Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum

American Journal of Physics

Volumn 72, Issue 12, 2004, Pages 1524-1529

  Alsing, Paul M ^a   Milonni, Peter W ^b  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 10844220707     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1761064     Document Type: Article

References (39)

1. S. W. Hawking, "Black hole explosions," Nature (London) 248, 30-31 (1974); "Particle creation by black holes," Commun. Math. Phys. 43, 199-220 (1975).
2. S. W. Hawking, "Black hole explosions," Nature (London) 248, 30-31 (1974); "Particle creation by black holes," Commun. Math. Phys. 43, 199-220 (1975).
3. W. G. Unruh, "Notes on black hole evaporation," Phys. Rev. D 14, 870-892 (1976).
4. P. C. W. Davies, "Scalar production in Schwarzschild and Rindler metrics," J. Phys. A 8, 609-616 (1975).
5. The literature on this subject is vast. For some articles directly relevant for the work presented here see, for instance, P. Candelas and D. W. Sciama, "Irreversible thermodynamics of black holes," Phys. Rev. Lett. 38, 1372-1375 (1977); T. H. Boyer, "Thermal effects of acceleration through random classical radiation," Phys. Rev. D 21, 2137-2148 (1980) and "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," 29, 1089-1095 (1984); D. W. Sciama, P. Candelas, and D. Deutsch, "Quantum field theory, horizons, and thermodynamics," Adv. Phys. 30, 327-366 (1981).
6. The literature on this subject is vast. For some articles directly relevant for the work presented here see, for instance, P. Candelas and D. W. Sciama, "Irreversible thermodynamics of black holes," Phys. Rev. Lett. 38, 1372-1375 (1977); T. H. Boyer, "Thermal effects of acceleration through random classical radiation," Phys. Rev. D 21, 2137-2148 (1980) and "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," 29, 1089-1095 (1984); D. W. Sciama, P. Candelas, and D. Deutsch, "Quantum field theory, horizons, and thermodynamics," Adv. Phys. 30, 327-366 (1981).
7. The literature on this subject is vast. For some articles directly relevant for the work presented here see, for instance, P. Candelas and D. W. Sciama, "Irreversible thermodynamics of black holes," Phys. Rev. Lett. 38, 1372-1375 (1977); T. H. Boyer, "Thermal effects of acceleration through random classical radiation," Phys. Rev. D 21, 2137-2148 (1980) and "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," 29, 1089-1095 (1984); D. W. Sciama, P. Candelas, and D. Deutsch, "Quantum field theory, horizons, and thermodynamics," Adv. Phys. 30, 327-366 (1981).
8. The literature on this subject is vast. For some articles directly relevant for the work presented here see, for instance, P. Candelas and D. W. Sciama, "Irreversible thermodynamics of black holes," Phys. Rev. Lett. 38, 1372-1375 (1977); T. H. Boyer, "Thermal effects of acceleration through random classical radiation," Phys. Rev. D 21, 2137-2148 (1980) and "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," 29, 1089-1095 (1984); D. W. Sciama, P. Candelas, and D. Deutsch, "Quantum field theory, horizons, and thermodynamics," Adv. Phys. 30, 327-366 (1981).
9. An extensive review is given by S. Takagi, "Vacuum noise and stress induced by uniform acceleration," Prog. Theor. Phys. 88, 1-142 (1986). See Chap. 2 for a review of the Davies-Unruh effect.
10. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge U.P., New York, 1982).
11. Note that in order for a detector to remain at a fixed location outside the horizon of a black hole, it must undergo constant acceleration just to remain in place.
12. W. Rindler, "Kruskal space and the uniformly accelerated observer," Am. J. Phys. 34, 1174-1178 (1966).
13. P. W. Milonni, The Quantum Vacuum (Academic, New York, 1994), pp. 60-64.
14. z-1, Re(z)>0 can be analytically continued in the complex plane and in fact remains well defined, in particular, for Re(z)→0, Im(z)≠0, which is the case in Eq. (8). See for for example, J. T. Cushing, Applied Analytical Mathematics for Physical Scientists (Wiley, New York, 1975), p. 343.
15. z-1, Re(z)>0 can be analytically continued in the complex plane and in fact remains well defined, in particular, for Re(z)→0, Im(z)≠0, which is the case in Eq. (8). See for for example, J. T. Cushing, Applied Analytical Mathematics for Physical Scientists (Wiley, New York, 1975), p. 343.
16. Reference 10, Sec. 8.332.
17. μ of the phase of a quantum mechanical particle in curved space-time. See L. Stodolsky "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979) and P. M. Alsing, J. C. Evans, and K. K. Nandi, "The phase of a quantum mechanical particle in curved spacetime," ibid. 33, 1459-1487 (2001), gr-qc/ 0010065.
18. μ of the phase of a quantum mechanical particle in curved space-time. See L. Stodolsky "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979) and P. M. Alsing, J. C. Evans, and K. K. Nandi, "The phase of a quantum mechanical particle in curved spacetime," ibid. 33, 1459-1487 (2001), gr-qc/ 0010065.
19. μ of the phase of a quantum mechanical particle in curved space-time. See L. Stodolsky "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979) and P. M. Alsing, J. C. Evans, and K. K. Nandi, "The phase of a quantum mechanical particle in curved spacetime," ibid. 33, 1459-1487 (2001), gr-qc/ 0010065.
20. μ of the phase of a quantum mechanical particle in curved space-time. See L. Stodolsky "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979) and P. M. Alsing, J. C. Evans, and K. K. Nandi, "The phase of a quantum mechanical particle in curved spacetime," ibid. 33, 1459-1487 (2001), gr-qc/ 0010065.
21. μ of the phase of a quantum mechanical particle in curved space-time. See L. Stodolsky "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979) and P. M. Alsing, J. C. Evans, and K. K. Nandi, "The phase of a quantum mechanical particle in curved spacetime," ibid. 33, 1459-1487 (2001), gr-qc/ 0010065.
22. A similar derivation in terms of Doppler shifts appears in the appendix of H. Kolbenstvedt, "The principle of equivalence and quantum detectors," Eur. J. Phys. 12, 119-121 (1991). T. Padmanabhan and coauthors also have derived Eq. (9) by similarly considering the power spectrum of Doppler shifted plane waves as detected by the accelerated observer. See K. Srinivasan, L. Sriramkumar, and T. Padmanabhan, "Plane waves viewed from an accelerated frame: Quantum physics in a classical setting." Phys. Rev. D 56, 6692-6694 (1997); T. Padmanabhan, "Gravity and the thermodynamics of horizons," gr-qc/0311036.
23. A similar derivation in terms of Doppler shifts appears in the appendix of H. Kolbenstvedt, "The principle of equivalence and quantum detectors," Eur. J. Phys. 12, 119-121 (1991). T. Padmanabhan and coauthors also have derived Eq. (9) by similarly considering the power spectrum of Doppler shifted plane waves as detected by the accelerated observer. See K. Srinivasan, L. Sriramkumar, and T. Padmanabhan, "Plane waves viewed from an accelerated frame: Quantum physics in a classical setting." Phys. Rev. D 56, 6692-6694 (1997); T. Padmanabhan, "Gravity and the thermodynamics of horizons," gr-qc/0311036.
24. A similar derivation in terms of Doppler shifts appears in the appendix of H. Kolbenstvedt, "The principle of equivalence and quantum detectors," Eur. J. Phys. 12, 119-121 (1991). T. Padmanabhan and coauthors also have derived Eq. (9) by similarly considering the power spectrum of Doppler shifted plane waves as detected by the accelerated observer. See K. Srinivasan, L. Sriramkumar, and T. Padmanabhan, "Plane waves viewed from an accelerated frame: Quantum physics in a classical setting." Phys. Rev. D 56, 6692-6694 (1997); T. Padmanabhan, "Gravity and the thermodynamics of horizons," gr-qc/0311036.
25. Kin Eq. (14), for instance.
26. 3. In other works, our volume V here is really just a length.
27. If we use the same gamma function integrals as in Ref. 10, we will have for the Dirac case μ= iΩc/a+ 1/2, with Re(μ)=1/2 clearly in their domain of definition 0
28. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), pp. 365-370.
29. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), Chap. 6, pp. 163-176.
30. 3ξ/2) where tanh ξ=v/c. See J. D. Bjorkin and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), pp. 28-30. From Eq. (3) we have v/c = tanh(aτ/c) so that ξ=aτ/c, yielding the spinor Lorentz transformation to the instantaneous rest frame of the accelerated observer.
31. 3ξ/2) where tanh ξ=v/c. See J. D. Bjorkin and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), pp. 28-30. From Eq. (3) we have v/c = tanh(aτ/c) so that ξ=aτ/c, yielding the spinor Lorentz transformation to the instantaneous rest frame of the accelerated observer.
32. For simplicity, we have chosen the spin up wave function as an eigenstate of Ŝ(τ) with eigenvalue exp(aτ/2c). The exact spatial dependence of the accelerated (Rindler) spin up wave function is more complicated than this simple form, although both have the same zero bispinor components. See W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics in Strong Fields (Springer, New York, 1985), Chap. 21.3, pp. 563-567; M. Soffel, B. Müller, and W. Greiner, "Dirac particles in Rindler spacetime," Phys. Rev. D 22, 1935-1937 (1980).
33. For simplicity, we have chosen the spin up wave function as an eigenstate of Ŝ(τ) with eigenvalue exp(aτ/2c). The exact spatial dependence of the accelerated (Rindler) spin up wave function is more complicated than this simple form, although both have the same zero bispinor components. See W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics in Strong Fields (Springer, New York, 1985), Chap. 21.3, pp. 563-567; M. Soffel, B. Müller, and W. Greiner, "Dirac particles in Rindler spacetime," Phys. Rev. D 22, 1935-1937 (1980).
34. The spinor Lorentz transformation Ŝ(τ) does not mix spin components. Thus, for example, a spin up Minkowski state remains a spin up accelerated (Rindler) state. We can therefore drop the constant spinor |↑〉 from our calculations and retain the essential, new time-dependent modification exp(aτ/2c) to the plane wave for our Dirac "wave function."
35. Reference 5, Sec. 2, in particular Eqs. (2.7.4) and (2.8.8).
36. S. A. Fulling, "Nonuniqueness of canonical field quantization in Riemannian spacetime," Phys. Rev. D 7, 2850-2862 (1973) and Aspects of Quantum Field Theory in Curved Space-Time (Cambridge U.P., New York, 1989).
37. S. A. Fulling, "Nonuniqueness of canonical field quantization in Riemannian spacetime," Phys. Rev. D 7, 2850-2862 (1973) and Aspects of Quantum Field Theory in Curved Space-Time (Cambridge U.P., New York, 1989).
38. This viewpoint is also taken in a different derivation of the Unruh-Davies effect in Ref. 9.
39. L) as the vacuum for the left Rindler wedge.