1 Introduction
$$\begin{array}{c}\delta M=\frac{\kappa}{8\pi G}\delta a+\Omega \delta J,\end{array}$$ | (1) |
$$\begin{array}{c}{R}_{ab}-\frac{1}{2}R{g}_{ab}+\Lambda {g}_{ab}=8\pi G{T}_{ab}.\end{array}$$ | (2) |
2 Basic Notions
2.1 Isolated horizons
2.1.1 Definitions
$$\begin{array}{c}{\Theta}_{(\ell )}:={q}^{ab}{\nabla}_{a}{\ell}_{b}.\end{array}$$ | (3) |
$$\begin{array}{c}{R}_{ab}{\ell}^{a}{\ell}^{b}+{\sigma}_{ab}{\sigma}^{ab}=0,\end{array}$$ | (4) |
$$\begin{array}{c}({\mathcal{\mathcal{L}}}_{\ell}{\mathcal{D}}_{a}-{\mathcal{D}}_{a}{\mathcal{\mathcal{L}}}_{\ell}){\ell}^{b}=0.\end{array}$$ | (5) |
$$\begin{array}{c}({\mathcal{\mathcal{L}}}_{\ell}{\mathcal{D}}_{a}-{\mathcal{D}}_{a}{\mathcal{\mathcal{L}}}_{\ell}){V}^{b}=0\end{array}$$ | (6) |
^{ $\text{2}$ } However, Condition ( 6 ) may be too strong in some problems, e.g., in the construction of quasi-equilibrium initial data sets, where the notion of WIH is more useful [125] (see Section 5.2 ).
2.1.2 Examples
2.1.3 Geometrical properties
$$\begin{array}{c}{V}^{a}{\nabla}_{a}{\ell}^{b}={V}^{a}{\omega}_{a}{\ell}^{b}.\end{array}$$ | (7) |
$$\begin{array}{c}{\kappa}_{(\ell )}:={\ell}^{a}{\omega}_{a}.\end{array}$$ | (8) |
$$\begin{array}{c}d\omega =2\left(\text{Im}{\Psi}_{2}\right)\epsilon ,\end{array}$$ | (9) |
$$\begin{array}{c}div\stackrel{~}{\omega}=-\stackrel{~}{\Delta}ln{\left|{\Psi}_{2}\right|}^{1/3},\end{array}$$ | (10) |
2.2 Dynamical horizons
2.2.1 Definitions
^{ $\text{3}$ } Indeed, the situation is similar for black holes in equilibrium. While it is physically reasonable to restrict oneself to IHs, most results require only the WIH boundary conditions. The distinction can be important in certain applications, e.g., in finding boundary conditions on the quasi-equilibrium initial data at inner horizons.
2.2.2 Examples
$$\begin{array}{c}{g}_{ab}=-\left(1-\frac{2GM\left(v\right)}{r}\right){\nabla}_{a}v{\nabla}_{b}v+2{\nabla}_{(a}v{\nabla}_{b)}r+{r}^{2}\left({\nabla}_{a}\theta {\nabla}_{b}\theta +{sin}^{2}\theta {\nabla}_{a}\phi {\nabla}_{b}\phi \right),\end{array}$$ | (11) |
$$\begin{array}{c}{T}_{ab}=\frac{\dot{M}\left(v\right)}{4\pi {r}^{2}}{\nabla}_{a}v{\nabla}_{b}v,\end{array}$$ | (12) |
2.2.3 Uniqueness
3 Area Increase Law
3.1 Preliminaries
3.2 Area increase law
$$\begin{array}{c}\stackrel{~}{K}={\stackrel{~}{q}}^{ab}{D}_{a}{\widehat{r}}_{b}=\frac{1}{2}{\stackrel{~}{q}}^{ab}{\nabla}_{a}({\ell}_{b}-{n}_{b})=-\frac{1}{2}{\Theta}_{\left(n\right)}>0,\end{array}$$ | (13) |
$$\begin{array}{ccc}{H}_{S}& :=& \mathcal{\mathcal{R}}+{K}^{2}-{K}^{ab}{K}_{ab}=16\pi G{\overline{T}}_{ab}{\widehat{\tau}}^{a}{\widehat{\tau}}^{b},\end{array}$$ | (14) |
$$\begin{array}{ccc}{H}_{V}^{a}& :=& {D}_{b}\left({K}^{ab}-K{q}^{ab}\right)=8\pi G{\overline{T}}^{bc}{\widehat{\tau}}_{c}{q}_{b}^{a},\end{array}$$ | (15) |
$$\begin{array}{c}{\overline{T}}_{ab}={T}_{ab}-\frac{1}{8\pi G}\Lambda {g}_{ab},\end{array}$$ | (16) |
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\text{matter}}^{\left(\xi \right)}:={\int}_{\Delta H}{T}_{ab}{\widehat{\tau}}^{a}{\xi}^{b}{d}^{3}V.\end{array}$$ | (17) |
$$\begin{array}{ccc}{\mathcal{\mathcal{F}}}_{\text{matter}}^{\left(\xi \right)}& =& \frac{1}{16\pi G}{\int}_{\Delta H}N\left({H}_{S}+2{\widehat{r}}_{a}{H}_{V}^{a}\right){d}^{3}V\end{array}$$ |
$$\begin{array}{ccc}& =& \frac{1}{16\pi G}{\int}_{\Delta H}N\left(\mathcal{\mathcal{R}}+{K}^{2}-{K}^{ab}{K}_{ab}+2{\widehat{r}}_{a}{D}_{b}({K}^{ab}-K{q}^{ab})\right){d}^{3}V.\end{array}$$ | (18) |
$$\begin{array}{c}{\stackrel{~}{q}}^{ab}({K}_{ab}+{\stackrel{~}{K}}_{ab})={\Theta}_{(\ell )}=0\end{array}$$ | (19) |
$$\begin{array}{c}{\sigma}_{ab}={\stackrel{~}{q}}^{ac}{\stackrel{~}{q}}^{bd}{\nabla}_{a}{\ell}_{b},{\zeta}^{a}={\stackrel{~}{q}}^{ab}{\widehat{r}}^{c}{\nabla}_{c}{\ell}_{b}.\end{array}$$ | (20) |
$$\begin{array}{c}{\int}_{\Delta H}N\stackrel{~}{\mathcal{\mathcal{R}}}{d}^{3}V=16\pi G{\int}_{\Delta H}{\overline{T}}_{ab}{\widehat{\tau}}^{a}{\xi}^{b}{d}^{3}V+{\int}_{\Delta H}N\left(|\sigma {|}^{2}+2|\zeta {|}^{2}\right){d}^{3}V.\end{array}$$ | (21) |
$$\begin{array}{c}N=|\partial R|\equiv {N}_{R}.\end{array}$$ | (22) |
$$\begin{array}{c}{\int}_{\Delta H}{N}_{R}\stackrel{~}{\mathcal{\mathcal{R}}}{d}^{3}V={\int}_{{R}_{1}}^{{R}_{2}}dR\oint \stackrel{~}{\mathcal{\mathcal{R}}}{d}^{2}V=\mathcal{\mathcal{I}}({R}_{2}-{R}_{1}),\end{array}$$ | (23) |
$$\begin{array}{c}\mathcal{\mathcal{I}}({R}_{2}-{R}_{1})=16\pi G{\int}_{\Delta H}\left({T}_{ab}-\frac{\Lambda}{8\pi G}{g}_{ab}\right){\widehat{\tau}}^{a}{\xi}_{\left(R\right)}^{b}{d}^{3}V+{\int}_{\Delta H}{N}_{R}\left(|\sigma {|}^{2}+2|\zeta {|}^{2}\right){d}^{3}V.\end{array}$$ | (24) |
$$\begin{array}{c}\frac{1}{2G}({R}_{2}-{R}_{1})={\int}_{\Delta H}{T}_{ab}{\widehat{\tau}}^{a}{\xi}_{\left(R\right)}^{b}{d}^{3}V+\frac{1}{16\pi G}{\int}_{\Delta H}{N}_{R}\left(|\sigma {|}^{2}+2|\zeta {|}^{2}\right){d}^{3}V.\end{array}$$ | (25) |
3.3 Energy flux due to gravitational waves
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\text{grav}}^{\left(R\right)}:=\frac{1}{16\pi G}{\int}_{\Delta H}{N}_{R}\left(|\sigma {|}^{2}+2|\zeta {|}^{2}\right){d}^{3}V.\end{array}$$ | (26) |
4 Black Hole Mechanics
4.1 Mechanics of weakly isolated horizons
4.1.1 The zeroth law
$$\begin{array}{c}{\mathcal{\mathcal{L}}}_{\ell}{\omega}_{a}=0.\end{array}$$ | (27) |
$$\begin{array}{c}d\omega =2\left(\text{Im}{\Psi}_{2}\right)\epsilon \end{array}$$ | (28) |
$$\begin{array}{c}0={\mathcal{\mathcal{L}}}_{\ell}\omega =d(\ell \cdot \omega )=d{\kappa}_{(\ell )}.\end{array}$$ | (29) |
4.1.2 Phase space, symplectic structure, and angular momentum
$$\begin{array}{c}\delta {J}^{\left(\phi \right)}=\Omega (\delta ,{X}_{\left(\phi \right)}).\end{array}$$ | (30) |
$$\begin{array}{c}\Omega (\delta ,{X}_{\left(\phi \right)})=-\frac{1}{8\pi G}\delta {\oint}_{S}\left[{\left({\phi}^{a}{\omega}_{a}\right)}^{2}\epsilon \right]-\delta {J}_{\text{ADM}}^{\left(\phi \right)}=:\delta {J}^{\left(\phi \right)}.\end{array}$$ | (31) |
$$\begin{array}{c}{J}_{\Delta}:=-\frac{1}{8\pi}{\oint}_{S}({\omega}_{a}{\phi}^{a}{)}^{2}\epsilon =-\frac{1}{4\pi}{\oint}_{S}fIm[{\Psi}_{2}{]}^{2}\epsilon \end{array}$$ | (32) |
4.1.3 Energy, mass, and the first law
$$\begin{array}{c}{t}^{a}={B}_{(\ell ,t)}{\ell}^{a}-{\Omega}_{\left(t\right)}{\phi}^{a},\end{array}$$ | (33) |
$$\begin{array}{c}{Y}_{\left(t\right)}\left(\delta \right):=\Omega (\delta ,{X}_{\left(t\right)}),\end{array}$$ | (34) |
$$\begin{array}{c}{Y}_{\left(t\right)}\left(\delta \right)=-\frac{{\kappa}_{\left(t\right)}}{8\pi G}\delta {a}_{\Delta}-{\Omega}_{\left(t\right)}\delta {J}_{\Delta}+\delta {E}_{\text{ADM}}^{\left(t\right)},\end{array}$$ | (35) |
$$\begin{array}{c}\delta {E}_{\Delta}^{\left(t\right)}=\frac{{\kappa}_{\left(t\right)}}{8\pi G}\delta {a}_{\Delta}+{\Omega}_{\left(t\right)}\delta {J}_{\Delta}.\end{array}$$ | (36) |
$$\begin{array}{c}\frac{\partial {\kappa}_{\left(t\right)}}{\partial {J}_{\Delta}}=8\pi G\frac{\partial {\Omega}_{\left(t\right)}}{\partial {a}_{\Delta}}.\end{array}$$ | (37) |
$$\begin{array}{c}{\kappa}_{0}({a}_{\Delta},{J}_{\Delta})=\frac{{R}_{\Delta}^{4}-4{J}_{\Delta}^{2}}{2{R}_{\Delta}^{3}\sqrt{{R}_{\Delta}^{4}+4{J}_{\Delta}^{2}}},\end{array}$$ | (38) |
$$\begin{array}{c}{\Omega}_{\left(t\right)}={\Omega}_{\text{Kerr}}({a}_{\Delta},{J}_{\Delta})=\frac{\sqrt{{R}_{\Delta}^{4}+4G{J}_{\Delta}^{2}}}{2G{R}_{\Delta}}.\end{array}$$ | (39) |
$$\begin{array}{c}{E}_{\Delta}^{\left({t}_{0}\right)}=\frac{1}{2G{R}_{\Delta}}\sqrt{{R}_{\Delta}^{4}+4{G}^{2}{J}_{\Delta}^{2}}.\end{array}$$ | (40) |
$$\begin{array}{c}{M}_{\Delta}:={E}_{\Delta}^{\left({t}_{0}\right)}\end{array}$$ | (41) |
4.2 Mechanics of dynamical horizons
4.2.1 Angular momentum balance
$$\begin{array}{c}\frac{1}{8\pi G}{\oint}_{{S}_{2}}{K}_{ab}{\phi}^{a}{\widehat{r}}^{b}{d}^{2}V-\frac{1}{8\pi G}{\oint}_{{S}_{1}}{K}_{ab}{\phi}^{a}{\widehat{r}}^{b}{d}^{2}V={\int}_{\Delta H}\left({T}_{ab}{\widehat{\tau}}^{a}{\phi}^{b}+\frac{1}{16\pi G}({K}^{ab}-K{q}^{ab}){\mathcal{\mathcal{L}}}_{\phi}{q}_{ab}\right){d}^{3}V.\end{array}$$ | (42) |
$$\begin{array}{c}{J}_{S}^{\phi}=-\frac{1}{8\pi G}{\oint}_{S}{K}_{ab}{\phi}^{a}{\widehat{r}}^{b}{d}^{2}V\equiv \frac{1}{8\pi G}{\oint}_{S}{j}^{\phi}{d}^{2}V,\end{array}$$ | (43) |
$$\begin{array}{c}{\mathcal{J}}_{\text{matter}}^{\phi}=-{\int}_{\Delta H}{T}_{ab}{\widehat{\tau}}^{a}{\phi}^{b}{d}^{3}V,{\mathcal{J}}_{\text{grav}=-\frac{1}{16\pi G}}^{\phi}{\int}_{\Delta H}{P}^{ab}{\mathcal{\mathcal{L}}}_{\phi}{q}_{ab}{d}^{3}V,\end{array}$$ | (44) |
$$\begin{array}{c}{J}_{{S}_{2}}^{\phi}-{J}_{{S}_{1}}^{\phi}={\mathcal{J}}_{\text{matter}}^{\phi}+{\mathcal{J}}_{\text{grav}}^{\phi}.\end{array}$$ | (45) |
$$\begin{array}{c}{\overline{J}}_{S}^{\phi}=-\frac{1}{8\pi G}{\oint}_{S}{\overline{K}}_{ab}{\phi}^{a}{\overline{r}}^{b}{d}^{2}V,\end{array}$$ | (46) |
^{ $\text{4}$ } Note that we could replace ${\overline{T}}_{ab}$ with ${T}_{ab}$ because ${g}_{ab}{\widehat{\tau}}^{a}{\phi}^{b}=0$ . Thus the cosmological constant plays no role in this section.
4.2.2 Integral form of the first law
$$\begin{array}{c}{t}^{a}:={N}_{r}{\ell}^{a}+\Omega {\phi}^{a}\equiv {N}_{r}{\widehat{\tau}}^{a}+({N}_{r}{\widehat{r}}^{a}+\Omega {\phi}^{a}),\end{array}$$ | (47) |
$$\begin{array}{ccc}& & \frac{{r}_{2}-{r}_{1}}{2G}+\frac{1}{8\pi G}\left({\oint}_{{S}_{2}}\Omega {j}^{\phi}{d}^{2}V-{\oint}_{{S}_{1}}\Omega {j}^{\phi}{d}^{2}V-{\int}_{{\Omega}_{1}}^{{\Omega}_{2}}d\Omega {\oint}_{S}{j}^{\phi}{d}^{2}V\right)=\end{array}$$ |
$$\begin{array}{ccc}& & {\int}_{\Delta H}{T}_{ab}{\widehat{\tau}}^{a}{t}^{b}{d}^{3}V+\frac{1}{16\pi G}{\int}_{\Delta H}{N}_{r}\left(|\sigma {|}^{2}+2|\zeta {|}^{2}\right){d}^{3}V-\frac{1}{16\pi G}{\int}_{\Delta H}\Omega {P}^{ab}{\mathcal{\mathcal{L}}}_{\phi}{q}_{ab}{d}^{3}V.\end{array}$$ | (48) |
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\Delta H}^{t}=\frac{{r}_{2}-{r}_{1}}{2G}+\frac{1}{8\pi G}\left({\oint}_{{S}_{2}}\Omega {j}^{\phi}{d}^{2}V-{\oint}_{{S}_{1}}\Omega {j}^{\phi}{d}^{2}V-{\int}_{{\Omega}_{1}}^{{\Omega}_{2}}d\Omega {\oint}_{S}{j}^{\phi}{d}^{2}V\right).\end{array}$$ | (49) |
$$\begin{array}{ccc}\delta {E}_{S}^{t}& =& \frac{1}{8\pi G}\left(\frac{1}{2R}\frac{dr}{dR}\right){|}_{S}\delta {a}_{s}+\Omega \delta {J}_{S}^{\phi}\end{array}$$ |
$$\begin{array}{ccc}& \equiv & \frac{\overline{\kappa}}{8\pi G}\delta {a}_{S}+\Omega \delta {J}_{S}^{\phi}.\end{array}$$ | (50) |
4.2.3 Horizon mass
$$\begin{array}{c}{\kappa}_{\text{Kerr}}(R,J):=\frac{{R}^{4}-4{G}^{2}{J}^{2}}{2{R}^{3}\sqrt{{R}^{4}+4{G}^{2}{J}^{2}}},{\Omega}_{\text{Kerr}}(R,J):=\frac{2GJ}{R\sqrt{{R}^{4}+4{G}^{2}{J}^{2}}}.\end{array}$$ | (51) |
$$\begin{array}{c}{\overline{\kappa}}_{S}:={\kappa}_{\text{Kerr}}({R}_{S},{J}_{S}^{\phi}),{\Omega}_{S}:={\Omega}_{\text{Kerr}}({R}_{S},{J}_{S}^{\phi})\end{array}$$ | (52) |
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\Delta H}^{{t}_{0}}=\frac{{r}_{2}^{0}-{r}_{1}^{0}}{2G}+\frac{1}{8\pi G}\left({\oint}_{{S}_{2}}{\Omega}^{0}{j}^{\phi}{d}^{2}V-{\oint}_{{S}_{1}}{\Omega}^{0}{j}^{\phi}{d}^{2}V-{\int}_{{\Omega}_{1}^{0}}^{{\Omega}_{2}^{0}}d{\Omega}^{0}{\oint}_{S}{j}^{\phi}{d}^{2}V\right).\end{array}$$ | (53) |
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\Delta H}^{{t}_{0}}={E}_{{S}_{2}}^{{t}_{0}}-{E}_{{S}_{1}}^{{t}_{0}}\end{array}$$ | (54) |
$$\begin{array}{c}M\left(R\right):={E}^{{t}_{0}}\left(R\right)=\frac{\sqrt{{R}^{4}+4{G}^{2}{J}^{2}}}{2GR}.\end{array}$$ | (55) |
$$\begin{array}{c}{M}_{{S}_{2}}-{M}_{{S}_{1}}={\int}_{\Delta H}{T}_{ab}{\widehat{\tau}}^{a}{t}_{0}^{b}{d}^{3}V+\frac{1}{16\pi G}{\int}_{\Delta H}{N}_{{r}^{0}}\left(|\sigma {|}^{2}+2|\zeta {|}^{2}\right){d}^{3}V\end{array}$$ | (56) |
$$\begin{array}{c}{M}_{{S}_{2}}-{M}_{{S}_{1}}=\frac{{r}_{2}^{0}-{r}_{1}^{0}}{2G}+\frac{1}{8\pi G}\left({\oint}_{{S}_{2}}{\Omega}^{0}{j}^{\phi}{d}^{2}V-{\oint}_{{S}_{1}}{\Omega}^{0}{j}^{\phi}{d}^{2}V-{\int}_{{\Omega}_{1}^{0}}^{{\Omega}_{2}^{0}}d{\Omega}^{0}{\oint}_{S}{j}^{\phi}{d}^{2}V\right).\end{array}$$ | (57) |
4.3 Passage of dynamical horizons to equilibrium
5 Applications in Numerical Relativity
5.1 Numerical computation of black hole mass and angular momentum
$$\begin{array}{c}{J}_{\Delta}=-\frac{1}{8\pi G}{\oint}_{S}{\phi}^{a}{R}^{b}{\overline{K}}_{ab}{d}^{2}V,\end{array}$$ | (58) |
$$\begin{array}{c}{J}_{S}^{\left(\phi \right)}=-\frac{1}{8\pi G}{\oint}_{S}{\phi}^{a}{R}^{b}{\overline{K}}_{ab}{d}^{2}V.\end{array}$$ | (59) |
$$\begin{array}{c}{M}_{S}=\frac{1}{2G{R}_{\Delta}}\sqrt{{R}_{\Delta}^{4}+4{G}^{2}{J}_{\Delta}^{2}}.\end{array}$$ | (60) |
^{ $\text{5}$ } This formula has a different sign from that given in [84] due to a difference in the sign convention in the definition of the extrinsic curvature.
5.2 Initial data
5.2.1 Boundary conditions at the inner boundary
$$\begin{array}{c}{t}^{a}=B{\ell}_{0}^{a}-{\Omega}_{\text{Kerr}}{\phi}^{a}=B{T}^{a}+(B{R}^{a}-{\Omega}_{\text{Kerr}}{\phi}^{a}),\end{array}$$ | (61) |
$$\begin{array}{c}\alpha =B,{\beta}^{a}=\alpha {R}^{a}-{\Omega}_{\text{Kerr}}{\phi}^{a}.\end{array}$$ | (62) |
$$\begin{array}{c}{\mathcal{\mathcal{L}}}_{t}{\stackrel{~}{q}}_{ab}=0,{\mathcal{\mathcal{L}}}_{\phi}{\stackrel{~}{q}}_{ab}=0\end{array}$$ | (63) |
^{ $\text{6}$ } Cook's boundary condition on the conformal factor $\psi $ (Equation (82) in [70] ) is equivalent to ${\Theta}_{(\ell )}=0$ which (in the co-rotating case, or more generally, when the 2-metric on $S$ is axi-symmetric) reduces to ${\mathcal{\mathcal{L}}}_{t}\psi =0$ on $S$ . The Yo et al. boundary condition on $\psi $ (Equation (48) of [190] ) is equivalent to ${\mathcal{\mathcal{L}}}_{\overline{t}}\psi =0$ on $S$ , where, however, the evolution vector field ${\overline{t}}^{a}$ is obtained by a superposition of two Kerr–Schild data.
5.2.2 Binding energy