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The Unruh Temperature

The Unruh Temperature. For a Uniformly Accelerated Observer. Cory Thornsberry December 10, 2012. The Unruh Effect. Two inertial observers in the Minkowski vacuum will agree on the vacuum state We add a non-inertial observer accelerating with constant acceleration, a .

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The Unruh Temperature

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  1. The Unruh Temperature For a Uniformly Accelerated Observer Cory Thornsberry December 10, 2012

  2. The Unruh Effect • Two inertial observers in the Minkowski vacuum will agree on the vacuum state • We add a non-inertial observer accelerating with constant acceleration, a. • The accelerating observer will “feel” a thermal bath of particles.

  3. The Unruh Effect contd. • “…an accelerated detector, even in flat spacetime, will detect particles in the vacuum”Unruh, 1976 • There is a physical temperature associated with the particle bath, Tu. • For simplicity, we assume… • Uniformly accelerated observer • Acceleration is only in the z-direction

  4. The Inertial Observer • The accelerating observer is moving through so-called Rindler Space, but first… We begin in Minkowski Space

  5. The Inertial Observer Thus, our Klein-Gordon equation becomes Allowing solutions of the form Where

  6. The Inertial Observer So, for the Inertial Observer, the massless scalar field becomes With and

  7. Rindler Space Our metric is invariant under a Lorentz boost We may Re-parameterize our coordinates as Our metric Becomes (the Rindler metric)

  8. RindlerSpace Now we make the transformation ,

  9. The Rindler Observer Based on the transfromedRindlermetric Is our new field equation, allowing Where and

  10. The Rindler Observer • Our trajectory (world) curves are restricted to Region I • We need to cover all of Rindler space for valid solutions • We may “extend” our solutions into the other regions • (t,z) may vary in all space. (τ,ξ) is restricted to RI Table 1: Values of z± vs. Region Fig 1: Rindler Space

  11. The Rindler Observer • We required that z± > 0 • We may analytically extend into region IV where z- > 0 • Additionally, we may extend into region II where z+ > 0 • z± is never positive in Region III • We may not extend the solutions into RIII. We do not have a complete set of solutions

  12. The Rindler Observer • We perform a time reversal and a parity flip, • This exchanges RI for RII and RIII for RIV We get two (Unruh) modes

  13. The Rindler Observer We now have all the parts of the Field equation for the Rindler observer We must now relate the Unruh modes to the modes of the Inertial observer

  14. The BogoliubovTransformation We define new solutions Leading to the updated scalar field

  15. The BogoliubovTransformation Now define We may re-write the Rindler modes as

  16. The BogoliubovTransformation • Those two modes are known as a Bogoliubov Transformation. They relate the modes of the inertial and Rindler observers.

  17. The Unruh Temperature • Assume the system is in the Minkowskivacuum, The number operator is given by We are interested in the expectation value of the number operator

  18. The Unruh Temperature We get The factor looks surprisingly like Planck's Law

  19. The Unruh Temperature We can compare the arguments of the exponentials in the denominator of both equations to find that...

  20. Conclusion • So, an observer moving at a constant acceleration through the vacuum, will experience thermal particles with temperature proportional to its acceleration! • This does not violate conservation of energy. Some of the energy from the accelerating force goes to creating the thermal bath. • The observer will even be able to "detect" those thermal particles in the vacuum!

  21. References • Bièvre, S., Merkli, M. “The Unruh effect revisited”. Class. Quant. Grav. 23, 2006 pp. 6525 – 6542 • Crispino, L., Higuchi, A., Matsas, G. “The Unruh effect and its applications”, Rev. Mod. Phys. 80, 1 July 2008 pp. 787 – 838 • Pringle, L. N. “Rindler observers, correlated states, boundary conditions, and the meaning of the thermal spectrum”. Phys. Rev. D. Volume 39, Number 8, 15 April 1989 pp. 2178 – 2186 • Siopsis, G. “Quantum Field Theory I: Unit 5.3, The Unruh effect”. University of Tennessee Knoxville. 2012 pp. 134 – 140 • Rindler, W. “Kruskal Space and the Uniformly Accelerating Frame”. American Journal of Physics. Volume 34, Issue 12, December 1966, pp. 1174 • Unruh, W. G. “Notes on black-hole Evaporation”. Phys. Rev. D. Volume 14, Number 4, 15 August 1976 pp. 870 – 892

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