aa r X i v : . [ g r- q c ] D ec The Horizon Energy of a Black Hole
Yuan K. HaDepartment of Physics, Temple UniversityPhiladelphia, Pennsylvania 19122 [email protected] 1, 2017
Abstract
We investigate the energy distribution of a black hole in various space-times as reckoned by a distant observer using the quasi-local energyapproach. In each case the horizon mass of a black hole: neutral,charged or rotating, is found to be twice the irreducible mass ob-served at infinity. This is known as the Horizon Mass Theorem. Asa consequence, the electrostatic energy and the rotational energy of ageneral black hole are all external quantities. Matter carrying chargesand spins could only lie outside the horizon. This result could resolveseveral long-standing paradoxes related to known black hole proper-ties; such as why entropy is proportional to area and not to volume,the information loss problem, the firewall problem, the internal struc-ture and the thin shell model of a black hole.
Keywords : Quasi-local Energy; Horizon Mass; Horizon Mass Theorem. . Quasi-local Energy A black hole has the strongest gravitational field of all gravitational systemsand the greatest gravitational potential energy. It is well known that grav-itational energy density cannot be defined consistently in general relativitysince gravitational field can be transformed away in a local inertial frame.Nevertheless, it is possible to consider the total energy contained in a surfaceenclosing a black hole at a given coordinate distance. This is based on thequasi-local energy approach [1] obtained from a Hamiltonian-Jacobi analysisof the Hilbert action in general relativity.The quasi-local energy expression is the most important development ingeneral relativity in recent years to understand the dynamics of the gravi-tational field, such as energy, momentum and angular momentum [2]. Forasymptotically flat spacetime, the quasi-local energy agrees with the Arnowitt-Deser-Misner energy [3] at spatial infinity and for spherically symmetricspacetime, it has the correct Newtonian limit, including negative contribu-tion to gravitational binding. It also agrees with the Komar energy [4] andBondi energy [5] at null infinity. The quasi-local energy approach is thereforenaturally suited for investigating the energy distribution of a black hole.The expression for the quasi-local energy is given in terms of the totalmean curvature of a surface bounding a volume for a gravitational system infour-dimensional spacetime. The Brown and York expression is given in theform of an integral [1] E = c πG Z B d x √ σ ( k − k ) , (1)where σ is the determinant of the metric defined on the two-dimensional sur-face B ; k is the trace of extrinsic curvature of the surface and k , the traceof curvature of a reference space. For asymptotically flat reference spacetime, k is zero. 2 . Horizon Mass Theorem The Horizon Mass Theorem is the final outcome of the quasi-local energyexpression applied to the black hole. The mass of a black hole depends onwhere the observer is. The closer one gets to a black hole the less gravi-tational energy one expects to see. As a result, the mass of a black holeincreases as one gets near the horizon. The Horizon Mass Theorem can bestated in the following [6]:
Theorem.
For all black holes: neutral, charge or rotating, the horizon massis always twice the irreducible mass observed at infinity.It is useful to introduce the following definitions of mass in order to under-stand the energy of a black hole:1. The asymptotic mass is the mass of a neutral, charged or rotating blackhole including electrostatic and rotatinal energy. It is the mass observedat infinity.2. The horizon mass is the mass which cannot escape from the horizonof a neutral, charged or rotating black hole. It is the mass observed atthe horizon.3. The irreducible mass is the final mass of a charged or rotating black holewhen its charge or angular momentum is removed by adding externalparticles to the black hole. It is the mass observed at infinity.
3. Schwarzschild Black Hole
The total energy contained in a sphere enclosing the black hole at a coordi-nate distance r is given by the expression [1,7,8] E ( r ) = rc G − s − GMrc , (2)3here M is the mass of the black hole observed at infinity, c is the speed oflight and G is the gravitational constant. At the horizon, the Schwarzschildradius is r = 2 GM/c . Evaluating the expression in Eq.(2), we find themetric coefficient g = (1 − GM/rc ) / vanishes identically and the energyat the horizon is therefore E ( r ) = (cid:18) GMc (cid:19) c G = 2 M c . (3)The horizon mass of the Schwarzschild black hole is simply twice the asymp-totic mass M observed at infinity. The negative gravitational energy outsidethe black hole is as great as the asymptotic mass.
4. Charged Black Hole
The total energy of a charged black hole contained within a radius at coor-dinate r is now given by [7] E ( r ) = rc G − s − GMrc + GQ r c , (4)where M is the mass of the black hole including electrostatic energy observedat infinity and Q is the electric charge. At the horizon radius r + = GMc + GMc s − Q GM , (5)the square root in Eq.(4) again vanishes and the horizon energy becomes E ( r + ) = r + c G = M c + M c s − Q GM . (6)When this is expressed in terms of the irreducible mass of the charged blackhole M irr = M M s − Q GM , (7)the horizon energy becomes exactly twice the irreducible energy E ( r + ) = 2 M irr c . (8)4he horizon mass therefore depends only on the energy of the black holewhen it is neutralized by adding oppositely charged particles. There is noelectrostatic energy inside the charged black hole.
5. Slowly Rotating Black Hole
The total energy of a slowly rotating black hole with angular momentum J and angular momentum parameter a = J/M c using the quasi-local energyapproach is given by the approximate expression [9], 0 < a ≪ E ( r ) = rc G − s − GMrc + a r + a c rG GMrc + (cid:18) GMrc (cid:19) s − GMrc + a r + · · · (9)Again, using the horizon radius in this case r h = GMc + s G M c − J M c (10)and the irreducible mass M irr = M M s − J c G M , (11)we arrive at a very good approximate relation for the horizon energy M h ≃ M irr + O ( a ) . (12)
6. Black Hole at Any Rotation
For general and fast rotations, the quasi-local energy approach has limitationbut the total energy can be obtained very accurately by numerical evalua-tion in the teleparallel formulation of general relativity [10]. The teleparallelgravity is an equivalent geometric formulation of general relativity in which5he action is constructed purely with torsion without curvature. It has agauge field approach. There is a perfectly well-defined gravitational energydensity and the result agrees very well with Eq.(12) at any rotation. Thesmall discrepancy is due to the axial symmetry of a rotating black hole com-pared with the exact spherical symmetry of a Schwarzschild black hole. Theresult shows that the rotational energy appears to reside almost completelyoutside the black hole.For an exact relationship, however, we have to employ a formula knownfor the area of a rotating black hole valid for all rotations in the Kerr metric[11], A = 4 π ( r h + a ) = 16 πG M irr c . (13)This area is exactly the same as that of a Schwarzschild black hole withasymptotic mass M irr . Now a local observer who is comoving with the ro-tating black hole at the event horizon will see only this Schwarzschild blackhole. Since the horizon mass of the Schwarzschild black hole is 2 M irr , there-fore the horizon mass of the rotating black hole is exactly M h = 2 M irr . Wehave shown in each case, the horizon mass of a black hole is always twice theirreducible mass observed at infinity.The Horizon Mass Theorem shows that the electrostatic energy and therotational energy of a general black hole are all external quantities. Theyare absent inside the black hole. A charged black hole does not have elec-tric charges inside. A rotating black hole does not rotate; only the exter-nal space is rotating. The conclusion is surprising. It could resolve severallong-standing paradoxes of black holes such as the entropy problem, the in-formation problem, the firewall problem and the gravastar thin shell modelsince matter carrying charges and spins could only stay outside the horizon.The quasi-local energy of black holes is one of the profound and fascinatingresults known recently in general relativity.6 eferences [1] J.D. Brown and J.W. York, Jr. Phys. Rev. D , 1407 (1993).[2] M.T. Wang and S.T. Yau, Phys. Rev. Lett. , 021101 (2009).[3] R. Arnowitt, S. Deser and C.W. Misner,
Phys. Rev. , 1595 (1960).[4] A. Komar,
Phys. Rev. , 934 (1959).[5] H. Bondi, M.G.J. van der Burg and A.W.K. Metzner,
Proc. R. Soc.London Ser. A , 21 (1962).[6] Y.K. Ha,
Int. J. Mod. Phys. D , 2219 (2005).[7] J.W. Maluf, J. Math. Phys. , 4242 (1995).[8] Y.K. Ha, Gen. Rel. Gra. , 2045 (2003).[9] E.A. Martinez, Phys. Rev. D , 4920 (1994).[10] J.W. Maluf, E.F. Martins and A. Kneip, J. Math. Phys. , 6302 (1996).[11] D. Christodoulou and R. Ruffini, Phys. Rev. D4