Eur. Phys. J. C (2019) 79:925 https://doi.org/10.1140/epjc/s10052-019-7423-y

Regular Article - Theoretical Physics

Curvature invariants and lower dimensional horizons

Daniele Gregoris1,2,a , Yen Chin Ong1,2,b , Bin Wang1,2,c 1 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou 225002, Jiangsu, People’s Republic of China 2 School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

Received: 16 July 2019 / Accepted: 24 October 2019 / Published online: 14 November 2019 © The Author(s) 2019

Abstract It is known that the of a black hole (for example through the keplerian motion of orbiting stars can often be identified from the zeroes of some [4]), they must be regarded as physical quantities and not invariants. The situation in lower dimensions has not been only as mathematical oddities entering the theory equations. thoroughly clarified. In this work we investigate both (2+1)- The existence of black holes in the actual Universe we live and (1 + 1)-dimensional black hole horizons of static, sta- in has been indirectly established through the study of the tionary and dynamical black holes, identified with the zeroes X-rays emitted by the matter falling into them before cross- of scalar polynomial and Cartan curvature invariants, with ing the horizon [5,6]. Much more recently, the direct detec- the purpose of discriminating the different roles played by tion of gravitational waves by the LIGO collaboration has the Weyl and Riemann curvature . The situations and brought to the first observational identification of black hole applicability of the methods are found to be quite different binary systems in the Universe [7]. However the prediction of from that in 4-dimensional spacetime. The suitable Cartan the energy spectra of the waves emitted by coalescing black invariants employed for detecting the horizon can be inter- holes is almost entirely based on numerical techniques [8– preted as a local extremum of the tidal force suggesting that 10] and analytical methods are still scarce according to the the horizon of a black hole is a genuine special hypersur- latest review on the experimental and theoretical methods for face within the full manifold, contrary to the usual claim that studying gravitational waves [11]. From the theoretical point there is nothing special at the horizon, which is said to be a of view it has been argued that the evolution of a black hole consequence of the equivalence principle. can be fully accounted for by four mechanical (or also known as “thermodynamical” [12]) laws which relate its mass, elec- tric charge, angular momentum, area of the horizon, surface 1 Introduction: locating the event horizon of a black gravity and temperature [13]. Note that the notion of the hole horizon is crucial in these laws: if, eventually, we have the technology to test these laws in the actual Universe, we would Black holes have been defined differently across different need to first correctly identify the horizon of a black hole. disciplines like astrophysics, analogue gravity and mathe- Moreover, cornerstone theorems by Hawking and Pen- matical relativity [1]. For example section 12.4 of [2]intro- rose proved that physical singularities (not to be confused duces black holes as regions of spacetime from which noth- with coordinate singularities which can be removed by a ing – not even light – can escape from, making them some change of the coordinate system) can occur in the theory of of the most studied solutions to the Einstein equations of classical [14–17]. These results have been general relativity [3]. After a few decades from their intro- checked explicitly with respect to the spacelike singularity duction as a pure mathematical concept, an interpretation that occurs in non- with no electric charge, as astrophysical object which – as far as we know – can be and timelike singularities for rotating and electrically charged fully characterized in terms of their mass, electric charge and black holes, exploiting the knowledge of such exact analyti- angular momentum has been proposed. Since techniques for cal solutions. Spacetimes admitting a physical singularity are estimating the values of these parameters became available geodesically incomplete, meaning that timelike and/or light- like geodesics cannot be extended after reaching the singular a e-mail: [email protected] point. Thus it is not possible to know the physical behav- b e-mail: [email protected] ior of such singularities, and in the case in which they are c e-mail: [email protected] 123 925 Page 2 of 13 Eur. Phys. J. C (2019) 79 :925 naked (that is they can be seen by an observer located at the are formed becomes smaller than the speed of sound within future null infinity), physical theories would lose predictabil- the fluid. More recently an experimental detection of a black ity. Thus a mechanism that prevents observers to directly see hole horizon has been proposed as realizable in a laboratory singularities is needed in a physically solid formulation of setting exploiting a Bose-Einstein condensate [31]. More- general relativity. The presence of an event horizon which over they provide as well an experimental realizable labora- hides curvature singularities seems the most convenient pro- tory for the study of the as yet undetected Hawking radia- posal to address this puzzle: this is the so-called cosmic cen- tion since they are supposed to emit its acoustic counterpart sorship conjecture [18]. Unfortunately it has been claimed and it is possible to measure their temperature, and surface that even in principle at least in the near future it will not be gravity with current available techniques [32–34]. Their ana- possible to detect black hole horizons by direct astrophysi- logue electromagnetic properties have been established as cal measurements because indirect measurements can only well in [35]. Once again theoretical techniques for locating claim the existence of a black hole by measuring its mass, their event horizons are playing an important role in such which must be higher than the mass of a emitting analyses. the same X-rays waves, and its [19]. More- The purpose of this note is to provide an algorithm answer- over, the theoretical interpretation of the shadow imaging of ing the question whether the horizon of lower dimensional a black hole observed by the Event Horizon Telescope has black holes can be located as the zero of an appropriate been done only with computer simulations [20]. Therefore at curvature invariant. The location of a black hole event hori- this stage, we must first focus on the analytical study of black zon can be searched in a non-local way by integrating the hole horizon, and the localization of it becomes an issue of equations for the geodesic motion [2]. Other possible tech- crucial importance. niques require approximating the horizon as a minimal Rie- On the other hand, modifications to the Hilbert-Einstein mannian surface or as a marginally outer trapped surface action have been proposed in the recent literature for many [36]. Recently it has been proved that it can be located by reasons ranging from the modeling of the late time accel- solving the zeroes of some invariants of the Riemann erated expansion of the Universe [21], to the analysis of its [37,38], and later the same argument has been extended to the first inflationary stages [22], to the modeling of the rota- use of the Cartan curvature invariants whose utility has been tion curve of galaxies [23], just to mention a few examples. explicitly demonstrated for (3 + 1)- and (4+1)-dimensional Lower dimensional – both (2 + 1)- and (1 + 1)-dimensional black holes [39] leading to the geometric horizon conjecture – theories for gravity have been formulated with a twofold [40,41], which has been tested as well in multi-black holes purpose of clarifying the mathematical properties of the Ein- configurations such as the Kastor-Traschen metric [42]. This stein equations (since they are a system of coupled nonlinear literature has also shown that those curvature invariants can partial derivative equations, their solution has been derived be used for extracting locally the values of the mass, electric so far only in simplified cases requiring strong symmetry charge and angular velocity of a black hole, whose mutual assumptions) in an analogue but formally simpler case, and relationships can then be compared with the ones predicted for developing possible frameworks for a quantum formula- by the previously mentioned mechanical laws governing a tion of the gravitational interaction [24–27]. Even though black hole, and in particular the scaling relation between lower-dimensional gravity is a pure topological theory, it mass and temperature, which for our lower dimensional case admits black hole solutions, the first example was found in is going to zero when the former is vanishing contrary to (2+1) dimensions in the presence of a negative cosmological the case of ordinary (3 + 1)-dimensional black holes [43]. constant [28,29]. It is therefore of interest to see whether the Thus, it is natural to hypothesize that this is the procedure mechanical and thermodynamical laws governing the evo- we are looking for. The interpretation of the modifications lution of (3 + 1)-dimensional black holes hold also for the we must apply to this procedure can be used for providing lower-dimensional case. Assuming that the thoughts leading new insights into the nature of the horizon of a black hole. to the cosmic censorship conjecture still apply, it is of crucial In fact, the basic ingredient of this method, the curvature, importance for this goal to correctly identify the black hole can be decomposed into two parts, only one of which can be horizon. determined by imposing the Einstein equations. The other Besides, black holes are an important tool in analogue is completely free in (3 + 1)-dimensions, but on the other gravity for deepening the understanding of physical systems hand is completely constrained to vanish in lower dimen- which seem completely unrelated to them at first sight [30]. sional frameworks. Thus, a comparison between our tech- For example their acoustic or sonic properties are the exact niques for locating the horizon can indeed provide further counterpart to their optical ones in astrophysics, that is they information on the nature of black holes in different dimen- admit a surface interpretable as an event horizon since sound sions. signals cannot escape from it, which can be determined by Our manuscript is organized as follows: in the next sec- the fact that the speed of flow of the fluid inside which they tion we will review the basic properties of a black hole 123 Eur. Phys. J. C (2019) 79 :925 Page 3 of 13 925 horizon, in the third section we will motivate more explic- horizons have been shown to be marginally outer trapped itly our interest in a lower dimensional theory, in section surfaces, which are defined as closed surfaces on which four we will comment on the peculiar properties of (2 + 1)- outward-pointing light rays moving inwards are converging, dimensional gravity focusing on the consequences of apply- their expansion being negative. They remain a good approx- ing our method, while in the fifth section we will present some imation even for dynamical configurations [47]. It is worth explicit applications to the linearly and non-linearly charged noticing that instead this criterion is based on the solution Banados–Teitelboim–Zanelli (BTZ) black hole which gener- of the Raychauduri equation which depends on the matter ally depends on five physical parameters. These examples of content generating the spacetime under examination through applications clarify the applicability of the method even for a its stress-energy tensor. dynamical configuration currently employed in the descrip- More recently the localization of the black hole horizon tion of the formation of a black hole. Then we will move has been connected to the vanishing of certain curvature to the same analysis but in (1 + 1)-dimensional gravity in invariants of the Riemann tensor [37,39]. Curvature invari- section six. Finally we will discuss a connection between ants are scalar quantities defined as the traces of products the Cartan invariants we constructed for locating the horizon of the curvature tensors, and in general of their derivatives. and the behavior of tidal forces on its proximity in section They are a useful technique for classifying known solutions seven, and we conclude in section eight summarizing the dif- of the Einstein field equations in reference to the so-called ferences between our analysis and the case of (3 + 1)- and equivalence problem, in which one has to check whether two (4 + 1)-dimensional spacetimes. very different looking solutions (derived in different coor- dinate systems) are in fact one and the same [48]. One of their limitations is that they cannot distinguish the vanish- 2 Basic properties of the black hole horizon: a very ing scalar invariant (VSI) spacetime from Minkowski [49]. short review Specifically, in this paper we are interested in locating the stationary horizon of a black hole which is a null hypersur- In this section we will review the basic definitions dealing face orthogonal to a Killing vector field that is null on it. with black horizons for the sake of completeness. The rigorous mathematical formulation of the methods we will invesigate has been proposed in [37,39]. However these Definition Given a spacetime manifold M, any submani- theorems constitute only a formal result on the existence of fold which is a spatial section of M which is topologically a curvature invariant which detects the stationary horizon a 2-sphere and a null surface, whose any normal has vanish- without delivering the explict curvature invariant which is ing expansion, and the projection of the stress-energy tensor required, nor providing a constructive algorithm. It is there- along any of its null future-directed normal is future-causal, fore necessary to check by hands how they behave for the is said to be a non-expanding horizon [36]. lower-dimensional black holes, which is the subject of our We notice that such requirements remain true after rescal- manuscript. It is interesting to notice again that in this algo- ing the null normals considered. This is the same as for the rithm geometrical and physical properties of the black hole Killing horizons, which are null hypersurfaces defined by the are mixed within the Riemann tensor which can be decom- vanishing of the norm of the time-translational Killing vector posed into the accounting for the former and the field, in which it is possible to rescale the Killing vector field Ricci tensor and Ricci scalar accounting for the latter. without affecting it. However the previous definition does not provide any practical technique for identifying these submanifolds once 3 Why lower dimensional gravity and black holes? the full spacetime manifold is provided, that is they do not come from the solution of any differential equation. With Relationships between (2 + 1)-dimensional gravity with this problem in mind, as a first attempt it has been proved other theoretical frameworks have been explored since its that black hole horizons are minimal surfaces, i.e. surfaces first appearance, the Chern–Simons theory being the most which locally minimize their area [44,45]. In this way it famous example. In this case the dynamics is derived from was possible to determine that their mean curvature van- an action principle in which the Lagrangian is provided by the ishes identically and that they can be obtained from the solu- Chern–Simons 3-form, meaning that it belongs to the class of tion of the Lagrange partial derivative equation derived from Schwarz-type quantum field theory because its observables an action principle [46]. It is worth noticing that now these are computed exploiting a metric-independent action func- are purely geometric methods, while instead the matter con- tional [50]. Moreover it admits connections to many other tent was present in the original definition of the black hole branches of physics ranging from condensed matter theory horizon through the stress-energy tensor entering the Ein- to . In the former case it was shown that the stein equations. Subsequently stationary and static black hole monopoles exploited for solving the dynamics are related 123 925 Page 4 of 13 Eur. Phys. J. C (2019) 79 :925 to quantum vortices which typically are topological defects where  is the cosmological constant, Tμν is the stress- exhibited by superfluids and superconductors. Its duality with energy tensor and units such that 8πG = c = 1 are used. 7 respect to M-theory on AdS4 × S has been explored as well Thus the Einstein field equations imply that in the absence [51]. In the latter case a supergravity theory has been for- of matter-energy content, Tμν = 0, the components of the mulated in which supersymmetry and general relativity are Riemann tensor are constant. Explicitly, we have combined in a unique model by imposing supersymmetry  Rμνρσ = 2(gμρ gνσ + gνσ gμρ − gνρ gμσ − gμσ gνρ ) to fulfill the locality requirement; its availability as effective  low-energy limit of string theory has been discussed [52–56]. −3(gμρ gνσ − gμσ gνρ ) . (3) An interesting aspect for our current analysis is that string theory admits a formulation in term of the so-called Born- Using the Leibniz rule for the , and the Infeld Lagrangian, which can be interpreted as representing a fact that the covariant derivative of the metric tensor and of non-linear version of the Maxwell equations of electromag- the cosmological constant vanish, we get netism [57–59]. In fact it has been proved that it exhibits a twofold application in gravity: on one hand it may account for Rμνρσ;η ≡ 0 , (4) the late time accelerated expansion of the Universe through the Chaplygin gas [60], while on the other, solutions pictur- ing charged black holes can be found exactly in this theory where a semicolon denotes a covariant derivative, imply- ing that both the scalar polynomial (Karlhede’s) invariant as well [61,62], providing an interesting arena for checking μνρσ;η I = Rμνρσ;η R whether the current ways of localizing a black hole horizon , where repeated indices are summed, J = Rμνρσ;η remain valid. Applying this way of reasoning with the goal and the Cartan invariant identically vanish in + of treating the mathematical non-linearities arising in general any (2 1)-dimensional spacetime point, and therefore they cannot relativity, dilaton-type gravity models have been developed be used for determining the horizon location. by Jackiw and Teitelboim [63], and by Callan–Giddings– Note in particular that this claim does not depend on the J Harvey–Strominger (CGHS in short) [64] exploring differ- frame used for computing , and of course it still holds in  → ent possibilities of coupling between the dilaton field and the the limit 0. Iterating this procedure and increasing one time and one space dimensions of gravity. These theories the order of differentiation we cannot produce any non-zero higher-order curvature invariant. Moreover using the rela- present black hole type solutions despite the fact of allowing μρ ρ ρ gμν g = δν δν only flat geometries. tionship , where denotes the Kronecker delta, we can also see that the scalar polynomial invariants constructed by the self-contraction of the Riemann tensor, μνρσ μναβ ρσ like Rμνρσ R , R Rαβρσ Rμν , etc..., are constant 4 On the use of curvature invariants for detecting a and thus they do not provide any information concerning the black hole horizon: the peculiar case of lower horizon location. The same conclusion holds trivially even dimensional Einstein theory when the invariants I and J are computed in terms of the Weyl tensor rather than of the Riemann tensor. This implies In this section we want to establish whether the argument that the argument proposed in [39], contrary to the cases proposed in [39] for locating the horizon applies also to lower of (3 + 1)- and higher dimensional gravity theories, cannot dimensional solutions, and/or if it requires any modification be used for locating the horizon of the corresponding lower due to the different number of independent components of the dimensional Schwarzschild, Kerr, and Schwarzschild-(anti- curvature tensor. In fact, it is well known that the Weyl tensor )de Sitter black holes because the geometry does not admit identically vanishes for any spacetime in (2+1)-dimensional enough degrees of freedom. The question now becomes if gravity, implying that the Riemann tensor Rμνρσ can be fully instead it remains a useful algorithm when the black hole written in terms of the Ricci tensor Rμν, the Ricci scalar R under examination carries an electrical charge which pro- and the metric tensor gμν as follows [43]: vides a non-trivial Ricci tensor. Exploiting the same arguments as in the previous compu- = + − − Rμνρσ gμρ Rνσ gνσ Rμρ gνρ Rμσ gμσ Rνρ tations, we get −1( − ) .   gμρ gνσ gμσ gνρ R (1) 1 2 Rμνρσ;η = gμρ Tνσ;η − gνσ T,η 2 The ingredients of (1) must be obtained from the Einstein    1 equations +gνσ Tμρ;η − gμρ T,η − gνρ Tμσ;η 2  1 −1 Rμν − gμν R = gμν + Tμν , (2) gμσ T,η 2 2 123 Eur. Phys. J. C (2019) 79 :925 Page 5 of 13 925   1 where W(x) denotes the Lambert function which is the −gμσ Tνρ;η − gνρ T,η x 2   inverse function of xe . 1 1 An explicit computation gives us the result − (gμρ gνσ − gμσ gνρ ) − T,η 2 2   4 2 2 = gμρ Tνσ;η + gνσ Tμρ;η − gνρ Tμσ;η − gμσ Tνρ;η 5Q (l + ω) (l − ω) r I = πr 2 + Q2(ω2 − l2) . 2 2 3 2 ln 1 2(πr l ) r0 − (gμρ gνσ − gμσ gνρ )T,η , (5) 4 (13) where T denotes the trace of the stress-energy tensor entering Solving the equation I = 0 we get one and only one solution the Einstein equations. With some algebraic manipulations r = rh implying that the scalar polynomial invariant (13) we get detects the black hole horizon without false positives.   For computing the Cartan invariant J we fix a “null” triad νσ;η ,η I = 4 Tνσ;ηT + T,ηT . (6) coframe following [66] N(r) la = √ dv, This computation may play a role in providing further restric- 2 √ tions on the matter content the configuration must obey for ( ) a N r 2r getting a useful non-identically zero scalar curvature invari- n = √ dv − dr, 2 N(r)K (r) ant I. K (r)N φ(r) K (r) ma = √ dv + √ dφ, (14) 2 2 5 Explicit cases in 2 + 1 dimensions such that

a a a a a a 5.1 A first explicit example: the case of the BTZ black hole mam = 1 =−lan , lal = nan = lam = nam = 0 , with electric charge (15)

For making the reasoning of the previous section more trans- in terms of which the metric (7) reads parent we apply it to the test solution (2.14) of [65]: g =−2l( n ) + 2m( m ) , (16) 2r ab a b a b s2 =−N 2(r) v2 + v r d d ( ) d d K r where round parentheses denote symmetrization. For elimi- φ +K 2(r)(dφ + N (r) dv)2, where (7)   nating possible ambiguities in the construction of the coframe r 2 r 2 l¯2 r we employ for computing the following Cartan invariants, we N 2(r) = − Q2 , 2 2 2 ln (8) K (r) l 2πl r0 begin by recalling that the metric tensor is invariant under ω r Lorentz transformations. Therefore (16) is invariant under N φ(r) =− Q2 , 2 ln (9) 2π K (r) r0 boosts ω2 r K 2(r) = r 2 + Q2 ln , (10) a → , a → 1 a, a → a 2π r l C1l n n m m (17) 0 C1 l¯2 = l2 − ω2, (11) and null rotations + which describes a (2 1)-dimensional black hole written a 1 a a 1 2 a 1 a a v φ l → l , n → C l + n + C2m , in Eddington–Finkelstein coordinates ( , r, ) with angu- 2 2 2 2 ω a a a lar velocity , electric charge Q and negative cosmological m → C2l + m , (18) 2 constant  =−1/l , while r0 is a reference length related to the mass M of the black hole for making the argument where C1 and C2 are arbitrary functions of the manifold of the logarithm dimensionless. As pointed out in [65], the coordinates. Thus we must fix the triad in such a way that black hole horizon is given by the surface N(r) = 0, which the curvature is in its canonical form, i.e. with the small- explicitly provides the solution est possible number of non-zero independent components, which can be done applying the Newman–Penrose formal-    ism. 3-dimensional spacetimes admits at most five indepen- 4πr2 Q (ω2 − l2)W 0 dent Ricci scalars defined as [66] Q2(ω2−l2) = √ , 1 a b rh (12) = Rabl l , (19) 2 π 00 2 123 925 Page 6 of 13 Eur. Phys. J. C (2019) 79 :925

2 1 a b r 22 = Rabn n , (20) f (r) = − m + 2r 2β2(1 − ) 2 l2   = √1 a b , r(1 + ) 10 Rabm l (21) +q2 1 − 2ln (28) 2 2 2l  1 a b 12 = √ Rabm n , (22) q2 2 2 (r) = 1 + , (29) r 2β2 1 a b a b 11 = (Rabm m + Rabn l ). (23) 6 where the mass m enters the metric directly, and the limit of Then, we construct (15) setting the appropriate components a in the Maxwell theory is obtained in to zero or equal (or proportional to each other) following eqs the limit β →+∞, while the effects of the non-linearities (5.44)–(5.51) of [66] according to the isotropy group of the become stronger for small β. Using polar coordinates we can black hole in hand. This procedures is the lower dimensional complement the previous section in which the Eddington– equivalent to Appendix B of [67]. Finkelstein coordinates were adopted instead. The location The only functionally independent frame component of of the horizon is given by f (rhor) = 0, which can be written the covariant derivative of the Riemann tensor reads implicitly as: √ r 2 2 2 2 r 2 = hor + 2 β2( − ( )) ω 2 Q (ω − l ) ln + 2r π m 2rhor 1 rhor 2 Q r0 l2 J = R2113;2 = · .   4l2r 2π Q2ω2 ln r + 2r 2π r (1 + (r )) r0 +q2 1 − 2ln hor hor . (30) (24) 2l By an explicit computation we obtain J = = Setting 0 we get the only solution r rh, implying that f (r)   the above mentioned Cartan invariant constitutes an appro- I = · r 4( f (r))2 + 4(r 2 + 1)( f (r))2 priate quantity for identifying the black hole horizon. r 4   We stress the fact once again that J is computationally less −8rf (r) f (r) (31) expensive to compute than I because it is a first degree quan- √ f (r)   tity. Moreover we observe that Eqs. (13) and (24) provide J = Rrθtr;θ = √ [rf (r) − f (r)], (32) 2r 2 where the latter was computed with respect to the coframe lim I = 0 = lim J (25) → → √ √ Q 0 Q 0 f (r) dr f (r) dr na = √ dt + √ , la = √ dt − √ , 2 f (r) 2 f (r) confirming the general discussion presented in the previous 2 2 ma = r dθ. (33) section. Finally, we stress that under a SO(1, 1) isotropy group, By direct inspection we see that I = 0 = J for f (r) = 0, the frame curvature computed using the Newman–Penrose i.e. they properly detect the horizon. It is straightforward to formalism admits only one independent component [66]: note as well that they vanish only at the horizon by looking 1 a b a b at the quantity within the square brackets which reads = (Rabm m + Rabn l ). (26) 11 6 2 4 Note that we are indeed under such a situation since the metric 384β q h1(r)h2(r) , where admits two vector fields related to its stationarity and axis- (β2r 2 + q2)7/2(rβ + β2r 2 + q2)6 symmetry (as easily recognized from its independence from ( ) = 5 4β4 + 2 2β2 + 4, v φ h1 r r 2r q q and the coordinates and ). 4  16 q6 h (r) = r 6β6 + 8q2r 4β4 + 3q4r 2β2 + 2 3 6 5.2 Second explicit example: non-linearly charged BTZ   4 black hole β2r 2 + q2 + (4β2r 2 + q2)β(β2r 2 + q2) r 2β2 + q2 r 3 The metric for a non-linearly charged BTZ black hole is (34) obtained by implementing the Born–Infeld theory within the for I and Einstein equations. It reads [61]:   4 2r 2β2 + 2βr β2r 2 + q2 + q2 q2β dr 2 ds2 =−f (r) dt2 + + r 2 dθ 2 (27) (35) f (r) (rβ + β2r 2 + q2)2 β2r 2 + q2 123 Eur. Phys. J. C (2019) 79 :925 Page 7 of 13 925 for J which are positive definite. It is clear again from these where F2 is another non-zero function of the metric coeffi- expressions that (25) still holds. In the strongly non-linear cients, their first and second derivativatives. The same con- limit we get respectively: sideration as for the scalar polynomial invariant applies.  64q2β2 2βl I  q2l2 + (q2 − m)l2 + r 2 5.4 Curvature invariants and the dynamical formation of a 4 2 2 ln r l q black hole +O(β3) for β → 0 √ | | 2 2 ( β2 2/ 2) − 2 + 2 The cases we studied so far can only account for static and J  2 2 q q l ln 4 l q ml r β 2 stationary configurations, and thus they assume the existence √lr 4 2q2l of the black hole without entering into the details on how − β2 it was formed. For addressing this latter problem we will 2 2 ( β2 2/ 2) − 2 + 2 r q l ln 4 l q ml r apply our algorithm to [69] which provides the solution for +O(β3) for β → 0 . (36) a dynamically evolving black hole in (2 + 1)-dimensional gravity. In this section we will see how the framework pro- 5.3 The most general solution in terms of five parameters vided by the curvature invariants allows one to follow the mechanism of formation of a black hole (i.e. when in time After the warming up of the previous sections, we can apply and where in space a singularity and a horizon appear). Using our method to [68] which delivers an original metric in terms an Eddington-Finkelstein-like coordinates system (u, r, ψ) of five physical parameters M, J, , qα, and qβ , which must the metric reads as follows: be regarded as more general than the ones applied in the ds2 =−f (u, r) du2 + 2du dr previous sections. In a coordinate system (t, r, φ) it reads: +g(u, r) dψ2, where (44) 2     dr 2 2 = − 2( ) + 2 2( ) 2 + 12αu ds N r r N1 r dt 8α tanh − 1 r K (r) r 2 q 2 2 2 f (u, r) = +   +r dφ − 2r N1(r) dt dφ where (37) l2 12αu   q tanh q J 2   N(r) = qα − M r + qβ K (r), (38) 12α α 12αu 2r 2 − − tanh , and (45)   2 3 qβ qα q q r q N1(r) = J + K (r) , (39)   2 2r 2 r α 3 ( , ) = 2 12 u . J 2 g u r r tanh (46) K (r) = − M − r 2 . (40) q 4r 2 Introducing the coframe An explicit computation delivers:    f (u, r) f (u, r) 2 la = du , na = du − dr , I = K (r) · F1 , (41) 2 2 f (u, r)  g(u, r) where F is a non-zero function of the metric coefficients, ma = dψ, (47) 1 2 their first and second derivatives. However without the need of expanding any further the computation, we can see that in terms of which the metric is written as (16), we can com- this scalar polynomial invariant vanishes when K (r) = grr pute: √ vanishes. Introducing the coframe 3 ( , ) ∂3 ( , ) 2 f 2 u r 2 g u r J = Rrψrr;ψ = · g (u, r) N(r) + rN1(r) r g3(u, r) ∂r 3 la = √ dt − √ dφ, 2 2 ∂2g(u, r) ∂g(u, r) ( ) − ( ) −2g(u, r) a N r rN1 r r ∂ 2 ∂ n = √ dt + √ dφ,  r r 2 2 ∂g(u, r) 3 1 + . (48) ma = √ dr (42) ∂r 2K (r) We note that this Cartan invariant vanishes for grr = in terms of which the metric is written as (16) we can com- f (u, r) = 0 detecting the horizon. Interestingly, substitut- pute: ing Eq. (46) into the square brackets we get zero as well. This seems to be a result following the dynamical property 3 J = Rtrtt;φ = K 2 (r) · F2 , (43) of this solution. 123 925 Page 8 of 13 Eur. Phys. J. C (2019) 79 :925

On the other hand, for this metric the scalar polynomial • A black hole in CGHS gravity whose asymptotic region invariant I does not seem a good tool for locating the horizon. is r →−∞:2 This is most easily recognized considering a numerical coun- √ terexample. For the case l = 1, u = 5, q = 2, α =−3, and 2C e Br ξ(r) = 1 + 0 . (55) r  0.041659 we get f (u, r) = 0, but I =−4880 + 1024i, B where i is the imaginary unit.1 The latter solutions picture black holes in asymptotically Minkowski, Rindler and (anti)-de Sitter spaces respectively. 6 Black holes in 1 + 1 dimensions In particular we can recognize the linear and quadratic terms in the fourth and fifth cases which are characteristic for Grumiller et al. [26] reviews some interesting black hole solu- Rindler and de Sitter backgrounds. We note that in this class tions in (1 + 1)-dimensional gravity theory known under the of theories the Weyl tensor is again trivially identical to zero, name of Jackiw−Teitelboim theory, which are exact solu- thus we will work in terms of the Riemann tensor. A straight- tions in the dilatonic gravity theory. If we adopt the coordi- forward computation delivers: nate system (u, r), the general line element we must consider  2 is: d3ξ(r) I = ξ(r). (56) dr 3 ds2 =−ξ(r) du2 + 2dr du . (49) Thus this scalar polynomial invariant correctly identifies the Relevant subcases are: Killing horizon, i.e. it vanishes where ξ(r) = 0. Intoducing the co-frame • A Schwarzschild-like solution written in the chiral gauge    [27]: ξ(r) ξ(r) 2 la = du , na = du − dr , (57) 2 2 ξ(r) 2M ξ(r) = 1 − (50) r in terms of which the metric reads as • A Reissner–Nordström-like black hole which constitutes =− , a generalization of the previous solution which accounts gab 2l(anb) (58) for an electric charge: we can compute the Cartan invariant 2M Q2 √ ξ(r) = − + 3 1 2 (51) 2ξ(r) d ξ(r) r r J = R ; = · , (59) ruru u 4 dr 3 • A black hole in an asymptotically Minkowski spacetime in the Eddington-Finkelstein gauge: which detects the killing horizon for the same reason stated above. Furthermore, we note that the fact of having or not   2−a a a B 2(a−1) having false positives depends on the behavior of the third ξ(r) = 2C |1 − a| a−1 r a−1 + 1(52) 0 a derivative of the non-trivial metric coefficient, which will assume different values depending on the particular solu- • A black hole in an asymptotically Rindler spacetime in tion considered. For the cases reported in this paper we get, the Eddington–Finkelstein gauge: respectively, d3ξ(r) 12M 1 a = , (60) ξ( ) = 2 − a−1 3 4 r rB2 Mr (53) dr r d3ξ(r) 12M 24Q2 • = − , (61) A black hole in an asymptotically (anti)-de Sitter space- dr 3 r 4 r 5 time in the Eddington–Finkelstein gauge: a 3−2a 2−a 3a−4 3ξ( ) ( − )| − | a−1 a−1 2(a−1) 2(a−1) d r =−2C0 a 2 a 1 r B a , a 3 ( − )3 ξ(r) = r 2 B − Mr a−1 (54) dr a 1 (62) 1 We stress that for spacetimes that are not stationary, one should search for scalar polynomial curvature invariant that not only vanishes on the 2 The CGHS black hole corresponds to the case with a = 1, and there- horizon of the stationary metric, but that also remains zero on the horizon fore a different system of coordinates is required for dealing with the under a conformal transformation of the metric [70]. singularity that occurs in (52). 123 Eur. Phys. J. C (2019) 79 :925 Page 9 of 13 925

3−2a d3ξ(r) Ma(a − 2)r a−1 the two methods. When we use the scalar polynomial invari- = , (63) dr 3 (a − 1)3 ant (67) we need further assumptions on the explicit form 3−2a of the metric coefficient K (r, u), information that are not d3ξ(r) Ma(a − 2)r a−1 = , (64) required when we try to locate the horizon as the zero of a 3 ( − )3 dr a 1 Cartan curvature invariant (69). 3 √ √ d ξ(r) Of some interest to us as well is the black hole solu- = 2C Ber B. (65) dr 3 tion arising from the Callan–Giddings–Harvey–Strominger Thus a false positive is possible in the third, fourth and fifth model [26,64]: case for the particular choice of the solution parameter a = 2.  − The fourth and fifth cases vanish as well for r = 0 which is M 1 s2 = G(u,v) u v, G(u,v)=− − λ2uv , another root of ξ(r) = 0 for the corresponding black holes. d d d where λ Moreover, a false positive can be seen√ also in the second (70) = 2 3Q case, in the particular case M 3 after substituting 2 2 rhor = M ± M − Q . written in the coordinate system (u, v). A direct computation Grumiller et al. [26] discusses as well the following shows that dynamical solution: I = 8λ3uvM2G3(u,v), (71) ds2 = 2dr du + K (r, u) du2 . (66) which vanishes for G(u,v)= 0. Fixing the coframe A direct computation shows that √ √   2 2 ∂3 K (r, u) ∂3 K (r, u) ∂3 K (r, u) la =− G(u,v)du , na = dv (72) I = 2 − K (r, u) . (67) 2 2 ∂r 3 ∂u∂r 2 ∂r 3 we can compute:

J = Rv vv; u u   ∂3 ( ,v) ∂2 ( ,v) ∂ ( ,v) ∂2 ( ,v) ∂ ( ,v) ∂ ( ,v) ∂ ( ,v) 2 √ G(u,v) G(u,v) G u − G u G u − G u G u + G u G u ∂v2∂u ∂v2 ∂u 3 ∂v∂u ∂v 3 ∂u ∂v = 2 2 4( ,v) √ G u =−2 2λ3uMG2(u,v), (73)

Defining where in the last equality we have used the explicit form of   the metric coefficient. Thus this seems to vanish only after an 1 la = √ − 1 K (r, u) du , appropriate choice of the metric. A false positive is present  2  for u = 0. 1 na = √ + 1 K (r, u) du 2 √ 2 7 Physical interpretation of the horizon-detecting + √ √ dr , (68) ( 2 − 1) K (r, u) Cartan invariants it is possible to compute The tidal tensor is defined as: √ √ √ 2( 2 − 1) K (r, u) ∂3 K (r, u) ∂ J = Rrrur;r = · , (69) Ei = (74) 4 ∂r 3 j ∂xi ∂x j ( , ) = which vanishes for K r u 0. Recalling our previous dis- where is the gravitational potential. Tidal forces affect the + cussion about static (1 1)-dimensional black holes, whose geodesic deviation equation for the displacement vector ξ i generalization is considered here, we can exclude the second via factor of this Cartan invariant to vanish in general, meaning that its only zeroes are indeed the zeroes of the metric coeffi- 2ξ i d i j ( , ) =−E j ξ , (75) cient K r u . Thus we can appreciate a difference between dt2 123 925 Page 10 of 13 Eur. Phys. J. C (2019) 79 :925 allowing us to rewrite the tidal tensor in terms of the curvature +32( f (r) − 1)2 √ Riemann tensor as [71]   f (r)   J = √ f (r) − rf (r) , (78) i i 2 E j = R tjt (76) 2 2r where a prime denotes a derivative with respect to r, and the where t denotes the time index. Therefore we can interpret latter invariant has been computed with respect to the frame two of the Cartan invariants we used for locating the horizon √ (24)–(43) as providing a local extremum with respect to a spa- f (r) 1 la = √ dt − √ dr, ( ) tial direction of the tidal force. The same interpretation can √ 2 2 f r be proposed for (59) but with respect to time, suggesting that f (r) a = √ + √ 1 , the black hole (i.e. its horizon) forms when a local extremum n dt dr 2 2 f (r) of the tidal forces is reached. Taking into account that Cartan r ir sin θ invariants are foliation-independent [72] this seems to sug- ma = √ dθ + √ dφ, 2 2 gest that the horizon can be physically characterized by a r ir sin θ local extremum of the tidal forces. Therefore a hypothetical m¯ a = √ dθ − √ dφ, (79) spaceship crossing the horizon should in principle be able to 2 2 recognize it from a change in the behavior of the measured where i denotes the imaginary unit. We note that in such tidal force. We propose that this behavior of the tidal forces a case I = 0 = J where f (r) = 0 = grr. This implies can be a valuable technique for distinguishing black holes that these two curvature invariants locate the horizon of from other star-like objects which do not have an horizon the black hole without invoking any assumption concern- and therefore according to [39] do not exhibit this property ing the Einstein field equations, that is they work for the of the Cartan invariants (that is having a spacetime point in Schwarzshild solution, the Schwarzshild (anti-)de Sitter met- which a Cartan invariant connected to the tidal tensor van- ric, the Schwarzschild-NUT and Reissner-Nördstrom black ishes). holes, etc. Interestingly they provide information about the Interests in lower-dimensional black holes is motivated horizon location also for vacuum spacetimes. This is a con- from the existence of physical systems whose motion is sequence of the non-triviality of the Weyl tensor in 3 + 1 known to be confined in lower dimensions, like cosmic dimensions. strings and domain walls which arise in the Polyakov model On the other hand, when we move to lower dimensional [73,74](seealso[75] for a review about applications of (2 + 1)-dimensional theory the Weyl tensor vanishes identi- BTZ black holes in the light of this connection). The exis- cally and the Riemann tensor becomes a constant in vacuum tence of hydrodynamic equilibrium has led to the proposal and for spacetimes characterized only by the cosmological of stellar models in (1 + 2) gravity [76–78]. Distinguish- constant. An even more drastic phenomenon takes place in ing a lower-dimenional black hole from a lower-dimensional the gravitational theories characterized by only one tempo- star is important in laboratory settings which investigate ral and one spatial dimensions since their geometries are all Bose-Einstein condensate [79], and the optical properties of necessarily flat. Thus we cannot take for guaranteed that this graphene [80] through analogue sonic BTZ black holes. The argument holds automatically for detecting the location of variations of the tidal forces detected using absolute gravime- lower-dimensional black hole horizons. The purpose of this ters [81], and interpreted following this section can provide work was to investigate qualitatively and quantitatively how an answer to this issue. the dimensionality of the theory for gravity adopted influence the values of the curvature invariants which have been used for locating the horizon of a black hole in [39]. In particular, 8 Discussion and conclusion we examined in detail the role of the matter content check- ing explicitly that when we are dealing with a non-vacuum When we consider a (3+1)-dimensional Schwarzschild-like solution the applicability of the general method is recovered metric in spherical coordinates (t, r, θ, φ) because of the non-triviality of the Riemann curvature. More- dr 2 over in Sect. (5.4) we discussed the reasons why the Cartan ds2 =−f (r) dt2 + + r 2 dθ 2 + r 2 sin2 θ dφ2 , (77) f (r) curvature invariant should be regarded as a more powerful method than the polynomial one for tracing the horizon of a we can compute straightforwardly the two curvature invari- dynamical black hole. ants In [2] explicitly states the necessity of knowing the entire f (r)     I = r 6( f (r))2 + 8r 4( f (r))2 − 16r 3 f (r) f (r) spacetime evolution for claiming if a black hole is present or r 6 not. Therefore this is why our technique is important: it does   +16r 2( f (r))2 − 32r( f (r) − 1) f (r) not require all this amount of information. We do not need to 123 Eur. Phys. J. C (2019) 79 :925 Page 11 of 13 925 evolve any initial data, making this computationally conve- vacuum solutions in general relativity, meaning that they have nient when numerical relativity studies are adopted. Belong- the same Ricci tensor, but they have different Weyl tensors, ing to the same line of thinking, [82] compares and contrasts which in turn makes their full curvature different, with the local and global aspects of black hole horizons, explaining Minkowski spacetime being flat and the other two not. This that they are located by propagating light rays. Again, our can be appreciated also considering cosmological solutions: method is intended to provide a reliable alternative. Also the one can have a dust Friedman model and a dust Bianchi I, state of the art of numerical techniques still relies on (expen- Bianchi V, or Bianchi IX models, with same matter content sive) non-local techniques: see for example [83–86] where (dust, i.e. same physics entering the Einstein equations) but in particular the last one focuses on horizon deformations different geometries (Weyl part of the curvature). However in lower dimensional geometries by numerical methods in this decomposition of the Riemann tensor does not hold any- the context of the AdS/CFT correspondence. To this purpose more in lower dimensional gravity because the Weyl tensor [87] investigates how new numerical relativity techniques can is completely determined by the number of dimensions of locate horizons already known in the literature. Our research the theory and it is identically zero. Thus, in our language intends to complement it from an analytical perspective. The the curvature is a purely “physical” property of the space- research looking for connections between the zeros of cer- time because the Riemann tensor can be fully eliminated in tain curvature invariants and the location of the horizon has terms of the stress-energy tensor as we used in Eq. (5), and begun because numerical relativity simulations can extract even all solutions in (1+1)-dimensions are flat due to stricter the mass and angular momentum of a black hole only by restrictions. Once we have established that the horizon of a locating its horizon and by computing its area through the black hole can be found exploiting some quantities derived excision method. Finding the location of the black hole hori- from the spacetime curvature, as studied in this paper, the zon provides information about which region must be excised previous decomposition of the Riemann tensor can provide [88]. a new characterization the properties of the horizon, and in It is curious to see how our research fits inside the broader particular it turns out that in lower dimensional theories it is context dealing with the determination of a black hole hori- reduced to a purely “physical” property, in our language, of zon shortly reviewed in the second section. The very first the solutions, because its curvature is fully determined by the technique for finding the horizon was based on the com- field equations. pletely geometrical concept of minimal surface; then the subsequent technique based on the ideas of marginally outer Acknowledgements Y.C.O. thanks NNSFC (Grant No. 11705162) trapped surfaces and curvature invariants involved a mixture and the Natural Science Foundation of Jiangsu Province (No. BK20170479) for funding support. of physical and geometrical properties of the black holes, while now we have shown that in lower dimensional gravity Data Availability Statement This manuscript has no associated data the black hole horizon seems to be a purely physical object or the data will not be deposited. [Authors’ comment: This is a purely theoretical work and not an experimental work (which involves data), because only information about the matter content of the nor it involves simulations.] spacetime were used for locating it arriving at a conceptually different method for finding it than the one we started from. In Open Access This article is distributed under the terms of the Creative light of this we have considered both the Maxwell theory of Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, electromagnetism and the non-linear Born-Infeld theory for and reproduction in any medium, provided you give appropriate credit the modeling of the matter content in (2 + 1)-dimensional to the original author(s) and the source, provide a link to the Creative gravity, as well as the case of a dilatonic field coupled to Commons license, and indicate if changes were made. 3 gravity in 1 + 1 dimension. 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