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4.2 Mechanics of dynamical horizons

The variations δ in the first law (36View Equation) represent infinitesimal changes in equilibrium states of horizon geometries. In the derivation of Section 4.1, these variations relate nearby but distinct space-times in each of which the horizon is in equilibrium. Therefore Equation (36View Equation) is interpreted as the first law in a passive form. Physically, it is perhaps the active form of the first law that is of more direct interest where a physical process, such as the one depicted in the right panel of Figure 1View Image causes a transition from one equilibrium state to a nearby one. Such a law can be established in the dynamical horizon framework. In fact, one can consider fully non-equilibrium situations, allowing physical processes in a given space-time in which there is a finite – rather than an infinitesimal – change in the state of the horizon. This leads to an integral version of the first law.

Our summary of the mechanics of DHs is divided in to three parts. In the first, we begin with some preliminaries on angular momentum. In the second, we extend the area balance law (25View Equation) by allowing more general lapse and shift functions, which leads to the integral version of the first law. In the third, we introduce the notion of horizon mass.

4.2.1 Angular momentum balance

As one might expect, the angular momentum balance law results from the momentum constraint (15View Equation) on the DH H. Fix any vector field ϕa on H which is tangential to all the cross-sections S of H, contract both sides of Equation (15View Equation) with ϕa, and integrate the resulting equation over the region ΔH to obtain4

1 ∮ 1 ∮ ∫ ( 1 ) ----- Kab ϕa^rb d2V − ----- Kabϕa ^rbd2V = Tab^τaϕb + ------(Kab − Kqab )ℒϕqab d3V.(42 ) 8πG S2 8πG S1 ΔH 16 πG
It is natural to identify the surface integrals with the generalized angular momentum Jϕ S associated with cross-sections S and set
∮ ∮ J ϕ = − -1--- K ϕa^rb d2V ≡ -1--- jϕd2V, (43 ) S 8πG S ab 8πG S
where the overall sign ensures compatibility with conventions normally used in the asymptotically flat context, and where we have introduced an angular momentum density ϕ a b j := − Kabϕ ^r for later convenience. The term ‘generalized’ emphasizes the fact that the vector field ϕa need not be an axial Killing field even on S; it only has to be tangential to our cross-sections. If ϕa happens to be the restriction of a space-time Killing field to S, then Jϕ S agrees with the Komar integral. If the pair (qab,Kab) is spherically symmetric on S, as one would expect, the angular momenta associated with the rotational Killing fields vanish.

Equation (42View Equation) is a balance law; the right side provides expressions of fluxes of the generalized angular momentum across ΔH. The contributions due to matter and gravitational waves are cleanly separated and given by

∫ ∫ 𝒥 ϕ = − Tab^τaϕb d3V, 𝒥ϕ = − --1--- Pabℒ ϕqabd3V, (44 ) matter ΔH grav 16πG ΔH
with ab ab ab P = K − Kq, so that
ϕ ϕ ϕ JS2 − JS1 = 𝒥matter + 𝒥gϕrav. (45 )
As expected, if a ϕ is a Killing vector of the three-metric qab, then the gravitational angular momentum flux vanishes: ϕ 𝒥 grav = 0.

As with the area balance law, here we worked directly with the constraint equations rather than with a Hamiltonian framework. However, we could also have used, e.g., the standard ADM phase space framework based on a Cauchy surface M with internal boundary S and the outer boundary at infinity. If φa is a vector field on M which tends to a ϕ on S and to an asymptotic rotational symmetry at infinity, we can ask for the phase space function which generates the canonical transformation corresponding to the rotation generated by φa. When the constraints are satisfied, as usual the value of this generating function is given by just surface terms. The term at infinity provides the total angular momentum and, as in Section 4.1.2, it is natural to interpret the surface term at S as the ϕ-angular momentum of S. This term can be expressed in terms of the Cauchy data ¯ (¯qab,Kab ) on M as

ϕ 1 ∮ a b 2 J¯S = − ----- K¯ab ϕ ¯r d V, (46 ) 8πG S
where ¯ra is the unit normal to S within M. However, since the right side involves the extrinsic curvature of M, in general the value of the integral is sensitive to the choice of M. Hence, the notion of the ϕ-angular momentum associated with an arbitrary cross-section is ambiguous. This ambiguity disappears if a ϕ is divergence-free on S. In particular, in this case, one has ϕ ¯ϕ JS = JS. Thus, although the balance law (42View Equation) holds for more general vector fields ϕa, it is robust only when ϕa is divergence-free on S. (These considerations shed some light on the interpretation of the field ζa in the area balance law (25View Equation). For, the form of the right side of Equation (43View Equation) implies that the field ζa vanishes identically on S if and only if ϕ J S vanishes for every divergence-free a ϕ on S. In particular then, if the horizon has non-zero angular momentum, the ζ-contribution to the energy flux can not vanish.)

Finally, for ϕ JS to be interpreted as ‘the’ angular momentum, a ϕ has to be a symmetry. An obvious possibility is that it be a Killing field of qab on S. A more general scenario is discussed in Section 8.

4.2.2 Integral form of the first law

To obtain the area balance law, in Section 3.2 we restricted ourselves to vector fields a a ξ(R ) = NR ℓ, i.e., to lapse functions NR = |∂R | and shifts N a = NR ^ra. We were then led to a conservation law for the Hawking mass. In the spherically symmetric context, the Hawking mass can be taken to be the physical mass of the horizon. However, as the Kerr space-time already illustrates, in presence of rotation this interpretation is physically incorrect. Therefore, although Equation (25View Equation) continues to dictate the dynamics of the Hawking mass even in presence of rotation, a more general procedure is needed to obtain physically interesting conservation laws in this case. In the case of WIHs, the first law incorporating rotations required us to consider suitable linear combinations of ℓa and the rotational symmetry field ϕa on the horizon. In the same spirit, on DHs, one has to consider more general vector fields than a ξ(R), i.e., more general choices of lapses and shifts.

As on WIHs, one first restricts oneself to situations in which the metric qab on H admits a Killing field ϕa so that Jϕ S can be unambiguously interpreted as the angular momentum associated with each S. In the case of a WIH, a t was given by a a a t = cℓ + Ω ϕ and the freedom was in the choice of constants c and Ω. On a DH, one must allow the corresponding coefficients to be ‘time-dependent’. The simplest generalization is to choose, in place of ξa(R ), vector fields

a a a a a a t := Nrℓ + Ω ϕ ≡ Nr ^τ + (Nr ^r + Ω ϕ ), (47 )
where Ω is an arbitrary function of R, and the lapse Nr is given by Nr = |∂r | for any function r of R. Note that one is free to rescale Nr and Ω by functions of R so that on each cross-section (‘instant of time’) one has the same rescaling freedom as on a WIH. One can consider even more general lapse-shift pairs to allow, e.g., for differential rotation (see [30Jump To The Next Citation Point]).

Using a t in place of a ξ(R), one obtains the following generalization of the area balance equation [30Jump To The Next Citation Point]:

r − r 1 ( ∮ ∮ ∫ Ω2 ∮ ) -2----1 + ----- Ωj ϕ d2V − Ωj ϕd2V − d Ω jϕ d2V = ∫ 2G 8πG S2 ∫ S1 Ω1 S ∫ ab 3 1 ( 2 2) 3 1 ab 3 Tab^τ t d V + 16-πG- Nr |σ | + 2|ζ| d V − 16-πG- ΩP ℒϕqabd V. (48 ) ΔH ΔH ΔH
Note that there is one balance equation for every vector field ta of the form (47View Equation); as in Section 4.1, we have an infinite number of relations, now ensured by the constraint part of Einstein’s equations.

The right side of Equation (48View Equation) can be naturally interpreted as the flux t ℱ ΔH of the ‘energy’ t E associated with the vector field a t across ΔH. Hence, we can rewrite the equation as

r − r 1 (∮ ∮ ∫ Ω2 ∮ ) ℱ tΔH = -2----1+ ----- Ωj ϕd2V − Ωjϕ d2V − dΩ jϕd2V . (49 ) 2G 8πG S2 S1 Ω1 S
If S1 and S2 are only infinitesimally separated, this integral equation reduces to the differential condition
( ) | δEt = --1-- -1-dr- || δa + Ω δJϕ S 8πG 2R dR |S s S ¯κ ≡ -----δaS + Ω δJϕS. (50 ) 8πG
Thus, the infinitesimal form of Equation (48View Equation) is a familiar first law, provided [(1∕2R )(dr∕dR )](S ) is identified as an effective surface gravity on the cross-section S. This identification can be motivated as follows. First, on a spherically symmetric DH, it is natural to choose r = R. Then the surface gravity reduces to 1∕(2R ), just as one would hope from one’s experience with the Schwarzschild metric and more generally with static but possibly distorted horizons (See Appendix A of [13]). Under the change R ↦→ r(R ), we have ¯κr = (dr∕dR )¯κR, which is the natural generalization of the transformation property κcℓ = cκ ℓ of surface gravity of WIHs under the change ℓ ↦→ cℓ. Finally, ¯κ r can also be regarded as a 2-sphere average of a geometrically defined surface gravity associated with certain vector fields on H [53Jump To The Next Citation Point30Jump To The Next Citation Point]; hence the adjective ‘effective’.

To summarize, Equation (48View Equation) represents an integral generalization of the first law of mechanics of weakly isolated horizons to dynamical situations in which the horizon is permitted to make a transition from a given state to one far away, not just nearby. The left side represents the flux t ℱΔH of the energy associated with the vector field a t, analogous to the flux of Bondi energy across a portion of null infinity. A natural question therefore arises: Can one integrate this flux to obtain an energy EtS which depends only on fields defined locally on the cross-section, as is possible at null infinity? As discussed in the next section, the answer is in the affirmative and the procedure leads to a canonical notion of horizon mass.

4.2.3 Horizon mass

In general relativity, the notion of energy always refers to a vector field. On DHs, the vector field is a t. Therefore, to obtain an unambiguous notion of horizon mass, we need to make a canonical choice of ta = Nr ℓa + Ω ϕa, i.e., of functions Nr and Ω on H. As we saw in Section 4.1.3, on WIHs of 4-dimensional Einstein–Maxwell theory, the pair (aΔ, JΔ) suffices to pick a canonical time translation field ta 0 on Δ. The associated horizon energy Et0 Δ is then interpreted as the mass M Δ. This suggests that the pair (ϕ aS, JS) be similarly used to make canonical choices Nr0 and 0 Ω on S. Thanks to the black hole uniqueness theorems of the 4-dimensional Einstein–Maxwell theory, this strategy is again viable.

Recall that the horizon surface gravity and the horizon angular velocity in a Kerr solution can be expressed as a function only of the horizon radius R and angular momentum J:

R4 − 4G2J 2 2GJ κKerr(R,J ) :=----√-------------, ΩKerr(R, J) := --√------------. (51 ) 2R3 R4 + 4G2J 2 R R4 + 4G2J 2
Given a cross section S of H, the idea is to consider the unique Kerr solution in which the horizon area is given by a S and angular momentum by J S, and assign to S effective surface gravity ¯κ S and angular velocity ΩS through
¯κS := κKerr(RS,J ϕS), ΩS := ΩKerr(RS, JϕS) (52 )
Repeating this procedure on every cross-section, one obtains functions 0 ¯κ (R ) and 0 Ω (R) on H, since J ϕ is a function of R alone. The definition of the effective surface gravity then determines a function r0 of R and hence Nr0 uniquely. Thus, using Equation (51View Equation), one can select a canonical vector field ta 0 and Equation (49View Equation) then provides a canonical balance law:
0 0 ( ∮ ∮ ∫ Ω02 ∮ ) ℱ t0 = r2-−-r1 + --1-- Ω0jϕ d2V − Ω0j ϕd2V − d Ω0 jϕ d2V . (53 ) ΔH 2G 8πG S2 S1 Ω01 S
The key question is whether this equation is integrable, i.e., if
ℱ tΔ0H = Et0S − Et0S (54 ) 2 1
for some t0 E S which depends locally on fields defined on S. The answer is in the affirmative. Furthermore, the expression of Et0 is remarkably simple and is identified with the horizon mass:
√ ------------ R4 + 4G2J 2 M (R ) := Et0(R ) = -------------. (55 ) 2GR
Thus, on any cross-section S, MS is just the mass of the Kerr space-time which has horizon area aS and angular momentum J ϕS: As far as the mass is concerned, one can regard the DH as an evolution through ‘a sequence of Kerr horizons’. The non-triviality of the result lies in the fact that, although this definition of mass is so ‘elementary’, thanks to the balance law (48View Equation) it obeys a Bondi-type flux formula,
∫ ∫ a b 3 --1--- ( 2 2) 3 MS2 − MS1 = ΔHTab^τ t0 d V + 16 πG ΔHNr0 |σ | + 2|ζ| d V (56 )
for a specific vector field a a 0 a t0 = Nr0ℓ + Ω ϕ, where each term on the right has a well-defined physical meaning. Thus, DHs admit a locally-defined notion of mass and an associated, canonical conservation law (56View Equation). The availability of a mass formula also provides a canonical integral version of the first law through Equation (48View Equation):
( ) r0− r0 1 ∮ ∮ ∫ Ω02 ∮ MS2 − MS1 = -2---1-+ ----- Ω0j ϕd2V − Ω0j ϕ d2V − dΩ0 jϕ d2V . (57 ) 2G 8πG S2 S1 Ω01 S
The infinitesimal version of this equation yields the familiar first law δM = (¯κ∕8πG )δa + Ω δJ.

On weakly isolated as well as dynamical horizons, area a and angular momentum J arise as the fundamental quantities and mass is expressed in terms of them. The fact that the horizon mass is the same function of a and J in both dynamical and equilibrium situations is extremely convenient for applications to numerical relativity [33Jump To The Next Citation Point]. Conceptually, this simplicity is a direct consequence of the first law and the non-triviality lies in the existence of a balance equation (48View Equation), which makes it possible to integrate the first law.

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