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."