and Equation ( 40 ) takes the form of Equation ( 5 ) with

q^{2} / (3 m)

replacing the factor of

2 e^{2} / (3 m)

. In this simple case Equation ( 41 ) reduces to

d m / d τ = 0

and the mass is in fact a constant. This property remains true in a conformally-flat spacetime when the wave equation is conformally invariant (

ξ = 1 / 6

): In this case the Green's function possesses only a light-cone part, and the right-hand side of Equation ( 41 ) vanishes. In generic situations the mass of a point scalar charge will vary with proper time.

^$2$ His analysis was restricted to a minimally-coupled scalar field, so that $ξ = 0$ in his expressions. The extension to an arbitrary coupling constant was carried out by myself for this review.

1.9 Motion of a point mass, or a black hole, in a background spacetime

The case of a point mass moving in a specified background spacetime presents itself with a serious conceptual challenge, as the fundamental equations of the theory are nonlinear and the very notion of a “point mass” is somewhat misguided. Nevertheless, to the extent that the perturbation

h_{α β} (x)

created by the point mass can be considered to be “small”, the problem can be formulated in close analogy with what was presented before.

We take the metric

g_{α β}

of the background spacetime to be a solution of the Einstein field equations in vacuum. (We impose this condition globally.) We describe the gravitational perturbation produced by a point particle of mass

m

in terms of trace-reversed potentials

γ_{α β}

defined by

\begin{matrix} γ_{α β} = h_{α β} - \frac{1}{2} (g^{γ δ} h_{γ δ}) g_{α β}, \end{matrix}

(42)

where

h_{α β}

is the difference between

g_{α β}

, the actual metric of the perturbed spacetime, and

g_{α β}

The potentials satisfy the wave equation

\begin{matrix} □ γ^{α β} + 2 R_{γ δ}^{α β} γ^{γ δ} = - 16 π T^{α β} \end{matrix}

(43)

together with the Lorenz gauge condition

γ_{; β}^{α β} = 0

. Here and below, covariant differentiation refers to a connection that is compatible with the background metric,

□ = g^{α β} \nabla_{α} \nabla_{β}

is the wave operator for the background spacetime, and

T^{α β}

is the stress-energy tensor of the point mass; this is given by a Dirac distribution supported on the particle's world line

γ

. The retarded solution is

\begin{matrix} γ^{α β} (x) = 4 m \int_{γ} G_{+ μ ν}^{α β} (x, z) u^{μ} u^{ν} d τ, \end{matrix}

(44)

where

G_{+ μ ν}^{α β} (x, z)

is the retarded Green's function associated with Equation ( 43 ). The perturbation

h_{α β} (x)

can be recovered by inverting Equation ( 42 ).

Equations of motion for the point mass can be obtained by formally demanding that the motion be geodesic in the perturbed spacetime with metric

g_{α β} = g_{α β} + h_{α β}

. After a mapping to the background spacetime, the equations of motion take the form of

\begin{matrix} a^{μ} = - \frac{1}{2} (g^{μ ν} + u^{μ} u^{ν}) (2 h_{ν λ; ρ} - h_{λ ρ; ν}) u^{λ} u^{ρ} . \end{matrix}

(45)

The acceleration is thus proportional to

m

; in the test-mass limit the world line of the particle is a geodesic of the background spacetime.

We now remove

h_{α β}^{S} (x)

from the retarded perturbation and postulate that it is the radiative field

h_{α β}^{S} (x)

that should act on the particle. (Note that

γ_{α β}^{S}

satisfies the same wave equation as the retarded potentials, but that

γ_{α β}^{R}

is a free gravitational field that satisfies the homogeneous wave equation.) On the world line we have

\begin{matrix} h_{μ ν; λ}^{R} = - 4 m (u_{(μ} R_{ν) ρ λ ξ} + R_{μ ρ ν ξ} u_{λ}) u^{ρ} u^{ξ} + h_{μ ν λ}^{t a i l}, \end{matrix}

(46)

where the tail term is given by

\begin{matrix} h_{μ ν λ}^{t a i l} = 4 m \int_{- \infty}^{τ^{-}} \nabla_{λ} (G_{+ μ ν μ^{'} ν^{'}} - \frac{1}{2} g_{μ ν} G_{+ ρ μ^{'} ν^{'}}^{ρ}) (z (τ), z (τ^{'})) u^{μ^{'}} u^{ν^{'}} d τ^{'} . \end{matrix}

(47)

When Equation ( 46 ) is substituted into Equation ( 45 ) we find that the terms that involve the Riemann tensor cancel out, and we are left with

\begin{matrix} a^{μ} = - \frac{1}{2} (g^{μ ν} + u^{μ} u^{ν}) (2 h_{ν λ ρ}^{t a i l} - h_{λ ρ ν}^{t a i l}) u^{λ} u^{ρ} . \end{matrix}

(48)

Only the tail integral appears in the final form of the equations of motion. It involves the current position

z (τ)

of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions

z (τ^{'})

, at which tensors with primed indices are evaluated. As before the integral is cut short at

τ^{'} = τ^{-} \equiv τ - 0^{+}

to avoid the singular behaviour of the retarded Green's function at coincidence.

The equations of motion of Equation ( 48 ) were first derived by Mino, Sasaki, and Tanaka [39] , and then reproduced with a different analysis by Quinn and Wald [49] . They are now known as the MiSaTaQuWa equations of motion. Detweiler and Whiting [23] have contributed the compelling interpretation that the motion is actually geodesic in a spacetime with metric

g_{α β} + h_{α β}^{R}

. This metric satisfies the Einstein field equations in vacuum and is perfectly smooth on the world line.

This spacetime can thus be viewed as the background spacetime perturbed by a free gravitational wave produced by the particle at an earlier stage of its history.

While Equation ( 48 ) does indeed give the correct equations of motion for a small mass

m

moving in a background spacetime with metric

g_{α β}

, the derivation outlined here leaves much to be desired – to what extent should we trust an analysis based on the existence of a point mass? Fortunately, Mino, Sasaki, and Tanaka [39] gave two different derivations of their result, and the second derivation was concerned not with the motion of a point mass, but with the motion of a small nonrotating black hole. In this alternative derivation of the MiSaTaQuWa equations, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: It must move according to Equation ( 48 ). This alternative derivation is entirely free of conceptual and technical pitfalls, and we conclude that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body's internal structure can be ignored).

It is important to understand that unlike Equations ( 33 ) and ( 40 ), which are true tensorial equations, Equation ( 48 ) reflects a specific choice of coordinate system and its form would not be preserved under a coordinate transformation. In other words, the MiSaTaQuWa equations are not gauge invariant, and they depend upon the Lorenz gauge condition

γ_{; β}^{α β} = 0

. Barack and Ori [8] have shown that under a coordinate transformation of the form

x^{α} \to x^{α} + ξ^{α}

, where

x^{α}

are the coordinates of the background spacetime and

ξ^{α}

is a smooth vector field of order

m

, the particle's acceleration changes according to

a^{μ} \to a^{μ} + a [ξ]^{μ}

, where

\begin{matrix} a [ξ]^{μ} = (δ_{ν}^{μ} + u^{μ} u_{ν}) (\frac{D^{2} ξ^{ν}}{d τ^{2}} + R_{ρ ω λ}^{ν} u^{ρ} ξ^{ω} u^{λ}) \end{matrix}

(49)

is the “gauge acceleration”;

D^{2} ξ^{ν} / d τ^{2} = (ξ_{; μ}^{ν} u^{μ})_{; ρ} u^{ρ}

is the second covariant derivative of

ξ^{ν}

in the direction of the world line. This implies that the particle's acceleration can be altered at will by a gauge transformation;

ξ^{α}

could even be chosen so as to produce

a^{μ} = 0

, making the motion geodesic after all. This observation provides a dramatic illustration of the following point:

The MiSaTaQuWa equations of motion are not gauge invariant and they cannot by themselves produce a meaningful answer to a well-posed physical question; to obtain such answers it shall always be necessary to combine the equations of motion with the metric perturbation

h_{α β}

so as to form gauge-invariant quantities that will correspond to direct observables. This point is very important and cannot be over-emphasized.

1.10 Evaluation of the self-force

To concretely evaluate the self-force, whether it be for a scalar charge, an electric charge, or a point mass, is a difficult undertaking. The difficulty resides mostly with the computation of the retarded Green's function for the spacetime under consideration. Because Green's functions are known for a very limited number of spacetimes, the self-force has so far been evaluated in a rather limited number of situations.

The first evaluation of the electromagnetic self-force was carried out by DeWitt and DeWitt [41] for a charge moving freely in a weakly-curved spacetime characterized by a Newtonian potential

Φ ≪ 1

. (This condition must be imposed globally, and requires the spacetime to contain a matter distribution.) In this context the right-hand side of Equation ( 33 ) reduces to the tail integral, since there is no external force acting on the charge. They found the spatial components of the self-force to be given by

\begin{matrix} f_{e m} = e^{2} \frac{M}{r^{3}} \hat{r} + \frac{2}{3} e^{2} \frac{d g}{d t}, \end{matrix}

(50)

where

M

is the total mass contained in the spacetime,

r = | x |

is the distance from the centre of mass,

\hat{r} = x / r

, and

g = - \nabla Φ

is the Newtonian gravitational field. (In these expressions the bold-faced symbols represent vectors in three-dimensional flat space.) The first term on the right-hand side of Equation ( 50 ) is a conservative correction to the Newtonian force

m g

. The second term is the standard radiation-reaction force; although it comes from the tail integral, this is the same result that would be obtained in flat spacetime if an external force

m g

were acting on the particle. This agreement is necessary, but remarkable!

A similar expression was obtained by Pfenning and Poisson [46] for the case of a scalar charge.

Here

\begin{matrix} f_{s c a l a r} = 2 ξ q^{2} \frac{M}{r^{3}} \hat{r} + \frac{1}{3} q^{2} \frac{d g}{d t}, \end{matrix}

(51)

where

ξ

is the coupling of the scalar field to the spacetime curvature; the conservative term disappears when the field is minimally coupled. Pfenning and Poisson also computed the gravitational self-force acting on a massive particle moving in a weakly curved spacetime. The expression they obtained is in complete agreement (within its domain of validity) with the standard post-Newtonian equations of motion.

The force required to hold an electric charge in place in a Schwarzschild spacetime was computed, without approximations, by Smith and Will [54] . As measured by a free-falling observer momentarily at rest at the position of the charge, the total force is

\begin{matrix} f = \frac{M m}{r^{2}} {(1 - \frac{2 M}{r})}^{- 1 / 2} - e^{2} \frac{M}{r^{3}}, \end{matrix}

(52)

and it is directed in the radial direction. Here,

m

is the mass of the charge,

M

is the mass of the black hole, and

r

is the charge's radial coordinate (the expression is valid in Schwarzschild coordinates). The first term on the right-hand side of Equation ( 52 ) is the force required to keep a neutral test particle stationary in a Schwarzschild spacetime; the second term is the negative of the electromagnetic self-force, and its expression agrees with the weak-field result of Equation ( 50 ).

Wiseman [62] performed a similar calculation for a scalar charge. He found that in this case the self-force vanishes. This result is not incompatible with Equation ( 51 ), even for nonminimal coupling, because the computation of the weak-field self-force requires the presence of matter, while Wiseman's scalar charge lives in a purely vacuum spacetime.

The intriguing phenomenon of mass loss by a scalar charge was studied by Burko, Harte, and Poisson [15] in the simple context of a particle at rest in an expanding universe. For the special cases of a de Sitter cosmology, or a spatially-flat matter-dominated universe, the retarded Green's function could be computed, and the action of the scalar field on the particle determined, without approximations. In de Sitter spacetime the particle is found to radiate all of its rest mass into monopole scalar waves. In the matter-dominated cosmology this happens only if the charge of the particle is sufficiently large; for smaller charges the particle first loses a fraction of its mass, but then regains it eventually.

In recent years a large effort has been devoted to the elaboration of a practical method to compute the (scalar, electromagnetic, and gravitational) self-force in the Schwarzschild spacetime.

This work originated with Barack and Ori [7] and was pursued by Barack [2, 3] until it was put in its definitive form by Barack, Mino, Nakano, Ori, and Sasaki [6, 9, 11, 38] . The idea is to take advantage of the spherical symmetry of the Schwarzschild solution by decomposing the retarded Green's function

G_{+} (x, x^{'})

into spherical-harmonic modes which can be computed individually. (To be concrete I refer here to the scalar case, but the method works just as well for the electromagnetic and gravitational cases.) From the mode-decomposition of the Green's function one obtains a mode-decomposition of the field gradient

\nabla_{α} Φ

, and from this subtracts a mode-decomposition of the singular field

\nabla_{α} Φ_{S}

, for which a local expression is known. This results in the radiative field

\nabla_{α} Φ_{R}

decomposed into modes, and since this field is well behaved on the world line, it can be directly evaluated at the position of the particle by summing over all modes. (This sum converges because the radiative field is smooth; the mode sums for the retarded or singular fields, on the other hand, do not converge.) An extension of this method to the Kerr spacetime has recently been presented [44, 34, 10] , and Mino [37] has devised a surprisingly simple prescription to calculate the time-averaged evolution of a generic orbit around a Kerr black hole.

The mode-sum method was applied to a number of different situations. Burko computed the self-force acting on an electric charge in circular motion in flat spacetime [12] , as well as on a scalar and electric charge kept stationary in a Schwarzschild spacetime [14] , in a spacetime that contains a spherical matter shell (Burko, Liu, and Soren [17] ), and in a Kerr spacetime (Burko and Liu [16] ). Burko also computed the scalar self-force acting on a particle in circular motion around a Schwarzschild black hole [13] , a calculation that was recently revisited by Detweiler, Messaritaki, and Whiting [21] . Barack and Burko considered the case of a particle falling radially into a Schwarzschild black hole, and evaluated the scalar self-force acting on such a particle [4] ; Lousto [33] and Barack and Lousto [5] , on the other hand, calculated the gravitational self-force.

1.11 Organization of this review

The main body of the review begins in Section 2 with a description of the general theory of bitensors, the name designating tensorial functions of two points in spacetime. I introduce Synge's world function

σ (x, x^{'})

and its derivatives in Section 2.1 , the parallel propagator

g_{α^{'}}^{α} (x, x^{'})

in Section 2.3 , and the van Vleck determinant

Δ (x, x^{'})

in Section 2.5 . An important portion of the theory (covered in Sections 2.2 and 2.4 ) is concerned with the expansion of bitensors when

x

is very close to

x^{'}

; expansions such as those displayed in Equations ( 23 ) and ( 24 ) are based on these techniques. The presentation in Section 2 borrows heavily from Synge's book [55] and the article by DeWitt and Brehme [24] . These two sources use different conventions for the Riemann tensor, and I have adopted Synge's conventions (which agree with those of Misner, Thorne, and Wheeler [40] ).

The reader is therefore warned that formulae derived in Section 2 may look superficially different from what can be found in DeWitt and Brehme. In Section 3 I introduce a number of coordinate systems that play an important role in later parts of the review. As a warmup exercise I first construct (in Section 3.1 ) Riemann normal coordinates in a neighbourhood of a reference point

x^{'}

. I then move on (in Section 3.2 ) to Fermi normal coordinates [36] , which are defined in a neighbourhood of a world line

γ

. The retarded coordinates, which are also based at a world line and which were briefly introduced in Section 1.5 , are covered systematically in Section 3.3 . The relationship between Fermi and retarded coordinates is worked out in Section 3.4 , which also locates the advanced point

z (v)

associated with a field point

x

. The presentation in Section 3 borrows heavily from Synge's book [55] . In fact, I am much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. I have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while I am interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.

In Section 4 I review the theory of Green's functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. I begin in Section 4.1 with a pedagogical introduction to the retarded and advanced Green's functions for a massive scalar field in flat spacetime; in this simple context the all-important Hadamard decomposition [28] of the Green's function into “light-cone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 4.2 along with its restrictions on the past and future null cones of a reference point

x^{'}

. The retarded, advanced, singular, and radiative Green's functions for the scalar wave equation are introduced in Section 4.3 . In Sections 4.4 and 4.5 I cover the vectorial and tensorial wave equations, respectively.

The presentation in Section 4 is based partly on the paper by DeWitt and Brehme [24] , but it is inspired mostly by Friedlander's book [27] . The reader should be warned that in one important aspect, my notation differs from the notation of DeWitt and Brehme: While they denote the tail part of the Green's function by

- v (x, x^{'})

, I have taken the liberty of eliminating the silly minus sign and I call it instead

+ V (x, x^{'})

. The reader should also note that all my Green's functions are normalized in the same way, with a factor of

- 4 π

multiplying a four-dimensional Dirac functional of the right-hand side of the wave equation. (The gravitational Green's function is sometimes normalized with a

- 16 π

on the right-hand side.) In Section 5 I compute the retarded, singular, and radiative fields associated with a point scalar charge (Section 5.1 ), a point electric charge (Section 5.2 ), and a point mass (Section 5.3 ). I provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: I follow Detweiler and Whiting [23] and postulate that only the radiative field exerts a force on the particle. In the second type of derivation I take guidance from Quinn and Wald [49] and postulate that the net force exerted on a point particle is given by an average of the retarded field over a surface of constant proper distance orthogonal to the world line – this rest-frame average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia. In Section 5.4 I present an alternative derivation of the MiSaTaQuWa equations of motion based on the method of matched asymptotic expansions [35, 32, 58, 19, 1, 20] ; the derivation applies to a small nonrotating black hole instead of a point mass. The ideas behind this derivation were contained in the original paper by Mino, Sasaki, and Tanaka [39] , but the implementation given here, which involves the retarded coordinates of Section 3.3 and displays explicitly the transformation between external and internal coordinates, is original work.

Concluding remarks are presented in Section 5.5 . Throughout this review I use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [40] .

2 General Theory of Bitensors

2.1 Synge's world function

2.1.1 Definition

In this and the following sections we will construct a number of bitensors, tensorial functions of two points in spacetime. The first is

x^{'}

, to which we refer as the “base point”, and to which we assign indices

α^{'}

β^{'}

, etc. The second is

x

, to which we refer as the “field point”, and to which we assign indices

α

β

, etc. We assume that

x

belongs to

N (x^{'})

, the normal convex neighbourhood of

x^{'}

; this is the set of points that are linked to

x^{'}

by a unique geodesic. The geodesic

β

that links

x

x^{'}

is described by relations

z^{μ} (λ)

in which

λ

is an affine parameter that ranges from

λ_{0}

λ_{1}

; we have

z (λ_{0}) \equiv x^{'}

and

z (λ_{1}) \equiv x

. To an arbitrary point

z

on the geodesic we assign indices

μ

ν

, etc. The vector

t^{μ} = d z^{μ} / d λ

is tangent to the geodesic, and it obeys the geodesic equation

D t^{μ} / d λ = 0

. The situation is illustrated in Figure 5 .

Figure 5 : The base point

x^{'}

, the field point

x

, and the geodesic

β

that links them. The geodesic is described by parametric relations

z^{μ} (λ)

, and

t^{μ} = d z^{μ} / d λ

is its tangent vector.

Synge's world function is a scalar function of the base point

x^{'}

and the field point

x

. It is defined by

\begin{matrix} σ (x, x^{'}) = \frac{1}{2} (λ_{1} - λ_{0}) \int_{λ_{0}}^{λ_{1}} g_{μ ν} (z) t^{μ} t^{ν} d λ, \end{matrix}

(53)

and the integral is evaluated on the geodesic

β

that links

x

x^{'}

. You may notice that

σ

is invariant under a constant rescaling of the affine parameter,

λ \to \bar{λ} = a λ + b

, where

a

and

b

are constants.

By virtue of the geodesic equation, the quantity

ɛ \equiv g_{μ ν} t^{μ} t^{ν}

is constant on the geodesic. The world function is therefore numerically equal to

\frac{1}{2} ɛ (λ_{1} - λ_{0})^{2}

. If the geodesic is timelike, then

λ

can be set equal to the proper time

τ

, which implies that

ɛ = - 1

and

σ = - \frac{1}{2} (Δ τ)^{2}

. If the geodesic is spacelike, then

λ

can be set equal to the proper distance

s

, which implies that

ɛ = 1

and

σ = \frac{1}{2} (Δ s)^{2}

. If the geodesic is null, then

σ = 0

. Quite generally, therefore, the world function is half the squared geodesic distance between the points

x^{'}

and

x

In flat spacetime, the geodesic linking

x

x^{'}

is a straight line, and

σ = \frac{1}{2} η_{α β} (x - x^{'})^{α} (x - x^{'})^{β}

in Lorentzian coordinates.

2.1.2 Differentiation of the world function

The world function

σ (x, x^{'})

can be differentiated with respect to either argument. We let

σ_{α} = \partial σ / \partial x^{α}

be its partial derivative with respect to

x

, and

σ_{α^{'}} = \partial σ / \partial x^{α^{'}}

its partial derivative with respect to

x^{'}

. It is clear that

σ_{α}

behaves as a dual vector with respect to tensorial operations carried out at

x

, but as a scalar with respect to operations carried out

x^{'}

. Similarly,

σ_{α^{'}}

is a scalar at

x

but a dual vector at

x^{'}

We let

σ_{α β} \equiv \nabla_{β} σ_{α}

be the covariant derivative of

σ_{α}

with respect to

x

; this is a rank-2 tensor at

x

and a scalar at

x^{'}

. Because

σ

is a scalar at

x

, we have that this tensor is symmetric:

σ_{β α} = σ_{α β}

Similarly, we let

σ_{α β^{'}} \equiv \partial_{β^{'}} σ_{α} = \partial^{2} σ / \partial x^{β^{'}} \partial x^{α}

be the partial derivative of

σ_{α}

with respect to

x^{'}

; this is a dual vector both at

x

and

x^{'}

. We can also define

σ_{α^{'} β} \equiv \partial_{β} σ_{α^{'}} = \partial^{2} σ / \partial x^{β} \partial x^{α^{'}}

to be the partial derivative of

σ_{α^{'}}

with respect to

x

. Because partial derivatives commute, these bitensors are equal:

σ_{β^{'} α} = σ_{α β^{'}}

. Finally, we let

σ_{α^{'} β^{'}} \equiv \nabla_{β^{'}} σ_{α^{'}}

be the covariant derivative of

σ_{α^{'}}

with respect to

x^{'}

; this is a symmetric rank-2 tensor at

x^{'}

and a scalar at

x

The notation is easily extended to any number of derivatives. For example, we let

σ_{α β γ δ^{'}} \equiv \nabla_{δ^{'}} \nabla_{γ} \nabla_{β} \nabla_{α} σ

, which is a rank-3 tensor at

x

and a dual vector at

x^{'}

. This bitensor is symmetric in the pair of indices

α

and

β

, but not in the pairs

α

and

γ

, nor

β

and

γ

. Because

\nabla_{δ^{'}}

is here an ordinary partial derivative with respect to

x^{'}

, the bitensor is symmetric in any pair of indices involving

δ^{'}

. The ordering of the primed index relative to the unprimed indices is therefore irrelevant: The same bitensor can be written as

σ_{δ^{'} α β γ}

σ_{α δ^{'} β γ}

σ_{α β δ^{'} γ}

, making sure that the ordering of the unprimed indices is not altered.

More generally, we can show that derivatives of any bitensor

Ω_{. . .} (x, x^{'})

satisfy the property

\begin{matrix} Ω_{. . .; β α^{'} . . .} = Ω_{. . .; α^{'} β . . .}, \end{matrix}

(54)

in which “

. . .

” stands for any combination of primed and unprimed indices. We start by establishing the symmetry of

Ω_{. . .; α β^{'}}

with respect to the pair

α

and

β^{'}

. This is most easily done by adopting Fermi normal coordinates (see Section 3.2 ) adapted to the geodesic

β

, and setting the connection to zero both at

x

and

x^{'}

. In these coordinates, the bitensor

Ω_{. . .; α}

is the partial derivative of

Ω_{. . .}

with respect to

x^{α}

, and

Ω_{. . .; α β^{'}}

is obtained by taking an additional partial derivative with respect to

x^{β^{'}}

. These two operations commute, and

Ω_{. . .; β^{'} α} = Ω_{. . .; α β^{'}}

follows as a bitensorial identity. Equation ( 54 ) then follows by further differentiation with respect to either

x

x^{'}

The message of Equation ( 54 ), when applied to derivatives of the world function, is that while the ordering of the primed and unprimed indices relative to themselves is important, their ordering with respect to each other is arbitrary. For example,

σ_{α^{'} β^{'} γ δ^{'} ε} = σ_{α^{'} β^{'} δ^{'} γ ε} = σ_{γ ε α^{'} β^{'} δ^{'}}

2.1.3 Evaluation of first derivatives

We can compute

σ_{α}

by examining how

σ

varies when the field point

x

moves. We let the new field point be

x + δ x

, and

δ σ \equiv σ (x + δ x, x^{'}) - σ (x, x^{'})

is the corresponding variation of the world function. We let

β + δ β

be the unique geodesic that links

x + δ x

x^{'}

; it is described by relations

z^{μ} (λ) + δ z^{μ} (λ)

, in which the affine parameter is scaled in such a way that it runs from

λ_{0}

λ_{1}

also on the new geodesic. We note that

δ z (λ_{0}) = δ x^{'} \equiv 0

and

δ z (λ_{1}) = δ x

Working to first order in the variations, Equation ( 53 ) implies

δ σ = Δ λ \int_{λ_{0}}^{λ_{1}} (g_{μ ν} {\dot{z}}^{μ} δ {\dot{z}}^{ν} + \frac{1}{2} g_{μ ν, λ} {\dot{z}}^{μ} {\dot{z}}^{ν} δ z^{λ}) d λ,

where

Δ λ = λ_{1} - λ_{0}

, an overdot indicates differentiation with respect to

λ

, and the metric and its derivatives are evaluated on

β

. Integrating the first term by parts gives

δ σ = Δ λ {[g_{μ ν} {\dot{z}}^{μ} δ z^{ν}]}_{λ_{0}}^{λ_{1}} - Δ λ \int_{λ_{0}}^{λ_{1}} (g_{μ ν} {\ddot{z}}^{ν} + Γ_{μ ν λ} {\dot{z}}^{ν} {\dot{z}}^{λ}) δ z^{μ} d λ .

The integral vanishes because

z^{μ} (λ)

satisfies the geodesic equation. The boundary term at

λ_{0}

is zero because the variation

δ z^{μ}

vanishes there. We are left with

δ σ = Δ λ g_{α β} t^{α} δ x^{β}

, or

\begin{matrix} σ_{α} (x, x^{'}) = (λ_{1} - λ_{0}) g_{α β} t^{β}, \end{matrix}

(55)

in which the metric and the tangent vector are both evaluated at

x

. Apart from a factor

Δ λ

, we see that

σ^{α} (x, x^{'})

is equal to the geodesic's tangent vector at

x

. If in Equation ( 55 ) we replace

x

by a generic point

z (λ)

β

, and if we correspondingly replace

λ_{1}

λ

, we obtain

σ^{μ} (z, x^{'}) = (λ - λ_{0}) t^{μ}

; we therefore see that

σ^{μ} (z, x^{'})

is a rescaled tangent vector on the geodesic.

A virtually identical calculation reveals how

σ

varies under a change of base point

x^{'}

. Here the variation of the geodesic is such that

δ z (λ_{0}) = δ x^{'}

and

δ z (λ_{1}) = δ x = 0

, and we obtain

δ σ = - Δ λ g_{α^{'} β^{'}} t^{α^{'}} δ x^{β^{'}}

. This shows that

\begin{matrix} σ_{α^{'}} (x, x^{'}) = - (λ_{1} - λ_{0}) g_{α^{'} β^{'}} t^{β^{'}}, \end{matrix}

(56)

in which the metric and the tangent vector are both evaluated at

x^{'}

. Apart from a factor

Δ λ

, we see that

σ^{α^{'}} (x, x^{'})

is minus the geodesic's tangent vector at

x^{'}

It is interesting to compute the norm of

σ_{α}

. According to Equation ( 55 ) we have

g_{α β} σ^{α} σ^{β} = (Δ λ)^{2} g_{α β} t^{α} t^{β} = (Δ λ)^{2} ɛ

. According to Equation ( 53 ), this is equal to

2 σ

. We have obtained

\begin{matrix} g^{α β} σ_{α} σ_{β} = 2 σ, \end{matrix}

(57)

and similarly

\begin{matrix} g^{α^{'} β^{'}} σ_{α^{'}} σ_{β^{'}} = 2 σ . \end{matrix}

(58)

These important relations will be the starting point of many computations to be described below.

We note that in flat spacetime,

σ_{α} = η_{α β} (x - x^{'})^{β}

and

σ_{α^{'}} = - η_{α β} (x - x^{'})^{β}

in Lorentzian coordinates. From this it follows that

σ_{α β} = σ_{α^{'} β^{'}} = - σ_{α β^{'}} = - σ_{α^{'} β} = η_{α β}

, and finally,

g^{α β} σ_{α β} = 4 = g^{α^{'} β^{'}} σ_{α^{'} β^{'}}

2.1.4 Congruence of geodesics emanating from $x^{'}$

If the base point

x^{'}

is kept fixed,

σ

can be considered to be an ordinary scalar function of

x

. According to Equation ( 57 ), this function is a solution to the nonlinear differential equation

\frac{1}{2} g^{α β} σ_{α} σ_{β} = σ

. Suppose that we are presented with such a scalar field. What can we say about it?

An additional differentiation of the defining equation reveals that the vector

σ^{α} \equiv σ^{; α}

satisfies

\begin{matrix} σ_{; β}^{α} σ^{β} = σ^{α}, \end{matrix}

(59)

which is the geodesic equation in a non-affine parameterization. The vector field is therefore tangent to a congruence of geodesics. The geodesics are timelike where

σ < 0

, they are spacelike where

σ > 0

, and they are null where

σ = 0

. Here, for concreteness, we shall consider only the timelike subset of the congruence.

The vector

\begin{matrix} u^{α} = \frac{σ^{α}}{| 2 σ |^{1 / 2}} \end{matrix}

(60)

is a normalized tangent vector that satisfies the geodesic equation in affine-parameter form:

u_{; β}^{α} u^{β} = 0

. The parameter

λ

is then proper time

τ

. If

λ^{*}

denotes the original parameterization of the geodesics, we have that

d λ^{*} / d τ = | 2 σ |^{- 1 / 2}

, and we see that the original parameterization is singular at

σ = 0

In the affine parameterization, the expansion of the congruence is calculated to be

\begin{matrix} θ = \frac{θ^{*}}{| 2 σ |^{1 / 2}}, θ^{*} = σ_{; α}^{α} - 1, \end{matrix}

(61)

where

θ^{*} = (δ V)^{- 1} (d / d λ^{*}) (δ V)

is the expansion in the original parameterization (

δ V

is the congruence's cross-sectional volume). While

θ^{*}

is well behaved in the limit

σ \to 0

(we shall see below that

θ^{*} \to 3

), we have that

θ \to \infty

. This means that the point

x^{'}

at which

σ = 0

is a caustic of the congruence: All geodesics emanate from this point.

These considerations, which all follow from a postulated relation

\frac{1}{2} g^{α β} σ_{α} σ_{β} = σ

, are clearly compatible with our preceding explicit construction of the world function.

2.2 Coincidence limits

It is useful to determine the limiting behaviour of the bitensors

σ_{. . .}

x

approaches

x^{'}

. We introduce the notation

[Ω_{. . .}] = {lim}_{x \to x^{'}} Ω_{. . .} (x, x^{'}) = a tensor at x^{'}

to designate the limit of any bitensor

Ω_{. . .} (x, x^{'})

x

approaches

x^{'}

; this is called the coincidence limit of the bitensor. We assume that the coincidence limit is a unique tensorial function of the base point

x^{'}

, independent of the direction in which the limit is taken. In other words, if the limit is computed by letting

λ \to λ_{0}

after evaluating

Ω_{. . .} (z, x^{'})

as a function of

λ

on a specified geodesic

β

, it is assumed that the answer does not depend on the choice of geodesic.

2.2.1 Computation of coincidence limits

From Equations ( 53 , 55 , 56 ) we already have

\begin{matrix} [σ] = 0, [σ_{α}] = [σ_{α^{'}}] = 0 . \end{matrix}

(62)

Additional results are obtained by repeated differentiation of the relations ( 57 ) and ( 58 ). For example, Equation ( 57 ) implies

σ_{γ} = g^{α β} σ_{α} σ_{β γ} = σ^{β} σ_{β γ}

, or

(g_{β γ} - σ_{β γ}) t^{β} = 0

after using Equation ( 55 ). From the assumption stated in the preceding paragraph,

σ_{β γ}

becomes independent of

t^{β}

in the limit

x \to x^{'}

, and we arrive at

[σ_{α β}] = g_{α^{'} β^{'}}

. By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are

\begin{matrix} [σ_{α β}] = [σ_{α^{'} β^{'}}] = g_{α^{'} β^{'}}, [σ_{α β^{'}}] = [σ_{α^{'} β}] = - g_{α^{'} β^{'}} . \end{matrix}

(63)

From these relations we infer that

[σ_{α}^{α}] = 4

, so that

[θ^{*}] = 3

, where

θ^{*}

was defined in Equation ( 61 ).

To generate coincidence limits of bitensors involving primed indices, it is efficient to invoke Synge's rule,

\begin{matrix} [σ_{. . . α^{'}}] = {[σ_{. . .}]}_{; α^{'}} - [σ_{. . . α}], \end{matrix}

(64)

in which “

. . .

” designates any combination of primed and unprimed indices; this rule will be established below. For example, according to Synge's rule we have

[σ_{α β^{'}}] = [σ_{α}]_{; β^{'}} - [σ_{α β}]

, and since the coincidence limit of

σ_{α}

is zero, this gives us

[σ_{α β^{'}}] = - [σ_{α β}] = - g_{α^{'} β^{'}}

, as was stated in Equation ( 63 ). Similarly,

[σ_{α^{'} β^{'}}] = [σ_{α^{'}}]_{; β^{'}} - [σ_{α^{'} β}] = - [σ_{β α^{'}}] = g_{α^{'} β^{'}}

. The results of Equation ( 63 ) can thus all be generated from the known result for

[σ_{α β}]

The coincidence limits of Equation ( 63 ) were derived from the relation

σ_{α} = σ_{α}^{δ} σ_{δ}

. We now differentiate this twice more and obtain

σ_{α β γ} = σ_{α β γ}^{δ} σ_{δ} + σ_{α β}^{δ} σ_{δ γ} + σ_{α γ}^{δ} σ_{δ β} + σ_{α}^{δ} σ_{δ β γ}

. At coincidence we have

[σ_{α β γ}] = [σ_{α β}^{δ}] g_{δ^{'} γ^{'}} + [σ_{α γ}^{δ}] g_{δ^{'} β^{'}} + δ_{α^{'}}^{δ^{'}} [σ_{δ β γ}],

[σ_{γ α β}] + [σ_{β α γ}] = 0

if we recognize that the operations of raising or lowering indices and taking the limit

x \to x^{'}

commute. Noting the symmetries of

σ_{α β}

, this gives us

[σ_{α γ β}] + [σ_{α β γ}] = 0

, or

2 [σ_{α β γ}] - [R_{α β γ}^{δ} σ_{δ}] = 0

, or

2 [σ_{α β γ}] = R_{α^{'} β^{'} γ^{'}}^{δ^{'}} [σ_{δ^{'}}]

. Since the last factor is zero, we arrive at

\begin{matrix} [σ_{α β γ}] = [σ_{α β γ^{'}}] = [σ_{α β^{'} γ^{'}}] = [σ_{α^{'} β^{'} γ^{'}}] = 0 . \end{matrix}

(65)

The last three results were derived from

[σ_{α β γ}] = 0

by employing Synge's rule.

We now differentiate the relation

σ_{α} = σ_{α}^{δ} σ_{δ}

three times and obtain

σ_{α β γ δ} = σ_{α β γ δ}^{ε} σ_{ε} + σ_{α β γ}^{ε} σ_{ε δ} + σ_{α β δ}^{ε} σ_{ε γ} + σ_{α γ δ}^{ε} σ_{ε β} + σ_{α β}^{ε} σ_{ε γ δ} + σ_{α γ}^{ε} σ_{ε β δ} + σ_{α δ}^{ε} σ_{ε β γ} + σ_{α}^{ε} σ_{ε β γ δ} .

At coincidence this reduces to

[σ_{α β γ δ}] + [σ_{α δ β γ}] + [σ_{α γ β δ}] = 0

. To simplify the third term we differentiate Ricci's identity

σ_{α γ β} = σ_{α β γ} - R_{α β γ}^{ε} σ_{ε}

with respect to

x^{δ}

and then take the coincidence limit. This gives us

[σ_{α γ β δ}] = [σ_{α β γ δ}] + R_{α^{'} δ^{'} β^{'} γ^{'}}

. The same manipulations on the second term give

[σ_{α δ β γ}] = [σ_{α β δ γ}] + R_{α^{'} γ^{'} β^{'} δ^{'}}

. Using the identity

σ_{α β δ γ} = σ_{α β γ δ} - R_{α γ δ}^{ε} σ_{ε β} - R_{β γ δ}^{ε} σ_{α ε}

and the symmetries of the Riemann tensor, it is then easy to show that

[σ_{α β δ γ}] = [σ_{α β γ δ}]

. Gathering the results, we obtain

3 [σ_{α β γ δ}] + R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}} = 0

, and Synge's rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are

\begin{matrix} \begin{matrix} [σ_{α β γ δ}] & = & - \frac{1}{3} (R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}}), \\ [σ_{α β γ δ^{'}}] & = & \frac{1}{3} (R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}}), \\ [σ_{α β γ^{'} δ^{'}}] & = & - \frac{1}{3} (R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}}), \\ [σ_{α β^{'} γ^{'} δ^{'}}] & = & - \frac{1}{3} (R_{α^{'} β^{'} γ^{'} δ^{'}} + R_{α^{'} γ^{'} β^{'} δ^{'}}), \\ [σ_{α^{'} β^{'} γ^{'} δ^{'}}] & = & - \frac{1}{3} (R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}}) . \end{matrix} \end{matrix}

(66)

2.2.2 Derivation of Synge's rule

We begin with any bitensor

Ω_{A B^{'}} (x, x^{'})

in which

A = α . . . β

is a multi-index that represents any number of unprimed indices, and

B^{'} = γ^{'} . . . δ^{'}

a multi-index that represents any number of primed indices. (It does not matter whether the primed and unprimed indices are segregated or mixed.) On the geodesic

β

that links

x

x^{'}

we introduce an ordinary tensor

P^{M} (z)

where

M

is a multi-index that contains the same number of indices as

A

. This tensor is arbitrary, but we assume that it is parallel transported on

β

; this means that it satisfies

P_{; α}^{A} t^{α} = 0

x

. Similarly, we introduce an ordinary tensor

Q^{N} (z)

in which

N

contains the same number of indices as

B^{'}

. This tensor is arbitrary, but we assume that it is parallel transported on

β

; at

x^{'}

it satisfies

Q_{; α^{'}}^{B^{'}} t^{α^{'}} = 0

With

Ω

P

, and

Q

we form a biscalar

H (x, x^{'})

defined by

H (x, x^{'}) = Ω_{A B^{'}} (x, x^{'}) P^{A} (x) Q^{B^{'}} (x^{'}) .

Having specified the geodesic that links

x

x^{'}

, we can consider

H

to be a function of

λ_{0}

and

λ_{1}

. If

λ_{1}

is not much larger than

λ_{0}

(so that

x

is not far from

x^{'}

), we can express

H (λ_{1}, λ_{0})

{H (λ_{1}, λ_{0}) = H (λ_{0}, λ_{0}) + (λ_{1} - λ_{0}) \frac{\partial H}{\partial λ_{1}} |}_{λ_{1} = λ_{0}} + . . . .

Alternatively,

{H (λ_{1}, λ_{0}) = H (λ_{1}, λ_{1}) - (λ_{1} - λ_{0}) \frac{\partial H}{\partial λ_{0}} |}_{λ_{0} = λ_{1}} + . . .,

and these two expressions give

{\frac{d}{d λ_{0}} H (λ_{0}, λ_{0}) = \frac{\partial H}{\partial λ_{0}} |}_{λ_{0} = λ_{1}} + {\frac{\partial H}{\partial λ_{1}} |}_{λ_{1} = λ_{0}},

because the left-hand side is the limit of

[H (λ_{1}, λ_{1}) - H (λ_{0}, λ_{0})] / (λ_{1} - λ_{0})

when

λ_{1} \to λ_{0}

. The partial derivative of

H

with respect to

λ_{0}

is equal to

Ω_{A B^{'}; α^{'}} t^{α^{'}} P^{A} Q^{B^{'}}

, and in the limit this becomes

[Ω_{A B^{'}; α^{'}}] t^{α^{'}} P^{A^{'}} Q^{B^{'}}

. Similarly, the partial derivative of

H

with respect to

λ_{1}

Ω_{A B^{'}; α} t^{α} P^{A} Q^{B^{'}}

, and in the limit

λ_{1} \to λ_{0}

this becomes

[Ω_{A B^{'}; α}] t^{α^{'}} P^{A^{'}} Q^{B^{'}}

. Finally,

H (λ_{0}, λ_{0}) = [Ω_{A B^{'}}] P^{A^{'}} Q^{B^{'}}

, and its derivative with respect to

λ_{0}

[Ω_{A B^{'}}]_{; α^{'}} t^{α^{'}} P^{A^{'}} Q^{B^{'}}

. Gathering the results we find that

{{[Ω_{A B^{'}}]}_{; α^{'}} - [Ω_{A B^{'}; α^{'}}] - [Ω_{A B^{'}; α}]} t^{α^{'}} P^{A^{'}} Q^{B^{'}} = 0,

and the final statement of Synge's rule,

\begin{matrix} {[Ω_{A B^{'}}]}_{; α^{'}} = [Ω_{A B^{'}; α^{'}}] + [Ω_{A B^{'}; α}], \end{matrix}

(67)

follows from the fact that the tensors

P^{M}

and

Q^{N}

, and the direction of the selected geodesic

β

, are all arbitrary. Equation ( 67 ) reduces to Equation ( 64 ) when

σ_{. . .}

is substituted in place of

Ω_{A B^{'}}

2.3 Parallel propagator

2.3.1 Tetrad on $β$

On the geodesic

β

that links

x

x^{'}

we introduce an orthonormal basis

e^{μ} a (z)

that is parallel transported on the geodesic. The frame indices^$3$

a

b

, . . . , run from 0 to 3, and the frame vectors satisfy

\begin{matrix} g_{μ ν} e^{μ} a e^{ν} b = η_{a b}, \frac{D e^{μ} a}{d λ} = 0, \end{matrix}

(68)

where

η_{a b} = diag (- 1, 1, 1, 1)

is the Minkowski metric (which we shall use to raise and lower frame indices). We have the completeness relations

\begin{matrix} g^{μ ν} = η^{a b} e^{μ} a e^{ν} b, \end{matrix}

(69)

and we define a dual tetrad

e^{a} μ (z)

\begin{matrix} e^{a} μ \equiv η^{a b} g_{μ ν} e^{ν} b; \end{matrix}

(70)

this is also parallel transported on

β

. In terms of the dual tetrad the completeness relations take the form

\begin{matrix} g_{μ ν} = η_{a b} e^{a} μ e^{b} ν, \end{matrix}

(71)

and it is easy to show that the tetrad and its dual satisfy

e^{a} μ e^{μ} b = δ_{b}^{a}

and

e^{a} ν e^{μ} a = δ_{ν}^{μ}

. Equations ( 68 , 69 , 70 , 71 ) hold everywhere on

β

. In particular, with an appropriate change of notation they hold at

x^{'}

and

x

; for example,

g_{α β} = η_{a b} e^{a} α e^{b} β

is the metric at

x

^$3$ Note that I use $sans-serif$ symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.

2.3.2 Definition and properties of the parallel propagator

Any vector field

A^{μ} (z)

β

can be decomposed in the basis

e^{μ} a

A^{μ} = A^{a} e^{μ} a

, and the vector's frame components are given by

A^{a} = A^{μ} e^{a} μ

. If

A^{μ}

is parallel transported on the geodesic, then the coefficients

A^{a}

are constants. The vector at

x

can then be expressed as

A^{α} = (A^{α^{'}} e^{a} α^{'}) e^{α} a

, or

\begin{matrix} A^{α} (x) = g_{α^{'}}^{α} (x, x^{'}) A^{α^{'}} (x^{'}), g_{α^{'}}^{α} (x, x^{'}) \equiv e^{α} a (x) e^{a} α^{'} (x^{'}) . \end{matrix}

(72)

The object

g_{α^{'}}^{α} = e^{α} a e^{a} α^{'}

is the parallel propagator : It takes a vector at

x^{'}

and parallel-transports it to

x

along the unique geodesic that links these points.

Similarly, we find that

\begin{matrix} A^{α^{'}} (x^{'}) = g_{α}^{α^{'}} (x^{'}, x) A^{α} (x), g_{α}^{α^{'}} (x^{'}, x) \equiv e^{α^{'}} a (x^{'}) e^{a} α (x), \end{matrix}

(73)

and we see that

g_{α}^{α^{'}} = e^{α^{'}} a e^{a} α

performs the inverse operation: It takes a vector at

x

and parallel-transports it back to

x^{'}

. Clearly,

\begin{matrix} g_{α^{'}}^{α} g_{β}^{α^{'}} = δ_{β}^{α}, g_{α}^{α^{'}} g_{β^{'}}^{α} = δ_{β^{'}}^{α^{'}}, \end{matrix}

(74)

and these relations formally express the fact that

g_{α}^{α^{'}}

is the inverse of

g_{α^{'}}^{α}

The relation

g_{α^{'}}^{α} = e^{α} a e^{a} α^{'}

can also be expressed as

g_{α}^{α^{'}} = e^{a} α e^{α^{'}} a

, and this reveals that

\begin{matrix} g_{α}^{α^{'}} (x, x^{'}) = g_{α}^{α^{'}} (x^{'}, x), g_{α^{'}}^{α} (x^{'}, x) = g_{α^{'}}^{α} (x, x^{'}) . \end{matrix}

(75)

The ordering of the indices, and the ordering of the arguments, are therefore arbitrary.

The action of the parallel propagator on tensors of arbitrary ranks is easy to figure out. For example, suppose that the dual vector

p_{μ} = p_{a} e^{a} μ

is parallel transported on

β

. Then the frame components

p_{a} = p_{μ} e^{μ} a

are constants, and the dual vector at

x

can be expressed as

p_{α} = (p_{α^{'}} e^{α^{'}} a) e^{α} a

, or

\begin{matrix} p_{α} (x) = g_{α}^{α^{'}} (x^{'}, x) p_{α^{'}} (x^{'}) . \end{matrix}

(76)

It is therefore the inverse propagator

g_{α}^{α^{'}}

that takes a dual vector at

x^{'}

and parallel-transports it to

x

. As another example, it is easy to show that a tensor

A^{α β}

x

obtained by parallel transport from

x^{'}

must be given by

\begin{matrix} A^{α β} (x) = g_{α^{'}}^{α} (x, x^{'}) g_{β^{'}}^{β} (x, x^{'}) A^{α^{'} β^{'}} (x^{'}) . \end{matrix}

(77)

Here we need two occurrences of the parallel propagator, one for each tensorial index. Because the metric tensor is covariantly constant, it is automatically parallel transported on

β

, and a special case of Equation ( 77 ) is therefore

g_{α β} = g_{α}^{α^{'}} g_{β}^{β^{'}} g_{α^{'} β^{'}}

Because the basis vectors are parallel transported on

β

, they satisfy

e_{a; β}^{α} σ^{β} = 0

x

and

e_{a; β^{'}}^{α^{'}} σ^{β^{'}} = 0

x^{'}

. This immediately implies that the parallel propagators must satisfy

\begin{matrix} g_{α^{'}; β}^{α} σ^{β} = g_{α^{'}; β^{'}}^{α} σ^{β^{'}} = 0, g_{α; β}^{α^{'}} σ^{β} = g_{α; β^{'}}^{α^{'}} σ^{β^{'}} = 0 . \end{matrix}

(78)

Another useful property of the parallel propagator follows from the fact that if

t^{μ} = d z^{μ} / d λ

is tangent to the geodesic connecting

x

x^{'}

, then

t^{α} = g_{α^{'}}^{α} t^{α^{'}}

. Using Equations ( 55 ) and ( 56 ), this observation gives us the relations

\begin{matrix} σ_{α} = - g_{α}^{α^{'}} σ_{α^{'}}, σ_{α^{'}} = - g_{α^{'}}^{α} σ_{α} . \end{matrix}

(79)

2.3.3 Coincidence limits

Equation ( 72 ) and the completeness relations of Equations ( 69 ) or ( 71 ) imply that

\begin{matrix} [g_{β^{'}}^{α}] = δ_{β^{'}}^{α^{'}} . \end{matrix}

(80)

Other coincidence limits are obtained by differentiation of Equations ( 78 ). For example, the relation

g_{β^{'}; γ}^{α} σ^{γ} = 0

implies

g_{β^{'}; γ δ}^{α} σ^{γ} + g_{β^{'}; γ}^{α} σ_{δ}^{γ} = 0

, and at coincidence we have

\begin{matrix} [g_{β^{'}; γ}^{α}] = [g_{β^{'}; γ^{'}}^{α}] = 0; \end{matrix}

(81)

the second result was obtained by applying Synge's rule on the first result. Further differentiation gives

g_{β^{'}; γ δ ε}^{α} σ^{γ} + g_{β^{'}; γ δ}^{α} σ_{ε}^{γ} + g_{β^{'}; γ ε}^{α} σ_{δ}^{γ} + g_{β^{'}; γ}^{α} σ_{δ ε}^{γ} = 0,

and at coincidence we have

[g_{β^{'}; γ δ}^{α}] + [g_{β^{'}; δ γ}^{α}] = 0

, or

2 [g_{β^{'}; γ δ}^{α}] + R_{β^{'} γ^{'} δ^{'}}^{α^{'}} = 0

. The coincidence limit for

g_{β^{'}; γ δ^{'}}^{α} = g_{β^{'}; δ^{'} γ}^{α}

can then be obtained from Synge's rule, and an additional application of the rule gives

[g_{β^{'}; γ^{'} δ^{'}}^{α}]

. Our results are

\begin{matrix} \begin{matrix} [g_{β^{'}; γ δ}^{α}] & = & - \frac{1}{2} R_{β^{'} γ^{'} δ^{'}}^{α^{'}}, & [g_{β^{'}; γ δ^{'}}^{α}] & = & \frac{1}{2} R_{β^{'} γ^{'} δ^{'}}^{α^{'}}, \\ [g_{β^{'}; γ^{'} δ}^{α}] & = & - \frac{1}{2} R_{β^{'} γ^{'} δ^{'}}^{α^{'}}, & [g_{β^{'}; γ^{'} δ^{'}}^{α}] & = & \frac{1}{2} R_{β^{'} γ^{'} δ^{'}}^{α^{'}} . \end{matrix} \end{matrix}

(82)

2.4 Expansion of bitensors near coincidence

2.4.1 General method

We would like to express a bitensor

Ω_{α^{'} β^{'}} (x, x^{'})

near coincidence as an expansion in powers of

- σ^{α^{'}} (x, x^{'})

, the closest analogue in curved spacetime to the flat-spacetime quantity

(x - x^{'})^{α}

For concreteness we shall consider the case of rank-2 bitensor, and for the moment we will assume that the bitensor's indices all refer to the base point

x^{'}

The expansion we seek is of the form

\begin{matrix} Ω_{α^{'} β^{'}} (x, x^{'}) = A_{α^{'} β^{'}} + A_{α^{'} β^{'} γ^{'}} σ^{γ^{'}} + \frac{1}{2} A_{α^{'} β^{'} γ^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}} + O (ε^{3}), \end{matrix}

(83)

in which the “expansion coefficients”

A_{α^{'} β^{'}}

A_{α^{'} β^{'} γ^{'}}

, and

A_{α^{'} β^{'} γ^{'} δ^{'}}

are all ordinary tensors at

x^{'}

; this last tensor is symmetric in the pair of indices

γ^{'}

and

δ^{'}

, and

ε

measures the size of a typical component of

σ^{α^{'}}

To find the expansion coefficients we differentiate Equation ( 83 ) repeatedly and take coincidence limits. Equation ( 83 ) immediately implies

[Ω_{α^{'} β^{'}}] = A_{α^{'} β^{'}}

. After one differentiation we obtain

Ω_{α^{'} β^{'}; γ^{'}} = A_{α^{'} β^{'}; γ^{'}} + A_{α^{'} β^{'} ε^{'}; γ^{'}} σ^{ε^{'}} + A_{α^{'} β^{'} ε^{'}} σ_{γ^{'}}^{ε^{'}} + \frac{1}{2} A_{α^{'} β^{'} ε^{'} ι^{'}; γ^{'}} σ^{ε^{'}} σ^{ι^{'}} + A_{α^{'} β^{'} ε^{'} ι^{'}} σ^{ε^{'}} σ_{γ^{'}}^{ι^{'}} + O (ε^{2})

, and at coincidence this reduces to

[Ω_{α^{'} β^{'}; γ^{'}}] = A_{α^{'} β^{'}; γ^{'}} + A_{α^{'} β^{'} γ^{'}}

. Taking the coincidence limit after two differentiations yields

[Ω_{α^{'} β^{'}; γ^{'} δ^{'}}] = A_{α^{'} β^{'}; γ^{'} δ^{'}} + A_{α^{'} β^{'} γ^{'}; δ^{'}} + A_{α^{'} β^{'} δ^{'}; γ^{'}} + A_{α^{'} β^{'} γ^{'} δ^{'}}

. The expansion coefficients are therefore

\begin{matrix} \begin{matrix} A_{α^{'} β^{'}} & = & [Ω_{α^{'} β^{'}}], \\ A_{α^{'} β^{'} γ^{'}} & = & [Ω_{α^{'} β^{'}; γ^{'}}] - A_{α^{'} β^{'}; γ^{'}}, \\ A_{α^{'} β^{'} γ^{'} δ^{'}} & = & [Ω_{α^{'} β^{'}; γ^{'} δ^{'}}] - A_{α^{'} β^{'}; γ^{'} δ^{'}} - A_{α^{'} β^{'} γ^{'}; δ^{'}} - A_{α^{'} β^{'} δ^{'}; γ^{'}} . \end{matrix} \end{matrix}

(84)

These results are to be substituted into Equation ( 83 ), and this gives us

Ω_{α^{'} β^{'}} (x, x^{'})

to second order in

ε

Suppose now that the bitensor is

Ω_{α^{'} β}

, with one index referring to

x^{'}

and the other to

x

. The previous procedure can be applied directly if we introduce an auxiliary bitensor

{\tilde{Ω}}_{α^{'} β^{'}} \equiv g_{β^{'}}^{β} Ω_{α^{'} β}

whose indices all refer to the point

x^{'}

. Then

{\tilde{Ω}}_{α^{'} β^{'}}

can be expanded as in Equation ( 83 ), and the original bitensor is reconstructed as

Ω_{α^{'} β} = g_{β}^{β^{'}} {\tilde{Ω}}_{α^{'} β^{'}}

, or

\begin{matrix} Ω_{α^{'} β} (x, x^{'}) = g_{β}^{β^{'}} (B_{α^{'} β^{'}} + B_{α^{'} β^{'} γ^{'}} σ^{γ^{'}} + \frac{1}{2} B_{α^{'} β^{'} γ^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}}) + O (ε^{3}) . \end{matrix}

(85)

The expansion coefficients can be obtained from the coincidence limits of

{\tilde{Ω}}_{α^{'} β^{'}}

and its derivatives.

It is convenient, however, to express them directly in terms of the original bitensor

Ω_{α^{'} β}

by substituting the relation

{\tilde{Ω}}_{α^{'} β^{'}} = g_{β^{'}}^{β} Ω_{α^{'} β}

and its derivatives. After using the results of Equation ( 80 , 81 , 82 ) we find

\begin{matrix} \begin{matrix} B_{α^{'} β^{'}} & = & [Ω_{α^{'} β}], \\ B_{α^{'} β^{'} γ^{'}} & = & [Ω_{α^{'} β; γ^{'}}] - B_{α^{'} β^{'}; γ^{'}}, \\ B_{α^{'} β^{'} γ^{'} δ^{'}} & = & [Ω_{α^{'} β; γ^{'} δ^{'}}] + \frac{1}{2} B_{α^{'} ε^{'}} R_{β^{'} γ^{'} δ^{'}}^{ε^{'}} - B_{α^{'} β^{'}; γ^{'} δ^{'}} - B_{α^{'} β^{'} γ^{'}; δ^{'}} - B_{α^{'} β^{'} δ^{'}; γ^{'}} . \end{matrix} \end{matrix}

(86)

The only difference with respect to Equation ( 85 ) is the presence of a Riemann-tensor term in

B_{α^{'} β^{'} γ^{'} δ^{'}}

Suppose finally that the bitensor to be expanded is

Ω_{α β}

, whose indices all refer to

x

. Much as we did before, we introduce an auxiliary bitensor

{\tilde{Ω}}_{α^{'} β^{'}} = g_{α^{'}}^{α} g_{β^{'}}^{β} Ω_{α β}

whose indices all refer to

x^{'}

, we expand

{\tilde{Ω}}_{α^{'} β^{'}}

as in Equation ( 83 ), and we then reconstruct the original bitensor. This gives us

\begin{matrix} Ω_{α β} (x, x^{'}) = g_{α}^{α^{'}} g_{β}^{β^{'}} (C_{α^{'} β^{'}} + C_{α^{'} β^{'} γ^{'}} σ^{γ^{'}} + \frac{1}{2} C_{α^{'} β^{'} γ^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}}) + O (ε^{3}), \end{matrix}

(87)

and the expansion coefficients are now

\begin{matrix} C_{α^{'} β^{'}} & = & [Ω_{α β}], \end{matrix}

\begin{matrix} C_{α^{'} β^{'} γ^{'}} & = & [Ω_{α β; γ^{'}}] - C_{α^{'} β^{'}; γ^{'}}, \end{matrix}

\begin{matrix} C_{α^{'} β^{'} γ^{'} δ^{'}} & = & [Ω_{α β; γ^{'} δ^{'}}] + \frac{1}{2} C_{α^{'} ε^{'}} R_{β^{'} γ^{'} δ^{'}}^{ε^{'}} + \frac{1}{2} C_{ε^{'} β^{'}} R_{α^{'} γ^{'} δ^{'}}^{ε^{'}} - C_{α^{'} β^{'}; γ^{'} δ^{'}} - C_{α^{'} β^{'} γ^{'}; δ^{'}} - C_{α^{'} β^{'} δ^{'}; γ^{'}} . \end{matrix}

This differs from Equation ( 86 ) by the presence of an additional Riemann-tensor term in

C_{α^{'} β^{'} γ^{'} δ^{'}}

2.4.2 Special cases

We now apply the general expansion method developed in the preceding Section 2.4.1 to the bitensors

σ_{α^{'} β^{'}}

σ_{α^{'} β}

, and

σ_{α β}

. In the first instance we have

A_{α^{'} β^{'}} = g_{α^{'} β^{'}}

A_{α^{'} β^{'} γ^{'}} = 0

, and

A_{α^{'} β^{'} γ^{'} δ^{'}} = - \frac{1}{3} (R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}})

. In the second instance we have

B_{α^{'} β^{'}} = - g_{α^{'} β^{'}}

B_{α^{'} β^{'} γ^{'}} = 0

, and

B_{α^{'} β^{'} γ^{'} δ^{'}} = - \frac{1}{3} (R_{β^{'} α^{'} γ^{'} δ^{'}} + R_{β^{'} γ^{'} α^{'} δ^{'}}) - \frac{1}{2} R_{α^{'} β^{'} γ^{'} δ^{'}} = - \frac{1}{3} R_{α^{'} δ^{'} β^{'} γ^{'}} - \frac{1}{6} R_{α^{'} β^{'} γ^{'} δ^{'}}

. In the third instance we have

C_{α^{'} β^{'}} = g_{α^{'} β^{'}}

C_{α^{'} β^{'} γ^{'}} = 0

, and

C_{α^{'} β^{'} γ^{'} δ^{'}} = - \frac{1}{3} (R_{α^{'} γ^{'} β^{'} δ^{'}} + R_{α^{'} δ^{'} β^{'} γ^{'}})

. This gives us the expansions

\begin{matrix} σ_{α^{'} β^{'}} & = & g_{α^{'} β^{'}} - \frac{1}{3} R_{α^{'} γ^{'} β^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}} + O (ε^{3}), \end{matrix}

(88)

\begin{matrix} σ_{α^{'} β} & = & - g_{β}^{β^{'}} (g_{α^{'} β^{'}} + \frac{1}{6} R_{α^{'} γ^{'} β^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}}) + O (ε^{3}), \end{matrix}

(89)

\begin{matrix} σ_{α β} & = & g_{α}^{α^{'}} g_{β^{'}}^{β^{'}} (g_{α^{'} β^{'}} - \frac{1}{3} R_{α^{'} γ^{'} β^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}}) + O (ε^{3}) . \end{matrix}

(90)

Taking the trace of the last equation returns

σ_{α}^{α} = 4 - \frac{1}{3} R_{γ^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}} + O (ε^{3})

, or

\begin{matrix} θ^{*} = 3 - \frac{1}{3} R_{α^{'} β^{'}} σ^{α^{'}} σ^{β^{'}} + O (ε^{3}), \end{matrix}

(91)

where

θ^{*} \equiv σ_{α}^{α} - 1

was shown in Section 2.1.4 to describe the expansion of the congruence of geodesics that emanate from

x^{'}

. Equation ( 91 ) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: When

R_{α^{'} β^{'}} σ^{α^{'}} σ^{β^{'}} > 0

we see that

θ^{*}

is smaller than 3, the value it would take in flat spacetime.

The expansion method can easily be extended to bitensors of other tensorial ranks. In particular, it can be adapted to give expansions of the first derivatives of the parallel propagator. The expansions

\begin{matrix} g_{β^{'}; γ^{'}}^{α} = \frac{1}{2} g_{α^{'}}^{α} R_{β^{'} γ^{'} δ^{'}}^{α^{'}} σ^{δ^{'}} + O (ε^{2}), g_{β^{'}; γ}^{α} = \frac{1}{2} g_{α^{'}}^{α} g_{γ}^{γ^{'}} R_{β^{'} γ^{'} δ^{'}}^{α^{'}} σ^{δ^{'}} + O (ε^{2}) \end{matrix}

(92)

and thus easy to establish, and they will be needed in Section 4 of this review.

2.4.3 Expansion of tensors

The expansion method can also be applied to ordinary tensor fields. For concreteness, suppose that we wish to express a rank-2 tensor

A_{α β}

at a point

x

in terms of its values (and that of its covariant derivatives) at a neighbouring point

x^{'}

. The tensor can be written as an expansion in powers of

- σ^{α^{'}} (x, x^{'})

, and in this case we have

\begin{matrix} A_{α β} (x) = g_{α}^{α^{'}} g_{β}^{β^{'}} (A_{α^{'} β^{'}} - A_{α^{'} β^{'}; γ^{'}} σ^{γ^{'}} + \frac{1}{2} A_{α^{'} β^{'}; γ^{'} δ^{'}} σ^{γ^{'}} σ^{δ^{'}}) + O (ε^{3}) . \end{matrix}

(93)

If the tensor field is parallel transported on the geodesic

β

that links

x

x^{'}

, then Equation ( 93 ) reduces to Equation ( 77 ). The extension of this formula to tensors of other ranks is obvious.

To derive this result we express

A_{μ ν} (z)

, the restriction of the tensor field on

β

, in terms of its tetrad components

A_{a b} (λ) = A_{μ ν} e^{μ} a e^{ν} b

. Recall from Section 2.3.1 that

e^{μ} a

is an orthonormal basis that is parallel transported on

β

; recall also that the affine parameter

λ

ranges from

λ_{0}

(its value at

x^{'}

) to

λ_{1}

(its value at

x

). We have

A_{α^{'} β^{'}} (x^{'}) = A_{a b} (λ_{0}) e^{a} α^{'} e^{b} β^{'}

A_{α β} (x) = A_{a b} (λ_{1}) e^{a} α e^{b} β

, and

A_{a b} (λ_{1})

can be expressed in terms of quantities at

λ = λ_{0}

by straightforward Taylor expansion. Since, for example,

(λ_{1} - λ_{0}) {\frac{d A_{a b}}{d λ} |}_{λ_{0}} = {(λ_{1} - λ_{0}) {(A_{μ ν} e^{μ} a e^{ν} b)}_{; λ} t^{λ} |}_{λ_{0}} = {(λ_{1} - λ_{0}) A_{μ ν; λ} e^{μ} a e^{ν} b t^{λ} |}_{λ_{0}} = - A_{α^{'} β^{'}; γ^{'}} e^{α^{'}} a e^{β^{'}} b σ^{γ^{'}},

where we have used Equation ( 56 ), we arrive at Equation ( 93 ) after involving Equation ( 73 ).

2.5 Van Vleck determinant

2.5.1 Definition and properties

The van Vleck biscalar

Δ (x, x^{'})

is defined by

\begin{matrix} Δ (x, x^{'}) \equiv det [Δ_{β^{'}}^{α^{'}} (x, x^{'})], Δ_{β^{'}}^{α^{'}} (x, x^{'}) \equiv - g_{α}^{α^{'}} (x^{'}, x) σ_{β^{'}}^{α} (x, x^{'}) . \end{matrix}

(94)

As we shall show below, it can also be expressed as

\begin{matrix} Δ (x, x^{'}) = - \frac{det [- σ_{α β^{'}} (x, x^{'})]}{\sqrt{- g} \sqrt{- g^{'}}}, \end{matrix}

(95)

where

g

is the metric determinant at

x

and

g^{'}

the metric determinant at

x^{'}

Equations ( 63 ) and ( 80 ) imply that at coincidence,

[Δ_{β^{'}}^{α^{'}}] = δ_{β^{'}}^{α^{'}}

and

[Δ] = 1

. Equation ( 89 ), on the other hand, implies that near coincidence

\begin{matrix} Δ_{β^{'}}^{α^{'}} = δ_{β^{'}}^{α^{'}} + \frac{1}{6} R_{γ^{'} β^{'} δ^{'}}^{α^{'}} σ^{γ^{'}} σ^{δ^{'}} + O (ε^{3}), \end{matrix}

(96)

so that

\begin{matrix} Δ = 1 + \frac{1}{6} R_{α^{'} β^{'}} σ^{α^{'}} σ^{β^{'}} + O (ε^{3}) . \end{matrix}

(97)

This last result follows from the fact that for a “small” matrix

a

det (1 + a) = 1 + tr (a) + O (a^{2})

We shall prove below that the van Vleck determinant satisfies the differential equation

\begin{matrix} \frac{1}{Δ} {(Δ σ^{α})}_{; α} = 4, \end{matrix}

(98)

which can also be written as

(ln Δ)_{, α} σ^{α} = 4 - σ_{α}^{α}

, or

\begin{matrix} \frac{d}{d λ^{*}} (ln Δ) = 3 - θ^{*} \end{matrix}

(99)

in the notation introduced in Section 2.1.4 . Equation ( 99 ) reveals that the behaviour of the van Vleck determinant is governed by the expansion of the congruence of geodesics that emanate from

x^{'}

. If

θ^{*} < 3

, then the congruence expands less rapidly than it would in flat spacetime, and

Δ

increases along the geodesics. If, on the other hand,

θ^{*} > 3

, then the congruence expands more rapidly than it would in flat spacetime, and

Δ

decreases along the geodesics. Thus,

Δ > 1

indicates that the geodesics are undergoing focusing, while

Δ < 1

indicates that the geodesics are undergoing defocusing. The connection between the van Vleck determinant and the strong energy condition is well illustrated by Equation ( 97 ): The sign of

Δ - 1

near

x^{'}

is determined by the sign of

R_{α^{'} β^{'}} σ^{α^{'}} σ^{β^{'}}

2.5.2 Derivations

To show that Equation ( 95 ) follows from Equation ( 94 ) we rewrite the completeness relations at

x

g^{α β} = η^{a b} e^{α} a e^{β} b

, in the matrix form

g^{- 1} = E η E^{T}

, where

E

denotes the

4 \times 4

matrix whose entries correspond to

e^{α} a

. (In this translation we put tensor and frame indices on equal footing.) With

e

denoting the determinant of this matrix, we have

1 / g = - e^{2}

, or

e = 1 / \sqrt{- g}

. Similarly, we rewrite the completeness relations at

x^{'}

g^{α^{'} β^{'}} = η^{a b} e^{α^{'}} a e^{β^{'}} b

, in the matrix form

{g^{'}}^{- 1} = E^{'} η {E^{'}}^{T}

, where

E^{'}

is the matrix corresponding to

e^{α^{'}} a

. With

e^{'}

denoting its determinant, we have

1 / g^{'} = - e^{' 2}

, or

e^{'} = 1 / \sqrt{- g^{'}}

. Now, the parallel propagator is defined by

g_{α^{'}}^{α} = η^{a b} g_{α^{'} β^{'}} e^{α} a e^{β^{'}} b

, and the matrix form of this equation is

\hat{g} = E η {E^{'}}^{T} {g^{'}}^{T}

. The determinant of the parallel propagator is therefore

\hat{g} = - e e^{'} g^{'} = \sqrt{- g^{'}} / \sqrt{- g}

. So we have

\begin{matrix} det [g_{α^{'}}^{α}] = \frac{\sqrt{- g^{'}}}{\sqrt{- g}}, det [g_{α}^{α^{'}}] = \frac{\sqrt{- g}}{\sqrt{- g^{'}}}, \end{matrix}

(100)

and Equation ( 95 ) follows from the fact that the matrix form of Equation ( 94 ) is

Δ = - {\hat{g}}^{- 1} g^{- 1} σ

, where

σ

is the matrix corresponding to

σ_{α β^{'}}

To establish Equation ( 98 ) we differentiate the relation

σ = \frac{1}{2} σ^{γ} σ_{γ}

twice and obtain

σ_{α β^{'}} = σ_{α}^{γ} σ_{γ β^{'}} + σ^{γ} σ_{γ α β^{'}}

. If we replace the last factor by

σ_{α β^{'} γ}

and multiply both sides by

- g^{α^{'} α}

we find

Δ_{β^{'}}^{α^{'}} = - g^{α^{'} α} (σ_{α}^{γ} σ_{γ β^{'}} + σ^{γ} σ_{α β^{'} γ}) .

In this expression we make the substitution

σ_{α β^{'}} = - g_{α α^{'}} Δ_{β^{'}}^{α^{'}}

, which follows directly from Equation ( 94 ). This gives us

\begin{matrix} Δ_{β^{'}}^{α^{'}} = g_{α}^{α^{'}} g_{γ^{'}}^{γ} σ_{γ}^{α} Δ_{β^{'}}^{γ^{'}} + Δ_{β^{'}; γ}^{α^{'}} σ^{γ}, \end{matrix}

(101)

where we have used Equation ( 78 ). At this stage we introduce an inverse

(Δ^{- 1})_{β^{'}}^{α^{'}}

to the van Vleck bitensor, defined by

Δ_{β^{'}}^{α^{'}} (Δ^{- 1})_{γ^{'}}^{β^{'}} = δ_{γ^{'}}^{α^{'}}

. After multiplying both sides of Equation ( 101 ) by

(Δ^{- 1})_{γ^{'}}^{β^{'}}

we find

δ_{β^{'}}^{α^{'}} = g_{α}^{α^{'}} g_{β^{'}}^{β} σ_{β}^{α} + (Δ^{- 1})_{β^{'}}^{γ^{'}} Δ_{γ^{'}; γ}^{α^{'}} σ^{γ},

and taking the trace of this equation yields

4 = σ_{α}^{α} + (Δ^{- 1})_{α^{'}}^{β^{'}} Δ_{β^{'}; γ}^{α^{'}} σ^{γ} .

We now recall the identity

δ ln det M = tr (M^{- 1} δ M)

, which relates the variation of a determinant to the variation of the matrix elements. It implies, in particular, that

(Δ^{- 1})_{α^{'}}^{β^{'}} Δ_{β^{'}; γ}^{α^{'}} = (ln Δ)_{, γ}

, and we finally obtain

\begin{matrix} 4 = σ_{α}^{α} + (ln Δ)_{, α} σ^{α}, \end{matrix}

(102)

which is equivalent to Equation ( 98 ) or Equation ( 99 ).

3 Coordinate Systems

3.1 Riemann normal coordinates

3.1.1 Definition and coordinate transformation

Given a fixed base point

x^{'}

and a tetrad

e^{α^{'}} a (x^{'})

, we assign to a neighbouring point

x

the four coordinates

\begin{matrix} {\hat{x}}^{a} = - e^{a} α^{'} (x^{'}) σ^{α^{'}} (x, x^{'}), \end{matrix}

(103)

where

e^{a} α^{'} = η^{a b} g_{α^{'} β^{'}} e^{β^{'}} b

is the dual tetrad attached to

x^{'}

. The new coordinates

{\hat{x}}^{a}

are called Riemann normal coordinates (RNC), and they are such that

η_{a b} {\hat{x}}^{a} {\hat{x}}^{b} = η_{a b} e^{a} α^{'} e^{b} β^{'} σ^{α^{'}} σ^{β^{'}} = g_{α^{'} β^{'}} σ^{α^{'}} σ^{β^{'}}

, or

\begin{matrix} η_{a b} {\hat{x}}^{a} {\hat{x}}^{b} = 2 σ (x, x^{'}) . \end{matrix}

(104)

Thus,

η_{a b} {\hat{x}}^{a} {\hat{x}}^{b}

is the squared geodesic distance between

x

and the base point

x^{'}

. It is obvious that

x^{'}

is at the origin of the RNC, where

{\hat{x}}^{a} = 0

If we move the point

x

x + δ x

, the new coordinates change to

{\hat{x}}^{a} + δ {\hat{x}}^{a} = - e^{a} α^{'} σ^{α^{'}} (x + δ x, x^{'}) = {\hat{x}}^{a} - e^{a} α^{'} σ_{β}^{α^{'}} δ x^{β}

, so that

\begin{matrix} d {\hat{x}}^{a} = - e^{a} α^{'} σ_{β}^{α^{'}} d x^{β} . \end{matrix}

(105)

The coordinate transformation is therefore determined by

\partial {\hat{x}}^{a} / \partial x^{β} = - e^{a} α^{'} σ_{β}^{α^{'}}

, and at coincidence we have

\begin{matrix} [\frac{\partial {\hat{x}}^{a}}{\partial x^{α}}] = e^{a} α^{'}, [\frac{\partial x^{α}}{\partial {\hat{x}}^{a}}] = e^{α^{'}} a; \end{matrix}

(106)

the second result follows from the identities

e^{a} α^{'} e^{α^{'}} b = δ_{b}^{a}

and

e^{α^{'}} a e^{a} β^{'} = δ_{β^{'}}^{α^{'}}

It is interesting to note that the Jacobian of the transformation of Equation ( 105 ),

J \equiv det (\partial {\hat{x}}^{a} / \partial x^{β})

, is given by

J = \sqrt{- g} Δ (x, x^{'})

, where

g

is the determinant of the metric in the original coordinates, and

Δ (x, x^{'})

is the van Vleck determinant of Equation ( 95 ). This result follows simply by writing the coordinate transformation in the form

\partial {\hat{x}}^{a} / \partial x^{β} = - η^{a b} e^{α^{'}} b σ_{α^{'} β}

and computing the product of the determinants. It allows us to deduce that in the RNC, the determinant of the metric is given by

\begin{matrix} \sqrt{- g (RNC)} = \frac{1}{Δ (x, x^{'})} . \end{matrix}

(107)

It is easy to show that the geodesics emanating from

x^{'}

are straight lines in the RNC. The proper volume of a small comoving region is then equal to

d V = Δ^{- 1} d^{4} \hat{x}

, and this is smaller than the flat-spacetime value of

d^{4} \hat{x}

Δ > 1

, that is, if the geodesics are focused by the spacetime curvature.

3.1.2 Metric near $x^{'}$

We now would like to invert Equation ( 105 ) in order to express the line element

d s^{2} = g_{α β} d x^{α} d x^{β}

in terms of the displacements

d {\hat{x}}^{a}

. We shall do this approximately, by working in a small neighbourhood of

x^{'}

. We recall the expansion of Equation ( 89 ),

σ_{β}^{α^{'}} = - g_{β}^{β^{'}} (δ_{β^{'}}^{α^{'}} + \frac{1}{6} R_{γ^{'} β^{'} δ^{'}}^{α^{'}} σ^{γ^{'}} σ^{δ^{'}}) + O (ε^{3}),

and in this we substitute the frame decomposition of the Riemann tensor,

R_{γ^{'} β^{'} δ^{'}}^{α^{'}} = R_{c b d}^{a} e^{α^{'}} a e^{c} γ^{'} e^{b} β^{'} e^{d} δ^{'}

, and the tetrad decomposition of the parallel propagator,

g_{β}^{β^{'}} = e^{β^{'}} b e^{b} β

, where

e^{b} β (x)

is the dual tetrad at

x

obtained by parallel transport of

e^{b} β^{'} (x^{'})

. After some algebra we obtain

σ_{β}^{α^{'}} = - e^{α^{'}} a e^{a} β - \frac{1}{6} R_{c b d}^{a} e^{α^{'}} a e^{b} β {\hat{x}}^{c} {\hat{x}}^{d} + O (ε^{3}),

where we have used Equation ( 103 ). Substituting this into Equation ( 105 ) yields

\begin{matrix} d {\hat{x}}^{a} = [δ_{b}^{a} + \frac{1}{6} R_{c b d}^{a} {\hat{x}}^{c} {\hat{x}}^{d} + O (x^{3})] e^{b} β d x^{β}, \end{matrix}

(108)

and this is easily inverted to give

\begin{matrix} e^{a} α d x^{α} = [δ_{b}^{a} - \frac{1}{6} R_{c b d}^{a} {\hat{x}}^{c} {\hat{x}}^{d} + O (x^{3})] d {\hat{x}}^{b} . \end{matrix}

(109)

This is the desired approximate inversion of Equation ( 105 ). It is useful to note that Equation ( 109 ), when specialized from the arbitrary coordinates

x^{α}

{\hat{x}}^{a}

, gives us the components of the dual tetrad at

x

in the RNC. We are now in a position to calculate the metric in the new coordinates. We have

d s^{2} = g_{α β} d x^{α} d x^{β} = (η_{a b} e^{a} α e^{b} β) d x^{α} d x^{β} = η_{a b} (e^{a} α d x^{α}) (e^{b} β d x^{β})

, and in this we substitute Equation ( 109 ).

The final result is

d s^{2} = g_{a b} d {\hat{x}}^{a} d {\hat{x}}^{b}

, with

\begin{matrix} g_{a b} = η_{a b} - \frac{1}{3} R_{a c b d} {\hat{x}}^{c} {\hat{x}}^{d} + O (x^{3}) . \end{matrix}

(110)

The quantities

R_{a c b d}

appearing in Equation ( 110 ) are the frame components of the Riemann tensor evaluated at the base point

x^{'}

\begin{matrix} R_{a c b d} = R_{α^{'} γ^{'} β^{'} δ^{'}} e^{α^{'}} a e^{γ^{'}} c e^{β^{'}} b e^{δ^{'}} d, \end{matrix}

(111)

and these are independent of

{\hat{x}}^{a}

. They are also, by virtue of Equation ( 106 ), the components of the (base-point) Riemann tensor in the RNC, because Equation ( 111 ) can also be expressed as

R_{a c d b} = R_{α^{'} γ^{'} β^{'} δ^{'}} [\frac{\partial x^{α}}{\partial {\hat{x}}^{a}}] [\frac{\partial x^{γ}}{\partial {\hat{x}}^{c}}] [\frac{\partial x^{β}}{\partial {\hat{x}}^{b}}] [\frac{\partial x^{δ}}{\partial {\hat{x}}^{d}}],

which is the standard transformation law for tensor components.

It is obvious from Equation ( 110 ) that

g_{a b} (x^{'}) = η_{a b}

and

Γ_{b c}^{a} (x^{'}) = 0

, where

Γ_{b c}^{a} = - \frac{1}{3} (R_{b c d}^{a} + R_{c b d}^{a}) {\hat{x}}^{d} + O (x^{2})

is the connection compatible with the metric

g_{a b}

. The Riemann normal coordinates therefore provide a constructive proof of the local flatness theorem.

3.2 Fermi normal coordinates

3.2.1 Fermi–Walker transport

Let

γ

be a timelike curve described by parametric relations

z^{μ} (τ)

in which

τ

is proper time. Let

u^{μ} = d z^{μ} / d τ

be the curve's normalized tangent vector, and let

a^{μ} = D u^{μ} / d τ

be its acceleration vector.

A vector field

v^{μ}

is said to be Fermi–Walker transported on

γ

if it is a solution to the differential equation

\begin{matrix} \frac{D v^{μ}}{d τ} = (v_{ν} a^{ν}) u^{μ} - (v_{ν} u^{ν}) a^{μ} . \end{matrix}

(112)

Notice that this reduces to parallel transport if

a^{μ} = 0

and

γ

is a geodesic.

The operation of Fermi–Walker (FW) transport satisfies two important properties. The first is that

u^{μ}

is automatically FW transported along

γ

; this follows at once from Equation ( 112 ) and the fact that

u^{μ}

is orthogonal to

a^{μ}

. The second is that if the vectors

v^{μ}

and

w^{μ}

are both FW transported along

γ

, then their inner product

v_{μ} w^{μ}

is constant on

γ

D (v_{μ} w^{μ}) / d τ = 0

; this also follows immediately from Equation ( 112 ).

3.2.2 Tetrad and dual tetrad on $γ$

Let

\bar{z}

be an arbitrary reference point on

γ

. At this point we erect an orthonormal tetrad

(u^{\bar{μ}}, e^{\bar{μ}} a)

where, contrary to former usage, the frame index

a

runs from 1 to 3. We then propagate each frame vector on

γ

by FW transport; this guarantees that the tetrad remains orthonormal everywhere on

γ

. At a generic point

z (τ)

we have

\begin{matrix} \frac{D e^{μ} a}{d τ} = (a_{ν} e^{ν} a) u^{μ}, g_{μ ν} u^{μ} u^{ν} = - 1, g_{μ ν} e^{μ} a u^{ν} = 0, g_{μ ν} e^{μ} a e^{ν} b = δ_{a b} . \end{matrix}

(113)

From the tetrad on

γ

we define a dual tetrad

(e^{0} μ, e^{a} μ)

by the relations

\begin{matrix} e^{0} μ = - u_{μ}, e^{a} μ = δ^{a b} g_{μ ν} e^{ν} b; \end{matrix}

(114)

this is also FW transported on

γ

. The tetrad and its dual give rise to the completeness relations

\begin{matrix} g^{μ ν} = - u^{μ} u^{ν} + δ^{a b} e^{μ} a e^{ν} b, g_{μ ν} = - e^{0} μ e^{0} ν + δ_{a b} e^{a} μ e^{b} ν . \end{matrix}

(115)

3.2.3 Fermi normal coordinates

To construct the Fermi normal coordinates (FNC) of a point

x

in the normal convex neighbourhood of

γ

, we locate the unique spacelike geodesic

β

that passes through

x

and intersects

γ

orthogonally.

We denote the intersection point by

\bar{x} \equiv z (t)

, with

t

denoting the value of the proper-time parameter at this point. To tensors at

\bar{x}

we assign indices

\bar{α}

\bar{β}

, and so on. The FNC of

x

are defined by

\begin{matrix} {\hat{x}}^{0} = t, {\hat{x}}^{a} = - e^{a} \bar{α} (\bar{x}) σ^{\bar{α}} (x, \bar{x}), σ_{\bar{α}} (x, \bar{x}) u^{\bar{α}} (\bar{x}) = 0; \end{matrix}

(116)

the last statement determines

\bar{x}

from the requirement that

- σ^{\bar{α}}

, the vector tangent to

β

\bar{x}

, be orthogonal to

u^{\bar{α}}

, the vector tangent to

γ

. From the definition of the FNC and the completeness relations of Equation ( 115 ) it follows that

\begin{matrix} s^{2} \equiv δ_{a b} {\hat{x}}^{a} {\hat{x}}^{b} = 2 σ (x, \bar{x}), \end{matrix}

(117)

so that

s

is the spatial distance between

\bar{x}

and

x

along the geodesic

β

. This statement gives an immediate meaning to

{\hat{x}}^{a}

, the spatial Fermi normal coordinates; and the time coordinate

{\hat{x}}^{0}

is simply proper time at the intersection point

\bar{x}

. The situation is illustrated in Figure 6 .

Figure 6 : Fermi normal coordinates of a point

x

relative to a world line

γ

. The time coordinate

t

selects a particular point on the word line, and the disk represents the set of spacelike geodesics that intersect

γ

orthogonally at

z (t)

. The unit vector

ω^{a} \equiv {\hat{x}}^{a} / s

selects a particular geodesic among this set, and the spatial distance

s

selects a particular point on this geodesic.

Suppose that

x

is moved to

x + δ x

. This typically induces a change in the spacelike geodesic

β

, which moves to

β + δ β

, and a corresponding change in the intersection point

\bar{x}

, which moves to

x^{''} \equiv \bar{x} + δ \bar{x}

, with

δ x^{\bar{α}} = u^{\bar{α}} δ t

. The FNC of the new point are then

{\hat{x}}^{0} (x + δ x) = t + δ t

and

{\hat{x}}^{a} (x + δ x) = - e^{a} α^{''} (x^{''}) σ^{α^{''}} (x + δ x, x^{''})

, with

x^{''}

determined by

σ_{α^{''}} (x + δ x, x^{''}) u^{α^{''}} (x^{''}) = 0

Expanding these relations to first order in the displacements, and simplifying using Equations ( 113 ), yields

\begin{matrix} d t = μ σ_{\bar{α} β} u^{\bar{α}} d x^{β}, d {\hat{x}}^{a} = - e^{a} \bar{α} (σ_{β}^{\bar{α}} + μ σ_{\bar{β}}^{\bar{α}} u^{\bar{β}} σ_{β \bar{γ}} u^{\bar{γ}}) d x^{β}, \end{matrix}

(118)

where

μ

is determined by

μ^{- 1} = - (σ_{\bar{α} \bar{β}} u^{\bar{α}} u^{\bar{β}} + σ_{\bar{α}} a^{\bar{α}})

3.2.4 Coordinate displacements near $γ$

The relations of Equation ( 118 ) can be expressed as expansions in powers of

s

, the spatial distance from

\bar{x}

x

. For this we use the expansions of Equations ( 88 ) and ( 89 ), in which we substitute

σ^{\bar{α}} = - e^{\bar{α}} a {\hat{x}}^{a}

and

g_{α}^{\bar{α}} = u^{\bar{α}} {\bar{e}}_{α}^{0} + e^{\bar{α}} a {\bar{e}}_{α}^{a}

, where

({\bar{e}}_{α}^{0}, {\bar{e}}_{α}^{a})

is a dual tetrad at

x

obtained by parallel transport of

(- u_{\bar{α}}, e^{a} \bar{α})

on the spacelike geodesic

β

. After some algebra we obtain

μ^{- 1} = 1 + a_{a} {\hat{x}}^{a} + \frac{1}{3} R_{0 c 0 d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3}),

where

a_{a} (t) \equiv a_{\bar{α}} e^{\bar{α}} a

are frame components of the acceleration vector, and

R_{0 c 0 d} (t) \equiv R_{\bar{α} \bar{γ} \bar{β} \bar{δ}} u^{\bar{α}} e^{\bar{γ}} c u^{\bar{β}} e^{\bar{δ}} d

are frame components of the Riemann tensor evaluated on

γ

. This last result is easily inverted to give

μ = 1 - a_{a} {\hat{x}}^{a} + {(a_{a} {\hat{x}}^{a})}^{2} - \frac{1}{3} R_{0 c 0 d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3}) .

Proceeding similarly for the other relations of Equation ( 118 ), we obtain

\begin{matrix} d t = [1 - a_{a} {\hat{x}}^{a} + {(a_{a} {\hat{x}}^{a})}^{2} - \frac{1}{2} R_{0 c 0 d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] ({\bar{e}}_{β}^{0} d x^{β}) + [- \frac{1}{6} R_{0 c b d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] ({\bar{e}}_{β}^{b} d x^{β}) \end{matrix}

\begin{matrix}  \end{matrix}

(119)

and

\begin{matrix} d {\hat{x}}^{a} = [\frac{1}{2} R_{c 0 d}^{a} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] ({\bar{e}}_{β}^{0} d x^{β}) + [δ_{b}^{a} + \frac{1}{6} R_{c b d}^{a} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] ({\bar{e}}_{β}^{b} d x^{β}), \end{matrix}

(120)

where

R_{a c 0 d} (t) \equiv R_{\bar{α} \bar{γ} \bar{β} \bar{δ}} e^{\bar{α}} a e^{\bar{γ}} c u^{\bar{β}} e^{\bar{δ}} d

and

R_{a c b d} (t) \equiv R_{\bar{α} \bar{γ} \bar{β} \bar{δ}} e^{\bar{α}} a e^{\bar{γ}} c e^{\bar{β}} b e^{\bar{δ}} d

are additional frame components of the Riemann tensor evaluated on

γ

. (Note that frame indices are raised with

δ^{a b}

.) As a special case of Equations ( 119 ) and ( 120 ) we find that

\begin{matrix} {\frac{\partial t}{\partial x^{α}} |}_{γ} = - u_{\bar{α}}, {\frac{\partial {\hat{x}}^{a}}{\partial x^{α}} |}_{γ} = e^{a} \bar{α}, \end{matrix}

(121)

because in the limit

x \to \bar{x}

the dual tetrad

({\bar{e}}_{α}^{0}, {\bar{e}}_{α}^{a})

x

coincides with the dual tetrad

(- u_{\bar{α}}, e^{a} \bar{α})

\bar{x}

. It follows that on

γ

, the transformation matrix between the original coordinates

x^{α}

and the FNC

(t, {\hat{x}}^{a})

is formed by the Fermi–Walker transported tetrad:

\begin{matrix} {\frac{\partial x^{α}}{\partial t} |}_{γ} = u^{\bar{α}}, {\frac{\partial x^{α}}{\partial {\hat{x}}^{a}} |}_{γ} = e^{\bar{α}} a . \end{matrix}

(122)

This implies that the frame components of the acceleration vector

a_{a} (t)

are also the components of the acceleration vector in the FNC; orthogonality between

u^{\bar{α}}

and

a^{\bar{α}}

means that

a_{0} = 0

. Similarly,

R_{0 c 0 d} (t)

R_{0 c b d} (t)

, and

R_{a c b d} (t)

are the components of the Riemann tensor (evaluated on

γ

) in the Fermi normal coordinates.

3.2.5 Metric near $γ$

Inversion of Equations ( 119 ) and ( 120 ) gives

\begin{matrix} {\bar{e}}_{α}^{0} d x^{α} = [1 + a_{a} {\hat{x}}^{a} + \frac{1}{2} R_{0 c 0 d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] d t + [\frac{1}{6} R_{0 c b d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] d {\hat{x}}^{b} \end{matrix}

(123)

and

\begin{matrix} {\bar{e}}_{α}^{a} d x^{α} = [δ_{b}^{a} - \frac{1}{6} R_{c b d}^{a} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] d {\hat{x}}^{b} + [- \frac{1}{2} R_{c 0 d}^{a} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})] d t . \end{matrix}

(124)

These relations, when specialized to the FNC, give the components of the dual tetrad at

x

They can also be used to compute the metric at

x

, after invoking the completeness relations

g_{α β} = - {\bar{e}}_{α}^{0} {\bar{e}}_{β}^{0} + δ_{a b} {\bar{e}}_{α}^{a} {\bar{e}}_{β}^{b}

. This gives

d s^{2} = g_{t t} d t^{2} + 2 g_{t a} d t d {\hat{x}}^{a} + g_{a b} d {\hat{x}}^{a} d {\hat{x}}^{b},

with

\begin{matrix} g_{t t} & = & - [1 + 2 a_{a} {\hat{x}}^{a} + {(a_{a} {\hat{x}}^{a})}^{2} + R_{0 c 0 d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3})], \end{matrix}

(125)

\begin{matrix} g_{t a} & = & - \frac{2}{3} R_{0 c a d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3}), \end{matrix}

(126)

\begin{matrix} g_{a b} & = & δ_{a b} - \frac{1}{3} R_{a c b d} {\hat{x}}^{c} {\hat{x}}^{d} + O (s^{3}) . \end{matrix}

(127)

This is the metric near

γ

in the Fermi normal coordinates. Recall that

a_{a} (t)

are the components of the acceleration vector of

γ

– the timelike curve described by

{\hat{x}}^{a} = 0

– while

R_{0 c 0 d} (t)

R_{0 c b d} (t)

, and

R_{a c b d} (t)

are the components of the Riemann tensor evaluated on

γ

Notice that on

γ

, the metric of Equations ( 125 , 126 , 127 ) reduces to

g_{t t} = - 1

and

g_{a b} = δ_{a b}

On the other hand, the nonvanishing Christoffel symbols (on

γ

) are

Γ_{t a}^{t} = Γ_{t t}^{a} = a_{a}

; these are zero (and the FNC enforce local flatness on the entire curve) when

γ

is a geodesic.

3.2.6 Thorne–Hartle coordinates

The form of the metric can be simplified if the Ricci tensor vanishes on the world line:

\begin{matrix} R_{μ ν} (z) = 0 . \end{matrix}

(128)

In such circumstances, a transformation from the Fermi normal coordinates

(t, {\hat{x}}^{a})

to the Thorne–Hartle coordinates

(t, {\hat{y}}^{a})

brings the metric to the form

\begin{matrix} g_{t t} & = & - [1 + 2 a_{a} {\hat{y}}^{a} + {(a_{a} {\hat{y}}^{a})}^{2} + R_{0 c 0 d} {\hat{y}}^{c} {\hat{y}}^{d} + O (s^{3})], \end{matrix}

(129)

\begin{matrix} g_{t a} & = & - \frac{2}{3} R_{0 c a d} {\hat{y}}^{c} {\hat{y}}^{d} + O (s^{3}), \end{matrix}

(130)

\begin{matrix} g_{a b} & = & δ_{a b} (1 - R_{0 c 0 d} {\hat{y}}^{c} {\hat{y}}^{d}) + O (s^{3}) . \end{matrix}

(131)

We see that the transformation leaves

g_{t t}

and

g_{t a}

unchanged, but that it diagonalizes

g_{a b}

. This metric was first displayed in [58] and the coordinate transformation was later produced by Zhang [64] .

The key to the simplification comes from Equation ( 128 ), which dramatically reduces the number of independent components of the Riemann tensor. In particular, Equation ( 128 ) implies that the frame components

R_{a c b d}

of the Riemann tensor are completely determined by

ℰ_{a b} \equiv R_{0 a 0 b}

, which in this special case is a symmetric-tracefree tensor. To prove this we invoke the completeness relations of Equation ( 115 ) and take frame components of Equation ( 128 ). This produces the three independent equations

δ^{c d} R_{a c b d} = ℰ_{a b}, δ^{c d} R_{0 c a d} = 0, δ^{c d} ℰ_{c d} = 0,

the last of which states that

ℰ_{a b}

has a vanishing trace. Taking the trace of the first equation gives

δ^{a b} δ^{c d} R_{a c b d} = 0

, and this implies that

R_{a c b d}

has five independent components. Since this is also the number of independent components of

ℰ_{a b}

, we see that the first equation can be inverted –

R_{a c b d}

can be expressed in terms of

ℰ_{a b}

. A complete listing of the relevant relations is

R_{1212} = ℰ_{11} + ℰ_{22} = - ℰ_{33}

R_{1213} = ℰ_{23}

R_{1223} = - ℰ_{13}

R_{1313} = ℰ_{11} + ℰ_{33} = - ℰ_{22}

R_{1323} = ℰ_{12}

, and

R_{2323} = ℰ_{22} + ℰ_{33} = - ℰ_{11}

These are summarized by

\begin{matrix} R_{a c b d} = δ_{a b} ℰ_{c d} + δ_{c d} ℰ_{a b} - δ_{a d} ℰ_{b c} - δ_{b c} ℰ_{a d}, \end{matrix}

(132)

and

ℰ_{a b} \equiv R_{0 a 0 b}

satisfies

δ^{a b} ℰ_{a b} = 0

We may also note that the relation

δ^{c d} R_{0 c a d} = 0

, together with the usual symmetries of the Riemann tensor, imply that

R_{0 c a d}

too possesses five independent components. These may thus be related to another symmetric-tracefree tensor

ℬ_{a b}

. We take the independent components to be

R_{0112} \equiv - ℬ_{13}

R_{0113} \equiv ℬ_{12}

R_{0123} \equiv - ℬ_{11}

R_{0212} \equiv - ℬ_{23}

, and

R_{0213} \equiv ℬ_{22}

, and it is easy to see that all other components can be expressed in terms of these. For example,

R_{0223} = - R_{0113} = - ℬ_{12}

R_{0312} = - R_{0123} + R_{0213} = ℬ_{11} + ℬ_{22} = - ℬ_{33}

R_{0313} = - R_{0212} = ℬ_{23}

, and

R_{0323} = R_{0112} = - ℬ_{13}

. These relations are summarized by

\begin{matrix} R_{0 a b c} = - ɛ_{b c d} ℬ_{a}^{d}, \end{matrix}

(133)

where

ɛ_{a b c}

is the three-dimensional permutation symbol. The inverse relation is

ℬ_{b}^{a} = \frac{1}{2} ɛ^{a c d} R_{0 b c d}

Substitution of Equation ( 132 ) into Equation ( 127 ) gives

g_{a b} = δ_{a b} (1 - \frac{1}{3} ℰ_{c d} {\hat{x}}^{c} {\hat{x}}^{d}) - \frac{1}{3} ({\hat{x}}_{c} {\hat{x}}^{c}) ℰ_{a b} + \frac{1}{3} {\hat{x}}_{a} ℰ_{b c} {\hat{x}}^{c} + \frac{1}{3} {\hat{x}}_{b} ℰ_{a c} {\hat{x}}^{c} + O (s^{3}),

and we have not yet achieved the simple form of Equation ( 131 ). The missing step is the transformation from the FNC

{\hat{x}}^{a}

to the Thorne–Hartle coordinates

{\hat{y}}^{a}

. This is given by

\begin{matrix} {\hat{y}}^{a} = {\hat{x}}^{a} + ξ^{a}, ξ^{a} = - \frac{1}{6} ({\hat{x}}_{c} {\hat{x}}^{c}) ℰ_{a b} {\hat{x}}^{b} + \frac{1}{3} {\hat{x}}_{a} ℰ_{b c} {\hat{x}}^{b} {\hat{x}}^{c} + O (s^{4}) . \end{matrix}

(134)

It is easy to see that this transformation affects neither

g_{t t}

nor

g_{t a}

at orders

s

and

s^{2}

. The remaining components of the metric, however, transform according to

g_{a b} (THC) = g_{a b} (FNC) - ξ_{a; b} - ξ_{b; a}

, where

ξ_{a; b} = \frac{1}{3} δ_{a b} ℰ_{c d} {\hat{x}}^{c} {\hat{x}}^{d} - \frac{1}{6} ({\hat{x}}_{c} {\hat{x}}^{c}) ℰ_{a b} - \frac{1}{3} ℰ_{a c} {\hat{x}}^{c} {\hat{x}}_{b} + \frac{2}{3} {\hat{x}}_{a} ℰ_{b c} {\hat{x}}^{c} + O (s^{3}) .

It follows that

g_{a b}^{T H C} = δ_{a b} (1 - ℰ_{c d} {\hat{y}}^{c} {\hat{y}}^{d}) + O ({\hat{y}}^{3})

, which is just the same statement as in Equation ( 131 ).

Alternative expressions for the components of the Thorne–Hartle metric are

\begin{matrix} g_{t t} & = & - [1 + 2 a_{a} {\hat{y}}^{a} + {(a_{a} {\hat{y}}^{a})}^{2} + ℰ_{a b} {\hat{y}}^{a} {\hat{y}}^{b} + O (s^{3})], \end{matrix}

(135)

\begin{matrix} g_{t a} & = & - \frac{2}{3} ɛ_{a b c} ℬ_{d}^{b} {\hat{y}}^{c} {\hat{y}}^{d} + O (s^{3}), \end{matrix}

(136)

\begin{matrix} g_{a b} & = & δ_{a b} (1 - ℰ_{c d} {\hat{y}}^{c} {\hat{y}}^{d}) + O (s^{3}) . \end{matrix}

(137)

3.3 Retarded coordinates

3.3.1 Geometrical elements

We introduce the same geometrical elements as in Section 3.2 : We have a timelike curve

γ

described by relations

z^{μ} (τ)

, its normalized tangent vector

u^{μ} = d z^{μ} / d τ

, and its acceleration vector

a^{μ} = D u^{μ} / d τ

. We also have an orthonormal triad

e^{μ} a

that is transported on the world line according to

\begin{matrix} \frac{D e^{μ} a}{d τ} = a_{a} u^{μ} + ω_{a}^{b} e^{μ} b, \end{matrix}

(138)

where

a_{a} (τ) = a_{μ} e^{μ} a

are the frame components of the acceleration vector and

ω_{a b} (τ) = - ω_{b a} (τ)

is a prescribed rotation tensor. Here the triad is not Fermi–Walker transported: For added generality we allow the spatial vectors to rotate as they are transported on the world line. While

ω_{a b}

will be set to zero in most sections of this paper, the freedom to perform such a rotation can be useful and will be exploited in Section 5.4 . It is easy to check that Equation ( 138 ) is compatible with the requirement that the tetrad

(u^{μ}, e^{μ} a)

be orthonormal everywhere on

γ

. Finally, we have a dual tetrad

(e^{0} μ, e^{a} μ)

, with

e^{0} μ = - u_{μ}

and

e^{a} μ = δ^{a b} g_{μ ν} e^{ν} b

. The tetrad and its dual give rise to the completeness relations

\begin{matrix} g^{μ ν} = - u^{μ} u^{ν} + δ^{a b} e^{μ} a e^{ν} b, g_{μ ν} = - e^{0} μ e^{0} ν + δ_{a b} e^{a} μ e^{b} ν, \end{matrix}

(139)

which are the same as in Equation ( 115 ).

The Fermi normal coordinates of Section 3.2 were constructed on the basis of a spacelike geodesic connecting a field point

x

to the world line. The retarded coordinates are based instead on a null geodesic going from the world line to the field point. We thus let

x

be within the normal convex neighbourhood of

γ

β

be the unique future-directed null geodesic that goes from the world line to

x

, and

x^{'} \equiv z (u)

be the point at which

β

intersects the world line, with

u

denoting the value of the proper-time parameter at this point.

From the tetrad at

x^{'}

we obtain another tetrad

(e^{α} 0, e^{α} a)

x

by parallel transport on

β

. By raising the frame index and lowering the vectorial index we also obtain a dual tetrad at

x

e^{0} α = - g_{α β} e^{β} 0

and

e^{a} α = δ^{a b} g_{α β} e^{β} b

. The metric at

x

can be then be expressed as

\begin{matrix} g_{α β} = - e^{0} α e^{0} β + δ_{a b} e^{a} α e^{b} β, \end{matrix}

(140)

and the parallel propagator from

x^{'}

x

is given by

\begin{matrix} g_{α^{'}}^{α} (x, x^{'}) = - e^{α} 0 u_{α^{'}} + e^{α} a e^{a} α^{'}, g_{α}^{α^{'}} (x^{'}, x) = u^{α^{'}} e^{0} α + e^{α^{'}} a e^{a} α . \end{matrix}

(141)

3.3.2 Definition of the retarded coordinates

The quasi-Cartesian version of the retarded coordinates are defined by

\begin{matrix} {\hat{x}}^{0} = u, {\hat{x}}^{a} = - e^{a} α^{'} (x^{'}) σ^{α^{'}} (x, x^{'}), σ (x, x^{'}) = 0; \end{matrix}

(142)

the last statement indicates that

x^{'}

and

x

are linked by a null geodesic. From the fact that

σ^{α^{'}}

is a null vector we obtain

\begin{matrix} r \equiv (δ_{a b} {\hat{x}}^{a} {\hat{x}}^{b})^{1 / 2} = u_{α^{'}} σ^{α^{'}}, \end{matrix}

(143)

and

r

is a positive quantity by virtue of the fact that

β

is a future-directed null geodesic – this makes

σ^{α^{'}}

past-directed. In flat spacetime,

σ^{α^{'}} = - (x - x^{'})^{α}

, and in a Lorentz frame that is momentarily comoving with the world line,

r = t - t^{'} > 0

; with the speed of light set equal to unity,

r

is also the spatial distance between

x^{'}

and

x

as measured in this frame. In curved spacetime, the quantity

r = u_{α^{'}} σ^{α^{'}}

can still be called the retarded distance between the point

x

and the world line. Another consequence of Equation ( 142 ) is that

\begin{matrix} σ^{α^{'}} = - r (u^{α^{'}} + Ω^{a} e^{α^{'}} a), \end{matrix}

(144)

where

Ω^{a} \equiv {\hat{x}}^{a} / r

is a spatial vector that satisfies

δ_{a b} Ω^{a} Ω^{b} = 1

A straightforward calculation reveals that under a displacement of the point

x

, the retarded coordinates change according to

\begin{matrix} d u = - k_{α} d x^{α}, d {\hat{x}}^{a} = - (r a^{a} - ω_{b}^{a} {\hat{x}}^{b} + e^{a} α^{'} σ_{β^{'}}^{α^{'}} u^{β^{'}}) d u - e^{a} α^{'} σ_{β}^{α^{'}} d x^{β}, \end{matrix}

(145)

where

k_{α} = σ_{α} / r

is a future-directed null vector at

x

that is tangent to the geodesic

β

. To obtain these results we must keep in mind that a displacement of

x

typically induces a simultaneous displacement of

x^{'}

because the new points

x + δ x

and

x^{'} + δ x^{'}

must also be linked by a null geodesic. We therefore have

0 = σ (x + δ x, x^{'} + δ x^{'}) = σ_{α} δ x^{α} + σ_{α^{'}} δ x^{α^{'}}

, and the first relation of Equation ( 145 ) follows from the fact that a displacement along the world line is described by

δ x^{α^{'}} = u^{α^{'}} δ u

3.3.3 The scalar field $r (x)$ and the vector field $k^{α} (x)$

If we keep

x^{'}

linked to

x

by the relation

σ (x, x^{'}) = 0

, then the quantity

\begin{matrix} r (x) = σ_{α^{'}} (x, x^{'}) u^{α^{'}} (x^{'}) \end{matrix}

(146)

can be viewed as an ordinary scalar field defined in a neighbourhood of

γ

. We can compute the gradient of

r

by finding how

r

changes under a displacement of

x

(which again induces a displacement of

x^{'}

). The result is

\begin{matrix} \partial_{β} r = - (σ_{α^{'}} a^{α^{'}} + σ_{α^{'} β^{'}} u^{α^{'}} u^{β^{'}}) k_{β} + σ_{α^{'} β} u^{α^{'}} . \end{matrix}

(147)

Similarly, we can view

\begin{matrix} k^{α} (x) = \frac{σ^{α} (x, x^{'})}{r (x)} \end{matrix}

(148)

as an ordinary vector field, which is tangent to the congruence of null geodesics that emanate from

x^{'}

. It is easy to check that this vector satisfies the identities

\begin{matrix} σ_{α β} k^{β} = k_{α}, σ_{α^{'} β} k^{β} = \frac{σ_{α^{'}}}{r}, \end{matrix}

(149)

from which we also obtain

σ_{α^{'} β} u^{α^{'}} k^{β} = 1

. From this last result and Equation ( 147 ) we deduce the important relation

\begin{matrix} k^{α} \partial_{α} r = 1 . \end{matrix}

(150)

In addition, combining the general statement

σ^{α} = - g_{α^{'}}^{α} σ^{α^{'}}

(cf. Equation ( 79 )) with Equation ( 144 ) gives

\begin{matrix} k^{α} = g_{α^{'}}^{α} (u^{α^{'}} + Ω^{a} e^{α^{'}} a); \end{matrix}

(151)

the vector at

x

is therefore obtained by parallel transport of

u^{α^{'}} + Ω^{a} e^{α^{'}} a

β

. From this and Equation ( 141 ) we get the alternative expression

\begin{matrix} k^{α} = e^{α} 0 + Ω^{a} e^{α} a, \end{matrix}

(152)

which confirms that

k^{α}

is a future-directed null vector field (recall that

Ω^{a} = {\hat{x}}^{a} / r

is a unit vector).

The covariant derivative of

k_{α}

can be computed by finding how the vector changes under a displacement of

x

. (It is in fact easier to first calculate how

r k_{α}

changes, and then substitute our previous expression for

\partial_{β} r

.) The result is

\begin{matrix} r k_{α; β} = σ_{α β} - k_{α} σ_{β γ^{'}} u^{γ^{'}} - k_{β} σ_{α γ^{'}} u^{γ^{'}} + (σ_{α^{'}} a^{α^{'}} + σ_{α^{'} β^{'}} u^{α^{'}} u^{β^{'}}) k_{α} k_{β} . \end{matrix}

(153)

From this we infer that

k^{α}

satisfies the geodesic equation in affine-parameter form,

k_{; β}^{α} k^{β} = 0

, and Equation ( 150 ) informs us that the affine parameter is in fact

r

. A displacement along a member of the congruence is therefore given by

d x^{α} = k^{α} d r

. Specializing to retarded coordinates, and using Equations ( 145 ) and ( 149 ), we find that this statement becomes

d u = 0

and

d {\hat{x}}^{a} = ({\hat{x}}^{a} / r) d r

, which integrate to

u = c o n s t .

and

{\hat{x}}^{a} = r Ω^{a}

, respectively, with

Ω^{a}

still denoting a constant unit vector. We have found that the congruence of null geodesics emanating from

x^{'}

is described by

\begin{matrix} u = c o n s t ., {\hat{x}}^{a} = r Ω^{a} (θ^{A}) \end{matrix}

(154)

in the retarded coordinates. Here, the two angles

θ^{A}

(

A = 1, 2

) serve to parameterize the unit vector

Ω^{a}

, which is independent of

r

. Equation ( 153 ) also implies that the expansion of the congruence is given by

\begin{matrix} θ = k_{; α}^{α} = \frac{σ_{α}^{α} - 2}{r} . \end{matrix}

(155)

Using the expansion for

σ_{α}^{α}

given by Equation ( 91 ), we find that this becomes

r θ = 2 - \frac{1}{3} R_{α^{'} β^{'}} σ^{α^{'}} σ^{β^{'}} + O (r^{3})

, or

\begin{matrix} r θ = 2 - \frac{1}{3} r^{2} (R_{00} + 2 R_{0 a} Ω^{a} + R_{a b} Ω^{a} Ω^{b}) + O (r^{3}) \end{matrix}

(156)

after using Equation ( 144 ). Here,

R_{00} = R_{α^{'} β^{'}} u^{α^{'}} u^{β^{'}}

R_{0 a} = R_{α^{'} β^{'}} u^{α^{'}} e^{β^{'}} a

, and

R_{a b} = R_{α^{'} β^{'}} e^{α^{'}} a e^{β^{'}} b

are the frame components of the Ricci tensor evaluated at

x^{'}

. This result confirms that the congruence is singular at

r = 0

, because

θ

diverges as

2 / r

in this limit; the caustic coincides with the point

x^{'}

Finally, we infer from Equation ( 153 ) that

k^{α}

is hypersurface orthogonal. This, together with the property that

k^{α}

satisfies the geodesic equation in affine-parameter form, implies that there exists a scalar field

u (x)

such that

\begin{matrix} k_{α} = - \partial_{α} u . \end{matrix}

(157)

This scalar field was already identified in Equation ( 145 ): It is numerically equal to the proper-time parameter of the world line at

x^{'}

. We conclude that the geodesics to which

k^{α}

is tangent are the generators of the null cone

u = c o n s t .

As Equation ( 154 ) indicates, a specific generator is selected by choosing a direction

Ω^{a}

(which can be parameterized by two angles

θ^{A}

), and

r

is an affine parameter on each generator. The geometrical meaning of the retarded coordinates is now completely clear; it is illustrated in Figure 7 .

Figure 7 : Retarded coordinates of a point

x

relative to a world line

γ

. The retarded time

u

selects a particular null cone, the unit vector

Ω^{a} \equiv {\hat{x}}^{a} / r

selects a particular generator of this null cone, and the retarded distance

r

selects a particular point on this generator. This figure is identical to Figure 4 .

3.3.4 Frame components of tensor fields on the world line

The metric at

x

in the retarded coordinates will be expressed in terms of frame components of vectors and tensors evaluated on the world line

γ

. For example, if

a^{α^{'}}

is the acceleration vector at

x^{'}

, then as we have seen,

\begin{matrix} a_{a} (u) = a_{α^{'}} e^{α^{'}} a \end{matrix}

(158)

are the frame components of the acceleration at proper time

u

Similarly,

\begin{matrix} \begin{matrix} R_{a 0 b 0} (u) & = & R_{α^{'} γ^{'} β^{'} δ^{'}} e^{α^{'}} a u^{γ^{'}} e^{β^{'}} b u^{δ^{'}}, \\ R_{a 0 b d} (u) & = & R_{α^{'} γ^{'} β^{'} δ^{'}} e^{α^{'}} a u^{γ^{'}} e^{β^{'}} b e^{δ^{'}} d, \\ R_{a c b d} (u) & = & R_{α^{'} γ^{'} β^{'} δ^{'}} e^{α^{'}} a e^{γ^{'}} c e^{β^{'}} b e^{δ^{'}} d \end{matrix} \end{matrix}

(159)

are the frame components of the Riemann tensor evaluated on

γ

. From these we form the useful combinations

\begin{matrix} S_{a b} (u, θ^{A}) & = & R_{a 0 b 0} + R_{a 0 b c} Ω^{c} + R_{b 0 a c} Ω^{c} + R_{a c b d} Ω^{c} Ω^{d} = S_{b a}, \end{matrix}

(160)

\begin{matrix} S_{a} (u, θ^{A}) & = & S_{a b} Ω^{b} = R_{a 0 b 0} Ω^{b} - R_{a b 0 c} Ω^{b} Ω^{c}, \end{matrix}

(161)

\begin{matrix} S (u, θ^{A}) & = & S_{a} Ω^{a} = R_{a 0 b 0} Ω^{a} Ω^{b}, \end{matrix}

(162)

in which the quantities

Ω^{a} \equiv {\hat{x}}^{a} / r

depend on the angles

θ^{A}

only – they are independent of

u

and

r

We have previously introduced the frame components of the Ricci tensor in Equation ( 156 ).

The identity

\begin{matrix} R_{00} + 2 R_{0 a} Ω^{a} + R_{a b} Ω^{a} Ω^{b} = δ^{a b} S_{a b} - S \end{matrix}

(163)

follows easily from Equations ( 160 , 161 , 162 ) and the definition of the Ricci tensor.

In Section 3.2 we saw that the frame components of a given tensor were also the components of this tensor (evaluated on the world line) in the Fermi normal coordinates. We should not expect this property to be true also in the case of the retarded coordinates: the frame components of a tensor are not to be identified with the components of this tensor in the retarded coordinates. The reason is that the retarded coordinates are in fact singular on the world line. As we shall see, they give rise to a metric that possesses a directional ambiguity at

r = 0

. (This can easily be seen in Minkowski spacetime by performing the coordinate transformation

u = t - \sqrt{x^{2} + y^{2} + z^{2}}

.) Components of tensors are therefore not defined on the world line, although they are perfectly well defined for

r \neq 0

. Frame components, on the other hand, are well defined both off and on the world line, and working with them will eliminate any difficulty associated with the singular nature of the retarded coordinates.

3.3.5 Coordinate displacements near $γ$

The changes in the quasi-Cartesian retarded coordinates under a displacement of

x

are given by Equation ( 145 ). In these we substitute the standard expansions for

σ_{α^{'} β^{'}}

and

σ_{α^{'} β}

, as given by Equations ( 88 ) and ( 89 ), as well as Equations ( 144 ) and ( 151 ). After a straightforward (but fairly lengthy) calculation, we obtain the following expressions for the coordinate displacements:

\begin{matrix} d u & = & (e^{0} α d x^{α}) - Ω_{a} (e^{b} α d x^{α}), \end{matrix}

(164)

\begin{matrix} d {\hat{x}}^{a} & = & - [r a^{a} - r ω_{b}^{a} Ω^{b} + \frac{1}{2} r^{2} S^{a} + O (r^{3})] (e^{0} α d x^{α}) \end{matrix}

\begin{matrix} + [δ_{b}^{a} + (r a^{a} - r ω_{c}^{a} Ω^{c} + \frac{1}{3} r^{2} S^{a}) Ω_{b} + \frac{1}{6} r^{2} S_{b}^{a} + O (r^{3})] (e^{b} α d x^{α}) . \end{matrix}

(165)

Notice that the result for

d u

is exact, but that

d {\hat{x}}^{a}

is expressed as an expansion in powers of

r

These results can also be expressed in the form of gradients of the retarded coordinates:

\begin{matrix} \partial_{α} u & = & e^{0} α - Ω_{a} e^{a} α, \end{matrix}

(166)

\begin{matrix} \partial_{α} {\hat{x}}^{a} & = & - [r a^{a} - r ω_{b}^{a} Ω^{b} + \frac{1}{2} r^{2} S^{a} + O (r^{3})] e^{0} α \end{matrix}

\begin{matrix} + [δ_{b}^{a} + (r a^{a} - r ω_{c}^{a} Ω^{c} + \frac{1}{3} r^{2} S^{a}) Ω_{b} + \frac{1}{6} r^{2} S_{b}^{a} + O (r^{3})] e^{b} α . \end{matrix}

(167)

Notice that Equation ( 166 ) follows immediately from Equations ( 152 ) and ( 157 ). From Equation ( 167 ) and the identity

\partial_{α} r = Ω_{a} \partial_{α} {\hat{x}}^{a}

we also infer

\begin{matrix} \partial_{α} r = - [r a_{a} Ω^{a} + \frac{1}{2} r^{2} S + O (r^{3})] e^{0} α + [(1 + r a_{b} Ω^{b} + \frac{1}{3} r^{2} S) Ω_{a} + \frac{1}{6} r^{2} S_{a} + O (r^{3})] e^{a} α, \end{matrix}

(168)

where we have used the facts that

S_{a} = S_{a b} Ω^{b}

and

S = S_{a} Ω^{a}

; these last results were derived in Equations ( 161 ) and ( 162 ). It may be checked that Equation ( 168 ) agrees with Equation ( 147 ).

3.3.6 Metric near $γ$

It is straightforward (but fairly tedious) to invert the relations of Equations ( 164 ) and ( 165 ) and solve for

e^{0} α d x^{α}

and

e^{a} α d x^{α}

. The results are

\begin{matrix} e^{0} α d x^{α} & = & [1 + r a_{a} Ω^{a} + \frac{1}{2} r^{2} S + O (r^{3})] d u + [(1 + \frac{1}{6} r^{2} S) Ω_{a} - \frac{1}{6} r^{2} S_{a} + O (r^{3})] d {\hat{x}}^{a}, \end{matrix}

(169)

\begin{matrix} e^{a} α d x^{α} & = & [r (a^{a} - ω_{b}^{a} Ω^{b}) + \frac{1}{2} r^{2} S^{a} + O (r^{3})] d u + [δ_{b}^{a} - \frac{1}{6} r^{2} S_{b}^{a} + \frac{1}{6} r^{2} S^{a} Ω_{b} + O (r^{3})] d {\hat{x}}^{b} . \end{matrix}

(170)

These relations, when specialized to the retarded coordinates, give us the components of the dual tetrad

(e^{0} α, e^{a} α)

x

. The metric is then computed by using the completeness relations of Equation ( 140 ). We find

d s^{2} = g_{u u} d u^{2} + 2 g_{u a} d u d {\hat{x}}^{a} + g_{a b} d {\hat{x}}^{a} d {\hat{x}}^{b},

with

\begin{matrix} g_{u u} & = & - {(1 + r a_{a} Ω^{a})}^{2} + r^{2} (a_{a} - ω_{a b} Ω^{b}) (a^{a} - ω_{c}^{a} Ω^{c}) - r^{2} S + O (r^{3}), \end{matrix}

(171)

\begin{matrix} g_{u a} & = & - (1 + r a_{b} Ω^{b} + \frac{2}{3} r^{2} S) Ω_{a} + r (a_{a} - ω_{a b} Ω^{b}) + \frac{2}{3} r^{2} S_{a} + O (r^{3}), \end{matrix}

(172)

\begin{matrix} g_{a b} & = & δ_{a b} - (1 + \frac{1}{3} r^{2} S) Ω_{a} Ω_{b} - \frac{1}{3} r^{2} S_{a b} + \frac{1}{3} r^{2} (S_{a} Ω_{b} + Ω_{a} S_{b}) + O (r^{3}) . \end{matrix}

(173)

We see (as was pointed out in Section 3.3.4 ) that the metric possesses a directional ambiguity on the world line: The metric at

r = 0

still depends on the vector

Ω^{a} = {\hat{x}}^{a} / r

that specifies the direction to the point

x

. The retarded coordinates are therefore singular on the world line, and tensor components cannot be defined on

γ

By setting

S_{a b} = S_{a} = S = 0

in Equations ( 171 , 172 , 173 ) we obtain the metric of flat spacetime in the retarded coordinates. This we express as

\begin{matrix} η_{u u} & = & - {(1 + r a_{a} Ω^{a})}^{2} + r^{2} (a_{a} - ω_{a b} Ω^{b}) (a^{a} - ω_{c}^{a} Ω^{c}), \end{matrix}

\begin{matrix} η_{u a} & = & - (1 + r a_{b} Ω^{b}) Ω_{a} + r (a_{a} - ω_{a b} Ω^{b}), \end{matrix}

(174)

\begin{matrix} η_{a b} & = & δ_{a b} - Ω_{a} Ω_{b} . \end{matrix}

In spite of the directional ambiguity, the metric of flat spacetime has a unit determinant everywhere, and it is easily inverted:

\begin{matrix} η^{u u} = 0, η^{u a} = - Ω^{a}, η^{a b} = δ^{a b} + r (a^{a} - ω_{c}^{a} Ω^{c}) Ω^{b} + r Ω^{a} (a^{b} - ω_{c}^{b} Ω^{c}) . \end{matrix}

(175)

The inverse metric also is ambiguous on the world line.

To invert the curved-spacetime metric of Equations ( 171 , 172 , 173 ) we express it as

g_{α β} = η_{α β} + h_{α β} + O (r^{3})

and treat

h_{α β} = O (r^{2})

as a perturbation. The inverse metric is then

g^{α β} = η^{α β} - η^{α γ} η^{β δ} h_{γ δ} + O (r^{3})

, or

\begin{matrix} g^{u u} & = & 0, \end{matrix}

(176)

\begin{matrix} g^{u a} & = & - Ω^{a}, \end{matrix}

(177)

\begin{matrix} g^{a b} & = & δ^{a b} + r (a^{a} - ω_{c}^{a} Ω^{c}) Ω^{b} + r Ω^{a} (a^{b} - ω_{c}^{b} Ω^{c}) + \frac{1}{3} r^{2} S^{a b} + \frac{1}{3} r^{2} (S^{a} Ω^{b} + Ω^{a} S^{b}) + O (r^{3}) . \end{matrix}

(178)

The results for

g^{u u}

and

g^{u a}

are exact, and they follow from the general relations

g^{α β} (\partial_{α} u) (\partial_{β} u) = 0

and

g^{α β} (\partial_{α} u) (\partial_{β} r) = - 1

that are derived from Equations ( 150 ) and ( 157 ).

The metric determinant is computed from

\sqrt{- g} = 1 + \frac{1}{2} η^{α β} h_{α β} + O (r^{3})

, which gives

\begin{matrix} \sqrt{- g} = 1 - \frac{1}{6} r^{2} (δ^{a b} S_{a b} - S) + O (r^{3}) = 1 - \frac{1}{6} r^{2} (R_{00} + 2 R_{0 a} Ω^{a} + R_{a b} Ω^{a} Ω^{b}) + O (r^{3}), \end{matrix}

(179)

where we have substituted the identity of Equation ( 163 ). Comparison with Equation ( 156 ) then gives us the interesting relation

\sqrt{- g} = \frac{1}{2} r θ + O (r^{3})

, where

θ

is the expansion of the generators of the null cones

u = c o n s t

3.3.7 Transformation to angular coordinates

Because the vector

Ω^{a} = {\hat{x}}^{a} / r

satisfies

δ_{a b} Ω^{a} Ω^{b} = 1

, it can be parameterized by two angles

θ^{A}

. A canonical choice for the parameterization is

Ω^{a} = (sin θ cos φ, sin θ sin φ, cos θ)

. It is then convenient to perform a coordinate transformation from

{\hat{x}}^{a}

(r, θ^{A})

, using the relations

{\hat{x}}^{a} = r Ω^{a} (θ^{A})

. (Recall from Section 3.3.3 that the angles

θ^{A}

are constant on the generators of the null cones

u = c o n s t .

, and that

r

is an affine parameter on these generators. The relations

{\hat{x}}^{a} = r Ω^{a}

therefore describe the behaviour of the generators.) The differential form of the coordinate transformation is

\begin{matrix} d {\hat{x}}^{a} = Ω^{a} d r + r Ω_{A}^{a} d θ^{A}, \end{matrix}

(180)

where the transformation matrix

\begin{matrix} Ω_{A}^{a} \equiv \frac{\partial Ω^{a}}{\partial θ^{A}} \end{matrix}

(181)

satisfies the identity

Ω_{a} Ω_{A}^{a} = 0

We introduce the quantities

\begin{matrix} Ω_{A B} = δ_{a b} Ω_{A}^{a} Ω_{B}^{b}, \end{matrix}

(182)

which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization,

Ω_{A B} = diag (1, {sin}^{2} θ)

. We use the inverse of

Ω_{A B}

, denoted

Ω^{A B}

, to raise upper-case latin indices. We then define the new object

\begin{matrix} Ω_{a}^{A} = δ_{a b} Ω^{A B} Ω_{B}^{b} \end{matrix}

(183)

which satisfies the identities

\begin{matrix} Ω_{a}^{A} Ω_{B}^{a} = δ_{B}^{A}, Ω_{A}^{a} Ω_{b}^{A} = δ_{b}^{a} - Ω^{a} Ω_{b} . \end{matrix}

(184)

The second result follows from the fact that both sides are simultaneously symmetric in

a

and

b

, orthogonal to

Ω_{a}

and

Ω^{b}

, and have the same trace.

From the preceding results we establish that the transformation from

{\hat{x}}^{a}

(r, θ^{A})

is accomplished by

\begin{matrix} \frac{\partial {\hat{x}}^{a}}{\partial r} = Ω^{a}, \frac{\partial {\hat{x}}^{a}}{\partial θ^{A}} = r Ω_{A}^{a}, \end{matrix}

(185)

while the transformation from

(r, θ^{A})

{\hat{x}}^{a}

is accomplished by

\begin{matrix} \frac{\partial r}{\partial {\hat{x}}^{a}} = Ω_{a}, \frac{\partial θ^{A}}{\partial {\hat{x}}^{a}} = \frac{1}{r} Ω_{a}^{A} . \end{matrix}

(186)

With these transformation rules it is easy to show that in the angular coordinates, the metric takes the form of

d s^{2} = g_{u u} d u^{2} + 2 g_{u r} d u d r + 2 g_{u A} d u d θ^{A} + g_{A B} d θ^{A} d θ^{B},

with

\begin{matrix} g_{u u} & = & - {(1 + r a_{a} Ω^{a})}^{2} + r^{2} (a_{a} - ω_{a b} Ω^{b}) (a^{a} - ω_{c}^{a} Ω^{c}) - r^{2} S + O (r^{3}), \end{matrix}

(187)

\begin{matrix} g_{u r} & = & - 1, \end{matrix}

(188)

\begin{matrix} g_{u A} & = & r [r (a_{a} - ω_{a b} Ω^{b}) + \frac{2}{3} r^{2} S_{a} + O (r^{3})] Ω_{A}^{a}, \end{matrix}

(189)

\begin{matrix} g_{A B} & = & r^{2} [Ω_{A B} - \frac{1}{3} r^{2} S_{a b} Ω_{A}^{a} Ω_{B}^{b} + O (r^{3})] . \end{matrix}

(190)

The results

g_{r u} = - 1

g_{r r} = 0

, and

g_{r A} = 0

are exact, and they follow from the fact that in the retarded coordinates,

k_{α} d x^{α} = - d u

and

k^{α} \partial_{α} = \partial_{r}

The nonvanishing components of the inverse metric are

\begin{matrix} g^{u r} & = & - 1, \end{matrix}

(191)

\begin{matrix} g^{r r} & = & 1 + 2 r a_{a} Ω^{a} + r^{2} S + O (r^{3}), \end{matrix}

(192)

\begin{matrix} g^{r A} & = & \frac{1}{r} [r (a^{a} - ω_{b}^{a} Ω^{b}) + \frac{2}{3} r^{2} S^{a} + O (r^{3})] Ω_{a}^{A}, \end{matrix}

(193)

\begin{matrix} g^{A B} & = & \frac{1}{r^{2}} [Ω^{A B} + \frac{1}{3} r^{2} S^{a b} Ω_{a}^{A} Ω_{b}^{B} + O (r^{3})] . \end{matrix}

(194)

The results

g^{u u} = 0

g^{u r} = - 1

, and

g^{u A} = 0

are exact, and they follow from the same reasoning as before.

Finally, we note that in the angular coordinates, the metric determinant is given by

\begin{matrix} \sqrt{- g} = r^{2} \sqrt{Ω} [1 - \frac{1}{6} r^{2} (R_{00} + 2 R_{0 a} Ω^{a} + R_{a b} Ω^{a} Ω^{b}) + O (r^{3})], \end{matrix}

(195)

where

Ω

is the determinant of

Ω_{A B}

; in the canonical parameterization,

\sqrt{Ω} = sin θ

3.3.8 Specialization to $a^{μ} = 0 = R_{μ ν}$

In this section we specialize our previous results to a situation where

γ

is a geodesic on which the Ricci tensor vanishes. We therefore set

a^{μ} = 0 = R_{μ ν}

everywhere on

γ

, and for simplicity we also set

ω_{a b}

to zero.

We have seen in Section 3.2.6 that when the Ricci tensor vanishes on

γ

, all frame components of the Riemann tensor can be expressed in terms of the symmetric-tracefree tensors

ℰ_{a b} (u)

and

ℬ_{a b} (u)

The relations are

R_{a 0 b 0} = ℰ_{a b}

R_{a 0 b c} = ɛ_{b c d} ℬ_{a}^{d}

, and

R_{a c b d} = δ_{a b} ℰ_{c d} + δ_{c d} ℰ_{a b} - δ_{a d} ℰ_{b c} - δ_{b c} ℰ_{a d}

These can be substituted into Equations ( 160 , 161 , 162 ) to give

\begin{matrix} S_{a b} (u, θ^{A}) & = & 2 ℰ_{a b} - Ω_{a} ℰ_{b c} Ω^{c} - Ω_{b} ℰ_{a c} Ω^{c} + δ_{a b} ℰ_{b c} Ω^{c} Ω^{d} + ɛ_{a c d} Ω^{c} ℬ_{b}^{d} + ɛ_{b c d} Ω^{c} ℬ_{a}^{d}, \end{matrix}

(196)

\begin{matrix} S_{a} (u, θ^{A}) & = & ℰ_{a b} Ω^{b} + ɛ_{a b c} Ω^{b} ℬ_{d}^{c} Ω^{d}, \end{matrix}

(197)

\begin{matrix} S (u, θ^{A}) & = & ℰ_{a b} Ω^{a} Ω^{b} . \end{matrix}

(198)

In these expressions the dependence on retarded time

u

is contained in

ℰ_{a b}

and

ℬ_{a b}

, while the angular dependence is encoded in the unit vector

Ω^{a}

It is convenient to introduce the irreducible quantities

\begin{matrix} ℰ^{*} & = & ℰ_{a b} Ω^{a} Ω^{b}, \end{matrix}

(199)

\begin{matrix} ℰ_{a}^{*} & = & (δ_{a}^{b} - Ω_{a} Ω^{b}) ℰ_{b c} Ω^{c}, \end{matrix}

(200)

\begin{matrix} ℰ_{a b}^{*} & = & 2 ℰ_{a b} - 2 Ω_{a} ℰ_{b c} Ω^{c} - 2 Ω_{b} ℰ_{a c} Ω^{c} + (δ_{a b} + Ω_{a} Ω_{b}) ℰ^{*}, \end{matrix}

(201)

\begin{matrix} ℬ_{a}^{*} & = & ɛ_{a b c} Ω^{b} ℬ_{d}^{c} Ω^{d}, \end{matrix}

(202)

\begin{matrix} ℬ_{a b}^{*} & = & ɛ_{a c d} Ω^{c} ℬ_{e}^{d} (δ_{b}^{e} - Ω^{e} Ω_{b}) + ɛ_{b c d} Ω^{c} ℬ_{e}^{d} (δ_{a}^{e} - Ω^{e} Ω_{a}) . \end{matrix}

(203)

These are all orthogonal to

Ω^{a}

ℰ_{a}^{*} Ω^{a} = ℬ_{a}^{*} Ω^{a} = 0

and

ℰ_{a b}^{*} Ω^{b} = ℬ_{a b}^{*} Ω^{b} = 0

. In terms of these Equations ( 196 , 197 , 198 ) become

\begin{matrix} S_{a b} & = & ℰ_{a b}^{*} + Ω_{a} ℰ_{b}^{*} + ℰ_{a}^{*} Ω_{b} + Ω_{a} Ω_{b} ℰ^{*} + ℬ_{a b}^{*} + Ω_{a} ℬ_{b}^{*} + ℬ_{a}^{*} Ω_{b}, \end{matrix}

(204)

\begin{matrix} S_{a} & = & ℰ_{a}^{*} + Ω_{a} ℰ^{*} + ℬ_{a}^{*}, \end{matrix}

(205)

\begin{matrix} S & = & ℰ^{*} . \end{matrix}

(206)

When Equations ( 204 , 205 , 206 ) are substituted into the metric tensor of Equations ( 171 , 172 , 173 ) – in which

a_{a}

and

ω_{a b}

are both set equal to zero – we obtain the compact expressions

\begin{matrix} g_{u u} & = & - 1 - r^{2} ℰ^{*} + O (r^{3}), \end{matrix}

(207)

\begin{matrix} g_{u a} & = & - Ω_{a} + \frac{2}{3} r^{2} (ℰ_{a}^{*} + ℬ_{a}^{*}) + O (r^{3}), \end{matrix}

(208)

\begin{matrix} g_{a b} & = & δ_{a b} - Ω_{a} Ω_{b} - \frac{1}{3} r^{2} (ℰ_{a b}^{*} + ℬ_{a b}^{*}) + O (r^{3}) . \end{matrix}

(209)

The metric becomes

\begin{matrix} g_{u u} & = & - 1 - r^{2} ℰ^{*} + O (r^{3}), \end{matrix}

(210)

\begin{matrix} g_{u r} & = & - 1, \end{matrix}

(211)

\begin{matrix} g_{u A} & = & \frac{2}{3} r^{3} (ℰ_{A}^{*} + ℬ_{A}^{*}) + O (r^{4}), \end{matrix}

(212)

\begin{matrix} g_{A B} & = & r^{2} Ω_{A B} - \frac{1}{3} r^{4} (ℰ_{A B}^{*} + ℬ_{A B}^{*}) + O (r^{5}) \end{matrix}

(213)

after transforming to angular coordinates using the rules of Equation ( 185 ). Here we have introduced the projections

\begin{matrix} ℰ_{A}^{*} & \equiv & ℰ_{a}^{*} Ω_{A}^{a} = ℰ_{a b} Ω_{A}^{a} Ω^{b}, \end{matrix}

(214)

\begin{matrix} ℰ_{A B}^{*} & \equiv & ℰ_{a b}^{*} Ω_{A}^{a} Ω_{B}^{b} = 2 ℰ_{a b} Ω_{A}^{a} Ω_{B}^{b} + ℰ^{*} Ω_{A B}, \end{matrix}

(215)

\begin{matrix} ℬ_{A}^{*} & \equiv & ℬ_{a}^{*} Ω_{A}^{a} = ɛ_{a b c} Ω_{A}^{a} Ω^{b} ℬ_{d}^{c} Ω^{d}, \end{matrix}

(216)

\begin{matrix} ℬ_{A B}^{*} & \equiv & ℬ_{a b}^{*} Ω_{A}^{a} Ω_{B}^{b} = 2 ɛ_{a c d} Ω^{c} ℬ_{b}^{d} Ω_{(A}^{a} Ω_{B)}^{b} . \end{matrix}

(217)

It may be noted that the inverse relations are

ℰ_{a}^{*} = ℰ_{A}^{*} Ω_{a}^{A}

ℬ_{a}^{*} = ℬ_{A}^{*} Ω_{a}^{A}

ℰ_{a b}^{*} = ℰ_{A B}^{*} Ω_{a}^{A} Ω_{b}^{B}

, and

ℬ_{a b}^{*} = ℬ_{A B}^{*} Ω_{a}^{A} Ω_{b}^{B}

, where

Ω_{a}^{A}

was introduced in Equation ( 183 ).

3.4 Transformation between Fermi and retarded coordinates; advanced point

A point

x

in the normal convex neighbourhood of a world line

γ

can be assigned a set of Fermi normal coordinates (as in Section 3.2 ), or it can be assigned a set of retarded coordinates (see Section 3.3 ). These coordinate systems can obviously be related to one another, and our first task in this section (which will occupy us in Sections 3.4.1 , 3.4.2 , and 3.4.3 ) will be to derive the transformation rules. We begin by refining our notation so as to eliminate any danger of ambiguity.

Figure 8 : The retarded, simultaneous, and advanced points on a world line

γ

. The retarded point

x^{'} \equiv z (u)

is linked to

x

by a future-directed null geodesic. The simultaneous point

\bar{x} \equiv z (t)

is linked to

x

by a spacelike geodesic that intersects

γ

orthogonally. The advanced point

x^{''} \equiv z (v)

is linked to

x

by a past-directed null geodesic.

The Fermi normal coordinates of

x

refer to a point

\bar{x} \equiv z (t)

γ

that is related to

x

by a spacelike geodesic that intersects

γ

orthogonally (see Figure 8 ). We refer to this point as

x

's simultaneous point, and to tensors at

\bar{x}

we assign indices

\bar{α}

\bar{β}

, etc. We let

(t, s ω^{a})

be the Fermi normal coordinates of

x

, with

t

denoting the value of

γ

's proper-time parameter at

\bar{x}

s = \sqrt{2 σ (x, \bar{x})}

representing the proper distance from

\bar{x}

x

along the spacelike geodesic, and

ω^{a}

denoting a unit vector (

δ_{a b} ω^{a} ω^{b} = 1

) that determines the direction of the geodesic. The Fermi normal coordinates are defined by

s ω^{a} = - e^{a} \bar{α} σ^{\bar{α}}

and

σ_{\bar{α}} u^{\bar{α}} = 0

. Finally, we denote by

({\bar{e}}_{0}^{α}, {\bar{e}}_{a}^{α})

the tetrad at

x

that is obtained by parallel transport of

(u^{\bar{α}}, e^{\bar{α}} a)

on the spacelike geodesic.

The retarded coordinates of

x

refer to a point

x^{'} \equiv z (u)

γ

that is linked to

x

by a future-directed null geodesic (see Figure 8 ). We refer to this point as

x

's retarded point, and to tensors at

x^{'}

we assign indices

α^{'}

β^{'}

, etc. We let

(u, r Ω^{a})

be the retarded coordinates of

x

, with

u

denoting the value of

γ

's proper-time parameter at

x^{'}

r = σ_{α^{'}} u^{α^{'}}

representing the affine-parameter distance from

x^{'}

x

along the null geodesic, and

Ω^{a}

denoting a unit vector (

δ_{a b} Ω^{a} Ω^{b} = 1

) that determines the direction of the geodesic. The retarded coordinates are defined by

r Ω^{a} = - e^{a} α^{'} σ^{α^{'}}

and

σ (x, x^{'}) = 0

. Finally, we denote by

(e^{α} 0, e^{α} a)

the tetrad at

x

that is obtained by parallel transport of

(u^{α^{'}}, e^{α^{'}} a)

on the null geodesic.

The reader not interested in following the details of this discussion can be informed that

∙ our results concerning the transformation from the retarded coordinates $(u, r, Ω^{a})$ to the Fermi normal coordinates $(t, s, ω^{a})$ are contained in Equations ( 218 , 219 , 220 ) below;
∙ our results concerning the transformation from the Fermi normal coordinates $(t, s, ω^{a})$ to the retarded coordinates $(u, r, Ω^{a})$ are contained in Equations ( 221 , 222 , 223 );
∙ the decomposition of each member of $({\bar{e}}_{0}^{α}, {\bar{e}}_{a}^{α})$ in the tetrad $(e^{α} 0, e^{α} a)$ is given in retarded coordinates by Equations ( 224 ) and ( 225 ); and
∙ the decomposition of each member of $(e^{α} 0, e^{α} a)$ in the tetrad $({\bar{e}}_{0}^{α}, {\bar{e}}_{a}^{α})$ is given in Fermi normal coordinates by Equations ( 226 ) and ( 227 ).

Our final task will be to define, along with the retarded and simultaneous points, an advanced point

x^{''}

on the world line

γ

(see Figure 8 ). This is taken on in Section 3.4.4 . Throughout this section we shall set

ω_{a b} = 0

, where

ω_{a b}

is the rotation tensor defined by Equation ( 138 ) – the tetrad vectors

e^{μ} a

will be assumed to be Fermi–Walker transported on

γ

3.4.1 From retarded to Fermi coordinates

Quantities at

\bar{x} \equiv z (t)

can be related to quantities at

x^{'} \equiv z (u)

by Taylor expansion along the world line

γ

. To implement this strategy we must first find an expression for

Δ \equiv t - u

. (Although we use the same notation, this should not be confused with the van Vleck determinant introduced in Section 2.5 .) Consider the function

p (τ)

of the proper-time parameter

τ

defined by

p (τ) = σ_{μ} (x, z (τ)) u^{μ} (τ),

in which

x

is kept fixed and in which

z (τ)

is an arbitrary point on the world line. We have that

p (u) = r

and

p (t) = 0

, and

Δ

can ultimately be obtained by expressing

p (t)

p (u + Δ)

and expanding in powers of

Δ

. Formally,

p (t) = p (u) + \dot{p} (u) Δ + \frac{1}{2} \ddot{p} (u) Δ^{2} + \frac{1}{6} p^{(3)} (u) Δ^{3} + O (Δ^{4}),

where overdots (or a number within brackets) indicate repeated differentiation with respect to

τ

We have

\begin{matrix} \dot{p} (u) & = & σ_{α^{'} β^{'}} u^{α^{'}} u^{β^{'}} + σ_{α^{'}} a^{α^{'}}, \end{matrix}

\begin{matrix} \ddot{p} (u) & = & σ_{α^{'} β^{'} γ^{'}} u^{α^{'}} u^{β^{'}} u^{γ^{'}} + 3 σ_{α^{'} β^{'}} u^{α^{'}} a^{β^{'}} + σ_{α^{'}} {\dot{a}}^{α^{'}}, \end{matrix}

\begin{matrix} p^{(3)} (u) & = & σ_{α^{'} β^{'} γ^{'} δ^{'}} u^{α^{'}} u^{β^{'}} u^{γ^{'}} u^{δ^{'}} + σ_{α^{'} β^{'} γ^{'}} (5 a^{α^{'}} u^{β^{'}} u^{γ^{'}} + u^{α^{'}} u^{β^{'}} a^{γ^{'}}) + σ_{α^{'} β^{'}} (3 a^{α^{'}} a^{β^{'}} + 4 u^{α^{'}} {\dot{a}}^{β^{'}}) + σ_{α^{'}} {\ddot{a}}^{α^{'}}, \end{matrix}

where

a^{μ} = D u^{μ} / d τ

{\dot{a}}^{μ} = D a^{μ} / d τ

, and

{\ddot{a}}^{μ} = D {\dot{a}}^{μ} / d τ

We now express all of this in retarded coordinates by invoking the expansion of Equation ( 88 ) for

σ_{α^{'} β^{'}}

(as well as additional expansions for the higher derivatives of the world function, obtained by further differentiation of this result) and the relation

σ^{α^{'}} = - r (u^{α^{'}} + Ω^{a} e^{α^{'}} a)

first derived in Equation ( 144 ). With a degree of accuracy sufficient for our purposes we obtain

\begin{matrix} \dot{p} (u) & = & - [1 + r a_{a} Ω^{a} + \frac{1}{3} r^{2} S + O (r^{3})], \end{matrix}

\begin{matrix} \ddot{p} (u) & = & - r ({\dot{a}}_{0} + {\dot{a}}_{a} Ω^{a}) + O (r^{2}), \end{matrix}

\begin{matrix} p^{(3)} (u) & = & {\dot{a}}_{0} + O (r), \end{matrix}

where

S = R_{a 0 b 0} Ω^{a} Ω^{b}

was first introduced in Equation ( 162 ), and where

{\dot{a}}_{0} \equiv {\dot{a}}_{α^{'}} u^{α^{'}}

{\dot{a}}_{a} \equiv {\dot{a}}_{α^{'}} e^{α^{'}} a

are the frame components of the covariant derivative of the acceleration vector. To arrive at these results we made use of the identity

a_{α^{'}} a^{α^{'}} + {\dot{a}}_{α^{'}} u^{α^{'}} = 0

that follows from the fact that

a^{μ}

is orthogonal to

u^{μ}

. Notice that there is no distinction between the two possible interpretations

{\dot{a}}_{a} \equiv d a_{a} / d τ

and

{\dot{a}}_{a} \equiv {\dot{a}}_{μ} e^{μ} a

for the quantity

{\dot{a}}_{a} (τ)

; their equality follows at once from the substitution of

D e^{μ} a / d τ = a_{a} u^{μ}

(which states that the basis vectors are Fermi–Walker transported on the world line) into the identity

d a_{a} / d τ = D (a_{ν} e^{ν} a) / d τ

Collecting our results we obtain

r = [1 + r a_{a} Ω^{a} + \frac{1}{3} r^{2} S + O (r^{3})] Δ + \frac{1}{2} r [{\dot{a}}_{0} + {\dot{a}}_{a} Ω^{a} + O (r)] Δ^{2} - \frac{1}{6} [{\dot{a}}_{0} + O (r)] Δ^{3} + O (Δ^{4}),

which can readily be solved for

Δ \equiv t - u

expressed as an expansion in powers of

r

. The final result is

\begin{matrix} t = u + r {1 - r a_{a} (u) Ω^{a} + r^{2} {[a_{a} (u) Ω^{a}]}^{2} - \frac{1}{3} r^{2} {\dot{a}}_{0} (u) - \frac{1}{2} r^{2} {\dot{a}}_{a} (u) Ω^{a} - \frac{1}{3} r^{2} R_{a 0 b 0} (u) Ω^{a} Ω^{b} + O (r^{3})}, \end{matrix}

\begin{matrix}  \end{matrix}

(218)

where we show explicitly that all frame components are evaluated at the retarded point

z (u)

To obtain relations between the spatial coordinates we consider the functions

p_{a} (τ) = - σ_{μ} (x, z (τ)) e^{μ} a (τ),

in which

x

is fixed and

z (τ)

is an arbitrary point on

γ

. We have that the retarded coordinates are given by

r Ω^{a} = p^{a} (u)

, while the Fermi coordinates are given instead by

s ω^{a} = p^{a} (t) = p^{a} (u + Δ)

This last expression can be expanded in powers of

Δ

, producing

s ω^{a} = p^{a} (u) + {\dot{p}}^{a} (u) Δ + \frac{1}{2} {\ddot{p}}^{a} (u) Δ^{2} + \frac{1}{6} p^{a (3)} (u) Δ^{3} + O (Δ^{4}),

with

\begin{matrix} {\dot{p}}_{a} (u) & = & - σ_{α^{'} β^{'}} e^{α^{'}} a u^{β^{'}} - (σ_{α^{'}} u^{α^{'}}) (a_{β^{'}} e^{β^{'}} a) \end{matrix}

\begin{matrix} = & - r a_{a} - \frac{1}{3} r^{2} S_{a} + O (r^{3}), \end{matrix}

\begin{matrix} {\ddot{p}}_{a} (u) & = & - σ_{α^{'} β^{'} γ^{'}} e^{α^{'}} a u^{β^{'}} u^{γ^{'}} - (2 σ_{α^{'} β^{'}} u^{α^{'}} u^{β^{'}} + σ_{α^{'}} a^{α^{'}}) (a_{γ^{'}} e^{γ^{'}} a) - σ_{α^{'} β^{'}} e^{α^{'}} a a^{β^{'}} - (σ_{α^{'}} u^{α^{'}}) ({\dot{a}}_{β^{'}} e^{β^{'}} a) \end{matrix}

\begin{matrix} = & (1 + r a_{b} Ω^{b}) a_{a} - r {\dot{a}}_{a} + \frac{1}{3} r R_{a 0 b 0} Ω^{b} + O (r^{2}), \end{matrix}

\begin{matrix} p_{a}^{(3)} (u) & = & - σ_{α^{'} β^{'} γ^{'} δ^{'}} e^{α^{'}} a u^{β^{'}} u^{γ^{'}} u^{δ^{'}} \end{matrix}

\begin{matrix} - (3 σ_{α^{'} β^{'} γ^{'}} u^{α^{'}} u^{β^{'}} u^{γ^{'}} + 6 σ_{α^{'} β^{'}} u^{α^{'}} a^{β^{'}} + σ_{α^{'}} {\dot{a}}^{α^{'}} + σ_{α^{'}} u^{α^{'}} {\dot{a}}_{β^{'}} u^{β^{'}}) (a_{δ^{'}} e^{δ^{'}} a) \end{matrix}

\begin{matrix} - σ_{α^{'} β^{'} γ^{'}} e^{α^{'}} a (2 a^{β^{'}} u^{γ^{'}} + u^{β^{'}} a^{γ^{'}}) - (3 σ_{α^{'} β^{'}} u^{α^{'}} u^{β^{'}} + 2 σ_{α^{'}} a^{α^{'}}) ({\dot{a}}_{γ^{'}} e^{γ^{'}} a) - σ_{α^{'} β^{'}} e^{α^{'}} a {\dot{a}}^{β^{'}} \end{matrix}

\begin{matrix} - (σ_{α^{'}} u^{α^{'}}) ({\ddot{a}}_{β^{'}} e^{β^{'}} a) \end{matrix}

\begin{matrix} = & 2 {\dot{a}}_{a} + O (r) . \end{matrix}

To arrive at these results we have used the same expansions as before and re-introduced

S_{a} = R_{a 0 b 0} Ω^{b} - R_{a b 0 c} Ω^{b} Ω^{c}

, as it was first defined in Equation ( 161 ).

Collecting our results we obtain

\begin{matrix} s ω^{a} & = & r Ω^{a} - r [a^{a} + \frac{1}{3} r S^{a} + O (r^{2})] Δ + \frac{1}{2} [(1 + r a_{b} Ω^{b}) a^{a} - r {\dot{a}}^{a} + \frac{1}{3} r R_{0 b 0}^{a} Ω^{b} + O (r^{2})] Δ^{2} \end{matrix}

\begin{matrix} + \frac{1}{3} [{\dot{a}}^{a} + O (r)] Δ^{3} + O (Δ^{4}), \end{matrix}

which becomes

\begin{matrix} s ω^{a} = r {Ω^{a} - \frac{1}{2} r [1 - r a_{b} (u) Ω^{b}] a^{a} (u) - \frac{1}{6} r^{2} {\dot{a}}^{a} (u) - \frac{1}{6} r^{2} R_{0 b 0}^{a} (u) Ω^{b} + \frac{1}{3} r^{2} R_{b 0 c}^{a} (u) Ω^{b} Ω^{c} + O (r^{3})} \end{matrix}

\begin{matrix}  \end{matrix}

(219)

after substituting Equation ( 218 ) for

Δ \equiv t - u

. From squaring Equation ( 219 ) and using the identity

δ_{a b} ω^{a} ω^{b} = 1

we can also deduce

\begin{matrix} s = r {1 - \frac{1}{2} r a_{a} (u) Ω^{a} + \frac{3}{8} r^{2} {[a_{a} (u) Ω^{a}]}^{2} - \frac{1}{8} r^{2} {\dot{a}}_{0} (u) - \frac{1}{6} r^{2} {\dot{a}}_{a} (u) Ω^{a} - \frac{1}{6} r^{2} R_{a 0 b 0} (u) Ω^{a} Ω^{b} + O (r^{3})} \end{matrix}

\begin{matrix}  \end{matrix}

(220)

for the spatial distance between

x

and

z (t)

3.4.2 From Fermi to retarded coordinates

The techniques developed in the preceding Section 3.4.2 can easily be adapted to the task of relating the retarded coordinates of

x

to its Fermi normal coordinates. Here we use

\bar{x} \equiv z (t)

as the reference point and express all quantities at

x^{'} \equiv z (u)

as Taylor expansions about

τ = t

We begin by considering the function

σ (τ) = σ (x, z (τ))

of the proper-time parameter

τ

γ

. We have that

σ (t) = \frac{1}{2} s^{2}

and

σ (u) = 0

, and

Δ \equiv t - u

is now obtained by expressing

σ (u)

σ (t - Δ)

and expanding in powers of

Δ

. Using the fact that

\dot{σ} (τ) = p (τ)

, we have

σ (u) = σ (t) - p (t) Δ + \frac{1}{2} \dot{p} (t) Δ^{2} - \frac{1}{6} \ddot{p} (t) Δ^{3} + \frac{1}{24} p^{(3)} (t) Δ^{4} + O (Δ^{5}) .

Expressions for the derivatives of

p (τ)

evaluated at

τ = t

can be constructed from results derived previously in Section 3.4.1 : it suffices to replace all primed indices by barred indices and then substitute the relation

σ^{\bar{α}} = - s ω^{a} e^{\bar{α}} a

that follows immediately from Equation ( 116 ). This gives

\begin{matrix} \dot{p} (t) & = & - [1 + s a_{a} ω^{a} + \frac{1}{3} s^{2} R_{a 0 b 0} ω^{a} ω^{b} + O (s^{3})], \end{matrix}

\begin{matrix} \ddot{p} (t) & = & - s {\dot{a}}_{a} ω^{a} + O (s^{2}), \end{matrix}

\begin{matrix} p^{(3)} (t) & = & {\dot{a}}_{0} + O (s), \end{matrix}

and then

s^{2} = [1 + s a_{a} ω^{a} + \frac{1}{3} s^{2} R_{a 0 b 0} ω^{a} ω^{b} + O (s^{3})] Δ^{2} - \frac{1}{3} s [{\dot{a}}_{a} ω^{a} + O (s)] Δ^{3} - \frac{1}{12} [{\dot{a}}_{0} + O (s)] Δ^{4} + O (Δ^{5})

after recalling that

p (t) = 0

. Solving for

Δ

as an expansion in powers of

s

returns

\begin{matrix} u = t - s {1 - \frac{1}{2} s a_{a} (t) ω^{a} + \frac{3}{8} s^{2} {[a_{a} (t) ω^{a}]}^{2} + \frac{1}{24} s^{2} {\dot{a}}_{0} (t) + \frac{1}{6} s^{2} {\dot{a}}_{a} (t) ω^{a} - \frac{1}{6} s^{2} R_{a 0 b 0} (t) ω^{a} ω^{b} + O (s^{3})}, \end{matrix}

\begin{matrix}  \end{matrix}

(221)

in which we emphasize that all frame components are evaluated at the simultaneous point

z (t)

An expression for

r = p</