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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. • The accelerating observer will “feel” a thermal bath of particles. 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… o Uniformly accelerated observer o Acceleration is only in the z-direction The Inertial Observer • The accelerating observer is moving through socalled Rindler Space, but first… We begin in Minkowski Space 2 = 2 − 2 The Inertial Observer Thus, our Klein-Gordon equation becomes = 2 − 2 = 0 Allowing solutions of the form , = 1 2 ± Where = = 2 + 2 = ± = ± The Inertial Observer So, for the Inertial Observer, the massless scalar field becomes ∞ , = 0 † (() + ∗ ) 2 With † , ′ = ( − ′ ) and |0 = 0 Rindler Space Our metric is invariant under a Lorentz boost → cosh + sinh → sinh + cosh We may Re-parameterize our coordinates as () = sinh = cosh Our metric Becomes (the Rindler metric) 2 = 2 2 − 2 Rindler Space Now we make the transformation = , = 1 1 = sinh = sinh 1 1 = cosh = cosh → 2 = ( 2 − 2 ) The Rindler Observer Based on the transfromed Rindler metric 2 − 2 = 0 Is our new field equation, allowing , = 1 2 ±∓ = 1 2 (∓ ) ± Where ± = ± = 1 ± and ± > 0 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 Region I II III IV z+ = z+t >0 >0 <0 <0 z- = z-t >0 <0 <0 >0 Table 1: Values of z± vs. Region Fig 1: Rindler Space 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 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 (1) (2) 1 = = 2 1 2 − 1 = 0 0 ′ −+ = 2 1 2 −(−) (′ − ′ ) , , The Rindler Observer We now have all the parts of the Field equation for the Rindler observer ∞ , = 0 ( 2 1 (1) () + 2 2 () + 1 † 1 ∗ () + 2 † We must now relate the Unruh modes to the modes of the Inertial observer 2 ∗ () ) The Bogoliubov Transformation We define new solutions (1) (2) = = 1 2 2 ∗ − 1 ∗ + + − Leading to the updated scalar field ∞ , = 0 ( 2 1 (1) () ′ , † + 2 2 () + 1 † 1 ∗ () + − = (2)( − ′ ) 2 sinh 2 † 2 ∗ () ) The Bogoliubov Transformation Now define = − 2 2 sinh We may re-write the Rindler modes as = 1 2 sinh 2 + − 2 † The Bogoliubov Transformation • Those two modes are known as a Bogoliubov Transformation. They relate the modes of the inertial and Rindler observers. The Unruh Temperature • Assume the system is in the Minkowski vacuum, |0 The number operator is given by = 1 † 1 We are interested in the expectation value of the number operator The Unruh Temperature We get − 2 0| 2 sinh 1 = 2 2 (0) −1 0 0 = 2 † |0 The factor looks surprisingly like Planck's Law 1 ~ ℎ − 1 The Unruh Temperature We can compare the arguments of the exponentials in the denominator of both equations to find that... ℎ ~ 2 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! 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