Hawking Radiation as Quantum Tunneling

Transcription

1 Hawking Radiation as Quantum Tunneling Sang Pyo Kim Department of Physics, Kunsan National University, Kunsan , Korea and Asia Pacific Center for Theoretical Physics, Pohang , Korea Taitung International School/Workshop on Cosmology and Gravitation: January 7-10, Taiwan 1

2 Contents Hawking Radiation and Hawking Radiation as Quantum Tunneling Schwinger Mechanism and Hawking Radiation Unruh Effect as Quantum Tunneling Tunneling Rate of Black Holes in Rindler Coordinate Based on S. P. Kim, JHEP 0711 (007) 048. S. P. Kim, Schwinger Mechanism and Hawking Radiation as Quantum Tunneling, arxiv: [hep-th], APPC10 talk.

3 I. Motivation Hawking discovered thermal radiations with the Hawking temperature from black holes by calculating the scattering amplitude of an incoming wave to an outgoing amplitude [Hawking, Comm. Math. Phys. 43 (1975)]. Recently, Parikh and Wilczek [Phys. Rev. Lett. 85 (000)] reinterpreted the Hawking radiation as quantum tunneling. As a particle has a negative energy just inside and a positive energy just outside the horizon, a virtual pair created near the horizon can materialize into a real pair with zero total energy, one particle on each side of the horizon, thus leading the Hawking radiation as tunneling through the horizon. Within quantum field theory, we substantiate the tunneling idea in black holes using the analogy with the Schwinger mechanism. As the Schwinger mechanism can be interpreted as the Unruh effect seen by a charged particle accelerated by the electric field, we may use the Rindler coordinate for the accelerated particle. In addition, as the Hawking radiation can be interpreted as the Unruh effect with the surface gravity, the Rindler coordinate may be used to describe the tunneling process of particles or quantum fields. II. Hawking Radiation and Quantum Tunneling Hawking Radiation Hawking discovered a thermal radiation from black hole [Hawking, Comm. Math. Phys. 43 (1975)]. The Bogoliubov transformation between the Fock basis near the horizon and the Fock basis at the spatial infinity ˆbk = α kk â k + β kk â k The Bogoliubov coefficient gives the Boltzmann factor β kk α kk = exp ( πω κ ) = exp ( ω T H ) where κ is the surface gravity and T H is the Hawking temperature. For instance, the Schwarzschild black hole has κ = f (r H ) and T H = f (r H ) 4π. 3

4 Hawking Radiation as Quantum Tunneling Parikh and Wilczek [Phys. Rev. Lett. 85 (000)] reinterpreted the Hawking radiation as quantum tunneling of particles or fields through the horizon. For a Schwarzschild black hole, the Painlevé coordinate is regular through the event horizon ds = ( 1 M r ) dt + M r dtdr + dr + r dω, where the Schwarzschild time t s is related with t = t s + r M Mr + M. r + M The radial null geodesics (s-waves) are given by ṙ = dr M dt = ±1 r, where the upper (lower) sign corresponds to outgoing (incoming) geodesics. The total mass of the black hole and an s-wave of radiation is conserved: M = (M ω) black hole + ω s wave. The imaginary part of the action for the s-wave outgoing positive energy particle which crosses the horizon outwards rout rout pr ImS = Im p r dr = Im dp rdr. r in r in 0 From the Hamilton equation, ṙ = (dh/dp r ) r, the imaginary part becomes M ω rout dr ω ImS = Im M r in ṙ dh = Im rout 0 r in The emission (tunneling) rate is dr 1 (M ω )/r ( dω ). P = e ImS = e 8πω(M ω ), ImS = 4πω ( M ω ). Quantum Field Theory Approach The Klein-Gordon equation or the motion of a massive particle in the black hole. 4

5 The background metric of the black hole with the event horizon (f(r H ) = g(r H ) = 0) ds = f(r)dt + dr g(r) + g ijdx i dx j. The classical action from the Hamilton-Jacobi equation g µν µ S ν S + m = 0, has the form (φ(t, r, x i ) = e is ) S = ωt + p(r)dr + J(x i ). The imaginary part of the action is given by ImS = Im dr ω fg(m + g ij J j J j ) = fg πω f (r H )g (r H ) = πω κ. The emission (tunneling) rate is the Boltzmann factor with twice the Hawking temperature P = e ImS = e ω/t, T = κ π = T H. The ratio of the emission rate to absorption rate is [Srinivasan and Padmanabhan, Phys. Rev. D 60, (1999)] P em = e 4ImS P ab = e ω/t H P ab. It was pointed out that the imaginary part is not invariant under canonical transformations [E. T Akhmedov, V. Akhmedova, and D. Singleton, Phys. Lett. B 64, 14 (006)]. III. Schwinger Mechanism and Hawking Radiation Analogy between Schwinger Mechanism and Hawking Radiation Brout, Parentani, and Spindel, Nucl. Phys. B 353, 09 (1991); Parentani and Brout, ibid. 388, 474 (199); Parentani and Massar, Phys. Rev. D 55, 3603 (1997); Brout, Massar, Parentani, and Spindel, Phys. Rep. 60, 39 (1995). Srinivasan and Padmanabhan, Phys. Rev. D 60, (1999). 5

6 A naive interpretation would be separation of virtual pairs by an electric field in the Schwinger mechanism and by the event horizon in the Hawking radiation. Another interpretation would be that virtual pairs accelerate by the electric field and these pairs would feel thermal state from the Minkowski vacuum with the Unruh temperature. T U = a π Likewise, the Hawking radiation may be interpreted as the Unruh effect of a static particle just outside the horizon with the acceleration of the surface gravity measured at the infinity [W. G. Unruh, Phys. Rev. D 14, 870 (1976)]. What is the Schwinger Mechanism? The one-loop effective action for charged fermions in the presence of a background electromagnetic field where D = γ µ µ + iqa µ. S (1) = i ln det(id m) = i ln det(d + m ) The one-loop effective action for fermions of spin 1/ in uniform electromagnetic fields by using the proper time method [Schwinger, Phys. Rev. 8, 664 (195)] where and L eff = F 1 [ s ds e m (es) Re cosh(esx) G 8π 0 s 3 Im cosh(esx) 1 ] 3 (es) F. F = 1 4 F µνf µν = 1 (B E ), G = 1 4 F µνf µν = E B, X = [(F + ig)] 1/ = X r + ix i. Simple poles at s n = (nπ)/(ex i ), (n = 1,, ) contribute the imaginary part ImL eff = 1 4π n=1 eg X i s n e m s n coth ( X r X i nπ ). 6

7 The imaginary part determines the vacuum persistence (vacuum-to-vacuum transition) 0, out 0, in = e V T ImL eff, Pair-production rate for fermions and bosons in a pure electric field (B = 0, X r = 0, X i = E) ImL eff = s + 1 8π 3 [( 1) (s+1)(n+1) ] ( ee n=1 n Tunneling Interpretation of Schwinger Mechanism ) e nπm ee. Casher, Neuberger, and Nussinov, Phys. Rev. D 0, 179 (1979); Neuberger, ibid. 0, 936 (1979). Dunne and Schubert, Phys. Rev. D 7, (005); Dunne, Wang, Gies, and Schubert, ibid. 73, (006). Virtual pairs of charged particles from vacuum fluctuations can be separated to become real pairs when the potential energy over the Compton wavelength is greater than or equal to the rest mass energy. (qe) [ h] m[c] m[c ] Complex Method for Schwinger Mechanism zero-temperature: SPK & D. N. Page, Phys. Rev. D 65, (00); 73, (006); 75, (007). finite-temperature: SPK & H. K. Lee, Phys. Rev. D 76, Two dimensional case: A constant electric field along the x-direction has the spacedependent gauge potential A µ = ( Ex, 0). The Fourier mode of the Klein-Gordon equation for charge e (e > 0) and mass µ takes the form [in units with h = c = 1 and with metric signature (, +)] [ x q(x)] ϕ ω (x) = 0, q(x) = (ω + eex) µ. In quantum mechanics, the equation is a tunneling problem with the inverted potential barrier q(x) and q(x) corresponds to p x in the WKB approximation. 7

8 Using the phase-integral formula [N. Fröman and P. O. Fröman, Nucl. Phys. A147, 606 (1970)], the wave function can be written as ϕ ω (x) = Af ω + Bfω, where f ω (x) is an asymptotic solution with unit incoming flux and A = (e Sω + 1) 1/, B = e Sω. The pair production rate, P = e Sω, is determined by the instanton action defined in the complex x-plane by the contour integral S ω = i q(x)dx = πm C ee, where the integral is along the contour in the figure. IV. Unruh Effect as Quantum Tunneling A Rindler spacetime is the spacetime covered by all time-like congruences of an accelerated particle. Classically, the Rindler spacetime has two horizons that separate the Minkowski spacetime into two causally disconnected regions for each Cauchy surface. In two dimensions the right wedge (R)/ left wedge (L) of the Rindler spacetime have the coordinates t = ρ R sinh(aτ), z = ρ R cosh(aτ), t = ρ L sinh(aτ) z = ρ L cosh(aτ), with ρ R 0 and ρ L 0. Here a is the acceleration of the particle. In both wedges the spacetime has the metric ds = (aρ) dτ + dρ. The right wedge is causally disconnected from the left wedge by horizons, t = ±z, which correspond to ρ = 0. The accelerated particle would detect a thermal spectrum with the so-called Unruh temperature, T U = a/(π), from the Minkowski vacuum. 8

9 Quantum mechanically, fields or particles can cross horizons with a certain probability. To cover both (R) and (L), we analytically continue the coordinate ρ L = ρ R e iπ. The massive scalar field in (R) and (L) obeys the equation [ 1 ] (aρ) τ + ρ m Φ(τ, ρ) = 0, and the spatial part, Φ = e iωτ ϕ(ρ), satisfies the equation [ ] ρ + ω (aρ) m ϕ(ρ) = 0. For a tunneling wave function crossing ρ = 0, we may use the solution in (L) ϕ L (ρ) = ρj iν (imρ), ν = ω a 1 4. The wave function in (R) tunneled from (L) may be found by analytically continuing as ϕ R (ρ) = (ρ R e iπ ) 1/ J iν (imρ R e iπ ) = e νπ e iπ/ (ρ R ) 1/ J iν (imρ R ), where the relation J α (ze inπ ) = e inαπ J ν (z) for an integer n is used. Therefore we find the tunneling (emission) rate as the ratio of the amplitude square of the tunneled wave function in (R) from (L) to the amplitude square of the outgoing wave function (ρ R ) 1/ J iν (imρ R ) with a given flux in (R): P = e νπ e πω/a. We further show that the tunneling rate, P = e Sω, can also be obtained from the action, ϕ = e is(ρ), where with a contour enclosing ρ = 0. ω S ω = ImS = i (aρ) m dρ = πω a, 9

10 Analogy with the Schwinger Mechanism In two dimensions, the scalar field equation for charge e (e > 0) and mass m minimally coupled with the Coulomb gauge, A µ = ( Ex, 0), takes the form The spatial part, Φ = e iωt ϕ(x), satisfies [( t ieex) x + m] Φ = 0. [ x + (ω + eex) m ] ϕ(x) = 0. In quantum mechanics, the spatial equation is a tunneling problem with energy m under the inverted harmonic potential. instanton action [SPK & D. N. Page, (00), (006), (007)] The tunneling rate is given by the WKB P = e S, S = i (ω + eex) m dx = πm ee. Here the contour integral is taken outside a contour in the complex x-plane. A similarity: the tunneling rate is given by the same formula P = e i p, where for the Rindler case p is p(ρ) = while for the Schwinger mechanism p(x) = ω /(aρ) m, (ω + eex) m. A caveat: For the Schwinger mechanism, the contour excludes a branch cut connecting two roots, x ± = ( ω ± m)/(ee), while for the Rindler case, the contour excludes branch cuts from a root ω/(am) to the positive infinity and from another root ω/(am) to the negative infinity. V. Tunneling Rate of Black Holes in the Rindler Coordinate 10

11 The spacetime region near the event horizon may be locally approximated by a Rindler spacetime ds = (κρ) dt + dρ, where the Rindler coordinate is used κρ = f, dr dρ = g. The emission rate for the Hawking radiation is given by P (ω) = e Sω = e ω/t H, T H = κ π, where the instanton action is the contour integral in the complex Rindler ρ-plane including the simple pole at ρ = 0, the event horizon, ω S ω = i f(ρ) m dρ = πω κ. The surface gravity is κ = f (r H ) g(r H) f(r H ) = f (r H )g (r H ). The last equality holds only for non-extremal black holes. Schwarzschild Black Hole The metric function of a Schwarzschild black hole f = g = 1 M r = r r H, (r H = M). r The the metric in Rindler coordinate where ds = (κρ) dt + (r Hκ) (1 (κρ) ) 4 dρ, f = g = (κρ). The surface gravity and the Hawking temperature κ = f (r H ) = 1 4M, T H = 1 8πM. 11

12 Reissner-Nordström Black Hole The non-extremal Reissner-Nordström black hole f = g = 1 M r + Q r = (r r +)(r r ) r, has the event horizon, r + = M + M Q, and the inner horizon, r = M M Q. The metric in the Rindler coordinate, f = g = r +(κρ) /r, ds The surface gravity at the event horizon r +, (r + ) = (r + + r + (r + r ) + 4(κr + ρ) ) (κρ) dt + (κr +) (r + + r + (r + r ) + 4(κr + ρ) ) (r + r ) + 4(κr + ρ) dρ κ = f (r + ) = r + r r+. The emission rate is valid for non-extremal Reissner-Nordström black holes with the Hawking temperature T H = κ/π. The extremal Reissner-Nordström black hole with Q = M ( ds = 1 M r ) dt dr + ( 1 M r cannot be approximated by the Rindler spacetime, since with r = M + cρ for any c, the metric has the form Charged Kerr Black Hole ds = (cρ) (M + cρ) dt + ) (M + cρ) ρ dρ. The metric for a charged Kerr black hole where ds = fdt + dr g + k(dφ ωdt) + Σdθ, f = Σ (r + a ) a sin θ, g = Σ, k = (r + a ) a sin θ, Σ ω = a sin θ(r + a ) (r + a ) a sin θ, 1

13 where = r Mr + a + Q, Σ = r + a cos θ. The event horizon is located at r + = M + M a Q and the inner horizon at r = M M a Q. The metric near the event horizon approximately takes the form in the Rindler coordinate, f = (κx), With the surface gravity ds = (κρ) dt + 4κ (r + + a ) (r + r ) dρ. κ = r + r f (r+ + a ) = (r + )g (r + ), the charged Kerr black hole can be written in the Rindler coordinate near the event horizon and has the emission rate. de Sitter Space The de Sitter spacetime has The event horizon at r H = l. f = g = 1 r l = (κρ) In the Rindler coordinate the de Sitter space becomes ds = (κρ) dt + (lκ) 1 (κρ) dρ, and locally a Rindler spacetime when κ = 1/l, which is the surface gravity. The Hawking temperature is T H = 1/πl. Schwarzschild-anti de Sitter Black Hole The Schwarzschild-anti de Sitter black hole with has the event horizon at [( r H = l M/l + f = g = 1 M r + r l ) 1/3 ( ) 1/3 ] 1/7 + (M/l) + M/l 1/7 + (M/l). 13

14 Near the event horizon, choosing the coordinate f = g (r r H ) ( (rh /l) + Ml/r H r H we may write the metric approximately as With the surface gravity ds (κρ) + κ = 1 l ) (lκ) (r H / + Ml/r H ) dρ. ( rh the metric becomes a Rindler spacetime. Schwarzschild-anti de Sitter black hole. Connection with Other Coordinates l = (κρ) + Ml ) rh, Thus the emission rate is also valid for the We discuss why the isotropic coordinate and the proper distance can recover the correct Hawking temperature and the Boltzmann factor [Angheben, Nadalini, Vanzo, and Zerbini, JHEP 0505, 014 (005); Nadalini, Vanzo, and Zerbini, J. Phys. A: Math. Gen. 39, 6601 (006)]. The spherically symmetric metric can be written in the isotropic coordinate as ds = f(ζ)dt + k(ζ)(dζ + ζ dω ), where dζ ζ = dr r g(r). For instance, the Schwarzschild black hole has the isotropic coordinate ds = ( 1 M/ζ 1 + M/ζ ) dt + ( 1 + M/ζ ) 4(dζ + ζ dω ), where r = ζ(1 + M/ζ) /4. The event horizon is located at ζ H = r H = M. The metric near the event horizon is approximately given by ds (ζ M) dt + (dζ + ζ dω ). (4M) Setting (ζ M)/4M = κρ and κ = 1/4M, the Schwarzschild black hole metric becomes a Rindler one. This is the reason why the isotropic coordinate leads to the correct result. 14

15 Another coordinate is the proper distance and its metric metric σ = dr g, ds = f(σ)dt + dσ. For non-extremal black holes, the leading terms are f = f (r H )(r r H ), g = g (r H )(r r H ). Now the proper distance from the event horizon σ = r rh, g (r H ) leads to f = f (r H ) ( g (r H )/ ) σ = (κσ). Therefore, the proper distance method gives the same result as the Rindler coordinate. VI. Summary In quantum field theory we showed the Hawking radiation as quantum tunneling in the Rindler coordinate: P (ω) = e Sω = e ω/t H, T H = κ π, where the instanton action is the contour integral in the complex Rindler ρ-plane including the simple pole at ρ = 0, the event horizon, ω S ω = i f(ρ) m dρ = πω κ. 15