# Derivation of the Planck spectrum for relativistic classical scalar radiation from thermal equilibrium in an accelerating frame

Timothy H. Boyer

Department of Physics, City College of the City University of New York, New York, New York 10031

(Received 12 February 2010; published 25 May 2010)

Phys. Rev D, May 15, 2010

"Although the temperature of thermal radiation is constant

throughout nonrelativistic systems in equilibrium,

this constancy is not true in relativistic gravitational physics,

and, in particular, it is not true in a Rindler frame.

There are clearly profound differences between the thermodynamics

of nonrelativistic and relativistic physics. ...

Many textbooks present Boltzmann’s derivation [1] of

the Maxwell velocity distribution for free particles in

thermal equilibrium in a box. In his analysis, Boltzmann

introduced a uniform gravitational field, followed the implications

of thermal equilibrium under gravity, and finally

took the zero-gravity limit. The derivation is striking because

it uses only the physics of free nonrelativistic particles

moving in a gravitational field. By the principle of

equivalence, the gravitational field can be replaced by an

accelerating coordinate frame. But then thermodynamic

consistency requires that the interactions of particles that

lead to equilibrium in an inertial frame must be consistent

with the equilibrium determined by the physics of free

particles in an accelerating frame. The natural question

arises as to whether the analogue of this procedure can

be applied to the much more complicated problem of

thermal equilibrium for relativistic radiation with its infinite

number of normal modes. In this paper we show that

an analogous derivation is indeed possible for relativistic

classical scalar radiation. We introduce a relativistic accelerating

coordinate frame (a Rindler frame, which is the

closest relativistic equivalent to a uniform gravitational

field), consider the implications for thermal radiation equilibrium,

make the assumption that thermal equilibrium

involves but a single correlation time, and finally take the

limit of zero acceleration to obtain the thermal radiation

spectrum in an inertial frame. The use of an accelerating

coordinate frame to obtain the thermal equilibrium spectrum

seems striking because only noninteracting free radiation

fields are needed for the derivation. However, we

expect that any other interaction that produces equilibrium

must be consistent with the equilibrium determined by the

accelerating frame. ...

There have been many indignant objections to work

involving ‘‘classical’’ zero-point radiation; the claim is

made that zero-point radiation is exclusively a ‘‘quantum’’

concept. ...

B. The Rindler frame

Following the analogy with Boltzmann’s work, we

would like to discuss radiation in a box undergoing uniform

acceleration. Since we are dealing with relativistic

classical radiation, we would like to consider a box undergoing

uniform acceleration through Minkowski spacetime.

In the frame of the box, the acceleration should be constant

in time, and the dimensions of the box should not change so

that the radiation pattern can be assumed steady state.

However, relativity introduces some complications which

are quite different from nonrelativistic kinematics. When

viewed from an inertial frame where the box is momentarily

at rest ..., the acceleration a of a point of

the box will appear to change according to the Lorentz

transformation for accelerations,... with the acceleration a (seen in the inertial

frame) becoming smaller as the velocity v of the box

becomes larger even though the acceleration ... in the

frame of the box is constant in time. Furthermore, in order

for the box to maintain a constant length in its own rest

frame, the box must be found to undergo a length contraction

in the inertial frame. But this requires that different

points of the box must undergo different accelerations as

seen in any inertial frame, and indeed, in any inertial frame

momentarily at rest with respect to the box. Thus, the

proper acceleration of each point of the box must vary

with height. This relativistic situation has been explored in

the literature [15] and the coordinate frame associated with

the box is termed a Rindler frame. ...

that no single acceleration can be

assigned to a Rindler frame. Rather the acceleration varies

with the coordinate ...

The analysis given here has ties to work appearing in

quantum field theory [20,21]. In connection with

Hawking’s ideas regarding the quantum evaporation of

black holes [22] and Fulling’s nonuniqueness of the field

quantization [23], Davies [24] and Unruh [25] noted the

appearance of the Planck correlation function when a point

was accelerated through the quantum vacuum of

Minkowski spacetime. Within the quantum literature, a

mechanical system accelerating through the vacuum is

often said to experience a thermal bath at temperature T ~

ha/ckB and to take on a thermal distribution. There

have been controversies as to whether or not the acceleration

turns the ‘‘virtual photons’’ of the vacuum into ‘‘real

photons.’’ In this paper, the analysis has been entirely

within classical physics."

Written by Jack Sarfatti

Published on Friday, 28 May 2010 20:50

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