Generalized Painlevé-Gullstrand metrics

Transcription

1 Generalized Painlevé-Gullstrand metrics Chopin Soo Dept. of Physics, Nat. Cheng Kung U., Taiwan. ef: C. Y. Lin and CS, arxiv: ; Phys. Lett. B631 (in press) APCTP-NCTS Int. School/Workshop on Gravitation and Cosmology (Jan. 0, 009).

2 Paul Painlevé ( ) mathematician and politician : Grand Prix des Sciences Mathématiques (1890) Prix Bordin(1894), Prix Poncelet (1896) Member, geometry section of the Académie des Sciences Prime minister of France (the 84 th : 1 Sept. 16 Nov.,1917; & the 9 nd : 17 Apr. 8 Nov.,195) Allvar Gullstrand ( ) Nobel Prize ( 1911) (in Physiology or Medicine) (for his work in in Physical & Physiological Optics) member, Nobel Physics Committee ( ), and its Chairman ( ). "Einstein must never receive a Nobel Prize, even if the whole world demands it"

3 Generalized Painlevé-Gullstrand (PG) metrics General spherically symmetric metric ds = N(, T)dT + f 1 (, T)d + dω Note: Lorentzian signature = N(, T)/f(, T) = e Φ(,T) > 0. PG time-coordinate: dt p e I(T,r) (dt + βdr), g(r) (r)/l ; with g (r) dg(r) dr ds = Ne I dt p + βne I dt p dr + ( L f 1 g β N ) dr + L g (r)dω For dt P to remain a perfect differential, introduce integrating factor e I, with the requirement r e I = T (e I β); uniquely specifies e I given its initial value e I(r,T o). Choice of β = L(Nf) 1 g f = Spatial metric on constant-t P 3-dim. hypersurfaces is: L ( dr + g (r)dω ) ; And 4-dim. spacetime metric (Generalized PG metric): ( ds GPG = Nf 1 g e I dt p + g 1 d + dt p N 1 1 f e I g f) + dω = Nf 1 g e I dt P + (Ldr + dt P N 1 1 f e I g f) + L g (r)dω 1

4 ds = Nf 1 g e I dt p + (g 1 d + dt p N 1 1 f e I g f) + dω emarks: Note: N(, T)/f(, T) = e Φ = metric can be regular even when f 0 at the "horizon(s)". Solutions of Einstein s Eqs. with N() = f(); e I = 1 Conventional choice of g(r): L g(r) = r; g (r) = 1 = dt p = dt + f 1 1 fd usual" PG metric with (spatially flat) 3-dim. hypersurfaces: ds PG = dt p + [ d + dt p 1 f ] + dω Special case: Schwarzschild solution: f() = 1 GM (original PG metric). A particular parametrization is: f(, T) = 1 GM(,T) with Misner-Sharp mass. FOLK THEOEM: Always possible to choose spatially flat slicings i.e. g(r) /L = r, g (r) = 1. Works even when N(, T) f(, T) i.e. A spherically symmetric metric can always be reduced to the spatially flat PG form. Proof": look at the metric above. What s the problem with the folk theorem?

5 Lorentz boost ds = N(, T)dT + f 1 (, T)d + dω = Nf 1 g e I dt p + (g 1 d + dt p N 1 1 f e I g f) + dω = η AB e A e B Vierbein 1-forms are related by a local radial Lorentz boost ( e 0 = g e I ) ( dt P e 3 = g 1 d + coshξ sinhξ g fe I = dt P sinhξ coshξ GPG )( e 0 = N 1/ dt e 3 = f 1/ d ) Standard tanhξ = g 1 g f ( f 0 = 1) ξ = rapidity of the boost. Criterion for EAL PG variables and PHYSICAL Lorentz boosts is: g f 0 Cannot be satisfied for generic spherically symmetric metrics if we also demand spatially flat slicings (g = 1). ( obstruction to spatial flatness). 3

6 Explicit examples of the problem with spatial flatness. Consider time-indpt. case. ds = Nf 1 g dt p + (g 1 d + dt p N 1 f 1 g f) + dω L g(r) = r, g = 1; N() = f() ds = dt p + [ d + dt p 1 f ] + dω Schwarzschild-anti-deSitter N = f = 1 GM Λ 3, Λ < 0. [ ] GM ds = dt p + d + dt p + Λ + dω 3 Unphysical" (complex) variables for > c = 3 6GM Λ eissner-nordström N = f = 1 GM ( ds = dt p + + Q, d + dt p GM ) Q + dω, Unphysical" (complex)variables for < c = Q GM Misner-Sharp mass function" becomes negative Note: No problem for original PG metric N = f = 1 GM for Schwarzschild soln. esolution: Spatial flatness is too strong a demand! Give up spatial flatness. Criterion (g f) 0 can be always satisfied by choosing appropriate g(r) to give troublefree GENEALIZED PG metrics which are however NOT always spatially flat. 4

7 Generalizations beyond spatially flat slicings:(explicit examples) Schwarzschild-(anti-)deSitter metrics N = f = 1 GM Λ 3 sinr (k = +1, elliptic) L g(r) = r (k = 0, flat) g (r) = 1 kl sinh r (k = 1, hyperbolic); ds = ( 1 kl ) dt p+ [ (1 kl ) 1/ d + dtp Constant curvature slicings with 3-dim. icci scalar 3 = 6k/L ( GM Λ + 3 k ) ] L + dω Criterion (g f) = GM + ( Λ 3 ) k L 0 can be guaranteed iff Λ 3 k L. for Λ 0 spatially flat (k = 0) slicings can be attained, but for the anti-desitter case, hyperbolic (k = 1) 3-geometry is needed. Note that for Λ > 0, all spatial topologies k = 0, ±1 are allowed, but for k = 1 the range of is governed by L = sinr 1, yielding L which can be as large as needed. N metric with f = 1 GM ) ds = (1 + O dt p + + Q, can choose g(r) = r L O = /L ( ) GM + O d + dt p + O Q + dω, Criterion holds for O > Q. Constant-t p 3-dim. hypersurfaces characterized by eigenvalues of 3-dim. icci tensor 3 i j : λ i=1,,3 = (0, 0, O ). (O parametrizes deviation 4 from spatial flatness). 5

8 Some applications/remarks: Classical G: Classical solutions and their Generalized PG forms. {Spherical symmetric metrics} {Axially symmetric metrics e.g. Kerr-Newmann-(anti)deSitter}... = PG form of axially symmetric metrics must go beyond spatially flat slicings as well. Canonical formulation and canonical quantization: Spherically symmetric sector cannot be trivialized as spatially flat by choosing PG coordinates. ather, spatial metric on constant-t P 3-dim. Cauchy hypersurfaces (even with choice of (Gen.)PG coordinates) is: L ( dr + g (r)dω ). Elimination of spurious contributions in computation of Hawking radiation as tunneling via Parikh-Wilczek method. Parikh-Wilczek treatment of Hawking radiation: Hawking radiation treated as tunneling across the horizon from i to f of massless semiclassical s-wave emission with energy ω, and the black hole with initial mass parameter M shrinks by an amount ω maintaining energy conservation (assume simple back reaction: M to M ω and the form of the metric is preserved). Use Gen. PG metric satisfying Einstein s equations with N() = f(): outgoing particles follow null geodesic: Ṙ d dt P = g (g ) g f. Decay rate comes from the imaginary part of the particle action which is associated with I = f i p d = f i ( p 0 dp )d = f i H0 ω H 0 dh Ṙ d. In the last step Hamilton s equation, dh dp = 6 Ṙ, for the semiclassical process is invoked.

9 Switching the order of integration, together with dh = dω, yields I = = ω 0 ω 0 f d Ṙ ( dω ) g + g f i g d( dω ), f i f f in the integrand is evaluated at M ω ; and the pole is located at the horizon through which the tunneling occurs i.e. at h with f( h ) M ω = 0. Integral over is defined by deforming the contour to go through infinitesimal semicircle = h + ǫe iθ around the pole a Imaginary part is then Im d g + g f i g f f π = lim dθǫe iθ g + ǫe iθ g + g f + ǫe iθ (g f) ǫ 0 π g f + ǫe iθ ( g f + g f) = π, f() h (ω ) h (ω ) provided g f remains real. Otherwise, spurious contributions whenever g f 0 is violated. Final result ω Im I = dω π f() 0 h (ω ) governing the decay rate is, remarkably, independent of g(r), and hence universal". (1) a A positive decay rate is associated with clockwise traversal of the semicircle in the contour. 7

10 Change of the Bekenstein-Hawking entropy from S = Im I (note I(ω)) yields, at the lowest order, the temperature from the first law T eff. S = ω agrees with the Hawking temperature T H κ π = 1 4π f h ; BUT exists deviations from pure thermal physics indicated by higher order corrections in Im I of (1) are displayed below. Schwarzschild eissner-nordström Schwarzschild (anti-)desitter f = 1 GM f = 1 GM + Q f = 1 GM Λ 3 dim I dω 8πG (M ω) π (G(M ω)+ ) G (M ω) Q G (M ω) Q 8πΛ 1/ sin[ 1 3 arcsin (3G(M ω) Λ)] 1+cos[ 3 arcsin (3G(M ω) Λ)] Table 1: In the table, values of dim I dω for various spherically symmetric spacetimes are shown for tunneling through the outer horizon of N metric, and through the black hole horizon for the others. For the SAdS case, the expression Λ should be understood as i Λ. Note also that S = ImI as evaluated from (1) coincides with the computation from the area law S = A 4G = π h G. 8