Comments on anomaly versus WKB methods for calculating Unruh radiation*

1. Comments on anomaly versus WKB methods for calculating Unruh radiation* Douglas Singleton, CSU Fresno and PFUR “String Field Theory and Related Aspects” Moscow, Russia April 16th, 2009 *Work in collaboration with V. Akhmedova, T. Pilling, and A. de Gill Physics Letters B 673 (2009) 227-231 (arXiv:08083413)

2. Reduction 3+1  1+1 and gravitational anomaly • A scalar field in some gravitational background has an action, S: • Expanding and integrating reduces this to 1+1 • Chiral theories in 1+1 have a gravitational anomaly* Consistent Anomaly Covariant Anomaly *Alvarez-Gaume and Witten Nucl. Phys. B 234, 269 (1984) Bertlmann and Kohlprath, Ann. Phys. 288, 137 (2001)

3. Cancellation of the anomaly via flux • Vary the 1+1 action with parameters λµ =(λt , λr) • The action is not invariant under this general variation because of the anomaly. • Break up energy-momentum tensor as • Combine this with the variation λµ =(λt , λr) and require δS=0 and you get

4. Cancellation of the anomaly via flux • If Ntr≠0 then one needs Tr(O)t≠0. Assume a 2D Planckian distribution. Then • Ntr = Φ yields the temperature for a given spacetime, but not the spectrum which is assumed.

5. Hawking temperature via anomalies • For a Schwarzschild black hole one finds a flux of • This gives the (correct) Hawking temperature of T=1/8πM

6. Rindler Spacetime • The Rindler metric has two well known forms • The two forms are related by the transformation

7. Rindler metrics: different temperatures • For the first set of coordinates (r,t) the flux Ntr is • Apparently give correct Unruh temperature T=a/2π • For the second set of coordinates (r’,t’) the flux Ntr is • Which gives an incorrect Unruh temperature of T=a/2π√2

8. Rindler metrics: zero temperatures • However (i) Ntr =constant  (ii) anomaly is zero  (iii) zero Unruh temperature • For the covariant anomaly this is even easier to see since the 2D Ricci scalar vanishes  R=0 • The anomaly method fails (in its simplest form) for Rindler

9. De Sitter spacetime: split result • De Sitter spacetime emits Gibbons-Hawking radiation • With temperature T=1/2πα • The consistent anomaly does give this temperature • The covariant anomaly is zero since R=const. • The two anomaly methods give different answers for de Sitter.

10. WKB/tunneling calculation of Unruh temperature • Use φ(x)~exp[i S(x)/h] one finds the Hamilton-Jacobi form of Klein-Gordon • Split action as S(x)=Et+S0(x). Solution S0=∫prdr. • Imaginary S0(x)  the quasi-classical decay and temperature given via

11. Im(S0) for 1st form of Rindler metric • For the first form of the Rindler metric S0 is [with (+) outgoing and (-) ingoing] • Imaginary contribution comes from contour integration around r=-1/2a. The contour is parametrized as r=-(1/2a)+εeiθ • A round trip gives iπE/a which gives twice the Unruh temperature (a/π instead of a/2π)

12. Im(S0) for 2nd form of Rindler metric • The second form of the Rindler metric appears to give the correct answer • This can’t be correct since the two metrics are related by a coordinate transformation • The contour is also transformed to a quarter circle r’=-(1/a)+√εeiθ/2

13. Resolution: temporal contribution • The Both forms of Rindler metric give twice the Unruh temperature • The Rindler spacetime is obtained from ds2 =- dT2 + dR2 via r>-1/2a r<-1/2a • Crossing the horizon involves an imaginary time change tt-iπ/2a so Im(EΔt)=-πE/2a. For a round trip Im(EΔt)=-πE/a

14. Spatial + temporal contribution • Spatial + temporal contribution gives correct Unruh temperature via

15. Emission/Absorption Probability • The probability for emissions/absorption is Pa,e~|φin,out|2~|exp[2iSin,out(x)]| • Need Pa=1 • Without temporal piece (will give Probability>1 for large enough E) • With temporal piece

16. Canonical Invariance • Physical quantities should be canonically invariant • Note: 2Im(S0)=2Im∫ p dr is not canonically invariant so that Γ~exp[2Im(S0)] is not a proper observable [B.D. Chowdhury, hep-th/0605197] • But is canonically invariant.

17. Summary/Conclusions • Neither anomaly method works for Rindler spacetime/Unruh radiation. • The gravitational WKB method works for Rindler spacetime/Unruh radiation, but has both spatial and time contributions. • The gravitational WKB/tunneling problem has some distinct features: time contribution and ingoing and outgoing probabilities for tunneling are not equal.

18. Acknowledgments • Work partially supported through a 2008-2009 Fulbright Scholars Grant

19. Canonical Invariance • The proper, observable decay rate is then • For the Rindler metric in=out so numerically both give the same answer • There are cases when there is a difference such as the Painleve-Gulstrand form of the Schwarzschild metric

20. Painleve-Gulstrand case • The spatial part of the action is now • The two integrals have the same magnitude imaginary contributions. • Thus the ingoing and outgoing probabilities are not equal in this case (or for any case if the temporal piece is taken into account).