Hawking Radiation from Black Holes of Constant Negative Curvature via Gravitational Anomalies

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1 Hawking Radiation from Black Holes of Constant Negative Curvature via Gravitational Anomalies Petros Skamagoulis work with E. Papantonopoulos Phys. Rev. D 79, 0840 (009) [arxiv: [hep-th]] Department of Physics National Technical University of Athens MG1 Meeting, Paris July 15, 009 Petros Skamagoulis (NTUA) MG1 Meeting, Paris 0 / 14

2 Outline 1 Introduction-Motivations Hawking radiation from topological black holes Topological black holes (TBHs) Wave equation in the background of a TBH Dimensional reduction Hawking radiation 3 Hawking radiation from a TBH conformally coupled to a scalar field 4 Conclusions Petros Skamagoulis (NTUA) MG1 Meeting, Paris 1 / 14

3 Introduction-Motivations Introduction-Motivations Hawking radiation from the cancellation of gauge and gravitational anomalies near the event horizon [S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, (005)] [S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. Lett. 96, (006)] Simplification for asymptotically flat spacetimes [R. Banerjee and S. Kulkarni, Phys. Lett. B 659, 87 (008)] Motivations 1 Black holes of non-spherical topology Black hole of non-spherical topology conformally coupled to a scalar field 3 Asymptotically non-flat spacetimes Petros Skamagoulis (NTUA) MG1 Meeting, Paris / 14

4 Hawking radiation from TBHs Topological Black Holes (TBHs) Topological black holes (TBHs) Action in asymptotically AdS 4. Black hole µ > l 3 3 and I = 1 16πG d 4 x g [R + 6l ], ds = f(r)dt + 1 f(r) dr + r dσ, f(r) = r l 1 µ r. is a constant related to the mass of the black hole as ( M = µ + l/3 ) 3 ( g 1), dσ = dθ + sinh θdϕ, with θ 0 and 0 ϕ < π, is the line element of the two-dimensional manifold Σ Σ = H /Γ. Σ is a compact surface of constant negative curvature and of genus g. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 3 / 14

5 Hawking radiation from TBHs Topological black holes (TBHs) The simplest manifold Σ is a compact surface of genus two Fig. from [R. B. Mann, arxiv:gr-qc/ ] The horizons f(r) = r l 1 µ r = 0. l 3 < µ < 0 : an inner horizon r and an outer horizon r+. 3 µ 0 : one horizon r h. The horizons have the non-trivial topology of the manifold Σ. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 4 / 14

6 Hawking radiation from TBHs Wave Equation of a Scalar Field in the Background of a TBH Wave equation of a massive scalar field Φ [ 1 f t + 1 ] ( r r r ) 1 f r + r sinh θ ( ) 1 θ sinh θ θ + r sinh θ ϕ Φ = m ΦΦ. Separation of variables Wave modes where Φ ξm = R ξ(t, r) P m r 1 ±iξ(cosh θ)eimϕ. m = 0, ±1, ±, ±3,... [N. L. Balazs and A. Voros, Phys. Rept. 143, 109 (1986)] Σ = H /Γ is a compact surface of genus g =. Discrete spectrum. ξ 0 takes discrete real values. Generally for genus g >, there are no analytical results for the angular eigenvalues and for the angular eigenfunctions. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 5 / 14

7 Dimensional Reduction Hawking radiation from TBHs Dimensional reduction Consider matter given by a complex scalar field φ(x) in the background of a TBH of genus g =. Action S = S free + S int. S int includes a mass term, potential terms and (self-)interaction terms. S free is the free part of the action Partial wave decomposition S free = 1 φ(x) = + m= d 4 x gφ φ. R ξm (t, r) Yξ m (θ, ϕ), r where Y m ξ (θ, ϕ) ( ) 1 π Γ(iξ + 1 ) ξ tanh(πξ) Γ(iξ + m + 1 ) P m 1 +iξ(cosh θ)eimϕ. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 6 / 14

8 Hawking radiation from TBHs Dimensional reduction Substitute the partial wave decomposition in S free, S int and use the properties of the Y m ξ. Transform to the tortoise coordinate dr dr = 1 f(r). Consider a region near the outer event horizon r H. In this region r(r ) Ae κr + r H, f(r(r )) κae κr. In the original coordinates S = m= 1 [ dtdrrξm 1 ( ) ] f t + r f r R ξm. Physics in a region near the event horizon can be described by an infinite collection of (1+1)-dimensional free massless complex scalar fields, each propagating in a (1+1)-dimensional spacetime ds = f(r)dt + 1 f(r) dr. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 7 / 14

9 Hawking Radiation Hawking radiation from TBHs Hawking radiation Ignore the ingoing (left-moving) modes in the region near the event horizon. A gravitational anomaly appears [L. Alvarez-Gaume and E. Witten, Nucl. Phys. B 34, 69 (1984)] [R. A. Bertlmann and E. Kohlprath, Annals Phys. 88, 137 (001)] [S. A. Fulling, Gen. Rel. Grav. 18, 609 (1986)] µ T µν = The ν = t component is written as r ( T r t ɛ νµ 96π g () µr (). ) Ñ t r = 0, where Solution Ñt r = 1 (ff f ). 96π T r t (r) = a H + Ñ r t (r) Ñ r t (r H). Boundary condition [S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74, (006)] T r t (r H) = 0. Therefore T r t (r) = Ñ r t (r) Ñ r t (r H). Petros Skamagoulis (NTUA) MG1 Meeting, Paris 8 / 14

10 Hawking radiation from TBHs Hawking radiation In the asymptotic limit r ɛ νµ 96π g () µr () 0, r Ñ r t 0, The energy flux at infinity is Ñ r t l 48π. F = T r t (r ) Ñ r t (r ) = Ñ r t (r H), which equals to F = 1 19π f (r H) = π ( κ ). 1 π A beam of massless blackbody radiation of temperature T moving in the positive r direction has a flux Φ = π 1 T. The flux is equivalent to blackbody radiation with a temperature T = κ/π. Identified with the Hawking temperature of the (3+1)-dimensional TBH of genus g = [L. Vanzo, Phys. Rev. D 56, 6475 (1997)]. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 9 / 14

11 Hawking radiation from the MTZ black hole TBH Conformally Coupled to a Scalar Field (MTZ) Action [C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 70, (004)] I[g µν, Ψ] = d 4 x [ R + 6l g 16πG 1 gµν µψ νψ 1 1 RΨ πg ] 3l Ψ4. Black hole with scalar hair ds = f(r)dt + 1 f(r) dr + r ( dθ + sinh θdϕ ), ( f(r) = r l 1 + Gµ ), r 3 Gµ Ψ(r) = 4πG r + Gµ, µ > l/4g. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 10 / 14

12 Hawking radiation from the MTZ black hole Horizons µ 0 l/4 < Gµ < 0 r + = l ( Gµ ). l r = l ( Gµ ) l r = l ( Gµ ), l r + = l ( Gµ ). l r and r + are event horizons, while 0 < r < Gµ < r < l/ < r +., Petros Skamagoulis (NTUA) MG1 Meeting, Paris 11 / 14

13 Hawking radiation from the MTZ black hole Dimensional Reduction for the MTZ black Hole Consider a scalar field φ(x) in the background of the MTZ black hole of genus g = S = 1 d 4 x gφ φ + S int, where there is no interaction of φ(x) with Ψ(r). Perform a partial wave decomposition of φ in terms of the Yξ m. The dimensional reduction procedure proceeds as previously. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 1 / 14

14 Hawking radiation from the MTZ black hole Hawking Radiation from the MTZ Black Hole Ignore the ingoing modes in the region near the event horizon. The ν = t component of the gravitational anomaly ( ) r T r t Ñ t r = 0, Boundary condition In the asymptotic limit r ɛ νµ 96π g () µr () 0, r Ñ r t 0, Ñ r t (r) l 48π. The energy flux at infinity is F = T r t (r ) Ñ r t (r ) = Ñ r t (r H), F = 1 19π f (r H) = π ( κ ). 1 π This form is equivalent to blackbody radiation with temperature T = κ/π. Identified with the Hawking temperature of the MTZ black hole. [C. Martinez, J. P. Staforelli and R. Troncoso, Phys. Rev. D 74, (006)] [M. Nadalini, L. Vanzo and S. Zerbini, Phys. Rev. D 77, (008)] Petros Skamagoulis (NTUA) MG1 Meeting, Paris 13 / 14

15 Conclusions Conclusions Modification of the covariant anomaly method for asymptotically non-flat spacetimes & application to a TBH of genus g =. Application to a TBH of genus g = conformally coupled to a scalar field. The Hawking flux is universally determined only by the value of the gravitational anomaly on the event horizon. The backscattering of the Hawking radiation is ignored and its thermal spectrum is not proved. Petros Skamagoulis (NTUA) MG1 Meeting, Paris 14 / 14