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)
(arXiv: )

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/ ]
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 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).