Charged black holes in Einsteinian cubic gravity and non-uniqueness

We analyze charged and static black holes in four-dimensional Einsteinian cubic gravity minimally coupled to a Maxwell field, uncovering some surprising features: (1) When the cubic coupling parameter lies above a certain strictly positive value, the theory contains black holes with charge greater than mass (in Einstein-Maxwell theory such solutions would lack an event horizon, thus representing naked singularities); (2) Under such conditions, there exists a finite interval in this over-extremal regime in which there are two asymptotically flat black hole solutions with the same mass and electric charge, thus entailing (discrete) non-uniqueness; (3) The charged black holes do not possess an inner horizon, in contrast with the usual Reissner-Nordstr\"om spacetime. Examination of these black hole's thermodynamics reveals that when two branches coexist only the larger black hole is thermodynamically stable, while the smaller branch has negative specific heat. In addition to black holes, there exists a 1-parameter family of naked singularity spacetimes sharing the same mass and charge as the former, but not continuously connected with them. These naked singularities exist in the under-extremal regime, being present even in pure (uncharged) Einsteinian cubic gravity.

Introduction.-A hallmark of the Einstein-Maxwell theory that combines General Relativity (GR) with Electrodynamics is the validity of the celebrated black hole uniqueness theorem [1][2][3]. It applies to stationary, asymptotically flat, four-dimensional (4D) electrovacuum spacetimes and asserts that all such black hole (BH) spacetimes are determined uniquely by their mass, angular momentum and electric charge. Of course, the theorem can be evaded either by coupling gravity with different matter or by considering higher dimensions. Concerning the first option, the Bartnik-McKinnon selfgravitating soliton in Einstein-Yang-Mills theory shows that it is sufficient to promote the Maxwell field to a non-Abelian gauge field [4]. Considering higher dimensions, the discovery by Emparan and Reall of black rings in 5D [5], when taken in conjunction with the existence of topologically spherical Myers-Perry black holes [6], vividly illustrated the inappropriateness of a straightforward application of the theorem.
A third possibility to evade the uniqueness theorem is to modify the gravitational sector of the theory. This is the main purpose of the present paper. We explicitly demonstrate black hole non-uniqueness in the context of the recently formulated Einsteinian cubic gravity (ECG) [7] -a higher derivative gravity theory which falls within the broader class of generalized quasi-topological gravities [8]-when coupled to a Maxwell field.
ECG incorporates several appealing features [7][8][9]. In 4D, ECG is the most general diffeomorphism-invariant metric theory of gravity up to cubic order in curvature, whose linearized spectrum on maximally symmetric backgrounds coincides with that of GR, and for which static spherically symmetric vacuum solutions are gov-erned by a single equation of motion. Moreover, the theory admits a well-defined limit to GR, thus providing a phenomenologically interesting extension [10,11].
Asymptotically flat black hole solutions of 4D ECG were examined in [12,13]. The solutions are described by a single metric function satisfying a non-linear secondorder differential equation that has to be solved numerically, but all thermodynamic quantities can be computed analytically since they follow from a local analysis around the horizon. However, some startling aspects went unnoticed, especially regarding non-uniqueness of charged black holes, as well as the existence of positive-energy horizonless solutions. We aim to fill this gap here.
A uniqueness theorem in higher derivative gravity was previously obtained in [14] but it applies only to a restricted class of f (R) theories, for which the Lagrangian is a polynomial in the Ricci scalar, leaving ECG (and any metric theory whose Lagrangian contains contractions of multiple Riemann tensors) outside its scope.
Accordingly, the non-uniqueness we uncover is not specific to ECG: it is known to occur in quadratic gravity, even for neutral static BHs [15][16][17]. In that case a second branch of BHs exists, in addition to the usual Schwarzschild solution, but they show unreasonable pathological behavior. Furthermore, they are necessarily small in Planck units and thus feature large curvatures near the horizon (indicating even higher derivative corrections should be included). In contrast, we will show that ECG-Maxwell theory with coupling constant above a certain mass-dependent bound contains two competing branches of BHs with the same conserved charges (both of which are regular on and outside the event horizon) -as long as the electric charge is greater than the mass. In other words, BH non-uniqueness occurs precisely in what would be called the over-extremal regime in Einstein-Maxwell theory.
Finally, by studying horizonless (but singular) solutions, we find continuously non-unique families of positive-energy naked singularities sharing the same global conserved charges.
Einsteinian cubic gravity coupled to a Maxwell field.-In 4D, ECG-Maxwell theory is determined by the action S = d 4 x √ −g L, where the Lagrangian is given by [18] Here, G is the Newton gravitational constant, R represents the Ricci scalar, and the cubic-in-curvature correction to the Einstein-Hilbert action is incorporated in the scalar quantity The coupling constant λ is chosen to be nonnegative, otherwise the existence of asymptotically flat Schwarzschild-like solutions is precluded [10]. Einstein-Maxwell theory is recovered for λ = 0. The matter sector is comprised only of an Abelian gauge field A a , whose field strength is From the action above one derives the Einstein field equations, where E abcd ≡ ∂L/∂R abcd . These are complemented by the standard Maxwell equations, ∇ a F ab = 0 , obtained by varying (1) with respect to the gauge field. Charged black holes.-Electrically charged spherically symmetric BHs of ECG were studied in [13], though not thoroughly. When λ = 0 one naturally recovers Reissner-Nordström (RN), but at finite (positive) λ there can be strikingly marked differences, namely the absence of a Cauchy horizon in the interior of the BH, the appearance of event horizons in parameter ranges that would naively correspond to over-extremal regimes, and the coexistence of two BHs with the same conserved charges under such circumstances, as we now show.
We take the line element to be of the form Generic static, spherically symmetric spacetimes need not obey g tt g rr = −1 necessarily, but Eqs. (3) admit solutions with this property [12,13], to which case we shall restrict our considerations. As for the Maxwell field, we take an electric ansatz, A a = A 0 (r)δ t a . Solving the Maxwell equation yields A 0 (r) = Q/ √ 4πr, where Q is the electric charge of the solution. Plugging this in the modified Einstein equations (3) results in a single equation to be satisfied by the blackening factor f (r), The parameter M appears as an integration constant and corresponds to the mass of the spacetime. This equation is not amenable to exact analytic treatment, so one either resorts to approximations [10] or to numerical integration. Here we highlight the most relevant aspects of this procedure. More details can be found in [10,13,19]. An analysis of the large-r asymptotic behavior of (5) reveals that, besides the perturbative corrections (in λ) to the RN solution f RN (r) = 1−2GM/r +GQ 2 /r 2 , there are also non-perturbative corrections [10,13]. While the former are rational functions of r which depend on the charge Q, the latter are either growing or decaying exponentials in r 5/2 / √ λ (actually modified Bessel functions), which take the same form as in the neutral case. Requiring the absence of the growing mode leaves a three-parameter family of possible asymptotically flat geometries. (Details in the Appendix.) On the other hand, roots of f (r) identify possible event horizons of the spacetime, which will be denoted by r h and happen to be singular points of (5). By Taylorexpanding around such a point (assuming f is regular there) f (r) = ∞ n=1 a n (r − r h ) n , and solving (5) order by order in powers of (r − r h ), the coefficients a n can be determined. The two lowest-order equations form an algebraic system that is used to fix r h and the surface gravity k g ≡ f (r h )/2 = a 1 /2 in terms of λ and Q [20], Here we have scaled out the mass M by using dimensionless quantities, defined according to For | Q| < 1, system (6) has a unique real solution with k g > 0, which is to be identified with the black hole's event horizon. But for λ > λ b ≡ 1/768 there exists a finite interval, 1 ≤ | Q| < Q max , where there are two real solutions with k g > 0, suggesting that charged BHs in ECG need not comply with the extremality bound Q ≤ √ GM of Einstein-Maxwell theory. For | Q| > Q max solutions with k g > 0 cease to exist. All this is illustrated in Fig. 1. Clearly, Q max is a function of λ, which can be Horizon radius r h as a function of the electric charge Q (both in units of the mass M ) for various choices of λ. The lowest red curve corresponds to GR (λ = 0), in which case the black dot at the end of the curve indicates the extremal solution, for which r h = GM = √ GQ. Observe that for λ > λ b there are always choices of Q > √ GM with two possible solutions for r h . The green diamonds indicate the horizon radius of two coexisting black holes with the same M and Q, whose profiles are shown explicitly in Fig. 2. easily determined by finding the root of dQ/dr h . We emphasize that the bound λ > λ b is exact: it follows from a perturbative study around the extremal point, which is in excellent agreement with the numerics. (See the Appendix.) The bound λ > λ b is mass-dependent, so assuming quantum gravity sets in at the Planck scale it translates into a 'uniqueness' bound, λ ≤ G 2 M 4 P l /48. To confirm that both local solutions for r h are actually associated with asymptotically flat spacetimes we must integrate (5) out to large radii and match the corresponding asymptotic behavior. This is done numerically by determining the coefficients a n up to some order and then using the truncated series (which has finite radius of convergence) to obtain initial conditions for the integration, starting slightly away from the singular point of the equation f (r h ) = 0. In practice, truncating at n = 10 is good enough, and an initial integration point r i displaced by 1% relative to r h falls well within the radius of convergence. The expansion has a single free parameter, a 2 , in terms of which all other coefficients are fixed. (See Appendix.) The generic expressions obtained for the expansion coefficients are only valid for non-extremal horizons, k g = 0. Indeed, it turns out that the series expansion for the extremal case, k g = 0, does not have any free parameter: a 2 is also fixed in that particular case. Focusing on the non-extremal cases, it is always possible to fine-tune a 2 at the horizon to obtain an asymptotically flat solution that approaches the RN profile asymptotically, even for the cases in which there are two possible values for r h , as shown in Fig. 2, corresponding to a larger and a smaller BH. The same behavior of f (r) is reproduced for other values of λ and appears to be generic. This entails discrete (two-fold) non-uniqueness of BHs in ECG-Maxwell theory. Note the absence of an inner (or Cauchy) horizon in both black hole spacetimes, so the causal structure of the interior of these BHs is strikingly different from that of the (subextremal) RN solutions, and instead is qualitatively similar to that of the Schwarzschild solution. This behavior appears to be generic for charged BHs in ECG, implying that they do not suffer from any form of mass inflation problem. In the extremal case, there is no free parameter to adjust and generically a regular asymptotically flat solution cannot be obtained.
Similarly to what happens in quadratic gravity [15], the first law of BH thermodynamics is satisfied by the two coexisting solutions [12,13]. We now check that the larger (smaller) black hole branch has positive (negative) specific heat. Therefore, in the regime of parameters where there is non-uniqueness only the larger black holes are thermodynamically stable. By using Wald's formula [21,22] it is possible to evaluate the entropy of these black hole solutions [13]. The expressions for the temperature T = k g /2π and the mass M = r 0 /(2G) can instead be obtained from system (6). These are given by whereq ≡ GQ 2 and ≡ 48G 2 λ r 2 h −q + r 6 h 1/6 . Useful information regarding the coexisting BHs can be extracted from the behavior of the specific heat at constant charge C = T (∂S/∂T ) Q shown in Fig. 3. It can be seen that for √ GM < Q the two competing BH branches have specific heats with opposite signs. Working in the canon- ical ensemble by keeping the charge fixed, it is of interest to evaluate the free energy F = M − T S. It turns out that the thermodynamically stable larger BHs are also the ones with lowest free energy [23].
Naked singularities.-In addition to the BHs discussed above, the theory admits horizonless solutions with positive mass but which are nevertheless singular, so they represent naked singularities. As for BHs, they are determined by a single metric function f (r), the only distinction being that for naked singularities this blackening factor has no roots. To demonstrate their existence we focus on the neutral case (i.e., vacuum ECG) but similar results are obtained for the charged case. Uncharged BH solutions can be obtained numerically following the same strategy. Consider now integrating, not from the horizon, but from the origin. The equation (5) also has a singular point at r = 0. Assuming f is analytic there, it admits an expansion in Taylor series f (r) = ∞ n=0 c n r n , and the c n can be determined by solving (5) order by order in powers of r. The result is that two of them, namely c 0 ≡ f (0) and c 2 ≡ 1 2 f (0), are free parameters, in terms of which all others can be expressed. In particular f (0) = 0, so that near the origin Therefore, there is one additional free parameter compared to the expansion around a horizon: the mass M and the value of f at the origin, c 0 < 1, (the parameter c 2 has to be fine-tuned to get asymptotically flat solutions). The existence of such horizonless solutions (shown in Fig. 4) does not seem to impose an upper bound on λ, but the range of c 0 -values that yield asymptotically flat solutions shrinks as λ decreases, and we were not able to find global solutions for G 2 λ < ∼ 0.62(GM ) 4 [24]. As the Kretschmann scalar near the origin behaves as ∼ 4(c 0 −1) 2 r −4 , the strong curvature region can be made arbitrarily small by fine-tuning c 0 close to 1. However, it cannot be made fully regular (as was done in [25] for quadratic gravity sourced with incompressible matter) without sending M to zero and retrieving flat spacetime.
Discussion.-We have demonstrated that static charged BHs in 4D Einsteinian cubic gravity differ notoriously from their counterparts in GR in several aspects: non-uniqueness of regular solutions, non-compliance with the extremality bound, and the absence of an inner horizon. Given that RN black holes in GR display many properties analogous to rotating neutral BHs, it is tempting to speculate that similar features might be present even for vacuum ECG BHs. Little is known about rotating solutions in ECG. The only results obtained so far refer to the small λ regime [26] or near-horizon geometries of extremal BHs [27], where similar discrete degeneracy of near-horizon solutions was found. Interestingly, the existence of 'small' over-extremal solutions has been related to the weak gravity conjecture [28], providing a decay channel for extremal RN BHs.
The horizonless solutions, despite being singular spacetimes, present a more drastic continuous type of nonuniqueness. Moreover, since the potential r −2 f (r) does not feature extrema for these solutions, such naked singularities do not have a photon sphere, so gravitational lensing signatures are markedly different from those of BHs [10,11]. An interesting question is whether a naked singularity can be formed under a dynamical process starting from some regular initial state, thus addressing the weak cosmic censorship conjecture [29,30] in the context of ECG.
We have focused on cubic gravity. It remains to be seen if our findings extend to higher derivative gravities, beyond cubic order [19]. The effect of adding a cosmological constant or the natural inclusion of cubic terms also in the Maxwell field strength require separate studies.
Acknowledgments.-We thank Roberto Emparan for many useful discussions and João L. Costa, Roberto Emparan and Robert Mann for valuable comments on a draft of the manuscript. We acknowledge financial support from the European Union's Horizon 2020 research and innovation programme under ERC Advanced Grant GravBHs-692951. Funding for this work was partially provided by the Spanish MINECO under projects FPA-2016-76005-C2-2-P and MDM-2014-0369 of ICCUB (Unidad de Excelencia 'María de Maeztu').

Perturbative analysis around the extremal point
The numerical results presented in the main textnamely the existence of two BH solutions in the overextremal regime Q > 1-are confirmed by a perturbative study of Eqs. (6) around the extremal point ( r h , Q, k g ) = ( 1 2 , 1, 0), by using the surface gravity as the small perturbation parameter. The outcome of this analysis is the following, up to cubic corrections in k g : This shows that charged BHs in ECG can make an excursion to the overextremal regime only when λ > λ b is satisfied. Of course, one can extend the perturbative analysis to arbitrarily high orders in k g . Including terms up to tenth order is enough to obtain excellent approximations to the numerical results near the extremal point, as illustrated in Fig. 5.

Negative coupling constant
The algebraic system (6) admits solutions for the horizon radius (with positive temperature) also when λ < 0. In fact, there can be up to three distinct solutions for r h , as shown in Fig. 6. However, these local horizons cannot incorporate an asymptotically flat spacetime, one that is regular on and outside the horizon. The reason is simple: according to Eq. (12) below, at large r the blackening factor would become oscillatory instead of approaching the RN behavior. This argument strictly applies only in the small (negative) coupling regime and is in agreement with numerical explorations. It does not seem to be possible to fine-tune the horizon free parameter in such a way that the integration proceeds to arbitrarily large radius when λ < 0: inevitably a singular point is met, where f diverges irrespective of the magnitude of λ.
Hence, an extremal horizon has no free parameter to adjust once the mass and the coupling are fixed. A regular, asymptotically flat solution is not expected to exist in this case.