We discuss universal properties of axisymmetric and stationary configurations
consisting of a central black hole and surrounding matter in Einstein-Maxwell
theory. In particular, we find that certain physical equations and inequalities
(involving angular momentum, electric charge and horizon area) are not
restricted to the Kerr-Newman solution but can be generalized to the situation
where the black hole is distorted by an arbitrary axisymmetric and stationary
surrounding matter distribution.Comment: 7 page

Carla Cederbaum

D. Christodoulou

G. Neugebauer

I. Booth

J. Hennig

Jörg Hennig

M. Ansorg

Marcus Ansorg

P.T. Chruściel

R. Penrose

S. Abdolrahimi

English

arXiv

We discuss universal properties of axisymmetric and stationary configurations consisting of a central black hole and surrounding matter in Einstein-Maxwell theory. In particular, we find that certain physical equations and inequalities (involving angular momentum, electric charge and horizon area) are not restricted to the Kerr-Newman solution but can be generalized to the situation where the black hole is distorted by an arbitrary axisymmetric and stationary surrounding matter distribution

Ansorg, M.

Hennig, J.

Cederbaum, C.

MPG.PuRe

Gen Relativ Gravit (2011) 43:1205–1210DOI 10.1007/s10714-010-1136-8REVIEW ARTICLEUniversal properties of distorted Kerr–Newmanblack holesMarcus Ansorg · Jörg Hennig · Carla CederbaumReceived: 30 July 2010 / Accepted: 8 December 2010 / Published online: 23 December 2010© Springer Science+Business Media, LLC 2010Abstract We discuss universal properties of axisymmetric and stationary configu-rations consisting of a central black hole and surrounding matter in Einstein–Maxwelltheory. In particular, we find that certain physical equations and inequalities (involv-ing angular momentum, electric charge and horizon area) are not restricted to theKerr–Newman solution but can be generalized to the situation where the black hole isdistorted by an arbitrary axisymmetric and stationary surrounding matter distribution.Keywords Black holes · Einstein–Maxwell spacetimes1 IntroductionOne of the most important exact solutions in General Relativity is the Kerr metricdescribing the spacetime around axisymmetric and stationary black holes in vacuum.This solution is particularly interesting because it is generally expected that it will beapproached asymptotically by any collapsing matter distribution—provided that allthe matter finally falls into the black hole. It is, however, conceivable that for a longtime period a substantial portion of the matter distribution moves around the blackhole in a quasi-stationary state. Such a configuration can be approximated well by anM. AnsorgTheoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,Max-Wien-Platz 1, 07743 Jena, Germanye-mail: Marcus.Ansorg@uni-jena.deJ. Hennig (B) · C. CederbaumMax Planck Institute for Gravitational Physics, Am Mühlenberg 1, 14476 Golm, Germanye-mail: Joerg.Hennig@aei.mpg.deC. Cederbaume-mail: Carla.Cederbaum@aei.mpg.de1231206 M. Ansorg et al.equilibrium system consisting of a central black hole and surrounding matter, e.g., afluid ring or disk (describing a black hole with an accretion disk or a galaxy with acentral black hole).Little is known about how such surrounding matter influences the central blackhole. In this essay we discuss several universal properties that black holes possessindependent of the nature of their environment. Our investigations apply to the moregeneral case in which additional electromagnetic fields are considered, i.e. we study“distorted Kerr–Newman black holes” within Einstein–Maxwell theory.We point out that for stellar black holes it is not expected that surrounding mat-ter contributes more then a few percent to the total mass of the system. Hence, forsuch configurations the influence of the matter on the black hole should be small oreven negligible. On the other hand, for a galactic black hole the surrounding matter(the galaxy) contributes the main part of the mass and dominates the behaviour ofthe spacetime in larger distances. However, in a vicinity of a galactic black hole, it isexpected that the deviation of the spacetime geometry from a Kerr metric is again verysmall. But even though astrophysically relevant black holes might typically be closeto the Kerr solution, the following universal properties are of fundamental theoreti-cal interest since they provide exact relations (equations and inequalities) which arecompletely independent of the amount and type of matter surrounding a black hole.2 A universal inequality for sub-extremal black holesIt is well-known that the three-parameter Kerr–Newman family of solutions describessingle rotating, electrically charged, axisymmetric and stationary black holes in elec-trovacuum. For the parameters one can choose, e.g., the mass M, the angular momen-tum J, and the electric charge Q of the black hole.1 For a solitary black hole, which ischaracterized by the existence of an event horizon, the relation(JM)2+ Q2 ≤ M2 (1)must be satisfied. Equivalently, expressing M through J, Q, and the area A of the eventhorizon, we find that the three parameters J, Q, and A have to obey(8π J )2 + (4π Q2)2 ≤ A2. (2)Now, a Kerr–Newman black hole with equality in (1) [or (2)] is called extremal,otherwise sub-extremal.A different definition of sub-extremality of black holes refers to the existence oftrapped surfaces in every sufficiently small interior neighborhood of their event hori-zons [5]. In the Kerr–Newman family, both notions of sub-extremality turn out to beequivalent. However, the concept of trapped surfaces is more general and applies to1 In the special case of the Kerr–Newman metric, these quantities can be read off from the asymptoticbehavior of the metric coefficients.123Universal properties of distorted Kerr–Newman black holes 1207arbitrary axisymmetric and stationary black holes. Hence it is appropriate to use thecorresponding definition of sub-extremality, and we will do so in the following.In [8,10], we have investigated such sub-extremal stationary and axisymmetricblack holes surrounded by matter in full Einstein–Maxwell theory and found thatinequality (2) strictly holds:(8π J )2 + (4π Q2)2 < A2. (3)Through this result we generalize the link between the two notions of sub-extremalityto the realm of distorted black holes, i.e. we establish a universal relation between thegeometric concept of existence of trapped surfaces and the inequality to be obeyed bythe physical black hole parameters J, Q, and A.2For the proof of (3) for sub-extremal distorted black holes it suffices to study theEinstein-Maxwell equations in an electrovacuum vicinity of the event horizon. Usingthese equations, it is possible to reformulate (3) in terms of a variational problem: Anappropriate functional depending on the horizon values of three metric and electro-magnetic potentials must always be greater than or equal to 1. As shown in [8,10], thisvariational problem can be analyzed with methods from the calculus of variations.A fascinating application of (3) is the solution of an old problem in General Rela-tivity, the balance problem for two black holes: Is it possible that two aligned axisym-metric and stationary black holes are in equilibrium, i.e. can the spin-spin repulsionof two rotating black holes compensate their gravitational attraction? The applicationof a particular soliton method (the inverse scattering method),3 provides a proof ofthe non-existence of such equilibrium situations, showing that at least one of the twoblack holes in this ensemble would violate inequality (3), cf. [14].3 The interior of the black holeHaving discussed universal properties on a black hole’s event horizon, we will nowdirect our attention to its interior. Again, we were able to generalize a well-knownfeature of Kerr–Newman solutions, namely the existence of a second horizon insidethe black hole—the inner Cauchy horizon—which is defined as the future boundaryof the domain of dependence of the event horizon. Since predictability breaks downbeyond the Cauchy horizon, its stability is an important issue. Based on an argumentby Penrose [15], it was assumed that generic perturbations grow infinitely near theCauchy horizon. However, for purely axisymmetric and stationary (arbitrarily large)perturbations, i.e. for our situation of a central black hole with surrounding matter, a2 Note that these quantities can be determined locally on the event horizon. We further remark that thedefinition of J (in contrast to that of Q and A) involves a choice: The total angular momentum of the space-time is composed of matter, electromagnetic field, and black hole contributions. While clearly the matterpart should be excluded from the definition of J, both a Komar integral and an appropriate electromagneticevent horizon integral is taken into account for our notion of the black hole’s local angular momentum(cf. [2] for a more thorough discussion).3 Soliton methods are based on the existence of a linear matrix problem which is equivalent to the nonlin-ear field equations via its integrability condition [12]. (See [13] for a sophisticated introduction to solitonmethods in General Relativity.)1231208 M. Ansorg et al.regular inner Cauchy horizon can be shown to exist provided the angular momentum Jand the charge Q of the black hole do not vanish simultaneously. Moreover, the regionbetween event horizon and Cauchy horizon is then completely regular.4 In contrast,the Cauchy horizon becomes singular and approaches a scalar curvature singularity inthe limit J → 0, Q → 0 (precisely as in the transition from a Kerr–Newmann blackhole to a Schwarzschild black hole).Remarkably, if the inner Cauchy horizon exists (i.e. if J and Q do not vanish simul-taneously), then the area ACH of the Cauchy horizon and the event horizon area A arerelated via(8π J )2 + (4π Q2)2 = ACH A. (4)Note that the left hand side of (4) is the same as in (3).The proof of these statements [3,4,9] utilizes again soliton methods. From an anal-ysis of the linear matrix problem (see footnote 3) on a closed path (along the eventhorizon, the Cauchy horizon and the two parts of the symmetry axis, which connectboth horizons), it is possible to deduce the existence of a Cauchy horizon. Moreover,explicit formulas for the metric and electromagnetic potentials on the Cauchy horizonemerge, and these are used to show (4). The second essential ingredient of the proofis a theorem by Chrus´ciel [7].Finally, we note that the interior spacetime region of axisymmetric and stationaryblack holes is closely related to Gowdy spacetimes (cosmological models with twospacelike Killing fields), making the above statements applicable in this cosmologicalcontext as well. They also imply the analogous existence and regularity statements forCauchy horizons [11].4 Configurations with degenerate black holesIn the degenerate limit of converging Cauchy and event horizons (expressed by van-ishing surface gravity κ), we obtain equality in Eq. (2), cf. [2]:(8π J )2 + (4π Q2)2 = A2. (5)Consistently, Eq. (5) also follows from Eq. (4) as the areas A and ACH coincide in thedegenerate limit.As a consequence of (5), the characterizing relation(JM)2+ Q2 = M2CR (6)of the extremal Kerr–Newman solution continues to hold in presence of surround-ing matter—in accordance with the fact that Kerr–Newman black holes aredegenerate precisely if they are extremal. (In this general situation, MCR denotes the4 A particular exact solution for the interior black hole region is discussed in [1].123Universal properties of distorted Kerr–Newman black holes 1209Christodoulou-Ruffini mass [6] of the black hole which agrees with the ADM massin the Kerr–Newman setting.)5 DiscussionWe have demonstrated that the following statements about the Kerr–Newman blackhole are more generally valid for any axisymmetric and stationary black hole withsurrounding matter in Einstein–Maxwell theory:• Physically relevant black holes are expected to be sub-extremal, where sub-extremality is defined by the existence of trapped surfaces in every sufficientlysmall interior neighborhood of the event horizon. Angular momentum J and elec-tric charge Q of a sub-extremal black hole are bounded in terms of its horizon areaA: (8π J )2 + (4π Q2)2 < A2.• The black hole possesses a second horizon—an inner Cauchy horizon—if and onlyif J and Q do not vanish simultaneously. The Cauchy horizon is regular wheneverit exists.• The spacetime region between event horizon and Cauchy horizon is completelyregular.• The horizon areas satisfy the universal relation (8π J )2 + (4π Q2)2 = ACH A.• The relations (8π J )2 + (4π Q2)2 = A2 and ( JM )2 + Q2 = M2CR hold for degen-erate black holes (MCR: Christodoulou–Ruffini mass). In that sense, degenerateblack holes could be named extremal.It would be interesting to identify further universal properties of the black hole con-figurations in question, e.g. black holes in Yang-Mills theory or in higher dimensions.Acknowledgments We would like to thank David Petroff for commenting on the manuscript. This workwas supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research CentreSFB/TR7 “Gravitational Wave Astronomy” and by the International Max Planck Research School (IMPRS)“Geometric Analysis and Gravitation”.References1. Abdolrahimi, S., Frolov, V.P., Shoom, A.A.: Interior of a charged distorted black hole. Phys. Rev. D80, 024011 (2009)2. Ansorg, M., Pfister, H.: A universal constraint between charge and rotation rate for degenerate blackholes surrounded by matter. Class. Quantum Grav. 25, 035009 (2008)3. Ansorg, M., Hennig, J.: The inner Cauchy horizon of axisymmetric and stationary black holes withsurrounding matter. Class. Quantum Grav. 25, 222001 (2008)4. Ansorg, M., Hennig, J.: The inner Cauchy horizon of axisymmetric and stationary black holes withsurrounding matter in Einstein–Maxwell theory. Phys. Rev. Lett. 102, 221102 (2009)5. Booth, I., Fairhurst, S.: Extremality conditions for isolated and dynamical horizons. Phys. Rev. D77, 084005 (2008)6. Christodoulou, D., Ruffini, R.: Reversible transformations of a charged black hole. Phys. Rev. D4, 3552 (1971)7. Chrus´ciel, P.T.: On space-times with U (1) × U (1) symmetric compact Cauchy surfaces. Ann. Phys.202, 100 (1990)1231210 M. Ansorg et al.8. Hennig, J., Ansorg, M., Cederbaum, C.: A universal inequality between the angular momentum andhorizon area for axisymmetric and stationary black holes with surrounding matter. Class. QuantumGrav. 25, 162002 (2008)9. Hennig, J., Ansorg, M.: The inner Cauchy horizon of axisymmetric and stationary black holes withsurrounding matter in Einstein–Maxwell theory: study in terms of soliton methods. Ann. HenriPoincaré 10, 1075 (2009)10. Hennig, J., Cederbaum, C., Ansorg, M.: A universal inequality for axisymmetric and stationary blackholes with surrounding matter in the Einstein–Maxwell theory. Commun. Math. Phys. 293, 449 (2010)11. Hennig, J., Ansorg, M.: Regularity of Cauchy horizons in S2 × S1 Gowdy spacetimes. Class. QuantumGrav. 27, 065010 (2010)12. Neugebauer, G., Kramer, D.: Einstein–Maxwell solitons. J. Phys. A: Math. Gen. 16, 1927 (1983)13. Neugebauer, G.: Gravitostatics and rotating bodies. In: Hall, G.S., Pulham, J.R. (eds.) Proceedings 46thScottish Universities Summer School in Physics (Aberdeen). Institute of Physics Publishing, London(1996)14. Neugebauer, G., Hennig, J.: Non-existence of stationary two-black-hole configurations. Gen. Relativ.Gravit. 41, 2113 (2009)15. Penrose, R.: Structure of space-time. In: de Witt, C.M., Wheeler, J.A. (eds.) Battelle Rencontres,W.A. Benjamin, New York (1968)123

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