(2 + 1)-dimensional black holes with a nonminimally coupled scalar hair

Transcription

1 (2 + 1)-dimensional black holes with a nonminimally coupled scalar hair Liu Zhao School of Physics, Nankai University Collaborators: Wei Xu, Kun Meng, Bin Zhu Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 1 / 52

2 1 Why (2 + 1)-dimensional gravity? 2 Hairy BH: why scalar hair? 3 A static charged hairy black hole 4 Rotating hairy black hole 5 Turning into minimal coupling 6 CFT dual in the minimal coupling picture 7 CFT and entropy in the non-minimal coupling picture Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 2 / 52

3 Why (2 + 1)-dimensional gravity? It contains less degrees of freedom (only boundary DOF exist in pure (2+1)-d AdS gravity), making it easier to cope with; The boundary degrees of freedom in (2+1)-d AdS gravity naturally lead to a simplest realization of AdS/CFT duality [Brown-Henneaux, CMP 1986]; It can be reformulated in terms of Chern-Simons gauge theory, making it completely integrable and easy to quantize [Witten, NPB 1988; Strings 2007]; The mathematical simplicity makes it easy to treat and extend, while its physical richness allows for studying some universal aspects of gravitation regardless of dimensionality. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 3 / 52

4 Why (2 + 1)-dimensional gravity? Brief history of (2+1)-d gravity: Deser-Jackiw, 1984, first model of AdS gravity in (2+1)-d; Brown-Henneaux, 1986, asymptotic Virasoro symmetry; Witten, 1988, Chern-Simons formulation and integrability; Banados, Teitelboim and Zanelli, 1992 (PRL), first black hole solution; Ida, 2000, No-go theorem for black holes in non-ads (2+1)-d gravity; Carlip, 2005, Near horizon conformal symmetry & black hole thermodynamics; Astorino, 2011 and LZ WX, 2012, Accelerating BH in (2+1)-d, possibly non-ads. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 4 / 52

5 Why (2 + 1)-dimensional gravity? Extensions to (2+1)-d gravity: Topological massive gravity (traces back to Deser-Jackiw 1982 topological gauge theories, E-H action + Chern-Simons); New massive gravity (Bergshoeff, Hohm, and Townsend 2009, R 2 corrections and SUGRA settings); Scalar-tensor theory (Jordan, Brans-Dickii, Kaluza-Klein Fujii-Maeda, The Scalar Tensor Theory of Gravitation, Cambridge, 2003); High spin extensions (Kraus and Perlmutter, 2011, allowing for finite truncations Bin Chen et al, many works done). Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 5 / 52

6 Hairy BH: why scalar hair? 41 years ago (1971), Carter and Isreal proposed the black hole no hair theorem; Nonetheless, hairy BHs with free parameters which does not obey Gauss s law are analogues of excited states of atoms in atomic physics, which are of course interesting object to study; Of all possible choices of extra hairs, a scalar field is the most economic choice, and is perhaps the most interesting choice, because of potential applications in testing AdS/CFT, gravity/fluid dual as well as holographic phase transitions; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 6 / 52

7 Hairy BH: why scalar hair? An alternative choice of hair: ideal fluid equation of state parameter. However, it turns out that there is NO static, spherically symmetric black hole solutions in Einstein gravity with an ideal fluid source with a constant equation of state parameter in (3+1)-dimensions (Semiz, 2008); In (2+1)-d, BH solutions seem to exist with ideal fluid source (with some complicated EoS, however W Xu). This will not be the main concern of this talk; The cases of other dimensions remains to be explored; The (2+1)-d version of BH with scalar hair is extremely easy to find, and yet its importance does not reduce in view of applications. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 7 / 52

8 Hairy BH: why scalar hair? Possible ways of including a scalar hair: Minimally coupled scalar hair; Conformally coupled scalar hair; General non-minimally coupled scalar hair; Conformal transformations may turn one way of coupling to another, however, the physics needs not be equivalent; The scalar potential is solely determined by the consistency of the dynamics up to some constant parameters. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 8 / 52

9 A static charged hairy black hole Consider the following action (ξ = 1 8 corresponds to conformal coupling): I = 1 d 3 x g [R g µν µ φ ν φ ξrφ 2 2V (φ) 14 ] 2 F µνf µν The scalar φ couples non-minimally through the Rφ 2 term to gravity; We did not put in a cosmological constant term by hand, but it will emerge naturally as a constant term in the scalar potential; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 9 / 52

10 A static charged hairy black hole Field equations: where V φ = φ V (φ), T [φ] g µν µ ν, G µν T [φ] µν T [A] µν + V (φ)g µν = 0, φ ξrφ V φ = 0, ν ( gf µν ) = 0, µν = µ φ ν φ 1 2 g µν ρ φ ρ φ + ξ ( ) g µν µ ν + G µν φ 2, T µν [A] = 1 (F µρ F ρ ν 14 ) 2 g µνf ρσ F ρσ. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 10 / 52

11 A static charged hairy black hole We assume the metric takes the following form, ds 2 = f(r)dt f(r) dr2 + r 2 dψ 2, where the coordinate ranges are given by < t <, r 0, π ψ π; The Maxwell equation is linear and can be easily solved in any static spherically symmetric spacetime. In (2+1) dimensions, the solution reads ( ) r A µ dx µ = Q ln dt, r 0 where Q R, r 0 specified the zero potential surface (which can be set to + ; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 11 / 52

12 A static charged hairy black hole The (tt) and (rr) components of the Einstein equation give rise to which yield 3 ( d dr φ (r) ) 2 φ (r) d2 1 φ(r) = ±. k r + b φ (r) = 0, dr2 This is determined without specifying the scalar potential! Inserting φ(r) into the rest of the field equations enables us to determine f(r) and V (φ) simultaneously; Unlike flat spacetime field theories, consistency of Einstein s equation imposes severe constraints on the allowed form of self-interactions for the scalar field! Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 12 / 52

13 A static charged hairy black hole Significant simplifications will occur if the two parameters k and b in φ(r) are merged into one: 8B φ(r) = ± r + B. In this case, we have ) ) ( f(r) = (3 β Q2 + (2β Q2 B r Q2 2 + B ) ln(r) + r2 3r l 2, where β and l are integration constants; The metric contains 4 parameters B, β, l, Q, which are related to the scalar field, mass, cosmological constant and electrostatic charge respectively. The mass M is given by β = 1 ( ) Q M. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 13 / 52

14 A static charged hairy black hole The scalar potential: The scalar potential turns out to be V (φ) = 1 l ( l 2 + β ) B 2 φ 6 1 ( ) Q 2 (192 φ B φ φ 6) + 1 ( ) [ ( Q 2 2φ 2 3 B 2 (8 φ 2 ) 2 1 ( B 8 φ 2 ) )] 1024 φ6 ln φ 2, The first term is clearly a (negative) cosmological constant term. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 14 / 52

15 A static charged hairy black hole We may split V (φ) into a sum V (φ) = 1 l 2 + U(φ), where U(φ) encodes the true self interaction of the scalar field φ; As φ 0, the leading term in the power series expansion of U(φ) behaves as O(φ 6 ), U(φ) 1 { l 2 + β B ( Q 2 B 2 ) [ 1 3 ( )]} 8B 2 ln φ 2 φ 6 + O(φ 8 ), where only even powers of φ are present and the coefficients of all the O(φ 8 ) terms are positive; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 15 / 52

16 A static charged hairy black hole If Q = 0, then U(φ) degenerates into a φ 6 potential, with coefficient ( ). β There is a single extremum and stability against small l 2 B 2 perturbations in φ requires β B2 ; l 2 If Q 0, U(φ) will possess more than one extrema: two minima at φ = ±φ min 0 and one maximum at φ = 0; The potential is perfectly regular at B = 0, even though it appears to be singular at B = 0. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 16 / 52

17 A static charged hairy black hole Figure : Plot of U(φ) versus φ, with B = 1, l = 1, β = 1 and Q = 1. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 17 / 52

18 A static charged hairy black hole Singularities: Essential singularity at r = 0 if Q 0: The Cotton tensor R = 36 r3 3 rq 2 l 2 + 2B Q 2 l 2 6l 2 r 3. C abc = c R ab b R ac ( br g ac c R g ab ) is nonvanishing if either B > 0 or Q 0, C trt = C ttr = 1 d3 f (r) 4 dr 3 f (r), C ψrψ = C ψψr = 1 ( ) d 3 4 dr 3 f (r) r 2. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 18 / 52

19 A static charged hairy black hole Special cases: BTZ black hole: ( No hair limit, i.e. B = 0, BTZ, 1992) f(r) = M Q2 r2 ln(r) + 2 l 2, ( ) r A µ dx µ = Q ln dt, V (φ) = 1 l 2, φ(r) = 0, r 0 Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 19 / 52

20 A static charged hairy black hole Static neutral hairy black hole: Q = 0, hence U(φ) φ 6 (Nadalini, Vanzo, and Zerbini, 2007) f(r) = ( 3 + 2B r A µ dx µ = 0, 8B φ(r) = ± r + B, V (φ) = 1 l ) β + r2 l 2, ( 1 l 2 + β ) B 2 φ 6. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 20 / 52

21 Conformally dressed black hole (Q = 0 and β = B2 l 2, U = 0, Martinez, 1996): ( f(r) = 3 + 2B r A µ dx µ = 0, 8B φ(r) = ± r + B, V (φ) = 1 l 2. ) B 2 l 2 + r2 l 2, Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 21 / 52

22 A static charged hairy black hole For Q = 0, f(r) possesses a single zero if β < 0. Recalling the stability requirement, we see that the physically acceptable range of β [ ] is β B2, 0 at Q = 0; l 2 If Q 0, the horizon structure is very difficult to analyze. For B = 0, the situation is a little simpler. Let r X = 1 2 Q l. If β < Q2 6 ln(r X), f(r) has two zeros, the outer horizon is the event horizon; If β = Q2 6 ln(r X), we have f(r X ) = 0, which corresponds to the horizon of a charged extremal AdS black hole without the scalar hair; If β > Q2 6 ln(r X), then f(r X ) > 0, there is no zeros for the function f(r), so the metric becomes an asymptotically AdS spacetime with a naked singularity at the origin, which is a physically uninteresting case. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 22 / 52

23 A static charged hairy black hole We can prove that provided β is negative enough, f(r) will always have two zeros (for any Q 0 and B > 0), so our solution indeed corresponds to a hairy black hole; The exact value of the horizon radius cannot be worked out analytically; Nonetheless, it can be seen that the horizon radius increases monotonically as B increases with fixed Q, or as Q 2 increases with fixed B. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 23 / 52

24 Rotating hairy black hole It is harder to get exact rotating solutions with electromagnetic source, so we omit the EM field: I = 1 d 3 x [ g R l 2 gµν µ φ ν φ 1 ] 8 Rφ2 2 U(φ) ; Field equations: G µν 1 l 2 g µν T [φ] µν + U(φ)g µν = 0, φ 1 8 Rφ φu(φ) = 0, T [φ] µν = µ φ ν φ 1 2 g µν ρ φ ρ φ + 1 8( gµν µ ν + G µν ) φ 2 ; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 24 / 52

25 Rotating hairy black hole Rotating metric ansatz: ds 2 = f(r)dt ( 2 f(r) dr2 + r 2 dψ + ω(r)dt) ; Consistency of Einstein equation yields (no constant φ is allowed!) which in turn gives ( ) 8B 1/2 φ(r) = ±, r + B f(r) = 3β + 2Bβ + (3r + 2B)2 a 2 r r 4 (3r + 2B)a ω(r) = r 3 ; + r2 l 2, Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 25 / 52

26 Rotating hairy black hole The scalar potential is determined by the dynamics itself: U(φ) = Xφ 6 + Y ( φ 6 40 φ φ ) φ 10 (φ 2 8) 5, where X = 1 ( l 2 + β ) B 2, Y = 1 ( ) a B 4. This potential behaves as φ 6 + O(φ 10 ) as φ 0; It is crucial to notice that the scalar potential cannot be prescribed arbitrarily: it is determined up to some constant parameters such as X and Y given above; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 26 / 52

27 Rotating hairy black hole In order that the scalar potential to be bounded from below, we may have the following choices for the constants X and Y : Y = 0, X 0; Y > 0, in which case X can be an arbitrary real number. In particular, If X 0, then U(φ) has only a single extremum, which is the minimum at φ = 0; If X < 0, then U(φ) has three extrema, i.e. one local maximum at φ = 0 with U(0) = 0, and two minima at φ = ±φ 0 for some φ 0, with U(±φ 0 ) < 0. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 27 / 52

28 Rotating hairy black hole Singularities: ( 30B 2 R = r B ) r 5 a 2 6 l 2, R µν R µν = 12 l 4 + 6B2 β 2 ( 324B 4 r 6 + r 12 ( 12B 3 β + r B2 β r B3 r B2 r 6 l B r 5 l B2 r 10 ) a 2 ; ) a 4 Conformal non-flatness: ( 12B 2 C ψrψ = r B ) r 4 a 2 + 3β B r 2. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 28 / 52

29 Rotating hairy black hole It β 0, then f(r) > 0 for all r [0, + ). So existence of horizon requires β < 0; For β < 0, f(r) + as both r 0 and r +. So f(r) must have at least one minimum, and the total number of extrema of f(r) must be odd; The condition for the extrema of f(r) is just the condition for the zeros of f (r), We see that f (r) = 2r l 2 2β B r 2 ( 18 r B ) r B2 r 5 a 2. f (0) =, f (+ ) = +. So, f (r) has to have some zero in the range r [0, + ); Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 29 / 52

30 Rotating hairy black hole We denote by r i the location of possible extrema of f (r) (which need not exist at all). At the hypothetic extrema, we should have f (r i ) = 2r i l 2 2β B ( 18 r 2 i r B ) i r B2 a 2, i r 5 i f (r i ) = 2 l 2 + 4β B ( B ) + 80B2 a 2 = 0. r 3 i r 5 i r 6 i It is easy to see that ( f (r i ) = f (r i ) + r ) i 2 f (r i ) r 4 i = 3r ( i 9 l 2 + r B ) i r B2 a 2 > 0. i r 5 i This means that if f (r) has some extrema, it must be positive at all extrema; f (r) might not has any extrema at all; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 30 / 52

31 Rotating hairy black hole In either cases the curve for f (r) will cross zero only once. Therefore f(r) has only one minimum at r min ; Depending on the values of the parameters, we may encounter one of the following three possibilities: an extremal rotating hairy black hole; a non-extremal rotating hairy black hole ; a naked singularity. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 31 / 52

32 Rotating hairy black hole Extremal black hole: (β = 2a l ) r ex = z + al z, z = [ al ( B + (B 2 al) 1/2)] 1/3, r ex (3al) 1/2 ; Non-extremal black hole : β < 2a l ; Naked singularity: β > 2a l ; Putting together, we have a bound of the mass-angular momentum ratio, β a 2 l, which is an analogue of the famous Kerr bound in (2+1) dimensions. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 32 / 52

33 Turning into minimal coupling Recall that the action reads I = 1 dx 3 [ g R l 2 gµν µ φ ν φ 1 ] 8 Rφ2 2 U(φ). Conformal transformation: ḡ µν = Ω 2 g µν, ) Ω = (1 φ2 8 tanh φ = φ 8, = r r + B, V ( φ) = 1 8 U(φ) ( φ2 ) 3 Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 33 / 52

34 Turning into minimal coupling Resulting action: Ī = 1 16πG d 3 x ḡ [ R 8 µ φ µ φ 16V ( φ) ], where we have restored 8G = 1; V ( φ) now takes the form V ( φ) = 64X sinh 6 1 φ 8l 2 cosh6 φ ( ) + 64Y sinh 10 φ tanh 6 φ 5 tanh 4 φ + 10 tanh 2 φ 9 ; No isolated constant term is present, however, V ( φ) φ 0 = 1 8l 2, so cosmological constant is still there; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 34 / 52

35 Turning into minimal coupling Solution: New radial coordinate: Scalar field: φ(ρ) = arctanh ρ = r2 r + B ; B H(ρ) + B, H(ρ) = 1 2 ( ρ + ) ρ 2 + 4Bρ ; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 35 / 52

36 Turning into minimal coupling The line element: ds 2 = Adt 2 + dρ2 B + ρ2 ( 2 dψ ω(h)dt), where r = H(ρ), ( ) H 2 A = f(h) H + B ( H + 2B B = f(h) H + B ) 2 f(h) = 3β + 2Bβ H + (3H + 2B)2 a 2 H 4 (3H + 2B)a ω(h) = H 3 ; + H2 l 2, Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 36 / 52

37 Turning into minimal coupling Horizon structure: Assume β 2a l. Write X 0 = 3H + 2B H 3. ρ=ρ+ Then f(h) = 0 = H2 ( 1 + X 0 β l 2 + X 0 2 a 2 l 2) l 2, leading to X 0 = 3H + 2B H 3 = 1 ρ=ρ+ 2 β l + β 2 l 2 4 a 2 a 2. l Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 37 / 52

38 Turning into minimal coupling The above cubic equation for H + = H ρ=ρ+ solutions: H (1) + = X X 0 X 1 H (2) + = 1 2 ( X1 X 1 = B + X X 1 ) ± I B 2 X 0 X 0 possesses analytic ( X1 X 0 1 X 1 2 X 0 At most two of these can be real and positive, the bigger one of these corresponds to the event horizon ρ + = H2 + H + + B. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 38 / 52 1/3. ),

39 CFT dual in the minimal coupling picture Asymptotic behaviors φ = B1/2 ρ 1/2 2 B 3/2 3 ρ 3/2 + O(ρ 5/2 ), g ρρ = l2 ρ 2 4l2 B ρ 3 + O(ρ 4 ) g tt = ρ2 l 2 + O(1) g tρ = O(ρ 2 ) g ψψ = ρ 2 + O(1) g ψρ = O(ρ 2 ) g tψ = 3a 4aB + 9aB2 + O(ρ 5 ) ρ 2 ρ 3 ρ 4 Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 39 / 52

40 CFT dual in the minimal coupling picture Regge-Teitelboim approach: { [ ] } 1 2 δq G (ξ) = dψ ξ 16πG lρ δg ψψ + ρg 1/2 g ρρ δg ρρ + 2ξ ψ δg ψt { } 1 1 = dψ 16πG lρ ξ δg ψψ 2ρξ δ(g 1/2 ) + 2ξ ψ δg ψt, δq φ (ξ) = 1 πg [ ] dψ ξ g 1/2 g ρρ ρ φδφ, (Asymptotically, ξ ξ t ρ/l). Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 40 / 52

41 CFT dual in the minimal coupling picture Total charge: Q = Q G + Q φ Q(ξ) = 1 { ( ξ dψ (g ψψ ρ 2 ) 2ρ 2 (lg 1/2 1) 16π ρl [ + 4ρ 2 φ ] ) } 3 φ4 + 2ξ ψ π ρ ψ ; Central charge: Mass and angular momentum: c = 3l 2G ; M = Q( t ) = 3β, J = Q( θ ) = 6a; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 41 / 52

42 CFT dual in the minimal coupling picture Temperature: T = 1 A B 4π ρ ρ = 1 A B H 4π H H ρ = 1 ( ) ( H ( 4π H H + B )2 f(h) ( H + 2B ) (H + B) 2 H H + B )2 f(h) H(H + 2B) Beckenstein-Hawking entropy: S = 2πρ + 4G = 4πρ + = 4π H2 H + B ρ=ρ+ 64l (Ml + J ) 64l (Ml J ) = π + π Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 42 / 52

43 CFT dual in the minimal coupling picture First law: Smarr relation: dm = T ds + Ω dj, Ω dψ/dt = ω(h) H=H+ M ΩJ = 1 2 T S = 3 2 β l2 B + 2 H β l 2 H (3 H + 2 B) l 2 The bound on mass-angular momentum ratio is actually M J 1 l 2.. ρ=ρ+ Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 43 / 52

44 CFT and entropy in the non-minimal coupling picture Padmanabhan s Noether current method: [ ] CFT arises from Noether current associated with the diffeomorphism invariants of the York-Gibbons-Hawking surface term of the gravitational action rather than associated with the Einstein-Hilbert action as usual; The result is solely determined by the surface action and the near horizon geometry and is insensitive to the scalar potential; The procedure is also insensitive to the dimension of spacetime; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 44 / 52

45 CFT and entropy in the non-minimal coupling picture Consider a general Lagrangian density on the boundary: gl = g a A a ; Under a diffeomorphism x a x a + ξ a the left hand side changes by: δ ξ ( gl) L ξ ( gl) = g a (Lξ a ) ; The variation of the right hand side is given by: δ ξ ( g a A a ) = g a [ b (A a ξ b) A b b ξ a] ; Equating both variations, we get J a [ξ] = Lξ a b ( A a ξ b) + A b b ξ a = b [ξ a A b ξ b A a]. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 45 / 52

46 CFT and entropy in the non-minimal coupling picture Pure gravity: Boundary action: I B = 1 8πG Noether current: M σd 3 xk = 1 gd 4 x a (KN a ) ; 8πG M J a [ξ] = a J ab = 1 ( 8πG b Kξ a N b Kξ b N a) ; Noether charge: Q [ξ] = 1 2 hdσab J ab ; Σ where dσ ab = d (d 2) x (N a M b N b M a ); Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 46 / 52

47 CFT and entropy in the non-minimal coupling picture Commutator: [Q 1, Q 2 ] 1 2 (δ ξ 1 Q[ξ a ] δ ξa Q[ξ 1 ]) = 1 [ ] hdσab ξ a 2 2J b [ξ 1 ] ξ1j a b [ξ 2 ] ; Σ Mode expansion and final results: Q m = A κ 8πG α δ m,0; [Q m, Q n ] = iκa 8πGα (m n)δ m+n,0 im 3 αa 16πGκ δ m+n,0; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 47 / 52

48 CFT and entropy in the non-minimal coupling picture Gravity with non-minimally coupled scalar ((3+1)-d as an example): Action: I = M Boundary term: d 4 x [ R 2Λ g 16πG 1 2 gµν µ φ ν φ 1 ] 12 Rφ2 V (φ) ; I B = 1 8πG M d 3 x γ (1 16 ) 8πGφ2 K; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 48 / 52

49 CFT and entropy in the non-minimal coupling picture Charge and commutator: Q m = πGφ2 (r h ) κa 8πG α δ m,0, [Q m, Q n ] = πGφ2 (r h ) 8πG [ iκa α δ m+n,0 im 3 αa 2κ Central charge and the zero mode energy: c 12 = A ( α 1 1 ) 16πG κ 6 8πGφ2 (r h ), Q 0 = A ( κ 1 1 ) 8πG α 6 8πGφ2 (r h ) ; ] ; Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 49 / 52

50 CFT and entropy in the non-minimal coupling picture Entropy via Cardy formula: CQ0 S = 2π 6 Wald entropy: where δl δr µνρσ S W ald = 2π L R µνρσ + = A ( 1 1 ) 4G 6 8πGφ2 (r h ) ; d 2 x h δl ɛ µν ɛ ρσ, δr µνρσ ( ( 1) n ξ1... ξn n=1 S = A ( 1 1 ) 4G 6 8πGφ2 (r h ). L (ξ1... ξn)r µνρσ CFT result supports Wald entropy, but not Beckenstein entropy! ) + Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 50 / 52

51 CFT and entropy in the non-minimal coupling picture Comments on the two realizations of CFT dual: Padmanabhan s method is fast, mostly model independent and so reveals the universality of the holographic dual of gravitation to CFT, however it relies on the use of a particular choice of near horizon geometry and hence its use is subject to the availability of such a choice of geometry; Regge-Teitelboim approach makes use of the asymptotic behavior of the metric and thus is dependent of an underlying background geometry. Nonetheless the approach makes no difference for static and rotating backgrounds and so is reliable when novel solution is involved when considering CFT dual. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 51 / 52

52 Thank you. Liu Zhao School of Physics, Nankai University (2 Collaborators: + 1)-dimensional Wei black Xu, Kun holes Meng, with Bin a nonminimally Zhu coupled scalar hair 52 / 52