E.T. Akhmedov, T. Pilling, D. Singleton, JMPD 17. (2008)

Transcription

1 L. Parker, S. A. Fulling, PD 9, (1974) L.H. Ford, PD 11, (1975) J. S. Dowker,. Critchley, PD 13, (1976) D. Hochberg,T. W.Kephart, PD 49, (1994) J. G. Demers,.Lafrance,.C.Myers, CM PD 5, (1995) E.T. Akhmedov, T. Pilling, D. Singleton, JMPD 17. (008)

2 Concepts The spectrum of the Laplace Beltrami operator and therefore also the zero point energy ω [ ] of a g μν n (scalar) quantum field depend on the background metric. (Casimir effect). The zero point energy diverges, but its change with certain variations in the background metric can (and physically ought to be) finite for instance for an Extremal eissner Nordström (E N) background with conformally coupled scalars (classical BBMB solution). Change in zero point energy of quantum fields enters the relation between mass and radius of the BH. We compute the leading semi classical (WKB) contribution to this zero point energy due to periodic classical orbits. To enhance the importance of the zero point energy consider N identical conformally coupled scalars formally avoids complications with G.

3 The Setup I will consider the change in vacuum (zero point) energy due to the formation of a black hole. What should the black hole be made of? Dust? Quantum fields? We should expect to encounter terrible divergences unless we follow a physically plausible, if idealized d scenario. I.e., if the BH forms from dust, we would have to address (renormalize) some of the properties of this material too complicated. Simplest consistent scenario: a black hole is formed by scalar fields Indeed: A self consistent BH solution of classical G was found in the 1970 s : Bochorova Bronnikov B Melnikov Bekenstein black hole

4 eissner NordströmBH ess e odstö The eissner Nordström metric: () / (), () (1 )(1 ), r r For general is a static and spherically + > symmetric vacuum solution of the Einstein Maxwell equations. It describes a BH of mass, 4 π M c = ( + ) = E, + ds = v r c dt dr v r r d Ω vr = BH κ + vanishing angular momentum and electric charge Q = 8 π κ + ; electric field BH E= r ˆ φ = ˆ Q r r, r

5 Classical BH Thermodynamics Bk Bekensteinand Hawking interpreted changes in the (classical) mass of a BH thermodynamically, 4ππ debh = ( d+ + d ) = TdS +ΦdQ κ With BH entropy: π k 4π k S = ABH = + cκ cκ Temperature: c + T = 4 k Potential: Q Φ = = π + 8 κ π + + No Hawking radiation from E-N + = = Note: Although T and S depend on, the free energy is entirely classical! Demers, Lafrance, Myers 1995: Free energy of E-N is not renormalized by scalars. Why?

6 G coupled to N (massless) scalars (0) 1 4 μν Γ = dx g g φ, μφ, ν ξφ κ 1 G = g = κ T φ = ξφ μν μν μν μν 1 ; α T μν = φ, μφ, ν gμνφ; αφ + ξ[ gμν g φ φ; μν + Gμνφ ] Global SO(N) symmetry : consider classical solution for which all but one, φ, of the N scalars vanish: ξ = 0 (minimal) Schwarzschild: = 0, φ=const. ξ= (conformal) Extremal N: = =, φ = g 6 κ r Th E N l G ith f l l d SCALA The E-N solves G with conformal coupled SCALA= (Bochorova-Bronnikov-Melnikov-Bekenstein) black hole.(1970 s)

7 1 loop effective action ln Det [ ] ct's with, (1) (0) 1/ Γ =Γ + N g + N = N + 4 t' ( 5 μν μνρσ = d x g ZΛ + Z+ Z ) μν + μνρσ ( ) ct's ( ) No logarithmic 1-loop divergence for backgrounds with μν μνρσ d x g + ) = 0! 4 5 ( μν μνρσ Makes sense? to compute (finite) Casimir contributions to the mass of a BBMB black hole, e.g. 8π ME Nc = + κ N c χ χ > Q:? < 0

8 Partial Wave Analysis = = 1/ 1 i 4 ln Det [ 0 ] = Tr ln[ 0] = d x d λ G( x, x; λ) Gxy i y x G xy G xy yx x y 4 (, ; λ) = = (, ; λ) 0(, ; λ) with = δ ( ) λ+ λ+ 0 PWA : l * dε () l (, '; λ ) = lm ( Ω ') lm ( Ω ) (, '; λ, ε )exp[ ε ( ' )] l= 0 m= l π G xx Y Y G () rr i ct t 1/ ic n πinl ( l) ln Det [ ] = t ( 1) dε e ldl r dr dλg ( r, r; λ, ε) π n= Where the Poisson identity l= 0 n (l+ 1) f( l( l+ 1)) = ( 1) e f( l ) dl was used. n= 1/ π in l Proceed with WKB-approximation

9 Semiclassics. The radial Green s functions satisfy: ε l 1 ( l ) iδδ ( r r ') ( λ + ( ) ) (, ';, ) rrvr r G rr λ ε = vr () r r r ( l) ( l) ( l) Writing: G ( r, r'; λ, ε) = A ( r, r'; λ, ε) exp[ is ( r, r'; λ, ε)] ε l ( l) 1 ( l) 1) λ = vr ( )( rs ) + ( ) ( ) rrvr ra l vr ( ) r A r ( l) ( l) ( l) ) A δ ( r r') ) = WKB-approx: r( A ) r v( r)( rs )) => HJ-equ. ( l ) 1 dz G ( r, r; λ, ε)~ exp ( ;,, ) i k z ε l λ r k( r; ε, l, λ ) v( z) r l k r l vr r ( ; ε, l, λ ) = ε + λ ( )

10 l l k ( r; ε, l, λ) ε = + λ v( r) [ α V ( ; )] = eff x= β r r ε λ with α =, β = l l V x = v x x + β =Unstable circular orbit eff ( ) ( )( ) β = β c = 1 8 Horizon E-N β = 0 Schwarzschild β = 0

11 E Cas dz iπ n + ir k ( z ; ε,, λ) r vz ( ) i c n e ~ ( 1) dε d dr dλ π kr ( ; ε,, λ) n= n Saddlepoint evaluation of integrals about (unstable) critical trajectories with λ = 0, r =, l = ε gives: χ = (preliminary) The smallest E-N black hole: M m N ~1 1. N μ g min P 35 min 0.05 P N ~ N cm

12 Conclusion and Speculation Zero point energy tends to increase the mass (energy) of small black holes and may prevent them from forming (much) below Planck scales. o The smallest mass E N black hole, here estimated semiclassically, compares well (qualitatively and quantitatively) with a Loop Quantum Gravity approach by L. Modesto (ArXive: ) who finds that a ~ 01m 0.1 m P Schwarzschild black hole is stabilized by graviton fluctuations ( ). N = o The accuracy of a semiclassical calculation could be doubted for such tiny BH. But dimensional i analysis determines the form of the quantum correction to the BH mass and the dimensionless coefficient may be estimated for large (or N), where semiclassics generally would be trusted. M μg o Could BH grow by emitting negative energy Hawking radiation thereby lowering the zero point energy of quantum fields.? Is Dark Matter composed of stable primordial μg BHs?