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arXiv:2002.01135v2 [hep-th] 6 Feb 2020 stems n h pno h lc hole such the properties, of spin other the some and ergo- define mass the to the and as as horizon basis well a fol- the as as of the surface suggested location [2] was the In detect scalars to curvature hole. for of black works Schwarzschild set which [3] the lowing of al. et case Karlhede extending the of significantly method charac- hole, invariant earlier black the an Kerr gave the of who of work terization [2] the Lake by have and initiated there Abdelqader developments recent method, some coordinate-dependent been this In of . stead in coordinates well-chosen requires clearly fasnl function single a of erhfra vn oio rcesa olw:oeas- one horizon event follows: as the proceeds that horizon sumes event an for 2+1 search a is which the horizon event of hypersurface. the cross-section dimensional two not codimension but horizon the black event as Horizon the well Event of as full environment the hole the respect, by the contains this obtained needs (EHT) In image priori Telescope it. the a locate example, to one for spacetime so horizon the event and of the knowledge such, non-local, As highly [1]. cannot is infinity geodesics null casual future which reach from of region hole boundary black the the constitutes which hypersurface, null one where † ∗ n h w oetr enn h attresaasare scalars three last the as defining given covectors two the and lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic ie ercrpeetn pctm,teusual the spacetime, a representing metric a Given codimension a is hole black a of horizon event The I I I 5 3 1 = = = C µναβ k ∇ C k µ µναβ µ ρ k C µ = h he iesoa smttclyfltbakhlsa exa numbers: as PACS holes black fou di flat three, three asymptotically in and dimensional efficient Myers-Perry three Ou most the dimensional five is spacetime. but Kerr, full dimensions dimensional u the generic hole of black in instead BTZ holes the hypersurfaces of are one invariants horizon mension event curvature the all of since characterization hole black (BTZ) Zanelli fbakhlsin holes used be black can scalars of curvature local chosen judiciously Some µναβ I , steWy Tensor, Weyl the is C −∇ .INTRODUCTION I. µναβ 6 ∇ µ = ρ I F C 1 I , l and , µ µναβ satisfying l µ I , I , l µ > D 2 = 4 ideEs ehia nvriy 60 naa Turkey Ankara, 06800 University, Technical East Middle 7 H = = = g −∇ iesos u hyfi o h he iesoa Bañado dimensional three the for fail they but dimensions, 3 ∗ vn oio eetn Invariants Detecting Horizon Event µν sasot ee set level smooth a is ∗ ∇ k C C µ ∂ µναβ ρ µ yi Tavlayan Aydin l µναβ µ C µ I ∂ F 2 µναβ odtc the detect To . ν C sislf dual left its is F µναβ ∇ which 0 = ρ 1 ∗ eateto Physics, of Department C , µναβ (1) 1, , - ∗ n armTekin Bayram and oaino h vn oio fteShazcidblack Schwarzschild the of horizon hole event the of location n te he iesoa rvt hoywihad- as which well theory as dimensional constant, mits three cosmological the- dimensional Einstein’s other negative of 2+1 any solution a the a with is of which ory [5], example hole critical black BTZ the for work spacetime. the of cohomogeneity local the h vn oio.Hnei 2,tefloignonlinear following the locate [2], to black found in enough was Kerr Hence combination are the invariants horizon. For above , the the event horizon. of the the none inside on hole, negative vanishes is horizon, and event the side ns h xcto fterpoeuei sflos the follows: of as product is wedge the procedure of their invari- norm squared curvature of execution from The built ants. functions with but scalar manifold) de- other spacetime functions the some of scalar subsets are stated open invariantly (which on hori- fined coordinates be event the can the with statement not on this vanishes However, hyper- metric null zon. determinant hypersurface a the such, is the boils as horizon of metric; idea event degenerate the their a with that of con- space fact crux of the Page way The to by systematic hori- down remedied a invariants. event was proposed such the affairs who structing detect of [4], state to Shoom hoc used and curva- ad a be This define can zon. judiciously that must scalar one ture hole, black of kind the metric. describe the invariantly of to the sufficient and spin, the are mass, among which found invariants four, were independent to hole the reducing black that black (1), Kerr invariants syzygies seven Kerr the three for the invariant, valid of this are constructing horizon In event the hole. on vanishes which ntehrzno n ttoaybakhl,where hole, black stationary any of vanish horizon would the invariants on curvature smooth local pendent h aeSommto spwru,bti osnot does it but powerful, is method Page-Shoom The scnb eue rmteaoedsuso,freach for discussion, above the from deduced be can As I AdS 3 oivratycaatrz vn horizons event characterize invariantly to osat eew rvd ninvariant an provide we Here constant. mples. ssffiin:Nml,ta clri oiieout- positive is scalar that Namely, sufficient: is esoa apdat-eSte,and Sitter, warped-anti-de mensional ,adfiedmnin.W iefour give We dimensions. five and r, igtecraueivrat fcodi- of invariants curvature the sing 3 ehdi loapial oblack to applicable also is method r saslto.Terao o hsfiueis failure this for reason The solution. a as , Q † 2 ,θ r, = oriae ftecohomogeneity the of coordinates 27 1 ( I I 1 2 5 I + 6 s-Teitelboim- I − 2 2 ) I 5 7 2 / 2 n rdet finde- of gradients n (2) is 2 simple to understand: in three dimensions, the Riemann tensor is algebraically related to the as where m and j are mass and angular momentum, re- σρ Rµανβ = ǫµασǫνβρG : hence whenever the Einstein ten- spectively; and ℓ is the AdS3 radius. In this coordinate sor vanishes, the Riemann tensor vanishes along with all system, the coordinates t, and φ have associated Killing the curvature invariants; and whenever the Einstein ten- symmetries, ∂t and ∂φ respectively. The event horizon sor is proportional to the metric tensor (as in the case can be easily found as the largest root of grr = 0 or of the BTZ ) the Riemann tensor is locally equivalently as the location at which the time-like Killing maximally symmetric as in AdS3 and all the curvature vector ξ = ∂t +Ω∂φ becomes null where Ω = gtφ/gφφ 2 − invariants are just constants. The conclusion is that the which becomes Ω = j/(2r+) at the event horizon. Both spacetime curvature scalars cannot be used to detect the results yield event horizon of the BTZ black hole. This prompted us to search for a new approach that we describe in this mℓ2 j2 r2 = 1+ 1 , (4) work. + 2 2 2 r − ℓ m ! with the positive root giving the event horizon. II. CURVATURE INVARIANTS OF THE Let us now find this in a coordinate independent way. HYPERSURFACE The cohomogeneity of this spacetime is one. Keeping this intact, we need to choose a hypersurface which has Our method, which works for the BTZ black hole and at least one non-constant curvature invariant. For this other black holes in three, four, and five dimensions, is a purpose we can choose the t = t0 or φ = φ0 to define the variant of the method of Page and Shoom [4], but only hypersurface (note again that it is not the event horizon uses one curvature invariant. Beyond five dimensions, itself). Both options are valid and yield the same result, our method will still be valid, but one needs more than let us choose the latter, then the induced metric is one curvature invariant as such, it is not more advan- 2 r tageous than the method of [4], which also generically m ℓ2 0 γ = − 1 (5) ij 2 requires more than one curvature invariant. 0 j r2 2 + 2 −m ! The main idea of our proposal is as follows: as was re- 4r ℓ alized in [4] in a spacetime with a stationary black hole, in the induced coordinates (t, r). This metric has a sym- there is always a symmetry that can be used to split the metry only in the t coordinate. So, the cohomogeneity is tangent space of the spacetime into two parts. Let G be still one, as expected. The Kretschmann invariant com- the local symmetry group with m dimensional maximal puted for the induced metric (5) becomes, orbits. Hence the local cohomogeneity has n = D m i − dimensions. Considering a set of gradients dS built 2 2 { } j2ℓ2 4 r2 ℓ2m from n functionally independent non-constant curvature Σ Σ Σ ijkl − − I1 = Rijkl R = , (6) scalars, their wedge product, say W , (which is propor-  4 (ℓ3m ℓr2)4   tional to the volume form on the cohomogeneity space) − will vanish on the event horizon since the Hodge dual of from which one can calculate the horizon detecting in- W vanishes there. This can be used to detect the event variant horizon. However, clearly, for the case of the BTZ black Σ Σ mΣ I5 = I1 I1 (7) hole, there is no such non-constant curvature scalar, as ∇m ∇ shown above. One possible way out of this problem is to reduce the local cohomogeneity. Especially, if we can re- which reads explicitly as duce the local cohomogeneity to one, we can use just one curvature invariant, which vanishes on the horizon. In 2 2 j4 j2ℓ2 4ℓ2mr2 +4r4 j2ℓ2 4 r2 ℓ2m order to reduce the local cohomogeneity, we embed the Σ − − − I5 = . D 1 dimensional surface Σ to the D-dimensional space- 6 2 2 10  − ℓ (r  ℓ m)  time. It is essential to understand that this Σ is not the − (8) event horizon . Let us carry out this procedure for the H Σ BTZ black hole first and then to other black holes. whose largest real root, I5(r = r+) = 0, is exactly the event horizon given as 4. In fact for the BTZ black hole, the hypersurface version III. BTZ BLACK HOLE of the curvature invariant given in[3] can also be used to detect the event horizon as a somewhat simpler way. One In (t, r, φ) coordinates, the BTZ metric is given as can show that for the hypersurface metric (5) one has

2 r j Σ Σ mΣ ijkl m ℓ2 0 2 I3 = m Rijkl R (9) − 1 − ∇ ∇ 0 2 2 0 4 2 2 2 2 4 gµν = j r (3)  2 + 2 −m  j j ℓ 4ℓ mr +4r 4r ℓ = − j 0 r2 2 2 2 6  2  ℓ (r ℓ m)   −  − 3 which vanishes only on the event horizon and on the in- curvature invariants. In fact, if we define the hypersur- ner horizon. Therefore, it is a very useful tool for event face as φ = φ0, we obtain a flat metric, that is not useful horizon detection. at all. The other option, that is choosing the hypersur- face defined by t = t0, yields a nontrivial result. The induced metric is IV. WARPED AdS3 BLACK HOLE ℓ2 1 ds2 = dr2 + ℓ2rfdφ2 (13) Σ (ν2 +3)(r p)(r q) 4 The method described above is geometric, in the sense − − that it relies on the metric and not on the underlying field in the induced coordinates (r, φ). equations. Therefore, it can be applied to other metrics Now, we can calculate the relevant invariants as given that solve field equations that are different from general by (7) and/or (9) as both invariants work as in the case relativity. Let us consider the case of the warped AdS3 of the BTZ metric. The explicit expressions are cumber- black hole in the (t, r, φ) coordinates [6–8]. some, hence we do not depict here; but note that, they both vanish on the event horizon, i.e. ℓ2 ds2 = ℓ2dt2 + dr2 + − (ν2 +3)(r p)(r q) ΣI (r = r )=0, (14) − − 5 + 1 Σ + ℓ2rfdφ2 (10) I3(r = r+)=0. (15) 4 ℓ2 pq (ν2 + 3) 2rν dtdφ. − − V. TWO EXAMPLES IN 2+1 DIMENSIONS p  where the function f is given as In 2+1 dimensions, there is a purely quadratic gravity theory (the so called K-gravity) defined by the action f := ν2 +3 (p + q) 4ν pq (ν2 +3)+3r ν2 1 . [10] − −  p  The event horizon is at r+ = p, and the inner horizon is 3 µν 3 2 S = d x√ g Rµν R R (16) at r− = q. This metric solves the topologically massive ˆ − − 8 ℓ   gravity [9] for ν = µ 3 that admits asymptotically flat and rotating black holes [11, 12]. Let us apply the above procedure to detect the 1 1 1 R Rg + C =0, (11) event horizons of these black holes. µν − 2 µν − ℓ2 µ µν where Cµν is the Cotton tensor and µ is the topological A. Asymptotically flat solution of K-gravity mass. Even though the Ricci tensor and the Riemann tensor of this spacetime are different forms of the max- The metric is given as imally symmetric AdS3, the curvature invariants of the full spacetime cannot detect the event horizon because 1 ds2 = (br µ) dt2 + dr2 + r2dφ2. (17) all the curvature scalars are constant. In fact, explicit − − br µ forms of some of the curvature invariants are given as − The usual method shows that there is an event horizon at 12 2ν4 4ν2 +3 r = µ/b, as is clear from the metric. This spacetime has R Rµνρσ = − µνρσ 4 two symmetries and has cohomogeneity one which ad- ℓ  6 ν4 2ν2 +3 mits two possible hypersurfaces. But one of these, that R Rµν = − µν ℓ4 is reducing along the φ coordinate, yields a flat space and  2 hence, it is not a viable option. So, we choose the hyper- 36ν2 ν2 1 σ µν surface defined by t = t0. Then the relevant invariants σRµν R = 6 − ∇ ∇ − ℓ  (7) and (9) read as 2 2 2 σ µνργ 144ν ν 1 4 σRµνργ R = 6 − (12) Σ 4b (br µ) ∇ ∇ − ℓ I5 = − , (18)  r6 which are not useful to detect the event horizon. There- fore we resort to our method, but there is a subtle issue and here. As in the case of the BTZ black hole, there are two 2 Σ b (br µ) Killing symmetries for the warped AdS black hole and I3 = − , (19) 3 r4 two possible ways of choosing the hypersurface by keep- ing the cohomogeneity constant. However, only one of which vanish on the horizon. these two ways yields a hypersurface with non-constant 4

B. Rotating solution of K-gravity The induced metric on the hypersurface can be found by the pulling it back from the spacetime metric. The com- The following metric which describes a rotating black ponents of it can be written with respect to the induced hole also solves the K-gravity [11] coordinates (t, r, φ) as 2 2 2 2mr 4amr(1 X ) ds = (µ br)du ds2 = (1 )dt2 − 0 dtdφ (25) − Σ − − ρ2 − ρ2 2 (a2b +8r) 2 2 2 2 2 2a mr(1 X0 ) 2 2 ρ 2 2 4 2 2 2 dudr +(r + a + − )(1 X )dφ + dr , − sa b + 16a µ + 64r ρ2 − 0 Σ +a(µ br)dudφ − a4b2 a2µ The Kretschmann scalar + + + r2 dφ2. (20) 64 4 Σ Σ Σ ijkl   I1 = Rijkl R . (26) The cohomogeneity of this spacetime is one. Therefore, we can detect the event horizon by using the curvature of the hypersurface can be found, from which one can Σ Σ invariants I5 (7) and/or , I3 (9). We can induce the compute spacetime on the constant φ0 hypersurface which has Σ Σ µΣ I5 = µ I1 I1, (27) 2 ∇ ∇ 16777216a8b4(br µ) a2b2 8br + 16µ Σ which for X0 = 0 yields I5 = − − 2 16 (a b +8r)  4 2 Σ 20736M a + r(r 2M) 2 2 2 4 2 2 2 − a b 4br + 12µ a b + 16a µ + 64r (21) I5 = 16 (28) × − r  and   that vanishes on the event horizon given as r+ = m + 4 2 2 2 2 √m2 a2, which matches the result of the coordinate Σ 65536a b (br µ) a b 4br + 12µ − I3 = − − dependent method. 2 10 (a b +8 r)  Let us stress that instead of the hypersurface construc- a4b2 + 16a2µ + 64r2 . (22) tion given above, if we had used the full spacetime invari- × ant given as Both invariants vanish, for real values of r, on the event µ horizon r = µ/b and change sign there. They also vanish I5 = µI1 I1, (29) ∇ ∇ for some other values of r but at these points there is no we could not have detected the event horizon with change of sign. this invariant alone. Therefore, in this approach, we have a simplified way to detect the horizon. An- VI. KERR BLACK HOLE other advantage of this approach is that, when it is calculated in the X(p) = X0 = 0 hypersurface, the invariant vanishes only on the event horizon of the Let us apply the above reasoning to the four dimen- black hole and there are no other roots at finite r. In sional Kerr black hole [13], which has two Killing sym- the method of [4], the norm of the wedge square have metries. Therefore, the cohomogeneity of this spacetime other roots in addition to the root indicating the horizon. is two. By choosing some special hypersurfaces, we can reduce the cohomogeneity to one. The components of the in the Boyer- Lindquist coordinates (t,r,X,φ) is VII. 4+1 MYERS-PERRY BLACK HOLE

2 2 2mr 2 4amr(1 X ) ds = (1 2 )dt 2− dtdφ Myers and Perry [14] found the rotating, massive, Ricci − − ρ − ρ flat black holes in generic D 5 dimensions. Here, as an 2a2mr(1 X2) ρ2 ≥ +(r2 + a2 + − )(1 X2)dφ2 + dr2 example, we consider the five dimensional solution that ρ2 − Σ can be studied with our method with the help of a single ρ2 curvature invariant. In this spacetime, the coordinates + dX2, (23) 1 X2 are chosen as (t,r,X,φ1, φ2) where X = cos θ. The met- − ric is then, where ρ2 = r2 +a2X2,Σ= r2 2mr+a2, X = cos θ, and 2 2 m 2 2 2 − ds = dt + dt + k(1 X )dφ1 + lX dφ2 m and a correspond to mass and the rotation parameter, − Σ − respectively. The hypersurface that will be embedded r2Σ Σ  into the spacetime is defined by, + dr2 + dX2 + (r2 + k2)(1 X2)dφ2 Π mr2 1 X2 − 1 2− 2 2 2 − p M, p Σ X(p)= X0 = constant. (24) +(r + l )X dφ (30) ∈ ∈ ⇐⇒ 2 5 where VIII. CONCLUSIONS

Σ= r2 + k2X2 + l2(1 X2) − Π = (r2 + k2)(r2 + l2), (31)

Here k,l are angular momentum parameters as there are Inspired by the recent progress [2] in defining the event two possible independent angular momenta in five dimen- horizon of black holes with curvature invariants in D =4 dimensions and D 4 dimensions [4], we have presented sions. In this coordinate system, the coordinates t, φ1, φ2 ≥ have associated symmetries. Hence, the cohomogeneity a method which also works for D = 3 spacetime dimen- of this spacetime is two. We can induce the spacetime sions. The method of Page and Shoom makes use of the invariants of full spacetime that works well unless onto the hypersurface of constant X, let say X0. The induced metric is the invariants are all constant. Our method adapts their idea, but instead of using the invariants of the full space- 2 2 m 2 2 2 time, we use the invariants of well-chosen hypersurfaces. dsΣ = dt + (dt + k(1 X0 )dφ1 + lX0 dφ2) − Σ − We have given several examples in three, four and five 2 r Σ 2 2 2 2 2 dimensions where we need only one invariant to detect + dr + (r + k )(1 X )dφ Π mr2 − 0 1 the black hole horizon. In three dimensional black holes, 2− 2 2 2 the hypersurface version of the invariant, suggested in [3] +(r + l )X0 dφ2 (32) originally for the Schwarzschild black hole, works. Be- in the induced coordinates (t, r, φ1, φ2). Again, one com- sides this invariant, we provided another one constructed putes the Kretschmann invariant of this hypersurface from the gradients of the induced Kretschmann invariant from which one computes (7). The explicit forms of that also works in four and five dimensions. Beyond five the expressions are complicated, but one can show that dimensions, just like the method of [4], generically with Σ I5 =0 at our method, we need more than one horizon detecting in- variant because usually the cohomogeneity is more than two. For more on the construction of horizon detecting (k2 + l2 m)2 4k2l2 k2 l2 + m − − − − invariants see [15] where Cartan invariants are suggested r+ = r . (33) q √2 and see [16] where they were employed for lower dimen- sional nonvacuum black holes such as the charged BTZ which is the location of the event horizon. metric.

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