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Classical physics of thermal scalar radiation in two spacetime dimensions

### Abstract

Thermal scalar radiation in two spacetime dimensions is treated within relativistic classical physics. We first consider an inertial frame in which we give the analogues of Boltzmann’s derivation of the Stefan–Boltzmann law and Wien’s derivation of the displacement theorem using the scaling appropriate to relativistic radiationtheory. The spectrum of classical scalar zero-point radiation in an inertial frame is derived both from scale invariance and from Lorentz invariance. We then consider the behavior of thermal radiation in a coordinate frame undergoing (relativistic) constant acceleration. The classical zero-point radiation of inertial frames is transformed to the coordinates of an accelerating frame. Although the two-field correlation function for zero-point radiation at different spatial points at a single time is the same for inertial and accelerating frames, the correlation function at two different times at a single spatial coordinate is different and, in an accelerating frame, has a natural extension to nonzero temperature. The thermal spectrum in the accelerating frame is then transferred back to an inertial frame, giving the familiar Planck result.

© 2011 American Association of Physics Teachers

Received 11 August 2010
Accepted 12 November 2010
Published online 13 May 2011

Acknowledgments:
The author wishes to thank an anonymous referee for a number of helpful suggestions for improving the manuscript.

Article outline:

I. INTRODUCTION
II. THERMAL RADIATION IN AN INERTIAL FRAME
A. Simplified model theory
B. Thermodynamics of radiation in a one-dimensional box
C. Scaling and fundamental constants
D. Radiation spectrum in a box
E. Classical zero-point radiation
F. Thermal radiation spectrum requirements in an inertial frame
III. THERMAL RADIATION IN A RINDLER FRAME
A. Introduction of a Rindler frame
B. -scale change in a Rindler frame
C. Zero-point radiation in a Rindler frame
D. From zero-point radiation to thermal radiation in a Rindler frame
E. Transferring results from a Rindler frame to an inertial frame
IV. DISCUSSION

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2011-05-13

2016-02-08

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