Abstract

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle.
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
${g}_{\alpha \beta}$
. 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
${A}^{\alpha}$
that satisfies the wave equation

together with the Lorenz gauge condition
${\partial}_{\alpha}{A}^{\alpha}=0$
. (On page 294 in [30] Jackson explains why the term “Lorenz gauge” is preferable to “Lorentz gauge”.) The vector
${j}^{\alpha}$
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.

$$\begin{array}{c}\square {A}^{\alpha}=-4\pi {j}^{\alpha}\end{array}$$ | (1) |

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
${A}_{ret}^{\alpha}$
, the retarded solution, as the physically-relevant solution. Choosing instead the advanced solution
${A}_{adv}^{\alpha}$
would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition

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.

$$\begin{array}{c}{A}_{S}^{\alpha}=\frac{1}{2}\left({A}_{ret}^{\alpha}+{A}_{adv}^{\alpha}\right)\end{array}$$ | (2) |

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
${A}_{ret}^{\alpha}$
,
${A}_{adv}^{\alpha}$
, and
${A}_{S}^{\alpha}$
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
${A}_{S}^{\alpha}$
is just as singular as
${A}_{ret}^{\alpha}$
, removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because
${A}_{S}^{\alpha}$
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

and postulate that it, and it alone, exerts a force on the particle. The subscript `R' stands for `regular', because
${A}_{R}^{\alpha}$
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 ),
$\square {A}_{R}^{\alpha}=0$
; there is no singular source to produce a singular behaviour on the world line. Since
${A}_{R}^{\alpha}$
satisfies the homogeneous wave equation, it can be thought of as a free radiation field, and the subscript `R' could also stand for `radiative'.

$$\begin{array}{c}{A}_{R}^{\alpha}={A}_{ret}^{\alpha}-{A}_{S}^{\alpha}=\frac{1}{2}\left({A}_{ret}^{\alpha}-{A}_{adv}^{\alpha}\right)\end{array}$$ | (3) |

The self-action of the charge's own field is now clarified: A singular potential
${A}_{S}^{\alpha}$
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
${A}_{R}^{\alpha}$
that must be solely responsible for the radiation reaction. From the radiative potential we form an electromagnetic field tensor
${F}_{\alpha \beta}^{R}={\partial}_{\alpha}{A}_{\beta}^{R}-{\partial}_{\beta}{A}_{\alpha}^{R}$
, and we take the particle's equations of motion to be

where
${u}^{\mu}=d{z}^{\mu}/d\tau $
is the charge's four-velocity (
${z}^{\mu}\left(\tau \right)$
gives the description of the world line and
$\tau $
is proper time),
${a}^{\mu}=d{u}^{\mu}/d\tau $
its acceleration,
$m$
its (renormalized) mass, and
${f}_{ext}^{\mu}$
an external force also acting on the particle. Calculation of the radiative field yields the more concrete expression

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] ^{
$\text{1}$
}
.

$$\begin{array}{c}m{a}_{\mu}={f}_{\mu}^{ext}+e{F}_{\mu \nu}^{R}{u}^{\nu},\end{array}$$ | (4) |

$$\begin{array}{c}m{a}^{\mu}={f}_{ext}^{\mu}+\frac{2{e}^{2}}{3m}\left({\delta}_{\nu}^{\mu}+{u}^{\mu}{u}_{\nu}\right)\frac{d{f}_{ext}^{\nu}}{d\tau},\end{array}$$ | (5) |

^{
$\text{1}$
}
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}^{\nu}/d\tau $
is replaced with
${m}^{-1}d{f}_{ext}^{\nu}/d\tau $
. 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

in terms of the retarded Green's function
${G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})={\delta}_{{\beta}^{\prime}}^{\alpha}\delta (t-{t}^{\prime}-|\mathit{x}-{\mathit{x}}^{\mathsf{\prime}}\left|\right)/|\mathit{x}-{\mathit{x}}^{\mathsf{\prime}}|$
. Here
$x=(t,\mathit{x})$
is an arbitrary field point,
${x}^{\prime}=({t}^{\prime},{\mathit{x}}^{\mathsf{\prime}})$
is a source point, and
$d{V}^{\prime}\equiv {d}^{4}{x}^{\prime}$
; tensors at
$x$
are identified with unprimed indices, while primed indices refer to tensors at
${x}^{\prime}$
. Similarly, the advanced solution can be expressed as

in terms of the advanced Green's function
${G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})={\delta}_{{\beta}^{\prime}}^{\alpha}\delta (t-{t}^{\prime}+|\mathit{x}-{\mathit{x}}^{\mathsf{\prime}}\left|\right)/|\mathit{x}-{\mathit{x}}^{\mathsf{\prime}}|$
. The retarded Green's function is zero whenever
$x$
lies outside of the future light cone of
${x}^{\prime}$
, and
${G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
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
${x}^{\prime}$
, and
${G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
is infinite at these points. The retarded and advanced Green's functions satisfy the reciprocity relation

this states that the retarded Green's function becomes the advanced Green's function (and vice versa) when
$x$
and
${x}^{\prime}$
are interchanged.

$$\begin{array}{c}{A}_{ret}^{\alpha}\left(x\right)=\int {G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime}){j}^{{\beta}^{\prime}}\left({x}^{\prime}\right)d{V}^{\prime}\end{array}$$ | (6) |

$$\begin{array}{c}{A}_{adv}^{\alpha}\left(x\right)=\int {G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime}){j}^{{\beta}^{\prime}}\left({x}^{\prime}\right)d{V}^{\prime}\end{array}$$ | (7) |

$$\begin{array}{c}{G}_{{\beta}^{\prime}\alpha}^{-}({x}^{\prime},x)={G}_{\alpha {\beta}^{\prime}}^{+}(x,{x}^{\prime});\end{array}$$ | (8) |

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

and a radiative Green's function by

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

$$\begin{array}{c}{G}_{S{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})=\frac{1}{2}\left[{G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})+{G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})\right]\end{array}$$ | (9) |

$$\begin{array}{c}{G}_{R{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})={G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})-{G}_{S{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})=\frac{1}{2}\left[{G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})-{G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})\right].\end{array}$$ | (10) |

${G}_{{\beta}^{\prime}\alpha}^{S}({x}^{\prime},x)={G}_{\alpha {\beta}^{\prime}}^{S}(x,{x}^{\prime})$
. The radiative Green's function, on the other hand, is antisymmetric.

The potential

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

satisfies the homogeneous equation
$\square {A}^{\alpha}=0$
and is well behaved on the world line.

$$\begin{array}{c}{A}_{S}^{\alpha}\left(x\right)=\int {G}_{S{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime}){j}^{{\beta}^{\prime}}\left({x}^{\prime}\right)d{V}^{\prime}\end{array}$$ | (11) |

$$\begin{array}{c}{A}_{R}^{\alpha}\left(x\right)=\int {G}_{R{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime}){j}^{{\beta}^{\prime}}\left({x}^{\prime}\right)d{V}^{\prime}\end{array}$$ | (12) |

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
$z\left(u\right)$
;
$u$
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
$z\left(v\right)$
;
$v$
is the advanced time, the value of the proper-time parameter at the advanced point.

1.4 Green's functions in curved spacetime

In a curved spacetime with metric
${g}_{\alpha \beta}$
the wave equation for the vector potential becomes

where
$\square ={g}^{\alpha \beta}{\nabla}_{\alpha}{\nabla}_{\beta}$
is the covariant wave operator and
${R}_{\alpha \beta}$
is the spacetime's Ricci tensor; the Lorenz gauge conditions becomes
${\nabla}_{\alpha}{A}^{\alpha}=0$
, and
${\nabla}_{\alpha}$
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
$d{V}^{\prime}$
now stands for
$\sqrt{-g\left({x}^{\prime}\right)}{d}^{4}{x}^{\prime}$
.

$$\begin{array}{c}\square {A}^{\alpha}-{R}_{\beta}^{\alpha}{A}^{\beta}=-4\pi {j}^{\alpha},\end{array}$$ | (13) |

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
${x}^{\prime}$
, in curved spacetime its support extends inside the light cone as well;
${G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
is therefore nonzero when
$x\in {I}^{+}\left({x}^{\prime}\right)$
, which denotes the chronological future of
${x}^{\prime}$
. 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
$\tau \le u$
(see Figure 2 ).
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
${x}^{\prime}$
, in curved spacetime its support extends inside the light cone, and
${G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
is nonzero when
$x\in {I}^{-}\left({x}^{\prime}\right)$
, which denotes the chronological past of
${x}^{\prime}$
. 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
$\tau \ge v$
.

The physically relevant solution to Equation ( 13 ) is obviously the retarded potential
${A}_{ret}^{\alpha}\left(x\right)$
, 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
$\tau $
outside the interval
$u<\tau <v$
; 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 ):

The two-point function
${H}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
is introduced specifically to cure the pathology described in the preceding paragraph. It is symmetric in its indices and arguments, so that
${G}_{\alpha {\beta}^{\prime}}^{S}(x,{x}^{\prime})$
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,
$\square {H}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})-{R}_{\gamma}^{\alpha}\left(x\right){H}_{{\beta}^{\prime}}^{\gamma}(x,{x}^{\prime})=0$
, so that the singular, retarded, and advanced Green's functions will all satisfy the same wave equation.

$$\begin{array}{c}{G}_{S{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})=\frac{1}{2}\left[{G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})+{G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})-{H}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})\right].\end{array}$$ | (14) |

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
${x}^{\prime}$
:
${H}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})={G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
when
$x\in {I}^{-}\left({x}^{\prime}\right)$
. This ensures that
${G}_{S{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
vanishes when
$x$
is in the chronological past of
${x}^{\prime}$
.

In fact, reciprocity implies that
${H}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})$
will also agree with the retarded Green's function when
$x$
is in the chronological future of
${x}^{\prime}$
, and it follows that the symmetric Green's function vanishes also when
$x$
is in the chronological future of
${x}^{\prime}$
.

The potential
${A}_{S}^{\alpha}\left(x\right)$
constructed from the singular Green's function can now be seen to depend on the particle's state of motion at times
$\tau $
restricted to the interval
$u\le \tau \le v$
(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.)
The Detweiler–Whiting [23] definition for the radiative Green's function is then

The potential
${A}_{R}^{\alpha}\left(x\right)$
constructed from this depends on the particle's state of motion at all times
$\tau $
prior to the advanced time
$v$
:
$\tau \le v$
. 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.

$$\begin{array}{c}{G}_{R{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})={G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})-{G}_{S{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})=\frac{1}{2}\left[{G}_{+{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})-{G}_{-{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})+{H}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime})\right].\end{array}$$ | (15) |

And because the singular potential
${A}_{S}^{\alpha}\left(x\right)$
can be shown to exert no force on the particle, we conclude that
${A}_{R}^{\alpha}\left(x\right)$
alone is responsible for the self-force.

From the radiative potential we form an electromagnetic field tensor
${F}_{\alpha \beta}^{R}={\nabla}_{\alpha}{A}_{\beta}^{R}-{\nabla}_{\beta}{A}_{\alpha}^{R}$
, and the curved-spacetime generalization to Equation ( 4 ) is

where
${u}^{\mu}=d{z}^{\mu}/d\tau $
is again the charge's four-velocity, but
${a}^{\mu}=D{u}^{\mu}/d\tau $
is now its covariant acceleration.

$$\begin{array}{c}m{a}_{\mu}={f}_{\mu}^{ext}+e{F}_{\mu \nu}^{R}{u}^{\nu},\end{array}$$ | (16) |

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
$\gamma $
be the world line of a point particle in a curved spacetime. It is described by parametric relations
${z}^{\mu}\left(\tau \right)$
in which
$\tau $
is proper time. Its tangent vector is
${u}^{\mu}=d{z}^{\mu}/d\tau $
and its acceleration is
${a}^{\mu}=D{u}^{\mu}/d\tau $
; we shall also encounter
${\dot{a}}^{\mu}\equiv D{a}^{\mu}/d\tau $
.

On
$\gamma $
we erect an orthonormal basis that consists of the four-velocity
${u}^{\mu}$
and three spatial vectors
${e}^{\mu}a$
labelled by a frame index
$a=(1,2,3)$
. These vectors satisfy the relations
${g}_{\mu \nu}{u}^{\mu}{u}^{\nu}=-1$
,
${g}_{\mu \nu}{u}^{\mu}{e}^{\nu}a=0$
, and
${g}_{\mu \nu}{e}^{\mu}a{e}^{\nu}b={\delta}_{ab}$
. We take the spatial vectors to be Fermi–Walker transported on the world line:
$D{e}^{\mu}a/d\tau ={a}_{a}{u}^{\mu}$
, where

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,

as well as frame components of the Ricci tensor,

We shall use
${\delta}_{ab}=diag(1,1,1)$
and its inverse
${\delta}^{ab}=diag(1,1,1)$
to lower and raise frame indices, respectively.

$$\begin{array}{c}{a}_{a}\left(\tau \right)={a}_{\mu}{e}^{\mu}a\end{array}$$ | (17) |

$$\begin{array}{c}{R}_{a0b0}\left(\tau \right)={R}_{\mu \lambda \nu \rho}{e}^{\mu}a{u}^{\lambda}{e}^{\nu}b{u}^{\rho},{R}_{a0bc}\left(\tau \right)={R}_{\mu \lambda \nu \rho}{e}^{\mu}a{u}^{\lambda}{e}^{\nu}b{e}^{\rho}c,{R}_{abcd}\left(\tau \right)={R}_{\mu \lambda \nu \rho}{e}^{\mu}a{e}^{\lambda}b{e}^{\nu}c{e}^{\rho}d,\end{array}$$ | (18) |

$$\begin{array}{c}{R}_{00}\left(\tau \right)={R}_{\mu \nu}{u}^{\mu}{u}^{\nu},{R}_{a0}\left(\tau \right)={R}_{\mu \nu}{e}^{\mu}a{u}^{\nu},{R}_{ab}\left(\tau \right)={R}_{\mu \nu}{e}^{\mu}a{e}^{\nu}b.\end{array}$$ | (19) |

Consider a point
$x$
in a neighbourhood of the world line
$\gamma $
. We assume that
$x$
is sufficiently close to the world line that a unique geodesic links
$x$
to any neighbouring point
$z$
on
$\gamma $
. The two-point function
$\sigma (x,z)$
, 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
$\sigma (x,z)$
is zero if
$x$
and
$z$
are linked by a null geodesic. We denote its gradient
$\partial \sigma /\partial {z}^{\mu}$
by
${\sigma}_{\mu}(x,z)$
, and
$-{\sigma}^{\mu}$
gives a meaningful notion of a separation vector (pointing from
$z$
to
$x$
).

To construct a coordinate system in this neighbourhood we locate the unique point
${x}^{\prime}\equiv z\left(u\right)$
on
$\gamma $
which is linked to
$x$
by a future-directed null geodesic (this geodesic is directed from
${x}^{\prime}$
to
$x$
); I shall refer to
${x}^{\prime}$
as the retarded point associated with
$x$
, and
$u$
will be called the retarded time.

To tensors at
${x}^{\prime}$
we assign indices
${\alpha}^{\prime}$
,
${\beta}^{\prime}$
, . . . ; this will distinguish them from tensors at a generic point
$z\left(\tau \right)$
on the world line, to which we have assigned indices
$\mu $
,
$\nu $
, . . . . We have
$\sigma (x,{x}^{\prime})=0$
, and
$-{\sigma}^{{\alpha}^{\prime}}(x,{x}^{\prime})$
is a null vector that can be interpreted as the separation between
${x}^{\prime}$
and
$x$
.
The retarded coordinates of the point
$x$
are
$(u,{\hat{x}}^{a})$
, where
${\hat{x}}^{a}=-{e}^{a}{\alpha}^{\prime}{\sigma}^{{\alpha}^{\prime}}$
are the frame components of the separation vector. They come with a straightforward interpretation (see Figure 4 ). The invariant quantity

is an affine parameter on the null geodesic that links
$x$
to
${x}^{\prime}$
; it can be loosely interpreted as the time delay between
$x$
and
${x}^{\prime}$
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

is constant on the null geodesic that links
$x$
to
${x}^{\prime}$
. Because
${\Omega}^{a}$
is a different constant on each null geodesic that emanates from
${x}^{\prime}$
, keeping
$u$
fixed and varying
${\Omega}^{a}$
produces a congruence of null geodesics that generate the future light cone of the point
${x}^{\prime}$
(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
${\Omega}^{a}$
, 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
$\gamma $
.

$$\begin{array}{c}r\equiv \sqrt{{\delta}_{ab}{\hat{x}}^{a}{\hat{x}}^{b}}={u}_{{\alpha}^{\prime}}{\sigma}^{{\alpha}^{\prime}}\end{array}$$ | (20) |

$$\begin{array}{c}{\Omega}^{a}={\hat{x}}^{a}/r\end{array}$$ | (21) |

To tensors at
$x$
we assign indices
$\alpha $
,
$\beta $
, . . . . These tensors will be decomposed in a tetrad
$({e}^{\alpha}0,{e}^{\alpha}a)$
that is constructed as follows: Given
$x$
we locate its associated retarded point
${x}^{\prime}$
on the world line, as well as the null geodesic that links these two points; we then take the tetrad
$({u}^{{\alpha}^{\prime}},{e}^{{\alpha}^{\prime}}a)$
at
${x}^{\prime}$
and parallel transport it to
$x$
along the null geodesic to obtain
$({e}^{\alpha}0,{e}^{\alpha}a)$
.

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

The retarded solution to Equation ( 13 ) is

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.

$$\begin{array}{c}{A}^{\alpha}\left(x\right)=e{\int}_{\gamma}{G}_{+\mu}^{\alpha}(x,z){u}^{\mu}d\tau ,\end{array}$$ | (22) |

From the vector potential we form the electromagnetic field tensor
${F}_{\alpha \beta}$
, which we decompose in the tetrad
$({e}^{\alpha}0,{e}^{\alpha}a)$
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

where

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

In these expressions, all tensors (or their frame components) are evaluated at the retarded point
${x}^{\prime}\equiv z\left(u\right)$
associated with
$x$
; for example,
${a}_{a}\equiv {a}_{a}\left(u\right)\equiv {a}_{{\alpha}^{\prime}}{e}^{{\alpha}^{\prime}}a$
. 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
$-\infty <\tau \le {u}^{-}\equiv u-{0}^{+}$
; 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
$z\left(\tau \right)$
coincides with
${x}^{\prime}$
; 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 ).

$$\begin{array}{ccc}{F}_{a0}(u,r,{\Omega}^{a})& \equiv & {F}_{\alpha \beta}\left(x\right){e}^{\alpha}a\left(x\right){e}^{\beta}0\left(x\right)\end{array}$$ |

$$\begin{array}{ccc}& =& \frac{e}{{r}^{2}}{\Omega}_{a}-\frac{e}{r}\left({a}_{a}-{a}_{b}{\Omega}^{b}{\Omega}_{a}\right)+\frac{1}{3}e{R}_{b0c0}{\Omega}^{b}{\Omega}^{c}{\Omega}_{a}-\frac{1}{6}e\left(5{R}_{a0b0}{\Omega}^{b}+{R}_{ab0c}{\Omega}^{b}{\Omega}^{c}\right)\end{array}$$ |

$$\begin{array}{ccc}& & +\frac{1}{12}e\left(5{R}_{00}+{R}_{bc}{\Omega}^{b}{\Omega}^{c}+R\right){\Omega}_{a}+\frac{1}{3}e{R}_{a0}-\frac{1}{6}e{R}_{ab}{\Omega}^{b}+{F}_{a0}^{tail}+\mathcal{O}\left(r\right),\end{array}$$ | (23) |

$$\begin{array}{ccc}{F}_{ab}(u,r,{\Omega}^{a})& \equiv & {F}_{\alpha \beta}\left(x\right){e}^{\alpha}a\left(x\right){e}^{\beta}b\left(x\right)\end{array}$$ |

$$\begin{array}{ccc}& =& \frac{e}{r}\left({a}_{a}{\Omega}_{b}-{\Omega}_{a}{a}_{b}\right)+\frac{1}{2}e\left({R}_{a0bc}-{R}_{b0ac}+{R}_{a0c0}{\Omega}_{b}-{\Omega}_{a}{R}_{b0c0}\right){\Omega}^{c}\end{array}$$ |

$$\begin{array}{ccc}& & -\frac{1}{2}e\left({R}_{a0}{\Omega}_{b}-{\Omega}_{a}{R}_{b0}\right)+{F}_{ab}^{tail}+\mathcal{O}\left(r\right),\end{array}$$ | (24) |

$$\begin{array}{c}{F}_{a0}^{tail}={F}_{{\alpha}^{\prime}{\beta}^{\prime}}^{tail}\left({x}^{\prime}\right){e}^{{\alpha}^{\prime}}a{u}^{{\beta}^{\prime}},{F}_{ab}^{tail}={F}_{{\alpha}^{\prime}{\beta}^{\prime}}^{tail}\left({x}^{\prime}\right){e}^{{\alpha}^{\prime}}a{e}^{{\beta}^{\prime}}b\end{array}$$ | (25) |

$$\begin{array}{c}{F}_{{\alpha}^{\prime}{\beta}^{\prime}}^{tail}\left({x}^{\prime}\right)=2e{\int}_{-\infty}^{{u}^{-}}{\nabla}_{[{\alpha}^{\prime}}{G}_{+{\beta}^{\prime}]\mu}({x}^{\prime},z){u}^{\mu}d\tau .\end{array}$$ | (26) |

The expansion of
${F}_{\alpha \beta}\left(x\right)$
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
${F}_{a0}$
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
${F}_{ab}$
contains a term proportional to
${R}_{a0bc}{\Omega}^{c}$
whose limit depends on the direction of approach.

This singularity structure is perfectly reproduced by the singular field
${F}_{\alpha \beta}^{S}$
obtained from the potential

where
${G}_{S\mu}^{\alpha}(x,z)$
is the singular Green's function of Equation ( 14 ). Near the world line the singular field is given by

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

$$\begin{array}{c}{A}_{S}^{\alpha}\left(x\right)=e{\int}_{\gamma}{G}_{S\mu}^{\alpha}(x,z){u}^{\mu}d\tau ,\end{array}$$ | (27) |

$$\begin{array}{ccc}{F}_{a0}^{S}(u,r,{\Omega}^{a})& \equiv & {F}_{\alpha \beta}^{S}\left(x\right){e}^{\alpha}a\left(x\right){e}^{\beta}0\left(x\right)\end{array}$$ |

$$\begin{array}{ccc}& =& \frac{e}{{r}^{2}}{\Omega}_{a}-\frac{e}{r}\left({a}_{a}-{a}_{b}{\Omega}^{b}{\Omega}_{a}\right)-\frac{2}{3}e{\dot{a}}_{a}+\frac{1}{3}e{R}_{b0c0}{\Omega}^{b}{\Omega}^{c}{\Omega}_{a}-\frac{1}{6}e\left(5{R}_{a0b0}{\Omega}^{b}+{R}_{ab0c}{\Omega}^{b}{\Omega}^{c}\right)\end{array}$$ |

$$\begin{array}{ccc}& & +\frac{1}{12}e\left(5{R}_{00}+{R}_{bc}{\Omega}^{b}{\Omega}^{c}+R\right){\Omega}_{a}-\frac{1}{6}e{R}_{ab}{\Omega}^{b}+\mathcal{O}\left(r\right),\end{array}$$ | (28) |

$$\begin{array}{ccc}{F}_{ab}^{S}(u,r,{\Omega}^{a})& \equiv & {F}_{\alpha \beta}^{S}\left(x\right){e}^{\alpha}a\left(x\right){e}^{\beta}b\left(x\right)\end{array}$$ |

$$\begin{array}{ccc}& =& \frac{e}{r}\left({a}_{a}{\Omega}_{b}-{\Omega}_{a}{a}_{b}\right)+\frac{1}{2}e\left({R}_{a0bc}-{R}_{b0ac}+{R}_{a0c0}{\Omega}_{b}-{\Omega}_{a}{R}_{b0c0}\right){\Omega}^{c}\end{array}$$ |

$$\begin{array}{ccc}& & -\frac{1}{2}e\left({R}_{a0}{\Omega}_{b}-{\Omega}_{a}{R}_{b0}\right)+\mathcal{O}\left(r\right).\end{array}$$ | (29) |

The difference between the retarded and singular fields defines the radiative field
${F}_{\alpha \beta}^{R}\left(x\right)$
. Its frame components are

and at
${x}^{\prime}$
the radiative field becomes

where
${\dot{a}}^{{\gamma}^{\prime}}=D{a}^{{\gamma}^{\prime}}/d\tau $
is the rate of change of the acceleration vector, and where the tail term was given by Equation ( 26 ). We see that
${F}_{\alpha \beta}^{R}\left(x\right)$
is a smooth tensor field, even on the world line.

$$\begin{array}{ccc}{F}_{a0}^{R}& =& \frac{2}{3}e{\dot{a}}_{a}+\frac{1}{3}e{R}_{a0}+{F}_{a0}^{tail}+\mathcal{O}\left(r\right),\end{array}$$ | (30) |

$$\begin{array}{ccc}{F}_{ab}^{R}& =& {F}_{ab}^{tail}+\mathcal{O}\left(r\right),\end{array}$$ | (31) |

$$\begin{array}{c}{F}_{{\alpha}^{\prime}{\beta}^{\prime}}^{R}=2e{u}_{[{\alpha}^{\prime}}\left({g}_{{\beta}^{\prime}]{\gamma}^{\prime}}+{u}_{{\beta}^{\prime}]}{u}_{{\gamma}^{\prime}}\right)\left(\frac{2}{3}{\dot{a}}^{{\gamma}^{\prime}}+\frac{1}{3}{R}_{{\delta}^{\prime}}^{{\gamma}^{\prime}}{u}^{{\delta}^{\prime}}\right)+{F}_{{\alpha}^{\prime}{\beta}^{\prime}}^{tail},\end{array}$$ | (32) |

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
$m{a}_{\mu}={f}_{\mu}^{ext}+e{F}_{\mu \nu}^{R}{u}^{\nu}$
, where
${f}_{\mu}^{ext}$
is an external force also acting on the particle. Substituting Equation ( 32 ) gives

in which all tensors are evaluated at
$z\left(\tau \right)$
, the current position of the particle on the world line.

$$\begin{array}{c}m{a}^{\mu}={f}_{ext}^{\mu}+{e}^{2}\left({\delta}_{\nu}^{\mu}+{u}^{\mu}{u}_{\nu}\right)\left(\frac{2}{3m}\frac{D{f}_{ext}^{\nu}}{d\tau}+\frac{1}{3}{R}_{\lambda}^{\nu}{u}^{\lambda}\right)+2{e}^{2}{u}_{\nu}{\int}_{-\infty}^{{\tau}^{-}}{\nabla}^{[\mu}{G}_{+{\lambda}^{\prime}}^{\nu ]}\left(z\left(\tau \right),z\left({\tau}^{\prime}\right)\right){u}^{{\lambda}^{\prime}}d{\tau}^{\prime},\end{array}$$ | (33) |

The primed indices in the tail integral refer to a point
$z\left({\tau}^{\prime}\right)$
which represents a prior position; the integration is cut short at
${\tau}^{\prime}={\tau}^{-}\equiv \tau -{0}^{+}$
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
${\dot{a}}^{\nu}$
with
${m}^{-1}{\dot{f}}_{ext}^{\nu}$
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
${F}_{\alpha \beta}^{R}$
; 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
$\Phi \left(x\right)$
, which satisfies a wave equation

that is very similar to Equation ( 13 ). Here,
$R$
is the spacetime's Ricci scalar, and
$\xi $
is an arbitrary coupling constant; the scalar charge density
$\mu \left(x\right)$
is given by a four-dimensional Dirac functional supported on the particle's world line
$\gamma $
. The retarded solution to the wave equation is

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

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

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.

$$\begin{array}{c}\left(\square -\xi R\right)\Phi =-4\pi \mu \end{array}$$ | (34) |

$$\begin{array}{c}\Phi \left(x\right)=q{\int}_{\gamma}{G}_{+}(x,z)d\tau ,\end{array}$$ | (35) |

$$\begin{array}{c}m{a}^{\mu}=q\left({g}^{\mu \nu}+{u}^{\mu}{u}^{\nu}\right){\nabla}_{\nu}\Phi ,\end{array}$$ | (36) |

$$\begin{array}{c}\frac{dm}{d\tau}=-q{u}^{\mu}{\nabla}_{\mu}\Phi ,\end{array}$$ | (37) |

The scalar field of Equation ( 35 ) diverges on the world line, and its singular part
${\Phi}_{S}\left(x\right)$
must be removed before Equations ( 36 ) and ( 37 ) can be evaluated. This procedure produces the radiative field
${\Phi}_{R}\left(x\right)$
, 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

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

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

$$\begin{array}{c}{\nabla}_{\mu}{\Phi}_{R}=-\frac{1}{12}(1-6\xi )qR{u}_{\mu}+q\left({g}_{\mu \nu}+{u}_{\mu}{u}_{\nu}\right)\left(\frac{1}{3}{\dot{a}}^{\nu}+\frac{1}{6}{R}_{\lambda}^{\nu}{u}^{\lambda}\right)+{\Phi}_{\mu}^{tail}\end{array}$$ | (38) |

$$\begin{array}{c}{\Phi}_{\mu}^{tail}=q{\int}_{-\infty}^{{\tau}^{-}}{\nabla}_{\mu}{G}_{+}\left(z\left(\tau \right),z\left({\tau}^{\prime}\right)\right)d{\tau}^{\prime},\end{array}$$ | (39) |

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
${f}_{ext}^{\mu}$
and reduce the order of the differential equation.) The acceleration is given by

and the mass changes according to

These equations were first derived by Quinn [48] ^{
$\text{2}$
}
.

$$\begin{array}{c}m{a}^{\mu}={f}_{ext}^{\mu}+{q}^{2}\left({\delta}_{\nu}^{\mu}+{u}^{\mu}{u}_{\nu}\right)\left[\frac{1}{3m}\frac{D{f}_{ext}^{\nu}}{d\tau}+\frac{1}{6}{R}_{\lambda}^{\nu}{u}^{\lambda}+{\int}_{-\infty}^{{\tau}^{-}}{\nabla}^{\nu}{G}_{+}\left(z\left(\tau \right),z\left({\tau}^{\prime}\right)\right)d{\tau}^{\prime}\right],\end{array}$$ | (40) |

$$\begin{array}{c}\frac{dm}{d\tau}=-\frac{1}{12}(1-6\xi ){q}^{2}R-{q}^{2}{u}^{\mu}{\int}_{-\infty}^{{\tau}^{-}}{\nabla}_{\mu}{G}_{+}\left(z\left(\tau \right),z\left({\tau}^{\prime}\right)\right)d{\tau}^{\prime}.\end{array}$$ | (41) |

In flat spacetime the Ricci-tensor term and the tail integral disappear, and Equation ( 40 ) takes the form of Equation ( 5 ) with
${q}^{2}/\left(3m\right)$
replacing the factor of
$2{e}^{2}/\left(3m\right)$
. In this simple case Equation ( 41 ) reduces to
$dm/d\tau =0$
and the mass is in fact a constant. This property remains true in a conformally-flat spacetime when the wave equation is conformally invariant (
$\xi =1/6$
): 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.

^{
$\text{2}$
}
His analysis was restricted to a minimally-coupled scalar field, so that
$\xi =0$
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
${h}_{\alpha \beta}\left(x\right)$
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
${g}_{\alpha \beta}$
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
${\gamma}_{\alpha \beta}$
defined by

where
${h}_{\alpha \beta}$
is the difference between
${\mathsf{g}}_{\alpha \beta}$
, the actual metric of the perturbed spacetime, and
${g}_{\alpha \beta}$
.

$$\begin{array}{c}{\gamma}_{\alpha \beta}={h}_{\alpha \beta}-\frac{1}{2}\left({g}^{\gamma \delta}{h}_{\gamma \delta}\right){g}_{\alpha \beta},\end{array}$$ | (42) |

The potentials satisfy the wave equation

together with the Lorenz gauge condition
${\gamma}_{;\beta}^{\alpha \beta}=0$
. Here and below, covariant differentiation refers to a connection that is compatible with the background metric,
$\square ={g}^{\alpha \beta}{\nabla}_{\alpha}{\nabla}_{\beta}$
is the wave operator for the background spacetime, and
${T}^{\alpha \beta}$
is the stress-energy tensor of the point mass; this is given by a Dirac distribution supported on the particle's world line
$\gamma $
. The retarded solution is

where
${G}_{+\mu \nu}^{\alpha \beta}(x,z)$
is the retarded Green's function associated with Equation ( 43 ). The perturbation
${h}_{\alpha \beta}\left(x\right)$
can be recovered by inverting Equation ( 42 ).

$$\begin{array}{c}\square {\gamma}^{\alpha \beta}+2{R}_{\gamma \delta}^{\alpha \beta}{\gamma}^{\gamma \delta}=-16\pi {T}^{\alpha \beta}\end{array}$$ | (43) |

$$\begin{array}{c}{\gamma}^{\alpha \beta}\left(x\right)=4m{\int}_{\gamma}{G}_{+\mu \nu}^{\alpha \beta}(x,z){u}^{\mu}{u}^{\nu}d\tau ,\end{array}$$ | (44) |

Equations of motion for the point mass can be obtained by formally demanding that the motion be geodesic in the perturbed spacetime with metric
${\mathsf{g}}_{\alpha \beta}={g}_{\alpha \beta}+{h}_{\alpha \beta}$
. After a mapping to the background spacetime, the equations of motion take the form of

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.

$$\begin{array}{c}{a}^{\mu}=-\frac{1}{2}\left({g}^{\mu \nu}+{u}^{\mu}{u}^{\nu}\right)\left(2{h}_{\nu \lambda ;\rho}-{h}_{\lambda \rho ;\nu}\right){u}^{\lambda}{u}^{\rho}.\end{array}$$ | (45) |

We now remove
${h}_{\alpha \beta}^{S}\left(x\right)$
from the retarded perturbation and postulate that it is the radiative field
${h}_{\alpha \beta}^{S}\left(x\right)$
that should act on the particle. (Note that
${\gamma}_{\alpha \beta}^{S}$
satisfies the same wave equation as the retarded potentials, but that
${\gamma}_{\alpha \beta}^{R}$
is a free gravitational field that satisfies the homogeneous wave equation.) On the world line we have

where the tail term is given by

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

Only the tail integral appears in the final form of the equations of motion. It involves the current position
$z\left(\tau \right)$
of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions
$z\left({\tau}^{\prime}\right)$
, at which tensors with primed indices are evaluated. As before the integral is cut short at
${\tau}^{\prime}={\tau}^{-}\equiv \tau -{0}^{+}$
to avoid the singular behaviour of the retarded Green's function at coincidence.

$$\begin{array}{c}{h}_{\mu \nu ;\lambda}^{R}=-4m\left({u}_{(\mu}{R}_{\nu )\rho \lambda \xi}+{R}_{\mu \rho \nu \xi}{u}_{\lambda}\right){u}^{\rho}{u}^{\xi}+{h}_{\mu \nu \lambda}^{tail},\end{array}$$ | (46) |

$$\begin{array}{c}{h}_{\mu \nu \lambda}^{tail}=4m{\int}_{-\infty}^{{\tau}^{-}}{\nabla}_{\lambda}\left({G}_{+\mu \nu {\mu}^{\prime}{\nu}^{\prime}}-\frac{1}{2}{g}_{\mu \nu}{G}_{+\rho {\mu}^{\prime}{\nu}^{\prime}}^{\rho}\right)\left(z\left(\tau \right),z\left({\tau}^{\prime}\right)\right){u}^{{\mu}^{\prime}}{u}^{{\nu}^{\prime}}d{\tau}^{\prime}.\end{array}$$ | (47) |

$$\begin{array}{c}{a}^{\mu}=-\frac{1}{2}\left({g}^{\mu \nu}+{u}^{\mu}{u}^{\nu}\right)\left(2{h}_{\nu \lambda \rho}^{tail}-{h}_{\lambda \rho \nu}^{tail}\right){u}^{\lambda}{u}^{\rho}.\end{array}$$ | (48) |

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
${g}_{\alpha \beta}+{h}_{\alpha \beta}^{R}$
. 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
${g}_{\alpha \beta}$
, 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
${\gamma}_{;\beta}^{\alpha \beta}=0$
. Barack and Ori [8] have shown that under a coordinate transformation of the form
${x}^{\alpha}\to {x}^{\alpha}+{\xi}^{\alpha}$
, where
${x}^{\alpha}$
are the coordinates of the background spacetime and
${\xi}^{\alpha}$
is a smooth vector field of order
$m$
, the particle's acceleration changes according to
${a}^{\mu}\to {a}^{\mu}+a[\xi {]}^{\mu}$
, where

is the “gauge acceleration”;
${D}^{2}{\xi}^{\nu}/d{\tau}^{2}=({\xi}_{;\mu}^{\nu}{u}^{\mu}{)}_{;\rho}{u}^{\rho}$
is the second covariant derivative of
${\xi}^{\nu}$
in the direction of the world line. This implies that the particle's acceleration can be altered at will by a gauge transformation;
${\xi}^{\alpha}$
could even be chosen so as to produce
${a}^{\mu}=0$
, making the motion geodesic after all. This observation provides a dramatic illustration of the following point:

$$\begin{array}{c}a[\xi {]}^{\mu}=\left({\delta}_{\nu}^{\mu}+{u}^{\mu}{u}_{\nu}\right)\left(\frac{{D}^{2}{\xi}^{\nu}}{d{\tau}^{2}}+{R}_{\rho \omega \lambda}^{\nu}{u}^{\rho}{\xi}^{\omega}{u}^{\lambda}\right)\end{array}$$ | (49) |

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
${h}_{\alpha \beta}$
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
$\Phi \ll 1$
. (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

where
$M$
is the total mass contained in the spacetime,
$r=\left|\mathit{x}\right|$
is the distance from the centre of mass,
$\hat{\mathit{r}}=\mathit{x}/r$
, and
$\mathit{g}=-\mathsf{\nabla}\Phi $
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
$m\mathit{g}$
. 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
$m\mathit{g}$
were acting on the particle. This agreement is necessary, but remarkable!

$$\begin{array}{c}{\mathit{f}}_{em}={e}^{2}\frac{M}{{r}^{3}}\hat{\mathit{r}}+\frac{2}{3}{e}^{2}\frac{d\mathit{g}}{dt},\end{array}$$ | (50) |

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

Here

where
$\xi $
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.

$$\begin{array}{c}{\mathit{f}}_{scalar}=2\xi {q}^{2}\frac{M}{{r}^{3}}\hat{\mathit{r}}+\frac{1}{3}{q}^{2}\frac{d\mathit{g}}{dt},\end{array}$$ | (51) |

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

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 ).

$$\begin{array}{c}f=\frac{Mm}{{r}^{2}}{\left(1-\frac{2M}{r}\right)}^{-1/2}-{e}^{2}\frac{M}{{r}^{3}},\end{array}$$ | (52) |

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
${G}_{+}(x,{x}^{\prime})$
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}_{\alpha}\Phi $
, and from this subtracts a mode-decomposition of the singular field
${\nabla}_{\alpha}{\Phi}_{S}$
, for which a local expression is known. This results in the radiative field
${\nabla}_{\alpha}{\Phi}_{R}$
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
$\sigma (x,{x}^{\prime})$
and its derivatives in Section 2.1 , the parallel propagator
${g}_{{\alpha}^{\prime}}^{\alpha}(x,{x}^{\prime})$
in Section 2.3 , and the van Vleck determinant
$\Delta (x,{x}^{\prime})$
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
${x}^{\prime}$
; 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
${x}^{\prime}$
. I then move on (in Section 3.2 ) to Fermi normal coordinates [36] , which are defined in a neighbourhood of a world line
$\gamma $
. 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
$z\left(v\right)$
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
${x}^{\prime}$
. 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
$-v(x,{x}^{\prime})$
, I have taken the liberty of eliminating the silly minus sign and I call it instead
$+V(x,{x}^{\prime})$
. The reader should also note that all my Green's functions are normalized in the same way, with a factor of
$-4\pi $
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\pi $
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
${x}^{\prime}$
, to which we refer as the “base point”, and to which we assign indices
${\alpha}^{\prime}$
,
${\beta}^{\prime}$
, etc. The second is
$x$
, to which we refer as the “field point”, and to which we assign indices
$\alpha $
,
$\beta $
, etc. We assume that
$x$
belongs to
$\mathcal{N}\left({x}^{\prime}\right)$
, the normal convex neighbourhood of
${x}^{\prime}$
; this is the set of points that are linked to
${x}^{\prime}$
by a unique geodesic. The geodesic
$\beta $
that links
$x$
to
${x}^{\prime}$
is described by relations
${z}^{\mu}\left(\lambda \right)$
in which
$\lambda $
is an affine parameter that ranges from
${\lambda}_{0}$
to
${\lambda}_{1}$
; we have
$z\left({\lambda}_{0}\right)\equiv {x}^{\prime}$
and
$z\left({\lambda}_{1}\right)\equiv x$
. To an arbitrary point
$z$
on the geodesic we assign indices
$\mu $
,
$\nu $
, etc. The vector
${t}^{\mu}=d{z}^{\mu}/d\lambda $
is tangent to the geodesic, and it obeys the geodesic equation
$D{t}^{\mu}/d\lambda =0$
. The situation is illustrated in Figure 5 .
Synge's world function is a scalar function of the base point
${x}^{\prime}$
and the field point
$x$
. It is defined by

and the integral is evaluated on the geodesic
$\beta $
that links
$x$
to
${x}^{\prime}$
. You may notice that
$\sigma $
is invariant under a constant rescaling of the affine parameter,
$\lambda \to \overline{\lambda}=a\lambda +b$
, where
$a$
and
$b$
are constants.

$$\begin{array}{c}\sigma (x,{x}^{\prime})=\frac{1}{2}({\lambda}_{1}-{\lambda}_{0}){\int}_{{\lambda}_{0}}^{{\lambda}_{1}}{g}_{\mu \nu}\left(z\right){t}^{\mu}{t}^{\nu}d\lambda ,\end{array}$$ | (53) |

By virtue of the geodesic equation, the quantity
$\varepsilon \equiv {g}_{\mu \nu}{t}^{\mu}{t}^{\nu}$
is constant on the geodesic. The world function is therefore numerically equal to
$\frac{1}{2}\varepsilon ({\lambda}_{1}-{\lambda}_{0}{)}^{2}$
. If the geodesic is timelike, then
$\lambda $
can be set equal to the proper time
$\tau $
, which implies that
$\varepsilon =-1$
and
$\sigma =-\frac{1}{2}(\Delta \tau {)}^{2}$
. If the geodesic is spacelike, then
$\lambda $
can be set equal to the proper distance
$s$
, which implies that
$\varepsilon =1$
and
$\sigma =\frac{1}{2}(\Delta s{)}^{2}$
. If the geodesic is null, then
$\sigma =0$
. Quite generally, therefore, the world function is half the squared geodesic distance between the points
${x}^{\prime}$
and
$x$
.

In flat spacetime, the geodesic linking
$x$
to
${x}^{\prime}$
is a straight line, and
$\sigma =\frac{1}{2}{\eta}_{\alpha \beta}(x-{x}^{\prime}{)}^{\alpha}(x-{x}^{\prime}{)}^{\beta}$
in Lorentzian coordinates.

2.1.2 Differentiation of the world function

The world function
$\sigma (x,{x}^{\prime})$
can be differentiated with respect to either argument. We let
${\sigma}_{\alpha}=\partial \sigma /\partial {x}^{\alpha}$
be its partial derivative with respect to
$x$
, and
${\sigma}_{{\alpha}^{\prime}}=\partial \sigma /\partial {x}^{{\alpha}^{\prime}}$
its partial derivative with respect to
${x}^{\prime}$
. It is clear that
${\sigma}_{\alpha}$
behaves as a dual vector with respect to tensorial operations carried out at
$x$
, but as a scalar with respect to operations carried out
${x}^{\prime}$
. Similarly,
${\sigma}_{{\alpha}^{\prime}}$
is a scalar at
$x$
but a dual vector at
${x}^{\prime}$
.

We let
${\sigma}_{\alpha \beta}\equiv {\nabla}_{\beta}{\sigma}_{\alpha}$
be the covariant derivative of
${\sigma}_{\alpha}$
with respect to
$x$
; this is a rank-2 tensor at
$x$
and a scalar at
${x}^{\prime}$
. Because
$\sigma $
is a scalar at
$x$
, we have that this tensor is symmetric:
${\sigma}_{\beta \alpha}={\sigma}_{\alpha \beta}$
.

Similarly, we let
${\sigma}_{\alpha {\beta}^{\prime}}\equiv {\partial}_{{\beta}^{\prime}}{\sigma}_{\alpha}={\partial}^{2}\sigma /\partial {x}^{{\beta}^{\prime}}\partial {x}^{\alpha}$
be the partial derivative of
${\sigma}_{\alpha}$
with respect to
${x}^{\prime}$
; this is a dual vector both at
$x$
and
${x}^{\prime}$
. We can also define
${\sigma}_{{\alpha}^{\prime}\beta}\equiv {\partial}_{\beta}{\sigma}_{{\alpha}^{\prime}}={\partial}^{2}\sigma /\partial {x}^{\beta}\partial {x}^{{\alpha}^{\prime}}$
to be the partial derivative of
${\sigma}_{{\alpha}^{\prime}}$
with respect to
$x$
. Because partial derivatives commute, these bitensors are equal:
${\sigma}_{{\beta}^{\prime}\alpha}={\sigma}_{\alpha {\beta}^{\prime}}$
. Finally, we let
${\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}\equiv {\nabla}_{{\beta}^{\prime}}{\sigma}_{{\alpha}^{\prime}}$
be the covariant derivative of
${\sigma}_{{\alpha}^{\prime}}$
with respect to
${x}^{\prime}$
; this is a symmetric rank-2 tensor at
${x}^{\prime}$
and a scalar at
$x$
.

The notation is easily extended to any number of derivatives. For example, we let
${\sigma}_{\alpha \beta \gamma {\delta}^{\prime}}\equiv {\nabla}_{{\delta}^{\prime}}{\nabla}_{\gamma}{\nabla}_{\beta}{\nabla}_{\alpha}\sigma $
, which is a rank-3 tensor at
$x$
and a dual vector at
${x}^{\prime}$
. This bitensor is symmetric in the pair of indices
$\alpha $
and
$\beta $
, but not in the pairs
$\alpha $
and
$\gamma $
, nor
$\beta $
and
$\gamma $
. Because
${\nabla}_{{\delta}^{\prime}}$
is here an ordinary partial derivative with respect to
${x}^{\prime}$
, the bitensor is symmetric in any pair of indices involving
${\delta}^{\prime}$
. The ordering of the primed index relative to the unprimed indices is therefore irrelevant: The same bitensor can be written as
${\sigma}_{{\delta}^{\prime}\alpha \beta \gamma}$
or
${\sigma}_{\alpha {\delta}^{\prime}\beta \gamma}$
or
${\sigma}_{\alpha \beta {\delta}^{\prime}\gamma}$
, making sure that the ordering of the unprimed indices is not altered.

More generally, we can show that derivatives of any bitensor
${\Omega}_{...}(x,{x}^{\prime})$
satisfy the property

in which “
$...$
” stands for any combination of primed and unprimed indices. We start by establishing the symmetry of
${\Omega}_{...;\alpha {\beta}^{\prime}}$
with respect to the pair
$\alpha $
and
${\beta}^{\prime}$
. This is most easily done by adopting Fermi normal coordinates (see Section 3.2 ) adapted to the geodesic
$\beta $
, and setting the connection to zero both at
$x$
and
${x}^{\prime}$
. In these coordinates, the bitensor
${\Omega}_{...;\alpha}$
is the partial derivative of
${\Omega}_{...}$
with respect to
${x}^{\alpha}$
, and
${\Omega}_{...;\alpha {\beta}^{\prime}}$
is obtained by taking an additional partial derivative with respect to
${x}^{{\beta}^{\prime}}$
. These two operations commute, and
${\Omega}_{...;{\beta}^{\prime}\alpha}={\Omega}_{...;\alpha {\beta}^{\prime}}$
follows as a bitensorial identity. Equation ( 54 ) then follows by further differentiation with respect to either
$x$
or
${x}^{\prime}$
.

$$\begin{array}{c}{\Omega}_{...;\beta {\alpha}^{\prime}...}={\Omega}_{...;{\alpha}^{\prime}\beta ...},\end{array}$$ | (54) |

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,
${\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}\gamma {\delta}^{\prime}\epsilon}={\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}{\delta}^{\prime}\gamma \epsilon}={\sigma}_{\gamma \epsilon {\alpha}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}$
.

2.1.3 Evaluation of first derivatives

We can compute
${\sigma}_{\alpha}$
by examining how
$\sigma $
varies when the field point
$x$
moves. We let the new field point be
$x+\delta x$
, and
$\delta \sigma \equiv \sigma (x+\delta x,{x}^{\prime})-\sigma (x,{x}^{\prime})$
is the corresponding variation of the world function. We let
$\beta +\delta \beta $
be the unique geodesic that links
$x+\delta x$
to
${x}^{\prime}$
; it is described by relations
${z}^{\mu}\left(\lambda \right)+\delta {z}^{\mu}\left(\lambda \right)$
, in which the affine parameter is scaled in such a way that it runs from
${\lambda}_{0}$
to
${\lambda}_{1}$
also on the new geodesic. We note that
$\delta z\left({\lambda}_{0}\right)=\delta {x}^{\prime}\equiv 0$
and
$\delta z\left({\lambda}_{1}\right)=\delta x$
.

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

in which the metric and the tangent vector are both evaluated at
$x$
. Apart from a factor
$\Delta \lambda $
, we see that
${\sigma}^{\alpha}(x,{x}^{\prime})$
is equal to the geodesic's tangent vector at
$x$
. If in Equation ( 55 ) we replace
$x$
by a generic point
$z\left(\lambda \right)$
on
$\beta $
, and if we correspondingly replace
${\lambda}_{1}$
by
$\lambda $
, we obtain
${\sigma}^{\mu}(z,{x}^{\prime})=(\lambda -{\lambda}_{0}){t}^{\mu}$
; we therefore see that
${\sigma}^{\mu}(z,{x}^{\prime})$
is a rescaled tangent vector on the geodesic.

$$\delta \sigma =\Delta \lambda {\int}_{{\lambda}_{0}}^{{\lambda}_{1}}\left({g}_{\mu \nu}{\dot{z}}^{\mu}\delta {\dot{z}}^{\nu}+\frac{1}{2}{g}_{\mu \nu ,\lambda}{\dot{z}}^{\mu}{\dot{z}}^{\nu}\delta {z}^{\lambda}\right)d\lambda ,$$

where
$\Delta \lambda ={\lambda}_{1}-{\lambda}_{0}$
, an overdot indicates differentiation with respect to
$\lambda $
, and the metric and its derivatives are evaluated on
$\beta $
. Integrating the first term by parts gives
$$\delta \sigma =\Delta \lambda {\left[{g}_{\mu \nu}{\dot{z}}^{\mu}\delta {z}^{\nu}\right]}_{{\lambda}_{0}}^{{\lambda}_{1}}-\Delta \lambda {\int}_{{\lambda}_{0}}^{{\lambda}_{1}}\left({g}_{\mu \nu}{\ddot{z}}^{\nu}+{\Gamma}_{\mu \nu \lambda}{\dot{z}}^{\nu}{\dot{z}}^{\lambda}\right)\delta {z}^{\mu}d\lambda .$$

The integral vanishes because
${z}^{\mu}\left(\lambda \right)$
satisfies the geodesic equation. The boundary term at
${\lambda}_{0}$
is zero because the variation
$\delta {z}^{\mu}$
vanishes there. We are left with
$\delta \sigma =\Delta \lambda {g}_{\alpha \beta}{t}^{\alpha}\delta {x}^{\beta}$
, or
$$\begin{array}{c}{\sigma}_{\alpha}(x,{x}^{\prime})=({\lambda}_{1}-{\lambda}_{0}){g}_{\alpha \beta}{t}^{\beta},\end{array}$$ | (55) |

A virtually identical calculation reveals how
$\sigma $
varies under a change of base point
${x}^{\prime}$
. Here the variation of the geodesic is such that
$\delta z\left({\lambda}_{0}\right)=\delta {x}^{\prime}$
and
$\delta z\left({\lambda}_{1}\right)=\delta x=0$
, and we obtain
$\delta \sigma =-\Delta \lambda {g}_{{\alpha}^{\prime}{\beta}^{\prime}}{t}^{{\alpha}^{\prime}}\delta {x}^{{\beta}^{\prime}}$
. This shows that

in which the metric and the tangent vector are both evaluated at
${x}^{\prime}$
. Apart from a factor
$\Delta \lambda $
, we see that
${\sigma}^{{\alpha}^{\prime}}(x,{x}^{\prime})$
is minus the geodesic's tangent vector at
${x}^{\prime}$
.

$$\begin{array}{c}{\sigma}_{{\alpha}^{\prime}}(x,{x}^{\prime})=-({\lambda}_{1}-{\lambda}_{0}){g}_{{\alpha}^{\prime}{\beta}^{\prime}}{t}^{{\beta}^{\prime}},\end{array}$$ | (56) |

It is interesting to compute the norm of
${\sigma}_{\alpha}$
. According to Equation ( 55 ) we have
${g}_{\alpha \beta}{\sigma}^{\alpha}{\sigma}^{\beta}=(\Delta \lambda {)}^{2}{g}_{\alpha \beta}{t}^{\alpha}{t}^{\beta}=(\Delta \lambda {)}^{2}\varepsilon $
. According to Equation ( 53 ), this is equal to
$2\sigma $
. We have obtained

and similarly

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

$$\begin{array}{c}{g}^{\alpha \beta}{\sigma}_{\alpha}{\sigma}_{\beta}=2\sigma ,\end{array}$$ | (57) |

$$\begin{array}{c}{g}^{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}_{{\alpha}^{\prime}}{\sigma}_{{\beta}^{\prime}}=2\sigma .\end{array}$$ | (58) |

We note that in flat spacetime,
${\sigma}_{\alpha}={\eta}_{\alpha \beta}(x-{x}^{\prime}{)}^{\beta}$
and
${\sigma}_{{\alpha}^{\prime}}=-{\eta}_{\alpha \beta}(x-{x}^{\prime}{)}^{\beta}$
in Lorentzian coordinates. From this it follows that
${\sigma}_{\alpha \beta}={\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}=-{\sigma}_{\alpha {\beta}^{\prime}}=-{\sigma}_{{\alpha}^{\prime}\beta}={\eta}_{\alpha \beta}$
, and finally,
${g}^{\alpha \beta}{\sigma}_{\alpha \beta}=4={g}^{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}$
.

2.1.4 Congruence of geodesics emanating from ${\mathit{x}}^{\mathsf{\prime}}$

If the base point
${x}^{\prime}$
is kept fixed,
$\sigma $
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}^{\alpha \beta}{\sigma}_{\alpha}{\sigma}_{\beta}=\sigma $
. 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
${\sigma}^{\alpha}\equiv {\sigma}^{;\alpha}$
satisfies

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
$\sigma <0$
, they are spacelike where
$\sigma >0$
, and they are null where
$\sigma =0$
. Here, for concreteness, we shall consider only the timelike subset of the congruence.

$$\begin{array}{c}{\sigma}_{;\beta}^{\alpha}{\sigma}^{\beta}={\sigma}^{\alpha},\end{array}$$ | (59) |

The vector

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

$$\begin{array}{c}{u}^{\alpha}=\frac{{\sigma}^{\alpha}}{|2\sigma {|}^{1/2}}\end{array}$$ | (60) |

${u}_{;\beta}^{\alpha}{u}^{\beta}=0$
. The parameter
$\lambda $
is then proper time
$\tau $
. If
${\lambda}^{*}$
denotes the original parameterization of the geodesics, we have that
$d{\lambda}^{*}/d\tau =|2\sigma {|}^{-1/2}$
, and we see that the original parameterization is singular at
$\sigma =0$
.

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

where
${\theta}^{*}=\left(\delta V{)}^{-1}\right(d/d{\lambda}^{*}\left)\right(\delta V)$
is the expansion in the original parameterization (
$\delta V$
is the congruence's cross-sectional volume). While
${\theta}^{*}$
is well behaved in the limit
$\sigma \to 0$
(we shall see below that
${\theta}^{*}\to 3$
), we have that
$\theta \to \infty $
. This means that the point
${x}^{\prime}$
at which
$\sigma =0$
is a caustic of the congruence: All geodesics emanate from this point.

$$\begin{array}{c}\theta =\frac{{\theta}^{*}}{|2\sigma {|}^{1/2}},{\theta}^{*}={\sigma}_{;\alpha}^{\alpha}-1,\end{array}$$ | (61) |

These considerations, which all follow from a postulated relation
$\frac{1}{2}{g}^{\alpha \beta}{\sigma}_{\alpha}{\sigma}_{\beta}=\sigma $
, 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
${\sigma}_{...}$
as
$x$
approaches
${x}^{\prime}$
. We introduce the notation

$$\left[{\Omega}_{...}\right]={lim}_{x\to {x}^{\prime}}{\Omega}_{...}(x,{x}^{\prime})=\text{a tensor at}{x}^{\prime}$$

to designate the limit of any bitensor
${\Omega}_{...}(x,{x}^{\prime})$
as
$x$
approaches
${x}^{\prime}$
; this is called the coincidence limit of the bitensor. We assume that the coincidence limit is a unique tensorial function of the base point
${x}^{\prime}$
, independent of the direction in which the limit is taken. In other words, if the limit is computed by letting
$\lambda \to {\lambda}_{0}$
after evaluating
${\Omega}_{...}(z,{x}^{\prime})$
as a function of
$\lambda $
on a specified geodesic
$\beta $
, 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

Additional results are obtained by repeated differentiation of the relations ( 57 ) and ( 58 ). For example, Equation ( 57 ) implies
${\sigma}_{\gamma}={g}^{\alpha \beta}{\sigma}_{\alpha}{\sigma}_{\beta \gamma}={\sigma}^{\beta}{\sigma}_{\beta \gamma}$
, or
$({g}_{\beta \gamma}-{\sigma}_{\beta \gamma}){t}^{\beta}=0$
after using Equation ( 55 ). From the assumption stated in the preceding paragraph,
${\sigma}_{\beta \gamma}$
becomes independent of
${t}^{\beta}$
in the limit
$x\to {x}^{\prime}$
, and we arrive at
$\left[{\sigma}_{\alpha \beta}\right]={g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
. By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are

From these relations we infer that
$\left[{\sigma}_{\alpha}^{\alpha}\right]=4$
, so that
$\left[{\theta}^{*}\right]=3$
, where
${\theta}^{*}$
was defined in Equation ( 61 ).

$$\begin{array}{c}\left[\sigma \right]=0,\left[{\sigma}_{\alpha}\right]=\left[{\sigma}_{{\alpha}^{\prime}}\right]=0.\end{array}$$ | (62) |

$$\begin{array}{c}\left[{\sigma}_{\alpha \beta}\right]=\left[{\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}\right]={g}_{{\alpha}^{\prime}{\beta}^{\prime}},\left[{\sigma}_{\alpha {\beta}^{\prime}}\right]=\left[{\sigma}_{{\alpha}^{\prime}\beta}\right]=-{g}_{{\alpha}^{\prime}{\beta}^{\prime}}.\end{array}$$ | (63) |

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

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
$\left[{\sigma}_{\alpha {\beta}^{\prime}}\right]=[{\sigma}_{\alpha}{]}_{;{\beta}^{\prime}}-[{\sigma}_{\alpha \beta}]$
, and since the coincidence limit of
${\sigma}_{\alpha}$
is zero, this gives us
$\left[{\sigma}_{\alpha {\beta}^{\prime}}\right]=-\left[{\sigma}_{\alpha \beta}\right]=-{g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
, as was stated in Equation ( 63 ). Similarly,
$\left[{\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}\right]=[{\sigma}_{{\alpha}^{\prime}}{]}_{;{\beta}^{\prime}}-[{\sigma}_{{\alpha}^{\prime}\beta}]=-[{\sigma}_{\beta {\alpha}^{\prime}}]={g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
. The results of Equation ( 63 ) can thus all be generated from the known result for
$\left[{\sigma}_{\alpha \beta}\right]$
.

$$\begin{array}{c}\left[{\sigma}_{...{\alpha}^{\prime}}\right]={\left[{\sigma}_{...}\right]}_{;{\alpha}^{\prime}}-\left[{\sigma}_{...\alpha}\right],\end{array}$$ | (64) |

The coincidence limits of Equation ( 63 ) were derived from the relation
${\sigma}_{\alpha}={\sigma}_{\alpha}^{\delta}{\sigma}_{\delta}$
. We now differentiate this twice more and obtain
${\sigma}_{\alpha \beta \gamma}={\sigma}_{\alpha \beta \gamma}^{\delta}{\sigma}_{\delta}+{\sigma}_{\alpha \beta}^{\delta}{\sigma}_{\delta \gamma}+{\sigma}_{\alpha \gamma}^{\delta}{\sigma}_{\delta \beta}+{\sigma}_{\alpha}^{\delta}{\sigma}_{\delta \beta \gamma}$
. At coincidence we have

The last three results were derived from
$\left[{\sigma}_{\alpha \beta \gamma}\right]=0$
by employing Synge's rule.

$$\left[{\sigma}_{\alpha \beta \gamma}\right]=\left[{\sigma}_{\alpha \beta}^{\delta}\right]{g}_{{\delta}^{\prime}{\gamma}^{\prime}}+\left[{\sigma}_{\alpha \gamma}^{\delta}\right]{g}_{{\delta}^{\prime}{\beta}^{\prime}}+{\delta}_{{\alpha}^{\prime}}^{{\delta}^{\prime}}\left[{\sigma}_{\delta \beta \gamma}\right],$$

or
$\left[{\sigma}_{\gamma \alpha \beta}\right]+\left[{\sigma}_{\beta \alpha \gamma}\right]=0$
if we recognize that the operations of raising or lowering indices and taking the limit
$x\to {x}^{\prime}$
commute. Noting the symmetries of
${\sigma}_{\alpha \beta}$
, this gives us
$\left[{\sigma}_{\alpha \gamma \beta}\right]+\left[{\sigma}_{\alpha \beta \gamma}\right]=0$
, or
$2\left[{\sigma}_{\alpha \beta \gamma}\right]-\left[{R}_{\alpha \beta \gamma}^{\delta}{\sigma}_{\delta}\right]=0$
, or
$2\left[{\sigma}_{\alpha \beta \gamma}\right]={R}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}^{{\delta}^{\prime}}\left[{\sigma}_{{\delta}^{\prime}}\right]$
. Since the last factor is zero, we arrive at
$$\begin{array}{c}\left[{\sigma}_{\alpha \beta \gamma}\right]=\left[{\sigma}_{\alpha \beta {\gamma}^{\prime}}\right]=\left[{\sigma}_{\alpha {\beta}^{\prime}{\gamma}^{\prime}}\right]=\left[{\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}\right]=0.\end{array}$$ | (65) |

We now differentiate the relation
${\sigma}_{\alpha}={\sigma}_{\alpha}^{\delta}{\sigma}_{\delta}$
three times and obtain

$${\sigma}_{\alpha \beta \gamma \delta}={\sigma}_{\alpha \beta \gamma \delta}^{\epsilon}{\sigma}_{\epsilon}+{\sigma}_{\alpha \beta \gamma}^{\epsilon}{\sigma}_{\epsilon \delta}+{\sigma}_{\alpha \beta \delta}^{\epsilon}{\sigma}_{\epsilon \gamma}+{\sigma}_{\alpha \gamma \delta}^{\epsilon}{\sigma}_{\epsilon \beta}+{\sigma}_{\alpha \beta}^{\epsilon}{\sigma}_{\epsilon \gamma \delta}+{\sigma}_{\alpha \gamma}^{\epsilon}{\sigma}_{\epsilon \beta \delta}+{\sigma}_{\alpha \delta}^{\epsilon}{\sigma}_{\epsilon \beta \gamma}+{\sigma}_{\alpha}^{\epsilon}{\sigma}_{\epsilon \beta \gamma \delta}.$$

At coincidence this reduces to
$\left[{\sigma}_{\alpha \beta \gamma \delta}\right]+\left[{\sigma}_{\alpha \delta \beta \gamma}\right]+\left[{\sigma}_{\alpha \gamma \beta \delta}\right]=0$
. To simplify the third term we differentiate Ricci's identity
${\sigma}_{\alpha \gamma \beta}={\sigma}_{\alpha \beta \gamma}-{R}_{\alpha \beta \gamma}^{\epsilon}{\sigma}_{\epsilon}$
with respect to
${x}^{\delta}$
and then take the coincidence limit. This gives us
$\left[{\sigma}_{\alpha \gamma \beta \delta}\right]=\left[{\sigma}_{\alpha \beta \gamma \delta}\right]+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}$
. The same manipulations on the second term give
$\left[{\sigma}_{\alpha \delta \beta \gamma}\right]=\left[{\sigma}_{\alpha \beta \delta \gamma}\right]+{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}$
. Using the identity
${\sigma}_{\alpha \beta \delta \gamma}={\sigma}_{\alpha \beta \gamma \delta}-{R}_{\alpha \gamma \delta}^{\epsilon}{\sigma}_{\epsilon \beta}-{R}_{\beta \gamma \delta}^{\epsilon}{\sigma}_{\alpha \epsilon}$
and the symmetries of the Riemann tensor, it is then easy to show that
$\left[{\sigma}_{\alpha \beta \delta \gamma}\right]=\left[{\sigma}_{\alpha \beta \gamma \delta}\right]$
. Gathering the results, we obtain
$3\left[{\sigma}_{\alpha \beta \gamma \delta}\right]+{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}=0$
, and Synge's rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are
$$\begin{array}{c}\begin{array}{ccc}\left[{\sigma}_{\alpha \beta \gamma \delta}\right]& =& -\frac{1}{3}\left({R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}\right),\\ \left[{\sigma}_{\alpha \beta \gamma {\delta}^{\prime}}\right]& =& \frac{1}{3}\left({R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}\right),\\ \left[{\sigma}_{\alpha \beta {\gamma}^{\prime}{\delta}^{\prime}}\right]& =& -\frac{1}{3}\left({R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}\right),\\ \left[{\sigma}_{\alpha {\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}\right]& =& -\frac{1}{3}\left({R}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}\right),\\ \left[{\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}\right]& =& -\frac{1}{3}\left({R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}\right).\end{array}\end{array}$$ | (66) |

2.2.2 Derivation of Synge's rule

We begin with any bitensor
${\Omega}_{A{B}^{\prime}}(x,{x}^{\prime})$
in which
$A=\alpha ...\beta $
is a multi-index that represents any number of unprimed indices, and
${B}^{\prime}={\gamma}^{\prime}...{\delta}^{\prime}$
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
$\beta $
that links
$x$
to
${x}^{\prime}$
we introduce an ordinary tensor
${P}^{M}\left(z\right)$
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
$\beta $
; this means that it satisfies
${P}_{;\alpha}^{A}{t}^{\alpha}=0$
at
$x$
. Similarly, we introduce an ordinary tensor
${Q}^{N}\left(z\right)$
in which
$N$
contains the same number of indices as
${B}^{\prime}$
. This tensor is arbitrary, but we assume that it is parallel transported on
$\beta $
; at
${x}^{\prime}$
it satisfies
${Q}_{;{\alpha}^{\prime}}^{{B}^{\prime}}{t}^{{\alpha}^{\prime}}=0$
.

With
$\Omega $
,
$P$
, and
$Q$
we form a biscalar
$H(x,{x}^{\prime})$
defined by

follows from the fact that the tensors
${P}^{M}$
and
${Q}^{N}$
, and the direction of the selected geodesic
$\beta $
, are all arbitrary. Equation ( 67 ) reduces to Equation ( 64 ) when
${\sigma}_{...}$
is substituted in place of
${\Omega}_{A{B}^{\prime}}$
.

$$H(x,{x}^{\prime})={\Omega}_{A{B}^{\prime}}(x,{x}^{\prime}){P}^{A}\left(x\right){Q}^{{B}^{\prime}}\left({x}^{\prime}\right).$$

Having specified the geodesic that links
$x$
to
${x}^{\prime}$
, we can consider
$H$
to be a function of
${\lambda}_{0}$
and
${\lambda}_{1}$
. If
${\lambda}_{1}$
is not much larger than
${\lambda}_{0}$
(so that
$x$
is not far from
${x}^{\prime}$
), we can express
$H({\lambda}_{1},{\lambda}_{0})$
as
$${H({\lambda}_{1},{\lambda}_{0})=H({\lambda}_{0},{\lambda}_{0})+({\lambda}_{1}-{\lambda}_{0})\frac{\partial H}{\partial {\lambda}_{1}}|}_{{\lambda}_{1}={\lambda}_{0}}+....$$

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

and these two expressions give
$${\frac{d}{d{\lambda}_{0}}H({\lambda}_{0},{\lambda}_{0})=\frac{\partial H}{\partial {\lambda}_{0}}|}_{{\lambda}_{0}={\lambda}_{1}}+{\frac{\partial H}{\partial {\lambda}_{1}}|}_{{\lambda}_{1}={\lambda}_{0}},$$

because the left-hand side is the limit of
$\left[H\right({\lambda}_{1},{\lambda}_{1})-H({\lambda}_{0},{\lambda}_{0}\left)\right]/({\lambda}_{1}-{\lambda}_{0})$
when
${\lambda}_{1}\to {\lambda}_{0}$
. The partial derivative of
$H$
with respect to
${\lambda}_{0}$
is equal to
${\Omega}_{A{B}^{\prime};{\alpha}^{\prime}}{t}^{{\alpha}^{\prime}}{P}^{A}{Q}^{{B}^{\prime}}$
, and in the limit this becomes
$\left[{\Omega}_{A{B}^{\prime};{\alpha}^{\prime}}\right]{t}^{{\alpha}^{\prime}}{P}^{{A}^{\prime}}{Q}^{{B}^{\prime}}$
. Similarly, the partial derivative of
$H$
with respect to
${\lambda}_{1}$
is
${\Omega}_{A{B}^{\prime};\alpha}{t}^{\alpha}{P}^{A}{Q}^{{B}^{\prime}}$
, and in the limit
${\lambda}_{1}\to {\lambda}_{0}$
this becomes
$\left[{\Omega}_{A{B}^{\prime};\alpha}\right]{t}^{{\alpha}^{\prime}}{P}^{{A}^{\prime}}{Q}^{{B}^{\prime}}$
. Finally,
$H({\lambda}_{0},{\lambda}_{0})=\left[{\Omega}_{A{B}^{\prime}}\right]{P}^{{A}^{\prime}}{Q}^{{B}^{\prime}}$
, and its derivative with respect to
${\lambda}_{0}$
is
$[{\Omega}_{A{B}^{\prime}}{]}_{;{\alpha}^{\prime}}{t}^{{\alpha}^{\prime}}{P}^{{A}^{\prime}}{Q}^{{B}^{\prime}}$
. Gathering the results we find that
$$\left\{{\left[{\Omega}_{A{B}^{\prime}}\right]}_{;{\alpha}^{\prime}}-\left[{\Omega}_{A{B}^{\prime};{\alpha}^{\prime}}\right]-\left[{\Omega}_{A{B}^{\prime};\alpha}\right]\right\}{t}^{{\alpha}^{\prime}}{P}^{{A}^{\prime}}{Q}^{{B}^{\prime}}=0,$$

and the final statement of Synge's rule,
$$\begin{array}{c}{\left[{\Omega}_{A{B}^{\prime}}\right]}_{;{\alpha}^{\prime}}=\left[{\Omega}_{A{B}^{\prime};{\alpha}^{\prime}}\right]+\left[{\Omega}_{A{B}^{\prime};\alpha}\right],\end{array}$$ | (67) |

2.3 Parallel propagator

2.3.1 Tetrad on $\mathit{\beta}$

On the geodesic
$\beta $
that links
$x$
to
${x}^{\prime}$
we introduce an orthonormal basis
${e}^{\mu}\mathsf{a}\left(z\right)$
that is parallel transported on the geodesic. The frame indices^{
$\text{3}$
}
$\mathsf{a}$
,
$\mathsf{b}$
, . . . , run from 0 to 3, and the frame vectors satisfy

where
${\eta}_{\mathsf{a}\mathsf{b}}=diag(-1,1,1,1)$
is the Minkowski metric (which we shall use to raise and lower frame indices). We have the completeness relations

and we define a dual tetrad
${e}^{\mathsf{a}}\mu \left(z\right)$
by

this is also parallel transported on
$\beta $
. In terms of the dual tetrad the completeness relations take the form

and it is easy to show that the tetrad and its dual satisfy
${e}^{\mathsf{a}}\mu {e}^{\mu}\mathsf{b}={\delta}_{\mathsf{b}}^{\mathsf{a}}$
and
${e}^{\mathsf{a}}\nu {e}^{\mu}\mathsf{a}={\delta}_{\nu}^{\mu}$
. Equations ( 68 , 69 , 70 , 71 ) hold everywhere on
$\beta $
. In particular, with an appropriate change of notation they hold at
${x}^{\prime}$
and
$x$
; for example,
${g}_{\alpha \beta}={\eta}_{\mathsf{a}\mathsf{b}}{e}^{\mathsf{a}}\alpha {e}^{\mathsf{b}}\beta $
is the metric at
$x$
.

$$\begin{array}{c}{g}_{\mu \nu}{e}^{\mu}\mathsf{a}{e}^{\nu}\mathsf{b}={\eta}_{\mathsf{a}\mathsf{b}},\frac{D{e}^{\mu}\mathsf{a}}{d\lambda}=0,\end{array}$$ | (68) |

$$\begin{array}{c}{g}^{\mu \nu}={\eta}^{\mathsf{a}\mathsf{b}}{e}^{\mu}\mathsf{a}{e}^{\nu}\mathsf{b},\end{array}$$ | (69) |

$$\begin{array}{c}{e}^{\mathsf{a}}\mu \equiv {\eta}^{\mathsf{a}\mathsf{b}}{g}_{\mu \nu}{e}^{\nu}\mathsf{b};\end{array}$$ | (70) |

$$\begin{array}{c}{g}_{\mu \nu}={\eta}_{\mathsf{a}\mathsf{b}}{e}^{\mathsf{a}}\mu {e}^{\mathsf{b}}\nu ,\end{array}$$ | (71) |

^{
$\text{3}$
}
Note that I use
$\text{sans-serif}$
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
${A}^{\mu}\left(z\right)$
on
$\beta $
can be decomposed in the basis
${e}^{\mu}\mathsf{a}$
:
${A}^{\mu}={A}^{\mathsf{a}}{e}^{\mu}\mathsf{a}$
, and the vector's frame components are given by
${A}^{\mathsf{a}}={A}^{\mu}{e}^{\mathsf{a}}\mu $
. If
${A}^{\mu}$
is parallel transported on the geodesic, then the coefficients
${A}^{\mathsf{a}}$
are constants. The vector at
$x$
can then be expressed as
${A}^{\alpha}=\left({A}^{{\alpha}^{\prime}}{e}^{\mathsf{a}}{\alpha}^{\prime}\right){e}^{\alpha}\mathsf{a}$
, or

The object
${g}_{{\alpha}^{\prime}}^{\alpha}={e}^{\alpha}\mathsf{a}{e}^{\mathsf{a}}{\alpha}^{\prime}$
is the parallel propagator : It takes a vector at
${x}^{\prime}$
and parallel-transports it to
$x$
along the unique geodesic that links these points.

$$\begin{array}{c}{A}^{\alpha}\left(x\right)={g}_{{\alpha}^{\prime}}^{\alpha}(x,{x}^{\prime}){A}^{{\alpha}^{\prime}}\left({x}^{\prime}\right),{g}_{{\alpha}^{\prime}}^{\alpha}(x,{x}^{\prime})\equiv {e}^{\alpha}\mathsf{a}\left(x\right){e}^{\mathsf{a}}{\alpha}^{\prime}\left({x}^{\prime}\right).\end{array}$$ | (72) |

Similarly, we find that

and we see that
${g}_{\alpha}^{{\alpha}^{\prime}}={e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{\mathsf{a}}\alpha $
performs the inverse operation: It takes a vector at
$x$
and parallel-transports it back to
${x}^{\prime}$
. Clearly,

and these relations formally express the fact that
${g}_{\alpha}^{{\alpha}^{\prime}}$
is the inverse of
${g}_{{\alpha}^{\prime}}^{\alpha}$
.

$$\begin{array}{c}{A}^{{\alpha}^{\prime}}\left({x}^{\prime}\right)={g}_{\alpha}^{{\alpha}^{\prime}}({x}^{\prime},x){A}^{\alpha}\left(x\right),{g}_{\alpha}^{{\alpha}^{\prime}}({x}^{\prime},x)\equiv {e}^{{\alpha}^{\prime}}\mathsf{a}\left({x}^{\prime}\right){e}^{\mathsf{a}}\alpha \left(x\right),\end{array}$$ | (73) |

$$\begin{array}{c}{g}_{{\alpha}^{\prime}}^{\alpha}{g}_{\beta}^{{\alpha}^{\prime}}={\delta}_{\beta}^{\alpha},{g}_{\alpha}^{{\alpha}^{\prime}}{g}_{{\beta}^{\prime}}^{\alpha}={\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}},\end{array}$$ | (74) |

The relation
${g}_{{\alpha}^{\prime}}^{\alpha}={e}^{\alpha}\mathsf{a}{e}^{\mathsf{a}}{\alpha}^{\prime}$
can also be expressed as
${g}_{\alpha}^{{\alpha}^{\prime}}={e}^{\mathsf{a}}\alpha {e}^{{\alpha}^{\prime}}\mathsf{a}$
, and this reveals that

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

$$\begin{array}{c}{g}_{\alpha}^{{\alpha}^{\prime}}(x,{x}^{\prime})={g}_{\alpha}^{{\alpha}^{\prime}}({x}^{\prime},x),{g}_{{\alpha}^{\prime}}^{\alpha}({x}^{\prime},x)={g}_{{\alpha}^{\prime}}^{\alpha}(x,{x}^{\prime}).\end{array}$$ | (75) |

The action of the parallel propagator on tensors of arbitrary ranks is easy to figure out. For example, suppose that the dual vector
${p}_{\mu}={p}_{a}{e}^{a}\mu $
is parallel transported on
$\beta $
. Then the frame components
${p}_{\mathsf{a}}={p}_{\mu}{e}^{\mu}\mathsf{a}$
are constants, and the dual vector at
$x$
can be expressed as
${p}_{\alpha}=({p}_{{\alpha}^{\prime}}{e}^{{\alpha}^{\prime}}\mathsf{a}){e}^{\alpha}\mathsf{a}$
, or

It is therefore the inverse propagator
${g}_{\alpha}^{{\alpha}^{\prime}}$
that takes a dual vector at
${x}^{\prime}$
and parallel-transports it to
$x$
. As another example, it is easy to show that a tensor
${A}^{\alpha \beta}$
at
$x$
obtained by parallel transport from
${x}^{\prime}$
must be given by

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
$\beta $
, and a special case of Equation ( 77 ) is therefore
${g}_{\alpha \beta}={g}_{\alpha}^{{\alpha}^{\prime}}{g}_{\beta}^{{\beta}^{\prime}}{g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
.

$$\begin{array}{c}{p}_{\alpha}\left(x\right)={g}_{\alpha}^{{\alpha}^{\prime}}({x}^{\prime},x){p}_{{\alpha}^{\prime}}\left({x}^{\prime}\right).\end{array}$$ | (76) |

$$\begin{array}{c}{A}^{\alpha \beta}\left(x\right)={g}_{{\alpha}^{\prime}}^{\alpha}(x,{x}^{\prime}){g}_{{\beta}^{\prime}}^{\beta}(x,{x}^{\prime}){A}^{{\alpha}^{\prime}{\beta}^{\prime}}\left({x}^{\prime}\right).\end{array}$$ | (77) |

Because the basis vectors are parallel transported on
$\beta $
, they satisfy
${e}_{\mathsf{a};\beta}^{\alpha}{\sigma}^{\beta}=0$
at
$x$
and
${e}_{\mathsf{a};{\beta}^{\prime}}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}=0$
at
${x}^{\prime}$
. This immediately implies that the parallel propagators must satisfy

Another useful property of the parallel propagator follows from the fact that if
${t}^{\mu}=d{z}^{\mu}/d\lambda $
is tangent to the geodesic connecting
$x$
to
${x}^{\prime}$
, then
${t}^{\alpha}={g}_{{\alpha}^{\prime}}^{\alpha}{t}^{{\alpha}^{\prime}}$
. Using Equations ( 55 ) and ( 56 ), this observation gives us the relations

$$\begin{array}{c}{g}_{{\alpha}^{\prime};\beta}^{\alpha}{\sigma}^{\beta}={g}_{{\alpha}^{\prime};{\beta}^{\prime}}^{\alpha}{\sigma}^{{\beta}^{\prime}}=0,{g}_{\alpha ;\beta}^{{\alpha}^{\prime}}{\sigma}^{\beta}={g}_{\alpha ;{\beta}^{\prime}}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}=0.\end{array}$$ | (78) |

$$\begin{array}{c}{\sigma}_{\alpha}=-{g}_{\alpha}^{{\alpha}^{\prime}}{\sigma}_{{\alpha}^{\prime}},{\sigma}_{{\alpha}^{\prime}}=-{g}_{{\alpha}^{\prime}}^{\alpha}{\sigma}_{\alpha}.\end{array}$$ | (79) |

2.3.3 Coincidence limits

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

Other coincidence limits are obtained by differentiation of Equations ( 78 ). For example, the relation
${g}_{{\beta}^{\prime};\gamma}^{\alpha}{\sigma}^{\gamma}=0$
implies
${g}_{{\beta}^{\prime};\gamma \delta}^{\alpha}{\sigma}^{\gamma}+{g}_{{\beta}^{\prime};\gamma}^{\alpha}{\sigma}_{\delta}^{\gamma}=0$
, and at coincidence we have

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

$$\begin{array}{c}\left[{g}_{{\beta}^{\prime}}^{\alpha}\right]={\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}.\end{array}$$ | (80) |

$$\begin{array}{c}\left[{g}_{{\beta}^{\prime};\gamma}^{\alpha}\right]=\left[{g}_{{\beta}^{\prime};{\gamma}^{\prime}}^{\alpha}\right]=0;\end{array}$$ | (81) |

$${g}_{{\beta}^{\prime};\gamma \delta \epsilon}^{\alpha}{\sigma}^{\gamma}+{g}_{{\beta}^{\prime};\gamma \delta}^{\alpha}{\sigma}_{\epsilon}^{\gamma}+{g}_{{\beta}^{\prime};\gamma \epsilon}^{\alpha}{\sigma}_{\delta}^{\gamma}+{g}_{{\beta}^{\prime};\gamma}^{\alpha}{\sigma}_{\delta \epsilon}^{\gamma}=0,$$

and at coincidence we have
$\left[{g}_{{\beta}^{\prime};\gamma \delta}^{\alpha}\right]+\left[{g}_{{\beta}^{\prime};\delta \gamma}^{\alpha}\right]=0$
, or
$2\left[{g}_{{\beta}^{\prime};\gamma \delta}^{\alpha}\right]+{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}=0$
. The coincidence limit for
${g}_{{\beta}^{\prime};\gamma {\delta}^{\prime}}^{\alpha}={g}_{{\beta}^{\prime};{\delta}^{\prime}\gamma}^{\alpha}$
can then be obtained from Synge's rule, and an additional application of the rule gives
$\left[{g}_{{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}^{\alpha}\right]$
. Our results are
$$\begin{array}{c}\begin{array}{cccccc}\left[{g}_{{\beta}^{\prime};\gamma \delta}^{\alpha}\right]& =& -\frac{1}{2}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}},& \left[{g}_{{\beta}^{\prime};\gamma {\delta}^{\prime}}^{\alpha}\right]& =& \frac{1}{2}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}},\\ \left[{g}_{{\beta}^{\prime};{\gamma}^{\prime}\delta}^{\alpha}\right]& =& -\frac{1}{2}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}},& \left[{g}_{{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}^{\alpha}\right]& =& \frac{1}{2}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}.\end{array}\end{array}$$ | (82) |

2.4 Expansion of bitensors near coincidence

2.4.1 General method

We would like to express a bitensor
${\Omega}_{{\alpha}^{\prime}{\beta}^{\prime}}(x,{x}^{\prime})$
near coincidence as an expansion in powers of
$-{\sigma}^{{\alpha}^{\prime}}(x,{x}^{\prime})$
, the closest analogue in curved spacetime to the flat-spacetime quantity
$(x-{x}^{\prime}{)}^{\alpha}$
.

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
${x}^{\prime}$
.

The expansion we seek is of the form

in which the “expansion coefficients”
${A}_{{\alpha}^{\prime}{\beta}^{\prime}}$
,
${A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}$
, and
${A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}$
are all ordinary tensors at
${x}^{\prime}$
; this last tensor is symmetric in the pair of indices
${\gamma}^{\prime}$
and
${\delta}^{\prime}$
, and
$\epsilon $
measures the size of a typical component of
${\sigma}^{{\alpha}^{\prime}}$
.

$$\begin{array}{c}{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime}}(x,{x}^{\prime})={A}_{{\alpha}^{\prime}{\beta}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}{\sigma}^{{\gamma}^{\prime}}+\frac{1}{2}{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}+\mathcal{O}\left({\epsilon}^{3}\right),\end{array}$$ | (83) |

To find the expansion coefficients we differentiate Equation ( 83 ) repeatedly and take coincidence limits. Equation ( 83 ) immediately implies
$\left[{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime}}\right]={A}_{{\alpha}^{\prime}{\beta}^{\prime}}$
. After one differentiation we obtain
${\Omega}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}={A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\epsilon}^{\prime};{\gamma}^{\prime}}{\sigma}^{{\epsilon}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\epsilon}^{\prime}}{\sigma}_{{\gamma}^{\prime}}^{{\epsilon}^{\prime}}+\frac{1}{2}{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\epsilon}^{\prime}{\iota}^{\prime};{\gamma}^{\prime}}{\sigma}^{{\epsilon}^{\prime}}{\sigma}^{{\iota}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\epsilon}^{\prime}{\iota}^{\prime}}{\sigma}^{{\epsilon}^{\prime}}{\sigma}_{{\gamma}^{\prime}}^{{\iota}^{\prime}}+\mathcal{O}\left({\epsilon}^{2}\right)$
, and at coincidence this reduces to
$\left[{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}\right]={A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}$
. Taking the coincidence limit after two differentiations yields
$\left[{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}\right]={A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime};{\delta}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\delta}^{\prime};{\gamma}^{\prime}}+{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}$
. The expansion coefficients are therefore

These results are to be substituted into Equation ( 83 ), and this gives us
${\Omega}_{{\alpha}^{\prime}{\beta}^{\prime}}(x,{x}^{\prime})$
to second order in
$\epsilon $
.

$$\begin{array}{c}\begin{array}{ccc}{A}_{{\alpha}^{\prime}{\beta}^{\prime}}& =& \left[{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime}}\right],\\ {A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}& =& \left[{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}\right]-{A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}},\\ {A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}& =& \left[{\Omega}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}\right]-{A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}-{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime};{\delta}^{\prime}}-{A}_{{\alpha}^{\prime}{\beta}^{\prime}{\delta}^{\prime};{\gamma}^{\prime}}.\end{array}\end{array}$$ | (84) |

Suppose now that the bitensor is
${\Omega}_{{\alpha}^{\prime}\beta}$
, with one index referring to
${x}^{\prime}$
and the other to
$x$
. The previous procedure can be applied directly if we introduce an auxiliary bitensor
${\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}\equiv {g}_{{\beta}^{\prime}}^{\beta}{\Omega}_{{\alpha}^{\prime}\beta}$
whose indices all refer to the point
${x}^{\prime}$
. Then
${\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}$
can be expanded as in Equation ( 83 ), and the original bitensor is reconstructed as
${\Omega}_{{\alpha}^{\prime}\beta}={g}_{\beta}^{{\beta}^{\prime}}{\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}$
, or

The expansion coefficients can be obtained from the coincidence limits of
${\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}$
and its derivatives.

$$\begin{array}{c}{\Omega}_{{\alpha}^{\prime}\beta}(x,{x}^{\prime})={g}_{\beta}^{{\beta}^{\prime}}\left({B}_{{\alpha}^{\prime}{\beta}^{\prime}}+{B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}{\sigma}^{{\gamma}^{\prime}}+\frac{1}{2}{B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}\right)+\mathcal{O}\left({\epsilon}^{3}\right).\end{array}$$ | (85) |

It is convenient, however, to express them directly in terms of the original bitensor
${\Omega}_{{\alpha}^{\prime}\beta}$
by substituting the relation
${\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}={g}_{{\beta}^{\prime}}^{\beta}{\Omega}_{{\alpha}^{\prime}\beta}$
and its derivatives. After using the results of Equation ( 80 , 81 , 82 ) we find

The only difference with respect to Equation ( 85 ) is the presence of a Riemann-tensor term in
${B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}$
.

$$\begin{array}{c}\begin{array}{ccc}{B}_{{\alpha}^{\prime}{\beta}^{\prime}}& =& \left[{\Omega}_{{\alpha}^{\prime}\beta}\right],\\ {B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}& =& \left[{\Omega}_{{\alpha}^{\prime}\beta ;{\gamma}^{\prime}}\right]-{B}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}},\\ {B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}& =& \left[{\Omega}_{{\alpha}^{\prime}\beta ;{\gamma}^{\prime}{\delta}^{\prime}}\right]+\frac{1}{2}{B}_{{\alpha}^{\prime}{\epsilon}^{\prime}}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\epsilon}^{\prime}}-{B}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}-{B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime};{\delta}^{\prime}}-{B}_{{\alpha}^{\prime}{\beta}^{\prime}{\delta}^{\prime};{\gamma}^{\prime}}.\end{array}\end{array}$$ | (86) |

Suppose finally that the bitensor to be expanded is
${\Omega}_{\alpha \beta}$
, whose indices all refer to
$x$
. Much as we did before, we introduce an auxiliary bitensor
${\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}={g}_{{\alpha}^{\prime}}^{\alpha}{g}_{{\beta}^{\prime}}^{\beta}{\Omega}_{\alpha \beta}$
whose indices all refer to
${x}^{\prime}$
, we expand
${\stackrel{~}{\Omega}}_{{\alpha}^{\prime}{\beta}^{\prime}}$
as in Equation ( 83 ), and we then reconstruct the original bitensor. This gives us

and the expansion coefficients are now

This differs from Equation ( 86 ) by the presence of an additional Riemann-tensor term in
${C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}$
.

$$\begin{array}{c}{\Omega}_{\alpha \beta}(x,{x}^{\prime})={g}_{\alpha}^{{\alpha}^{\prime}}{g}_{\beta}^{{\beta}^{\prime}}\left({C}_{{\alpha}^{\prime}{\beta}^{\prime}}+{C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}{\sigma}^{{\gamma}^{\prime}}+\frac{1}{2}{C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}\right)+\mathcal{O}\left({\epsilon}^{3}\right),\end{array}$$ | (87) |

$$\begin{array}{ccc}{C}_{{\alpha}^{\prime}{\beta}^{\prime}}& =& \left[{\Omega}_{\alpha \beta}\right],\end{array}$$ |

$$\begin{array}{ccc}{C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}& =& \left[{\Omega}_{\alpha \beta ;{\gamma}^{\prime}}\right]-{C}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}},\end{array}$$ |

$$\begin{array}{ccc}{C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}& =& \left[{\Omega}_{\alpha \beta ;{\gamma}^{\prime}{\delta}^{\prime}}\right]+\frac{1}{2}{C}_{{\alpha}^{\prime}{\epsilon}^{\prime}}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\epsilon}^{\prime}}+\frac{1}{2}{C}_{{\epsilon}^{\prime}{\beta}^{\prime}}{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\epsilon}^{\prime}}-{C}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}-{C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime};{\delta}^{\prime}}-{C}_{{\alpha}^{\prime}{\beta}^{\prime}{\delta}^{\prime};{\gamma}^{\prime}}.\end{array}$$ |

2.4.2 Special cases

We now apply the general expansion method developed in the preceding Section 2.4.1 to the bitensors
${\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}$
,
${\sigma}_{{\alpha}^{\prime}\beta}$
, and
${\sigma}_{\alpha \beta}$
. In the first instance we have
${A}_{{\alpha}^{\prime}{\beta}^{\prime}}={g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
,
${A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}=0$
, and
${A}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}=-\frac{1}{3}({R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}})$
. In the second instance we have
${B}_{{\alpha}^{\prime}{\beta}^{\prime}}=-{g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
,
${B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}=0$
, and
${B}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}=-\frac{1}{3}({R}_{{\beta}^{\prime}{\alpha}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}+{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\alpha}^{\prime}{\delta}^{\prime}})-\frac{1}{2}{R}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}=-\frac{1}{3}{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}-\frac{1}{6}{R}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}$
. In the third instance we have
${C}_{{\alpha}^{\prime}{\beta}^{\prime}}={g}_{{\alpha}^{\prime}{\beta}^{\prime}}$
,
${C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}}=0$
, and
${C}_{{\alpha}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}=-\frac{1}{3}({R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}+{R}_{{\alpha}^{\prime}{\delta}^{\prime}{\beta}^{\prime}{\gamma}^{\prime}})$
. This gives us the expansions

Taking the trace of the last equation returns
${\sigma}_{\alpha}^{\alpha}=4-\frac{1}{3}{R}_{{\gamma}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}+\mathcal{O}\left({\epsilon}^{3}\right)$
, or

where
${\theta}^{*}\equiv {\sigma}_{\alpha}^{\alpha}-1$
was shown in Section 2.1.4 to describe the expansion of the congruence of geodesics that emanate from
${x}^{\prime}$
. Equation ( 91 ) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: When
${R}_{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}>0$
we see that
${\theta}^{*}$
is smaller than 3, the value it would take in flat spacetime.

$$\begin{array}{ccc}{\sigma}_{{\alpha}^{\prime}{\beta}^{\prime}}& =& {g}_{{\alpha}^{\prime}{\beta}^{\prime}}-\frac{1}{3}{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}+\mathcal{O}\left({\epsilon}^{3}\right),\end{array}$$ | (88) |

$$\begin{array}{ccc}{\sigma}_{{\alpha}^{\prime}\beta}& =& -{g}_{\beta}^{{\beta}^{\prime}}\left({g}_{{\alpha}^{\prime}{\beta}^{\prime}}+\frac{1}{6}{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}\right)+\mathcal{O}\left({\epsilon}^{3}\right),\end{array}$$ | (89) |

$$\begin{array}{ccc}{\sigma}_{\alpha \beta}& =& {g}_{\alpha}^{{\alpha}^{\prime}}{g}_{{\beta}^{\prime}}^{{\beta}^{\prime}}\left({g}_{{\alpha}^{\prime}{\beta}^{\prime}}-\frac{1}{3}{R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}\right)+\mathcal{O}\left({\epsilon}^{3}\right).\end{array}$$ | (90) |

$$\begin{array}{c}{\theta}^{*}=3-\frac{1}{3}{R}_{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}+\mathcal{O}\left({\epsilon}^{3}\right),\end{array}$$ | (91) |

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

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

$$\begin{array}{c}{g}_{{\beta}^{\prime};{\gamma}^{\prime}}^{\alpha}=\frac{1}{2}{g}_{{\alpha}^{\prime}}^{\alpha}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}{\sigma}^{{\delta}^{\prime}}+\mathcal{O}\left({\epsilon}^{2}\right),{g}_{{\beta}^{\prime};\gamma}^{\alpha}=\frac{1}{2}{g}_{{\alpha}^{\prime}}^{\alpha}{g}_{\gamma}^{{\gamma}^{\prime}}{R}_{{\beta}^{\prime}{\gamma}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}{\sigma}^{{\delta}^{\prime}}+\mathcal{O}\left({\epsilon}^{2}\right)\end{array}$$ | (92) |

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
${A}_{\alpha \beta}$
at a point
$x$
in terms of its values (and that of its covariant derivatives) at a neighbouring point
${x}^{\prime}$
. The tensor can be written as an expansion in powers of
$-{\sigma}^{{\alpha}^{\prime}}(x,{x}^{\prime})$
, and in this case we have

If the tensor field is parallel transported on the geodesic
$\beta $
that links
$x$
to
${x}^{\prime}$
, then Equation ( 93 ) reduces to Equation ( 77 ). The extension of this formula to tensors of other ranks is obvious.

$$\begin{array}{c}{A}_{\alpha \beta}\left(x\right)={g}_{\alpha}^{{\alpha}^{\prime}}{g}_{\beta}^{{\beta}^{\prime}}\left({A}_{{\alpha}^{\prime}{\beta}^{\prime}}-{A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}{\sigma}^{{\gamma}^{\prime}}+\frac{1}{2}{A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}{\delta}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}\right)+\mathcal{O}\left({\epsilon}^{3}\right).\end{array}$$ | (93) |

To derive this result we express
${A}_{\mu \nu}\left(z\right)$
, the restriction of the tensor field on
$\beta $
, in terms of its tetrad components
${A}_{\mathsf{a}\mathsf{b}}\left(\lambda \right)={A}_{\mu \nu}{e}^{\mu}\mathsf{a}{e}^{\nu}\mathsf{b}$
. Recall from Section 2.3.1 that
${e}^{\mu}\mathsf{a}$
is an orthonormal basis that is parallel transported on
$\beta $
; recall also that the affine parameter
$\lambda $
ranges from
${\lambda}_{0}$
(its value at
${x}^{\prime}$
) to
${\lambda}_{1}$
(its value at
$x$
). We have
${A}_{{\alpha}^{\prime}{\beta}^{\prime}}\left({x}^{\prime}\right)={A}_{\mathsf{a}\mathsf{b}}\left({\lambda}_{0}\right){e}^{\mathsf{a}}{\alpha}^{\prime}{e}^{\mathsf{b}}{\beta}^{\prime}$
,
${A}_{\alpha \beta}\left(x\right)={A}_{\mathsf{a}\mathsf{b}}\left({\lambda}_{1}\right){e}^{\mathsf{a}}\alpha {e}^{\mathsf{b}}\beta $
, and
${A}_{\mathsf{a}\mathsf{b}}\left({\lambda}_{1}\right)$
can be expressed in terms of quantities at
$\lambda ={\lambda}_{0}$
by straightforward Taylor expansion. Since, for example,

$$({\lambda}_{1}-{\lambda}_{0}){\frac{d{A}_{\mathsf{a}\mathsf{b}}}{d\lambda}|}_{{\lambda}_{0}}={({\lambda}_{1}-{\lambda}_{0}){\left({A}_{\mu \nu}{e}^{\mu}\mathsf{a}{e}^{\nu}\mathsf{b}\right)}_{;\lambda}{t}^{\lambda}|}_{{\lambda}_{0}}={({\lambda}_{1}-{\lambda}_{0}){A}_{\mu \nu ;\lambda}{e}^{\mu}\mathsf{a}{e}^{\nu}\mathsf{b}{t}^{\lambda}|}_{{\lambda}_{0}}=-{A}_{{\alpha}^{\prime}{\beta}^{\prime};{\gamma}^{\prime}}{e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{{\beta}^{\prime}}\mathsf{b}{\sigma}^{{\gamma}^{\prime}},$$

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
$\Delta (x,{x}^{\prime})$
is defined by

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

where
$g$
is the metric determinant at
$x$
and
${g}^{\prime}$
the metric determinant at
${x}^{\prime}$
.

$$\begin{array}{c}\Delta (x,{x}^{\prime})\equiv det\left[{\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}(x,{x}^{\prime})\right],{\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}(x,{x}^{\prime})\equiv -{g}_{\alpha}^{{\alpha}^{\prime}}({x}^{\prime},x){\sigma}_{{\beta}^{\prime}}^{\alpha}(x,{x}^{\prime}).\end{array}$$ | (94) |

$$\begin{array}{c}\Delta (x,{x}^{\prime})=-\frac{det\left[-{\sigma}_{\alpha {\beta}^{\prime}}(x,{x}^{\prime})\right]}{\sqrt{-g}\sqrt{-{g}^{\prime}}},\end{array}$$ | (95) |

Equations ( 63 ) and ( 80 ) imply that at coincidence,
$\left[{\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}\right]={\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}$
and
$[\Delta ]=1$
. Equation ( 89 ), on the other hand, implies that near coincidence

so that

This last result follows from the fact that for a “small” matrix
$\mathit{a}$
,
$det(1+\mathit{a})=1+tr\left(\mathit{a}\right)+\mathcal{O}\left({\mathit{a}}^{2}\right)$
.

$$\begin{array}{c}{\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}={\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}+\frac{1}{6}{R}_{{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}+\mathcal{O}\left({\epsilon}^{3}\right),\end{array}$$ | (96) |

$$\begin{array}{c}\Delta =1+\frac{1}{6}{R}_{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}+\mathcal{O}\left({\epsilon}^{3}\right).\end{array}$$ | (97) |

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

which can also be written as
$(ln\Delta {)}_{,\alpha}{\sigma}^{\alpha}=4-{\sigma}_{\alpha}^{\alpha}$
, or

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
${x}^{\prime}$
. If
${\theta}^{*}<3$
, then the congruence expands less rapidly than it would in flat spacetime, and
$\Delta $
increases along the geodesics. If, on the other hand,
${\theta}^{*}>3$
, then the congruence expands more rapidly than it would in flat spacetime, and
$\Delta $
decreases along the geodesics. Thus,
$\Delta >1$
indicates that the geodesics are undergoing focusing, while
$\Delta <1$
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
$\Delta -1$
near
${x}^{\prime}$
is determined by the sign of
${R}_{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}$
.

$$\begin{array}{c}\frac{1}{\Delta}{\left(\Delta {\sigma}^{\alpha}\right)}_{;\alpha}=4,\end{array}$$ | (98) |

$$\begin{array}{c}\frac{d}{d{\lambda}^{*}}(ln\Delta )=3-{\theta}^{*}\end{array}$$ | (99) |

2.5.2 Derivations

To show that Equation ( 95 ) follows from Equation ( 94 ) we rewrite the completeness relations at
$x$
,
${g}^{\alpha \beta}={\eta}^{\mathsf{a}\mathsf{b}}{e}^{\alpha}\mathsf{a}{e}^{\beta}\mathsf{b}$
, in the matrix form
${\mathit{g}}^{-1}=\mathit{E}\mathit{\eta}{\mathit{E}}^{T}$
, where
$\mathit{E}$
denotes the
$4\times 4$
matrix whose entries correspond to
${e}^{\alpha}\mathsf{a}$
. (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
${x}^{\prime}$
,
${g}^{{\alpha}^{\prime}{\beta}^{\prime}}={\eta}^{\mathsf{a}\mathsf{b}}{e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{{\beta}^{\prime}}\mathsf{b}$
, in the matrix form
${{\mathit{g}}^{\mathsf{\prime}}}^{-1}={\mathit{E}}^{\mathsf{\prime}}\mathit{\eta}{{\mathit{E}}^{\mathsf{\prime}}}^{T}$
, where
${\mathit{E}}^{\mathsf{\prime}}$
is the matrix corresponding to
${e}^{{\alpha}^{\prime}}\mathsf{a}$
. With
${e}^{\prime}$
denoting its determinant, we have
$1/{g}^{\prime}=-{e}^{\prime 2}$
, or
${e}^{\prime}=1/\sqrt{-{g}^{\prime}}$
. Now, the parallel propagator is defined by
${g}_{{\alpha}^{\prime}}^{\alpha}={\eta}^{\mathsf{a}\mathsf{b}}{g}_{{\alpha}^{\prime}{\beta}^{\prime}}{e}^{\alpha}\mathsf{a}{e}^{{\beta}^{\prime}}\mathsf{b}$
, and the matrix form of this equation is
$\hat{\mathit{g}}=\mathit{E}\mathit{\eta}{{\mathit{E}}^{\mathsf{\prime}}}^{T}{{\mathit{g}}^{\mathsf{\prime}}}^{T}$
. The determinant of the parallel propagator is therefore
$\hat{g}=-e{e}^{\prime}{g}^{\prime}=\sqrt{-{g}^{\prime}}/\sqrt{-g}$
. So we have

and Equation ( 95 ) follows from the fact that the matrix form of Equation ( 94 ) is
$\mathbf{\Delta}=-{\hat{\mathit{g}}}^{-1}{\mathit{g}}^{-1}\mathit{\sigma}$
, where
$\mathit{\sigma}$
is the matrix corresponding to
${\sigma}_{\alpha {\beta}^{\prime}}$
.

$$\begin{array}{c}det\left[{g}_{{\alpha}^{\prime}}^{\alpha}\right]=\frac{\sqrt{-{g}^{\prime}}}{\sqrt{-g}},det\left[{g}_{\alpha}^{{\alpha}^{\prime}}\right]=\frac{\sqrt{-g}}{\sqrt{-{g}^{\prime}}},\end{array}$$ | (100) |

To establish Equation ( 98 ) we differentiate the relation
$\sigma =\frac{1}{2}{\sigma}^{\gamma}{\sigma}_{\gamma}$
twice and obtain
${\sigma}_{\alpha {\beta}^{\prime}}={\sigma}_{\alpha}^{\gamma}{\sigma}_{\gamma {\beta}^{\prime}}+{\sigma}^{\gamma}{\sigma}_{\gamma \alpha {\beta}^{\prime}}$
. If we replace the last factor by
${\sigma}_{\alpha {\beta}^{\prime}\gamma}$
and multiply both sides by
$-{g}^{{\alpha}^{\prime}\alpha}$
we find

where we have used Equation ( 78 ). At this stage we introduce an inverse
$({\Delta}^{-1}{)}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}$
to the van Vleck bitensor, defined by
${\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}({\Delta}^{-1}{)}_{{\gamma}^{\prime}}^{{\beta}^{\prime}}={\delta}_{{\gamma}^{\prime}}^{{\alpha}^{\prime}}$
. After multiplying both sides of Equation ( 101 ) by
$({\Delta}^{-1}{)}_{{\gamma}^{\prime}}^{{\beta}^{\prime}}$
we find

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

$${\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}=-{g}^{{\alpha}^{\prime}\alpha}\left({\sigma}_{\alpha}^{\gamma}{\sigma}_{\gamma {\beta}^{\prime}}+{\sigma}^{\gamma}{\sigma}_{\alpha {\beta}^{\prime}\gamma}\right).$$

In this expression we make the substitution
${\sigma}_{\alpha {\beta}^{\prime}}=-{g}_{\alpha {\alpha}^{\prime}}{\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}$
, which follows directly from Equation ( 94 ). This gives us
$$\begin{array}{c}{\Delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}={g}_{\alpha}^{{\alpha}^{\prime}}{g}_{{\gamma}^{\prime}}^{\gamma}{\sigma}_{\gamma}^{\alpha}{\Delta}_{{\beta}^{\prime}}^{{\gamma}^{\prime}}+{\Delta}_{{\beta}^{\prime};\gamma}^{{\alpha}^{\prime}}{\sigma}^{\gamma},\end{array}$$ | (101) |

$${\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}={g}_{\alpha}^{{\alpha}^{\prime}}{g}_{{\beta}^{\prime}}^{\beta}{\sigma}_{\beta}^{\alpha}+({\Delta}^{-1}{)}_{{\beta}^{\prime}}^{{\gamma}^{\prime}}{\Delta}_{{\gamma}^{\prime};\gamma}^{{\alpha}^{\prime}}{\sigma}^{\gamma},$$

and taking the trace of this equation yields
$$4={\sigma}_{\alpha}^{\alpha}+({\Delta}^{-1}{)}_{{\alpha}^{\prime}}^{{\beta}^{\prime}}{\Delta}_{{\beta}^{\prime};\gamma}^{{\alpha}^{\prime}}{\sigma}^{\gamma}.$$

We now recall the identity
$\delta lndet\mathit{M}=tr\left({\mathit{M}}^{-1}\delta \mathit{M}\right)$
, which relates the variation of a determinant to the variation of the matrix elements. It implies, in particular, that
$({\Delta}^{-1}{)}_{{\alpha}^{\prime}}^{{\beta}^{\prime}}{\Delta}_{{\beta}^{\prime};\gamma}^{{\alpha}^{\prime}}=(ln\Delta {)}_{,\gamma}$
, and we finally obtain
$$\begin{array}{c}4={\sigma}_{\alpha}^{\alpha}+(ln\Delta {)}_{,\alpha}{\sigma}^{\alpha},\end{array}$$ | (102) |

3 Coordinate Systems

3.1 Riemann normal coordinates

3.1.1 Definition and coordinate transformation

Given a fixed base point
${x}^{\prime}$
and a tetrad
${e}^{{\alpha}^{\prime}}\mathsf{a}\left({x}^{\prime}\right)$
, we assign to a neighbouring point
$x$
the four coordinates

where
${e}^{\mathsf{a}}{\alpha}^{\prime}={\eta}^{\mathsf{a}\mathsf{b}}{g}_{{\alpha}^{\prime}{\beta}^{\prime}}{e}^{{\beta}^{\prime}}\mathsf{b}$
is the dual tetrad attached to
${x}^{\prime}$
. The new coordinates
${\hat{x}}^{\mathsf{a}}$
are called Riemann normal coordinates (RNC), and they are such that
${\eta}_{\mathsf{a}\mathsf{b}}{\hat{x}}^{\mathsf{a}}{\hat{x}}^{\mathsf{b}}={\eta}_{\mathsf{a}\mathsf{b}}{e}^{\mathsf{a}}{\alpha}^{\prime}{e}^{\mathsf{b}}{\beta}^{\prime}{\sigma}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}={g}_{{\alpha}^{\prime}{\beta}^{\prime}}{\sigma}^{{\alpha}^{\prime}}{\sigma}^{{\beta}^{\prime}}$
, or

Thus,
${\eta}_{\mathsf{a}\mathsf{b}}{\hat{x}}^{\mathsf{a}}{\hat{x}}^{\mathsf{b}}$
is the squared geodesic distance between
$x$
and the base point
${x}^{\prime}$
. It is obvious that
${x}^{\prime}$
is at the origin of the RNC, where
${\hat{x}}^{\mathsf{a}}=0$
.

$$\begin{array}{c}{\hat{x}}^{\mathsf{a}}=-{e}^{\mathsf{a}}{\alpha}^{\prime}\left({x}^{\prime}\right){\sigma}^{{\alpha}^{\prime}}(x,{x}^{\prime}),\end{array}$$ | (103) |

$$\begin{array}{c}{\eta}_{\mathsf{a}\mathsf{b}}{\hat{x}}^{\mathsf{a}}{\hat{x}}^{\mathsf{b}}=2\sigma (x,{x}^{\prime}).\end{array}$$ | (104) |

If we move the point
$x$
to
$x+\delta x$
, the new coordinates change to
${\hat{x}}^{\mathsf{a}}+\delta {\hat{x}}^{\mathsf{a}}=-{e}^{\mathsf{a}}{\alpha}^{\prime}{\sigma}^{{\alpha}^{\prime}}(x+\delta x,{x}^{\prime})={\hat{x}}^{\mathsf{a}}-{e}^{\mathsf{a}}{\alpha}^{\prime}{\sigma}_{\beta}^{{\alpha}^{\prime}}\delta {x}^{\beta}$
, so that

The coordinate transformation is therefore determined by
$\partial {\hat{x}}^{\mathsf{a}}/\partial {x}^{\beta}=-{e}^{\mathsf{a}}{\alpha}^{\prime}{\sigma}_{\beta}^{{\alpha}^{\prime}}$
, and at coincidence we have

the second result follows from the identities
${e}^{\mathsf{a}}{\alpha}^{\prime}{e}^{{\alpha}^{\prime}}\mathsf{b}={\delta}_{\mathsf{b}}^{\mathsf{a}}$
and
${e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{\mathsf{a}}{\beta}^{\prime}={\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}$
.

$$\begin{array}{c}d{\hat{x}}^{\mathsf{a}}=-{e}^{\mathsf{a}}{\alpha}^{\prime}{\sigma}_{\beta}^{{\alpha}^{\prime}}d{x}^{\beta}.\end{array}$$ | (105) |

$$\begin{array}{c}\left[\frac{\partial {\hat{x}}^{\mathsf{a}}}{\partial {x}^{\alpha}}\right]={e}^{\mathsf{a}}{\alpha}^{\prime},\left[\frac{\partial {x}^{\alpha}}{\partial {\hat{x}}^{\mathsf{a}}}\right]={e}^{{\alpha}^{\prime}}\mathsf{a};\end{array}$$ | (106) |

It is interesting to note that the Jacobian of the transformation of Equation ( 105 ),
$J\equiv det(\partial {\hat{x}}^{\mathsf{a}}/\partial {x}^{\beta})$
, is given by
$J=\sqrt{-g}\Delta (x,{x}^{\prime})$
, where
$g$
is the determinant of the metric in the original coordinates, and
$\Delta (x,{x}^{\prime})$
is the van Vleck determinant of Equation ( 95 ). This result follows simply by writing the coordinate transformation in the form
$\partial {\hat{x}}^{\mathsf{a}}/\partial {x}^{\beta}=-{\eta}^{\mathsf{a}\mathsf{b}}{e}^{{\alpha}^{\prime}}\mathsf{b}{\sigma}_{{\alpha}^{\prime}\beta}$
and computing the product of the determinants. It allows us to deduce that in the RNC, the determinant of the metric is given by

It is easy to show that the geodesics emanating from
${x}^{\prime}$
are straight lines in the RNC. The proper volume of a small comoving region is then equal to
$dV={\Delta}^{-1}{d}^{4}\hat{x}$
, and this is smaller than the flat-spacetime value of
${d}^{4}\hat{x}$
if
$\Delta >1$
, that is, if the geodesics are focused by the spacetime curvature.

$$\begin{array}{c}\sqrt{-g\left(\text{RNC}\right)}=\frac{1}{\Delta (x,{x}^{\prime})}.\end{array}$$ | (107) |

3.1.2 Metric near ${\mathit{x}}^{\mathsf{\prime}}$

We now would like to invert Equation ( 105 ) in order to express the line element
$d{s}^{2}={g}_{\alpha \beta}d{x}^{\alpha}d{x}^{\beta}$
in terms of the displacements
$d{\hat{x}}^{\mathsf{a}}$
. We shall do this approximately, by working in a small neighbourhood of
${x}^{\prime}$
. We recall the expansion of Equation ( 89 ),

and this is easily inverted to give

This is the desired approximate inversion of Equation ( 105 ). It is useful to note that Equation ( 109 ), when specialized from the arbitrary coordinates
${x}^{\alpha}$
to
${\hat{x}}^{\mathsf{a}}$
, 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
$d{s}^{2}={g}_{\alpha \beta}d{x}^{\alpha}d{x}^{\beta}=\left({\eta}_{\mathsf{a}\mathsf{b}}{e}^{\mathsf{a}}\alpha {e}^{\mathsf{b}}\beta \right)d{x}^{\alpha}d{x}^{\beta}={\eta}_{\mathsf{a}\mathsf{b}}\left({e}^{\mathsf{a}}\alpha d{x}^{\alpha}\right)\left({e}^{\mathsf{b}}\beta d{x}^{\beta}\right)$
, and in this we substitute Equation ( 109 ).

$${\sigma}_{\beta}^{{\alpha}^{\prime}}=-{g}_{\beta}^{{\beta}^{\prime}}\left({\delta}_{{\beta}^{\prime}}^{{\alpha}^{\prime}}+\frac{1}{6}{R}_{{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}{\sigma}^{{\gamma}^{\prime}}{\sigma}^{{\delta}^{\prime}}\right)+\mathcal{O}\left({\epsilon}^{3}\right),$$

and in this we substitute the frame decomposition of the Riemann tensor,
${R}_{{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}^{{\alpha}^{\prime}}={R}_{\mathsf{c}\mathsf{b}\mathsf{d}}^{\mathsf{a}}{e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{\mathsf{c}}{\gamma}^{\prime}{e}^{\mathsf{b}}{\beta}^{\prime}{e}^{\mathsf{d}}{\delta}^{\prime}$
, and the tetrad decomposition of the parallel propagator,
${g}_{\beta}^{{\beta}^{\prime}}={e}^{{\beta}^{\prime}}\mathsf{b}{e}^{\mathsf{b}}\beta $
, where
${e}^{\mathsf{b}}\beta \left(x\right)$
is the dual tetrad at
$x$
obtained by parallel transport of
${e}^{\mathsf{b}}{\beta}^{\prime}\left({x}^{\prime}\right)$
. After some algebra we obtain
$${\sigma}_{\beta}^{{\alpha}^{\prime}}=-{e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{\mathsf{a}}\beta -\frac{1}{6}{R}_{\mathsf{c}\mathsf{b}\mathsf{d}}^{\mathsf{a}}{e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{\mathsf{b}}\beta {\hat{x}}^{\mathsf{c}}{\hat{x}}^{\mathsf{d}}+\mathcal{O}\left({\epsilon}^{3}\right),$$

where we have used Equation ( 103 ). Substituting this into Equation ( 105 ) yields
$$\begin{array}{c}d{\hat{x}}^{\mathsf{a}}=\left[{\delta}_{\mathsf{b}}^{\mathsf{a}}+\frac{1}{6}{R}_{\mathsf{c}\mathsf{b}\mathsf{d}}^{\mathsf{a}}{\hat{x}}^{\mathsf{c}}{\hat{x}}^{\mathsf{d}}+\mathcal{O}\left({x}^{3}\right)\right]{e}^{\mathsf{b}}\beta d{x}^{\beta},\end{array}$$ | (108) |

$$\begin{array}{c}{e}^{\mathsf{a}}\alpha d{x}^{\alpha}=\left[{\delta}_{\mathsf{b}}^{\mathsf{a}}-\frac{1}{6}{R}_{\mathsf{c}\mathsf{b}\mathsf{d}}^{\mathsf{a}}{\hat{x}}^{\mathsf{c}}{\hat{x}}^{\mathsf{d}}+\mathcal{O}\left({x}^{3}\right)\right]d{\hat{x}}^{\mathsf{b}}.\end{array}$$ | (109) |

The final result is
$d{s}^{2}={g}_{\mathsf{a}\mathsf{b}}d{\hat{x}}^{\mathsf{a}}d{\hat{x}}^{\mathsf{b}}$
, with

The quantities
${R}_{\mathsf{a}\mathsf{c}\mathsf{b}\mathsf{d}}$
appearing in Equation ( 110 ) are the frame components of the Riemann tensor evaluated at the base point
${x}^{\prime}$
,

and these are independent of
${\hat{x}}^{\mathsf{a}}$
. 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

$$\begin{array}{c}{g}_{\mathsf{a}\mathsf{b}}={\eta}_{\mathsf{a}\mathsf{b}}-\frac{1}{3}{R}_{\mathsf{a}\mathsf{c}\mathsf{b}\mathsf{d}}{\hat{x}}^{\mathsf{c}}{\hat{x}}^{\mathsf{d}}+\mathcal{O}\left({x}^{3}\right).\end{array}$$ | (110) |

$$\begin{array}{c}{R}_{\mathsf{a}\mathsf{c}\mathsf{b}\mathsf{d}}={R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}{e}^{{\alpha}^{\prime}}\mathsf{a}{e}^{{\gamma}^{\prime}}\mathsf{c}{e}^{{\beta}^{\prime}}\mathsf{b}{e}^{{\delta}^{\prime}}\mathsf{d},\end{array}$$ | (111) |

$${R}_{\mathsf{a}\mathsf{c}\mathsf{d}\mathsf{b}}={R}_{{\alpha}^{\prime}{\gamma}^{\prime}{\beta}^{\prime}{\delta}^{\prime}}\left[\frac{\partial {x}^{\alpha}}{\partial {\hat{x}}^{\mathsf{a}}}\right]\left[\frac{\partial {x}^{\gamma}}{\partial {\hat{x}}^{\mathsf{c}}}\right]\left[\frac{\partial {x}^{\beta}}{\partial {\hat{x}}^{\mathsf{b}}}\right]\left[\frac{\partial {x}^{\delta}}{\partial {\hat{x}}^{\mathsf{d}}}\right],$$

which is the standard transformation law for tensor components.
It is obvious from Equation ( 110 ) that
${g}_{\mathsf{a}\mathsf{b}}\left({x}^{\prime}\right)={\eta}_{\mathsf{a}\mathsf{b}}$
and
${\Gamma}_{\mathsf{b}\mathsf{c}}^{\mathsf{a}}\left({x}^{\prime}\right)=0$
, where
${\Gamma}_{\mathsf{b}\mathsf{c}}^{\mathsf{a}}=-\frac{1}{3}({R}_{\mathsf{b}\mathsf{c}\mathsf{d}}^{\mathsf{a}}+{R}_{\mathsf{c}\mathsf{b}\mathsf{d}}^{\mathsf{a}}){\hat{x}}^{\mathsf{d}}+\mathcal{O}\left({x}^{2}\right)$
is the connection compatible with the metric
${g}_{\mathsf{a}\mathsf{b}}$
. 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
$\gamma $
be a timelike curve described by parametric relations
${z}^{\mu}\left(\tau \right)$
in which
$\tau $
is proper time. Let
${u}^{\mu}=d{z}^{\mu}/d\tau $
be the curve's normalized tangent vector, and let
${a}^{\mu}=D{u}^{\mu}/d\tau $
be its acceleration vector.

A vector field
${v}^{\mu}$
is said to be Fermi–Walker transported on
$\gamma $
if it is a solution to the differential equation

Notice that this reduces to parallel transport if
${a}^{\mu}=0$
and
$\gamma $
is a geodesic.

$$\begin{array}{c}\frac{D{v}^{\mu}}{d\tau}=\left({v}_{\nu}{a}^{\nu}\right){u}^{\mu}-\left({v}_{\nu}{u}^{\nu}\right){a}^{\mu}.\end{array}$$ | (112) |

The operation of Fermi–Walker (FW) transport satisfies two important properties. The first is that
${u}^{\mu}$
is automatically FW transported along
$\gamma $
; this follows at once from Equation ( 112 ) and the fact that
${u}^{\mu}$
is orthogonal to
${a}^{\mu}$
. The second is that if the vectors
${v}^{\mu}$
and
${w}^{\mu}$
are both FW transported along
$\gamma $
, then their inner product
${v}_{\mu}{w}^{\mu}$
is constant on
$\gamma $
:
$D\left({v}_{\mu}{w}^{\mu}\right)/d\tau =0$
; this also follows immediately from Equation ( 112 ).

3.2.2 Tetrad and dual tetrad on $\mathit{\gamma}$

Let
$\overline{z}$
be an arbitrary reference point on
$\gamma $
. At this point we erect an orthonormal tetrad
$({u}^{\overline{\mu}},{e}^{\overline{\mu}}a)$
where, contrary to former usage, the frame index
$a$
runs from 1 to 3. We then propagate each frame vector on
$\gamma $
by FW transport; this guarantees that the tetrad remains orthonormal everywhere on
$\gamma $
. At a generic point
$z\left(\tau \right)$
we have