Unruh Temperature and Rindler Space
Explore the fascinating concept of the Unruh effect and the Unruh temperature for uniformly accelerated observers in physics. Delve into the intricacies of Rindler Space and understand the implications of different observer perspectives on particle behavior and temperature perception.
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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 so- called 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 ?? 2?( ?(?)??+ ? With = ?(? ? ) and ? ? | ) ? ?,? = ? ?? 0 ??, ?? 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 ??sinh? =1 ? =1 ????sinh?? ??cosh? =1 ????cosh?? ??2= ???(??2 ??2)
The Rindler Observer Based on the transfromed Rindler metric ??2 ??2? = 0 Is our new field equation, allowing 1 2?? ??? = Where 1 2?(?? ) ?? ???,? = ? 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. ( , ) 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)= ?? , 0 0 ?????? ? ?????? ??? (2)= 1 2?? ???+ 1 2????(? ? ) , ?? =
The Rindler Observer We now have all the parts of the Field equation for the Rindler observer ?? 2?(?1(?) ?? (1)+ ?2(?) ?? 2+ ?1 1 + ?2 2 ) ? ?,? = (?) ?? (?) ?? 0 We must now relate the Unruh modes to the modes of the Inertial observer
The Bogoliubov Transformation We define new solutions ?? ?? ??? ?? ??? ?? (1)= ?? (2)= ?? 1+ ? 2+ ? 2 1 Leading to the updated scalar field ?? 2?(?1(?)?? (1)+ ?2(?)?? 2+ ?1 1 + ?2 2 ) ? ?,? = (?)?? (?)?? 0 ??? ??? ? ??? ,?? ???(2?)?(? ? ) ? = 2sinh
The Bogoliubov Transformation Now define ??2? 2sinh ?????? ??? = ? We may re-write the Rindler modes as 1 ? ?? 2???? + ? ?? ??? = 2??? ? 2sinh?? ?
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, The number operator is given by |0 ? ? = ?1 ? ?1? We are interested in the expectation value of the number operator
The Unruh Temperature We get ? ?? ? 2sinh?? ? 1 2??? 1 0|?2? ?2 ? | 0 ? ? 0 = 0 = 2? ?(0) ? The factor looks surprisingly like Planck's Law 1 ? ? ~ ???? 1 ?
The Unruh Temperature We can compare the arguments of the exponentials in the denominator of both equations to find that... ? ??~ 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 Bi vre, 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
