Dynamical horizons are considered in full, non-linear general relativity.
Expressions of fluxes of energy and angular momentum carried by gravitational
waves across these horizons are obtained. Fluxes are local, the energy flux is
positive and change in the horizon area is related to these fluxes. The flux
formulae also give rise to balance laws analogous to the ones obtained by Bondi
and Sachs at null infinity and provide generalizations of the first and second
laws of black hole mechanics.Comment: 4 pages, RevTeX4. Minor typos corrected. To appear in Phys.Rev.Let

Abhay Ashtekar

Badri Krishnan

R. M. Wald

English

arXiv

Crossref

Dynamical Horizons: Energy, Angular Momentum, Fluxes, and Balance Laws

Dynamical horizons are considered in full, non-linear general relativity. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics

Ashtekar, A.

Krishnan, B.

MPG.PuRe

Dynamical Horizons: Energy, Angular Momentum, Fluxes and Balance Laws

arXiv:gr-qc/0207080 v3   3 Nov 2002Dynamical Horizons: Energy, Angular Momentum, Fluxes and Balance LawsAbhay Ashtekar∗ and Badri Krishnan†Center for Gravitational Physics and Geometry and Center for Gravitational Wave Physics,Department of Physics, Penn State University, University Park, PA 16802, USA(Dated: November 3, 2002)Dynamical horizons are considered in full, non-linear general relativity. Expressions of fluxes ofenergy and angular momentum carried by gravitational waves across these horizons are obtained.Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes.The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachsat null infinity and provide generalizations of the first and second laws of black hole mechanics.PACS numbers: 04.25.Dm, 04.70.BwBlack holes which are themselves in equilibrium but inpossibly time-dependent space-times can be modelled byisolated horizons [1]. Over the past three years, proper-ties of isolated horizons were studied in detail. In par-ticular, the framework enabled one to extend the laws ofblack hole mechanics [2, 3] and has been used to extractphysics from initial data of widely separated black holes[4] and from numerical simulations of the final phases ofblack hole collisions [1, 5]. The purpose of this letter isto outline a generalization of these ideas to fully dynami-cal situations in which matter and gravitational radiationcan fall into black holes.Our definition of a dynamical horizon is geared to prac-tical applications in astrophysical situations, particularlythose considered in numerical relativity.I. DEFINITION AND NOTATIONDefinition: A smooth, three-dimensional, space-likesub-manifold H in a space-time is said to be a dynamicalhorizon if it is foliated by a preferred family of 2-spheressuch that, on each leaf S, the expansion θ(`) of a nullnormal `a vanishes and the expansion θ(n) of the othernull normal na is strictly negative.Thus, a dynamical horizon H is a 3-manifold whichis foliated by marginally trapped 2-spheres. As shownbelow, the area of these 2-spheres necessarily increases.An example is provided by continuous segments of worldtubes of apparent horizons that naturally arise in numer-ical evolutions of black holes. In contrast to event hori-zons, dynamical horizons can be located quasi-locally;knowledge of the full space-time is not required. Thecondition that H be space-like is implied by a strongerbut physically reasonable restriction that the derivativeof θ(`) along na be negative [6]. Finally, the requirementthat the leaves be topologically S2 can be replaced by∗Electronic address: ashtekar@gravity.psu.edu†Present address: Max-Planck-Institut für Gravitationsphysik,Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Ger-many; Electronic address: badkri@aei-potsdam.mpg.dethe weaker condition that they be compact. One canshow that the topology of S is necessarily S2 if the fluxof matter or gravitational energy across H is non-zero. Ifthese fluxes were to vanish identically, H would becomeisolated and replaced by a null, non-expanding horizon[2].Dynamical horizons are closely related to Hayward’strapping horizons [6]. However, while the definition oftrapping horizons imposes a condition on the derivativeof θ(`) off H , our conditions refer only to geometric quan-tities which are intrinsically defined on H . But in cases ofphysical interest, the additional condition would be satis-fied and dynamical horizons will be future, outer trappinghorizons. Nonetheless, our analysis and results differ con-siderably from those of Hayward’s. While his frameworkis based on a 2+2 decomposition, ours will be based onthe ADM 3+1 decomposition. Our discussion includesangular momentum, our flux formulae are new and ourgeneralization of black hole mechanics is different. Ouranalysis is geared to providing tools to extract physicsand perform checks on numerical simulations of dynam-ical black holes. Therefore we will restrict ourselves todynamical horizons with zero charge.Let us begin by fixing notation. Let τ̂a be the unittime-like normal to H and denote by ∇a the space-timederivative operator. The metric and extrinsic curvatureof H are denoted by qab and Kab := qacqbd∇cτ̂d respec-tively; Da is the derivative operator on H compatiblewith qab and Rab its Ricci tensor. Leaves of the preferredfoliation of H will be called cross-sections of H . The unitspace-like vector orthogonal to S and tangent to H is de-noted by r̂ a. Quantities intrinsic to S will be generallywritten with a tilde. Thus, the two-metric on S is q̃ab, theextrinsic curvature of S ⊂ H is K̃ab := q̃ ca q̃ db Dcr̂d, thederivative operator on (S, q̃ab) is D̃a and its Ricci tensoris R̃ab. Finally, we will fix the rescaling freedom in thechoice of null normals via `a := τ̂ a+r̂ a and na := τ̂ a−r̂ a.We first note an immediate consequence of the defini-tion. Since θ(`) = 0 and θ(n) < 0, it follows that K̃ > 0.Hence the area aS of S increases monotonically along r̂a.Thus the second law of black hole mechanics holds on H .We will obtain an explicit expression for the change ofarea in part III.2Our main analysis is based on the fact that, since His a space-like surface, the Cauchy data (qab, Kab) on Hmust satisfy the usual scalar and vector constraintsHS := R + K2 − KabKab = 16πGTabτ̂ aτ̂ b (1)HaV := Db(Kab − Kqab)= 8πGT bcτ̂ cqab . (2)We will often focus our attention on a portion ∆H ⊂ Hwhich is bounded by two cross-sections S1 and S2.II. ANGULAR MOMENTUMThe angular momentum analysis is rather straight for-ward and is, in fact, applicable to an arbitrary space-likehypersurface. Fix any vector field ϕa on H which is tan-gential to the cross-sections of H . Contract ϕa with bothsides of eqn. (2). Integrate the resulting equation overthe region ∆H ⊂ H , perform an integration by parts anduse the identity L ϕqab = 2D(aϕb) to obtain18πG∮S2Kabϕar̂ b d2V − 18πG∮S1Kabϕar̂ b d2V=∫∆H(Tabτ̂aϕb +116πGP abL ϕqab)d3V (3)where P ab := Kab − Kqab. It is natural to identify thesurface integrals with the generalized angular momentumJ (ϕ) associated with those surfaces and set:J(ϕ)S = −18πG∮SKabϕar̂ b d2V (4)where we have chosen the overall sign to ensure compat-ibility with conventions normally used in the asymptoti-cally flat context. The term ‘generalized’ emphasizes thefact that the vector field ϕa need not be an axial Killingfield even on S; it only has to be tangential to our cross-sections.The flux of this angular momentum due to matterfields and gravitational waves are respectivelyJ (ϕ)m = −∫∆HTabτ̂aϕb d3V , (5)J (ϕ)g = −116πG∫∆HP abL ϕqab d3V , (6)and we get the balance equationJ(ϕ)2 − J(ϕ)1 = J (ϕ)m + J (ϕ)g . (7)As expected, if ϕa is a Killing vector of the three-metricqab, then the gravitational flux vanishes: J (ϕ)g = 0.It is convenient to introduce the angular momentumcurrent jϕ := −Kabϕar̂ b so that (4) becomes J (ϕ)S =(8πG)−1∮Sjϕ d2V .III. ENERGY FLUXES AND AREA BALANCEAs is usual in general relativity, the notion of energy istied to a choice of a vector field. Here, we will considervector fields ξa = N`a where the lapse N is constructedas follows. Let r be a radial coordinate on H defined suchthat the cross-sections of H are level surfaces of r. Thenr̂a ∝ Dar. It turns out that in order to get the balancelaw for energy, we must tie our lapse functions N to radialcoordinates such that Dar = Nr r̂a. (Since ξa = N`a,as usual the term ‘lapse’ refers to space-time evolution;not to ‘evolution’ along r̂a.) Thus each r determines apermissible lapse function Nr. If we use a different radialcoordinate r′, then the lapse is rescaled according to therelationNr′ = Nrdr′dr. (8)Thus, although the lapse itself will in general be a func-tion of all three coordinates on H , the relative factor be-tween any two permissible lapses can be a function onlyof r. We denote the resulting permissible vector fields byξa(r) := Nr`a. Recall that, on an isolated horizon, physi-cal fields are time independent and null normals can berescaled by a positive constant [2]. Now the horizon fieldsare ‘dynamical’, i.e., r-dependent, and the rescaling free-dom is by a positive function of r.We are interested in calculating the flux of energyassociated with ξa(r) for any radial coordinate r. De-note the flux of matter energy across ∆H by F (r)m :=∫∆H Tabτ̂aξb(r)d3V . By taking the appropriate combina-tion of (1) and (2) we obtainF (r)m =116πG∫∆HNr {HS + 2r̂aHaV } d3V . (9)Since H is foliated by two-spheres, we can perform a 2+1split of the various quantities on H . Using the GaussCodazzi relation we rewrite R in terms of quantities onS:R = R̃ + K̃2 − K̃abK̃ab + 2Daαa (10)where αa = r̂ bDbr̂a − r̂ aDbr̂ b. Next, the fact that theexpansion θ(`) of `a vanishes leads to the relationK + K̃ = Kabr̂ar̂ b . (11)Using (10) and (11) in eqn. (9) and simplifying, we obtainthe result∫∆HNrR̃ d3V = 16πG∫∆HTabτ̂aξb(r) d3V+∫∆HNr{|σ|2 + 2|ζ|2}d3V (12)where |σ|2 = σabσab with σab := q̃ ca q̃ db ∇c`d −12 q̃abq̃cd∇c`d, the shear of `a, and |ζ|2 = ζaζa with3ζa := q̃ abr̂ c∇c`b. Both σab and ζa are tensors intrin-sic to S. To simplify the left side of this equation, notethat the volume element d3V on H can be written asd3V = N−1r dr d2V where d2V is the area element on S.Using the Gauss-Bonnet theorem, the integral of NrR̃can then be written as∫∆HNrR̃ d3V =∫ r2r1dr(∮SR̃ d2V)= 8π(r2 − r1) .(13)Substituting this result in eqn. (12) we finally obtain( r22G− r12G)=∫∆HTabτ̂aξb(r) d3V+116πG∫∆HNr{|σ|2 + 2|ζ|2}d3V. (14)This is the key result we were looking for. Let us nowinterpret the various terms appearing in this equation.The first integral on the right side of this equation isthe flux F (r)m of matter energy associated with the vectorfield ξa(r). Since ξa(r) is null and τ̂ time-like, if Tab sat-isfies, say, the dominant energy condition, this quantityis guaranteed to be non-negative. It is natural to inter-pret the second term as the flux F (r)g of ξa(r)-energy in thegravitational radiation:F (r)g :=116πG∫∆HNr{|σ|2 + 2|ζ|2}d3V . (15)This expression shares four desirable features with theBondi-Sachs energy flux at null infinity. First, it doesnot refer to any coordinates or tetrads; it refers only tothe given dynamical horizon H and the evolution vec-tor field ξa(r). Second, the energy flux is manifestly non-negative. Third, all fields used in it are local; we did nothave to perform, e.g., a radial integration to define anyof them. Finally, the expression vanishes in the spheri-cally symmetric case: if the Cauchy data (qab, Kab) andthe foliation on H is spherically symmetric, σab = 0 andζa = 0.To conclude this section, let us choose for our radialcoordinate the area radius R :=√a/4π. Then,R22G− R12G= F (R)m + F (R)g (16)Thus, as promised in part I, we have obtained an explicitformula relating the change in the area of the horizon tofluxes of matter and gravitational ξ(R)-energy.IV. MASS AND THE FIRST LAWLet us now combine the results of parts II and III toobtain the physical process version of the first law for Hand a mass formula for an arbitrary cross-section of H .Denote by Eξ(R) the ξ(R)-energy of cross-sections S ofH . While we do not yet have the explicit expression forit, we can assume that, because of the influx of matterand gravitational energy, Eξ(R) will change by an amount∆ Eξ(R) = F (R)m +F (R)g as we move from one cross sectionto another. Therefore, if we define effective surface grav-ity kR associated with ξa(R) as kR := 1/2R, the infinites-imal form of (16) implies (kR/8πG)da = dEξ(R) . For ageneral choice of the radial coordinate r, (14) yields ageneralized first law:kr8πGda = dEξ(r) (17)where the effective surface gravity kr of ξa(R) is given bykr =drdRkR where ξa(r) =drdRξa(R) . (18)This rescaling freedom in surface gravity is analogous tothe rescaling freedom which exists for Killing horizons,or more generally, isolated horizons. The new feature inthe present case is that we have the freedom to rescalethe surface gravity (and `a) by a positive function of theradius instead of just by a constant. This is just whatone would expect in a dynamical situation. Finally, notethat the differentials appearing in (17) are actual varia-tions along the dynamical horizon due to an infinitesimalchange in r and are not variations in phase space as insome of the formulations [2, 3, 7] of the first law.To include rotation, pick a vector field ϕa on H suchthat ϕa is tangent to the cross-sections of H , has closedorbits and has affine length 2π.( At this point, ϕa neednot be a Killing vector of qab.) Consider time evolutionvector fields ta which are of the form ta = Nr`a − Ωϕawhere Nr is a permissible lapse associated with a radialcoordinate r and Ω an arbitrary function of r. Evaluatethe quantity∫∆H Tabτ̂atb d3V using (3) and (14):r2 − r12G+18πG{∮S2Ωjϕ d2V −∮S1Ωjϕ d2V−∫ Ω2Ω1dΩ∮Sjϕ d2V}=∫∆HTabτ̂atb d3V+116πG∫∆HNr(|σ|2 + 2|ζ|2)d3V− 116πG∫∆HΩP abL ϕqab d3V . (19)Again, if we denote by Et the t-energy associated withcross-sections S of H , the right side of (19) can be inter-preted as ∆Et. If we now restrict ourselves to infinites-imal ∆H , the three terms in the curly brackets combineto give d(ΩJ) − JdΩ and we obtaindr2G+ ΩdJ =kr8πGda + ΩdJ = dEt. (20)This equation is our generalization of the first law for dy-namical horizons. Since the differentials in this equationare variations along H , this can be viewed as a ‘physi-cal process version of the first law’. Note that for each4allowed choice of lapse Nr, angular velocity Ω(r) and vec-tor field ϕa on H , we obtain a permissible time evolutionvector field ta = Nr`a − Ωϕa and a corresponding firstlaw. This situation is very similar to what happens in theisolated horizon framework where we obtain a first law foreach permissible time translation on the horizon. Again,the generalization from that time independent situationconsists of allowing the lapse and the angular velocity tobecome r-dependent, i.e., ‘dynamical’.For every allowed choice of (Nr, Ω(r), ϕa), we can in-tegrate eqn. (20) on H to obtain a formula for Et on anycross section but, in general, the result may not be ex-pressible just in terms of geometric quantities defined lo-cally on that cross-section. However, in some physicallyinteresting cases, the expression is local. For example,In the case of spherical symmetry, it is natural to chooseΩ = 0 and R as the radial coordinate in which case weobtain Et = R/2G. This is just the irreducible (or Hawk-ing) mass of the cross-section. Even in this simple case,(19) provides a useful balance law, with clear-cut inter-pretation. Physically, perhaps the most interesting caseis the one in which qab is only axi-symmetric with ϕaas its axial Killing vector. In this case we can naturallyapply, at each cross-section S of H , the strategy usedin the isolated horizon framework to select a preferredta: Calculate the angular momentum J defined by theaxial Killing field ϕ, choose the radial coordinate r (orequivalently, the lapse Nr) such thatkr = ko(R) :=R4 − 4G2J22R3√R4 + 4G2J2(21)and choose Ω such thatΩ = Ωo(R) :=2GJR√R4 + 4G2J2. (22)This functional dependence of kr on R and J is exactlythat of the Kerr family. (The condition on surface gravitycan always be implemented provided the right side of (21)is positive, which in the kerr family corresponds to non-extremal horizons. The resulting r and Nr are unique.)With this choice of Nr and Ω, the energy EtS is given bythe well known Smarr formulaEto = 2 (koa8πG+ ΩoJ) =√R4 + 4G2J22GR; (23)as a function of its angular momentum and area, eachcross-section is assigned simply that mass which it wouldhave in the Kerr family. However, there is still a balanceequation in which the flux of gravitational energy F (to)g islocal and positive definite (see (19)). (The gravitationalangular momentum flux which, in general, has indeter-minate sign vanishes due to axi-symmetry.) Motivatedby the isolated horizon framework, we will refer to thiscanonical Eto as the mass associated with cross-sectionsS of H and denote it simply by M . Thus, among theinfinitely many first laws (20), there is a canonical one:dM =ko8πGda + ΩodJ . (24)We conclude with three remarks.i) Note that the mass and angular momentum dependonly on local fields on each cross section S and changesin these quantities over finite regions ∆H of H have beenrelated to matter and gravitational radiation fluxes, de-termined by the local geometry of H .ii) Unlike the vector fields ξa(r) = Nr`a, general permis-sible vector fields ta is not necessarily causal. Thereforethe matter flux∫∆H Tabtaτ̂ bd3V need not be positive.Similarly, if ϕa is not a Killing field of qab, the gravita-tional flux need not be positive. Therefore, although thearea a always increase along r̂a, Et can decrease. Thisis the analog of the Penrose process in which ‘rotationalenergy’ is extracted from the dynamical horizon.iii) While the infinitesimal version eq (20) of the firstlaw is conceptually more interesting, the finite balanceequation (19) is likely to be more directly useful in theanalysis of astrophysical situations. In particular, thepresence of an infinite number of these balance equationscan provide useful checks on numerical simulations in thestrong field regime.Acknowledgements We would like to thank ChrisBeetle, Steve Fairhurst and Jerzy Lewandowski for stim-ulating discussions. This work was supported in partby the NSF grant PHY-0090091, the NSF CooperativeAgreement PHY-0114375 and the Eberly research fundsof Penn State. BK was also supported through Duncanand Roberts fellowships.[1] A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krish-nan, J. Lewandowski and J. Wisniewski, Phys. Rev. Lett.85, 3564-3567 (2000).[2] A. Ashtekar, S. Fairhurst, and B. Krishnan, Phys. Rev.D62 104025 (2000).[3] A. Ashtekar, C. Beetle, and J. Lewandowski, Phys. Rev.D64, 044016 (2001).[4] B. Krishnan, Thesis dissertation, Penn State (2002); A.Ashtekar and B. Krishnan (in preparation).[5] O. Dreyer, B. Krishnan, E. Schnetter and D. Shoemaker,gr-qc/0206008.[6] S. Hayward, Phys. Rev. D49, 6467-6474 (1994).[7] R. M. Wald, Quantum Field Theory in Curved Spacetimeand Black Hole Thermodynamics (University of ChicagoPress) (1994).

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