Charged black hole and radiating solutions in entangled relativity
aa r X i v : . [ g r- q c ] F e b Charged black hole and radiating solutions in entangled relativity
Olivier Minazzoli ∗ Artemis, Universit´e Cˆote d’Azur, CNRS, Observatoire Cˆote d’Azur, BP4229, 06304, Nice Cedex 4, France
Edison Santos † PPGCosmo, CCE, Universidade Federal do Esp´ırito Santo, Vit´oria, ES, CEP29075-910, Brazil
In this manuscript, we show that the external Schwarzschild metric can be a good approximationof exact black hole solutions of entangled relativity. Since entangled relativity cannot be definedfrom vacuum, the demonstrations need to rely on the definition of matter fields. The electromagneticfield being the easiest (and perhaps the only) existing matter field with infinite range to consider, westudy the case of a charged black hole—for which the solution of entangled relativity and a dilatontheory agree—as well as the case of a pure radiation—for which the solution of entangled relativityand general relativity seem to agree, despite an apparent ambiguity in the field equations. Basedon these results, we argue that the external Schwarzschild metric is an appropriate mathematicalidealization of a spherical black hole in entangled relativity. The extension to rotating cases is brieflydiscussed.
I. INTRODUCTION
Unlike in general relativity, the action of entangled rel-ativity cannot be defined without matter fields [1–4]—therefore satisfying the Einstein 1918 definition of Mach’sprinciple [5] . Nevertheless, black hole solutions in vac-uum of general relativity—such as the Schwarzschild andthe Kerr metrics—play an important role in explainingmany different phenomena, from the observations of theEvent Horizon Telescope [6, 7] to the detection of gravi-tational waves [8, 9]. Therefore, it is important to checkwhether or not the usual vacuum solutions of generalrelativity—which are good mathematical idealization ofastrophysical black holes—are also good approximationsof black holes in entangled relativity.While vacuum solutions should not exist in entangledrelativity, nothing prevents the density of matter fieldoutside the event horizon to be arbitrarily small—notablyrecovering some usual astrophysical conditions. In whatfollows, we shall name such a condition a near vacuumsituation .In the present manuscript, we shall present an exactspherical solution of entangled relativity that can be ap-proximated by the Schwarzschild metric in a near vacuumsituation. We shall then argue that this result could ac-tually come from a general property that makes that vac-uum solutions of general relativity good approximationsof near vacuum solutions of entangled relativity. It istherefore argued that astrophysical black holes of entan-gled relativity are likely indistinguishable from the onesof general relativity in many cases. ∗ [email protected] † edison [email protected] A translation in English of the origi-nal paper in German is available online athttps://einsteinpapers.press.princeton.edu/vol7-trans/49.
II. ACTION AND FIELD EQUATION
The action of entangled relativity is given by [2] S = − ξ c Z d x √− g L m R , (1)where the constant ξ has the dimension of the usual cou-pling constant of general relativity κ ≡ πG/c —where G is the Newtonian constant and c the speed of light. R is the usual Ricci scalar constructed upon the space-timemetric g αβ , with determinant g ; while L m is a scalar La-grangian representing the matter fields. ξ defines a novelfundamental scale that is relevant at the quantum levelonly, and therefore is notably not related to the size ofblack holes; whereas the Planck scale—defined from κ and which is related to the size of black holes—no longeris fundamental, nor constant, in entangled relativity [2]. The impossible existence of space-time without mat-ter, and vice versa, is obvious from the action. It comesfrom the fact that one has replaced the usual additivecoupling between matter and geometry by a pure multi-plicative coupling. The metric field equation reads R µν − g µν R = − R L m T µν + R L m ( ∇ µ ∇ ν − g µν (cid:3) ) L m R , (2)with T µν ≡ − √− g δ ( √− g L m ) δg µν . (3) As also discussed in [4, 10], this might be a way out of the on-tological paradox of conventional quantum gravity that, in Free-man Dyson’s words [11], “nature conspires to forbid any measure-ment [through the creation of black holes] of distance with errorsmaller than the Planck length”, because the effective Planckscale [2]—which fixes the size of a black hole’s event horizon fora given mass—depends on the field equations in entangled rela-tivity.
Note that the trace of Eq. (2) reads3 R L m (cid:3) L m R = − R L m ( T − L m ) . (4)Also note that the stress-energy tensor is no longer con-served in general, as one has ∇ σ (cid:18) L m R T ασ (cid:19) = L m ∇ α (cid:18) L m R (cid:19) . (5)But entangled relativity is more easily understood inits dilaton equivalent form that reads [1, 2] S = 1 c ξ ˜ κ Z d x √− g (cid:20) φR κ + p φ L m (cid:21) , (6)where ˜ κ is a positive effective coupling constant betweenmatter and geometry, with the dimension of κ . ˜ κ takesits value from the asymptotic behavior of the effectivescalar degree of freedom in Eq. (1) [1, 2], as well as theconsidered normalisation of φ . ˜ κ/ √ φ , which defines aneffective Planck scale, notably fixes the size of black holeswith a given mass. The equivalence between the two ac-tions is similar to the equivalence between f ( R ) theoriesand the corresponding specific scalar-tensor theories [12].From this alternative action, one can easily see why en-tangled relativity reduces to general relativity when thevariation of the scalar-field degree of freedom vanishes.The dilaton field equations read G αβ = ˜ κ T αβ √ φ + 1 φ [ ∇ α ∇ β − g αβ (cid:3) ] φ, (7) p φ = − ˜ κ L m /R, (8)where G αβ is the Einstein tensor and the conservationequation reads ∇ σ (cid:16)p φT ασ (cid:17) = L m ∇ α p φ, (9)The trace of the metric field equation can therefore berewritten as follows3 φ (cid:3) φ = ˜ κ √ φ ( T − L m ) . (10)The equivalence between Eqs. (7-10) and (2-5) is prettystraightforward to check. This simply means that, in-deed, the action (1) possesses an additional gravitationalscalar degree of freedom with respect to general relativ-ity. This dilaton theory is equivalent, at least at the classical level,as long as L m /R <
0. Notably, it seems that one must alwaysconsider cases such that ( R, L m ) = 0 when one uses the dilatonform of entangled relativity, although R and L m can be arbitrar-ily small in principle. We shall come back on this point in themanuscript. The good thing with this extra degree of freedom isthat it is not excited in all situations where L m ∼ T , thisleads to a phenomenology that closely resembles the oneof general relativity [2, 4, 13–15]; whereas it is expectedto differ from the one of general relativity in all othersituations—see e.g. [3] or [10].The electromagnetic field being the easiest (and per-haps the only) matter field with infinite range to con-sider, we will only study the case of the electromagneticLagrangian L m = − F / (2 µ ) in what follows, where µ is the magnetic permeability. III. CHARGED BLACK HOLE
In its scalar-tensor form (6), entangled relativity isjust a specific case of a dilaton theory, for which thesolution for charged black hole has been investigatedby many authors during the first superstring revolution[17, 18]. Indeed, defining the Einstein frame metric by˜ g αβ = e − ϕ/ √ g αβ , with φ = e − ϕ/ √ , the action in theEinstein frame reads S = 1 c ξκ Z d x p − ˜ g (cid:20) κ (cid:16) ˜ R − g αβ ∂ α ϕ∂ β ϕ (cid:17) − e − ϕ/ √ ˜ F µ , (11)where ˜ F = ˜ g ασ ˜ g βǫ ˜ F σǫ ˜ F αβ , where ˜ F αβ := F αβ . Onehas used the conformal invariance of the electromagneticaction. From now on, in order to follow the literature,we use natural units. This action corresponds exactly tothe one considered in [16, 19–21] with a = (2 √ − . Thespherical solution therefore reads [16, 19–21]d˜ s = − ˜ λ d t + ˜ λ − d r + ˜ ρ (cid:0) d θ + sin θ d ψ (cid:1) , (12)with ˜ λ = (cid:16) − r + r (cid:17) (cid:16) − r − r (cid:17) ( − a ) / ( a ) , (13)and ˜ ρ = r (cid:16) − r − r (cid:17) a / ( a ) , (14)whereas the field solutions read˜ F = − Qe aϕ ˜ ρ dt ∧ dr = − Qr dt ∧ dr, (15) In natural units, we consider L m = − F / L m = − F /
4, in order to follow the definition used in the literature[16]. In particular, it means that the electromagnetic stress-energy tensor reads T µν = 2 (cid:0) F µα F να − g µν F αβ F αβ (cid:1) in nat-ural units. for an electric charge, and e aϕ = (cid:16) − r − r (cid:17) a / ( a ) , (16)where we normalized the scalar-field such that its back-ground value ϕ corresponds ϕ = 0. r + is an eventhorizon, whereas r − is a curvature singularity for a = 0.They are related to the mass and charge, M and Q , by2 M = r + + (cid:18) − a a (cid:19) r − , (17)and Q = r − r + a . (18)Performing the inverse conformal transformation g αβ = e ϕ/ √ ˜ g αβ in order to have the solution of the action Eq.(6), one getsd s = − λ d t + λ − r d r + ρ (cid:0) d θ + sin θ d ϕ (cid:1) , (19)with λ = (cid:16) − r + r (cid:17) (cid:16) − r − r (cid:17) / , (20) λ r = (cid:16) − r + r (cid:17) (cid:16) − r − r (cid:17) / , (21) ρ = r (cid:16) − r − r (cid:17) / . (22)The scalar-field solution on the other hands reads φ = (cid:16) − r − r (cid:17) − / . (23)The solution (19-23) has been verified via Mathematica,the code is accessible on GitHub [22]. It is therefore thefirst known black hole solution of entangled relativity.In a near vacuum situation, one has Q →
0, such thatone gets r − →
0. Hence, the scalar-field tends to aconstant field and the metric in Eq. (19) tends to theSchwarzschild solution in a near vacuum situation.This represents the first example for which an exactsolution of entangled relativity is shown to be well ap-proximated in a near vacuum situation by the usualSchwarzschild solution in vacuum. This is an indica-tion that the outside metric of the Schwarzschild solutioncan be an accurate mathematical idealisation of a non-rotating astrophysical black hole in entangled relativity.
A. Discussion on the validity of the solutionbeyond the event horizon
While the solution (19-23) seems perfectly well be-haved within the event horizon at a mathematical level,we would like to argue that this region might not corre-spond to the solution after the collapse of an astrophys-ical object, therefore only the region outside the event horizon might be relevant at the physical level. The rea-son being that nothing guarantees that singularities oc-cur after the collapse of compact objects in entangledrelativity.Indeed, the effective coupling between matter and cur-vature in the metric field equation −L m /R is not nec-essarily positive everywhere since it notably depends onthe on-shell value of matter fields L m . As a consequence,gravity is potentially not attractive everywhere in en-tangled relativity. In particular, if one assumes that L m = K − V , where K and V are the kinetic and poten-tial energy densities, it seems plausible that L m flips itssign at high enough energy, when kinetic energy shoulddominate matter dynamics. It may not mean that theeffective coupling between matter and curvature in themetric field equation becomes negative. But if it does—such that gravity can indeed become repulsive at a givenhigh energy threshold—one can genuinely assume thatthe solution will not look like (19-23) within the eventhorizon. Unfortunately, investigating this issue seems torequire an accurate description of matter fields at (arbi-trarily) high energy; while it is believed that the stan-dard model of particles is not accurate at (arbitrarily)high energy. In any case, this difficult topic is left forfurther studies. IV. PURE ELECTROMAGNETIC RADIATION
The case of pure electromagnetic radiation is of interestbecause radiating solutions of general relativity seems tosatisfy entangled relativity as well, despite an apparentambiguity in the field equations.Indeed, from the trace of Einstein’s equation of generalrelativity, and from the conformal invariance of electro-magnetism, one deduces that any purely radiative solu-tion of general relativity must be such that R = T = 0.Also, even though one has T µν = 0, the electromagneticLagrangian L m = B − E vanishes on-shell in entan-gled relativity as well, since E = B for pure radiation.Therefore, assuming any purely radiative solution of gen-eral relativity, one has L m = R = T = 0.Nevertheless, φ = ˜ κ L m /R = φ , where φ is a con-stant, is consistent with all the field equations of entan-gled relativity, because in that case, they reduce exactlyto the one of the Einstein-Maxwell theory. Hence, anypurely radiative solution of general relativity seems likelyto be solution in entangled relativity as well. The divi-sion L m /R , however, is ambiguous, despite it being aconstant.At this stage, we do not conclude that purely radiativesolutions of general relativity are also solutions of entan-gled relativity, but that it seems that it might very wellbe. In any case, these solutions, such as Vaidya’s radiat-ing Schwarzschild solution [23, 24], may be used in orderto study the behavior of entangled relativity in the limit( L m , T, R ) →
0. Another possibility to study such casesmight be achieved by analysing a solution that is bothcharged and radiating, and then taking the limit when itscharge goes to zero. This issue is left for further studies.Nevertheless, note that even if it turns out that a pureradiative field cannot be solution in entangled relativity(1), this might not be a fundamental issue for the theory,as the quantum trace anomaly of self-interacting fields ina curved background should induce small, but non-null,values of the Ricci scalar, the trace of the stress-energytensor and the Lagrangian that appears in the field equa-tions [4, 25]. Quantum trace anomalies may thereforeimply that the theory is well behaved everywhere. Inves-tigation of this aspect is left for further studies.
V. DISCUSSION
Now, we argue that a near vacuum black hole solutionof entangled relativity can always be well approximatedby a vacuum solution of general relativity. Indeed, in anear vacuum situation—that is T µν ∼ This means that while black holes in entangled rela-tivity are not entirely the same as in general relativity,their differences might be insignificant in all the situa-tions that correspond to a scalar-field which equation ismostly source-less.Otherwise, it is known that black holes might growsome hair due to a variation (either temporal or spatial)of the background value of the scalar-field in scalar-tensortheories [26]. Let us note that, whether or not this maybe true in entangled relativity as well, the scalar-fieldis not expected to vary significantly neither temporallynor spatially. Indeed, with respect to the former, thescalar-field is attracted toward a constant in entangledrelativity during the expansion of the universe [4, 14, 15];whereas, because the scalar-field is also not sourced bypressure-less matter fields in the weak field regime [13],one does not expect a significant spatial variation of thescalar-field either. Both cases follow from the intrinsicdecoupling of the scalar-field at the level of the scalar-fieldequation for L m ∼ T [4, 13–15, 27].In particular, this argument seems to indicate that anastrophysical rotating black hole in entangled relativityshould be well approximated by the external Kerr metricof general relativity.Before concluding, we would like to stress again thatone should not take seriously the exact solutions pre-sented in this manuscript beyond the event horizon. In- deed, in order to describe any compact object inside theblack hole in this model, one must have a high energydescription of matter fields in order to tell what happensthere in entangled relativity. The reason being that grav-ity becomes repulsive in entangled relativity for matterfields that are such that L m /R > and one cannotexclude the possibility that this situation could happenafter a phase transition of matter fields at high energy.In particular, this may be a way to avoid black hole sin-gularities [28] without the absolute need of a quantumfield description of gravity [10]. VI. CONCLUSION
Black holes in entangled relativity are somewhat morecomplex to study than in general relativity, given thatvacuum does not seem to be allowed by the theory.Therefore one has to study solutions that involve matterfields, before contingently taking the limit toward vac-uum in order to have a more realistic representation ofastrophysical black holes—which are usually thought toevolve in a near vacuum environment. In this manuscript,using previous results developed in the framework ofstring theory, we presented an exact spherically chargedsolution of entangled relativity in Eqs. (19-23). As onewould expect, the solution tends to the Schwarzschild’ssolution in a near vacuum limit—that is, when the chargeof the black hole goes to zero.Additionally, we argued that any solution of pure ra-diation in general relativity, such as Vaidya’s solution,might also be solution of entangled relativity, althoughmore careful analyses are required to pin the argumenton a more firm mathematical ground.In any case, both Vaidya’s and the solutions in Eqs.(19-23) are well approximated by the external solutionof the Schwarzschild metric in a near vacuum situation,providing evidence that an astrophysical spherical blackhole in entangled relativity can be approximated by aSchwarzschild black hole.Otherwise, we have argued that this result is likelygeneric in near vacuum situations, such that an astro-physical rotating black hole in entangled relativity canalso likely be approximated by a Kerr black hole.
ACKNOWLEDGMENTS
O.M. acknowledges support from the
Fondation desfr`eres Louis et Max Principale . E. S. thanks the PhDfellowship conceded from the
Funda¸c˜ao de Amparo `aPesquisa e Inova¸c˜ao do Esp´ırito Santo (FAPES). Given the small value of the inferred cosmological constant ingeneral relativity from the apparent acceleration of the expansionof the universe, black hole solutions of general relativity with andwithout a cosmological constant are alike on scales well below the Hubble scale. Hence, we will not enter into such details. See the first term of the right hand side of Eq. (2). [1] Hendrik Ludwig, Olivier Minazzoli, and SalvatoreCapozziello, “Merging matter and geometry in the sameLagrangian,” Physics Letters B , 576–578 (2015),arXiv:1506.03278 [gr-qc].[2] Olivier Minazzoli, “Rethinking the link between mat-ter and geometry,” Phys. Rev. D , 124020 (2018),arXiv:1811.05845 [gr-qc].[3] Denis Arruga, Olivier Rousselle, and OlivierMinazzoli, “Compact objects in entangledrelativity,” Phys. Rev. D , 024034 (2021),arXiv:2011.14629 [gr-qc].[4] Olivier Minazzoli, “De Sitter space-times in EntangledRelativity,” arXiv e-prints , arXiv:2011.14633 (2020),arXiv:2011.14633 [gr-qc].[5] A. Einstein, “Prinzipielles zurallgemeinen Relativit¨atstheorie,”Annalen der Physik , 241–244 (1918).[6] Event Horizon Telescope Collaboration, “First M87Event Horizon Telescope Results. I. The Shadow ofthe Supermassive Black Hole,” ApJ , L1 (2019),arXiv:1906.11238 [astro-ph.GA].[7] Dimitrios Psaltis et al. (EHT Collaboration), “Grav-itational test beyond the first post-newtonian or-der with the shadow of the m87 black hole,”Phys. Rev. Lett. , 141104 (2020).[8] B. P. Abbott et al., LIGO Scientific Collabora-tion, and Virgo Collaboration, “Observation ofGravitational Waves from a Binary Black HoleMerger,” Phys. Rev. Lett. , 061102 (2016),arXiv:1602.03837 [gr-qc].[9] B. P. Abbott et al. (LIGO Scientific Collaboration andVirgo Collaboration), “Gwtc-1: A gravitational-wavetransient catalog of compact binary mergers observedby ligo and virgo during the first and second observingruns,” Phys. Rev. X , 031040 (2019).[10] Olivier Minazzoli, “Spacetime might not be doomed afterall,” in preparation.[11] Freeman Dyson, “Is a Graviton Detectable?”International Journal of Modern Physics A , 1330041 (2013).[12] Salvatore Capozziello and Mariafelicia DeLaurentis, “F(R) theories of gravitation,”Scholarpedia , 31422 (2015).[13] Olivier Minazzoli and Aur´elien Hees, “Intrinsic So-lar System decoupling of a scalar-tensor theory witha universal coupling between the scalar field and thematter Lagrangian,” Phys. Rev. D , 041504 (2013),arXiv:1308.2770 [gr-qc].[14] Olivier Minazzoli, “On the cosmic con-vergence mechanism of the massless dila-ton,” Physics Letters B , 119–121 (2014),arXiv:1312.4357 [gr-qc].[15] Olivier Minazzoli and Aur´elien Hees, “Late-time cos-mology of a scalar-tensor theory with a universal mul-tiplicative coupling between the scalar field and thematter Lagrangian,” Phys. Rev. D , 023017 (2014),arXiv:1404.4266 [gr-qc].[16] Christoph F. E. Holzhey and Frank Wilczek, “Black holes as elementary parti-cles,” Nuclear Physics B , 447–477 (1992),arXiv:hep-th/9202014 [hep-th].[17] G. W. Gibbons and Kei-Ichi Maeda, “Black holes andmembranes in higher-dimensional theories with dilatonfields,” Nuclear Physics B , 741–775 (1988).[18] David Garfinkle, Gary T. Horowitz, and AndrewStrominger, “Charged black holes in string theory,”Phys. Rev. D , 3140–3143 (1991).[19] James H. Horne and Gary T. Horowitz, “Rotating dila-ton black holes,” Phys. Rev. D , 1340–1346 (1992),arXiv:hep-th/9203083 [hep-th].[20] P. H. Cox, B. Harms, and Y. Leblanc, “Dila-ton black holes, naked singularities and strings.”EPL (Europhysics Letters) , 321–326 (1994),arXiv:hep-th/9207079 [hep-th].[21] Ju Ho Kim and Sei-Hoon Moon, “Electric charge ininteraction with magnetically charged black holes,”Journal of High Energy Physics , 088 (2007),arXiv:0707.4183 [gr-qc].[22] E. Santos, “Verifying entangled bh solution.nb,” https://github.com/Edison-Santos/Mathematica_github.git (2021).[23] P. C. Vaidya, “A Radiation-absorbing Cen-tre in a Non-statical Homogeneous Universe,”Nature , 565 (1950).[24] Jerry B. Griffiths and Jir´ı Podolsk´y, Exact Space-Timesin Einstein’s General Relativity (2009).[25] Ralf Sch¨utzhold, “Small Cosmologi-cal Constant from the QCD TraceAnomaly?” Phys. Rev. Lett. , 081302 (2002),arXiv:gr-qc/0204018 [gr-qc].[26] Emanuele Berti, Enrico Barausse, Vitor Cardoso,Leonardo Gualtieri, Paolo Pani, Ulrich Sperhake, Leo C.Stein, Norbert Wex, Kent Yagi, Tessa Baker, C. P.Burgess, Fl´avio S. Coelho, Daniela Doneva, AntonioDe Felice, Pedro G. Ferreira, Paulo C. C. Freire, JamesHealy, Carlos Herdeiro, Michael Horbatsch, BurkhardKleihaus, Antoine Klein, Kostas Kokkotas, Jutta Kunz,Pablo Laguna, Ryan N. Lang, Tjonnie G. F. Li, TysonLittenberg, Andrew Matas, Saeed Mirshekari, HirotadaOkawa, Eugen Radu, Richard O’Shaughnessy, Banga-lore S. Sathyaprakash, Chris Van Den Broeck, Hans A.Winther, Helvi Witek, Mir Emad Aghili, Justin Alsing,Brett Bolen, Luca Bombelli, Sarah Caudill, LiangChen, Juan Carlos Degollado, Ryuichi Fujita, CaixiaGao, Davide Gerosa, Saeed Kamali, Hector O. Silva,Jo˜ao G. Rosa, Laleh Sadeghian, Marco Sampaio, HajimeSotani, and Miguel Zilhao, “Testing general relativitywith present and future astrophysical observations,”Classical and Quantum Gravity , 243001 (2015),arXiv:1501.07274 [gr-qc].[27] Olivier Minazzoli and Aur´elien Hees, “Dilatons withintrinsic decouplings,” Phys. Rev. D , 064038 (2016),arXiv:1512.05232 [gr-qc].[28] Roger Penrose, “Gravitational Collapse and Space-TimeSingularities,” Phys. Rev. Lett.14