(where ss = v/c)
In this note we consider the historical and
theoretical origin of relativistic thermodynamics in the recent theory of
black-body radiation in relation to a recent paper by Liu1.
1.
The Planck-Einstein transformation of temperature
In 1952 Einstein turned from the Planck-Einstein
transformation formulas (1907) to the heat Q and therefore the temperature T.
Instead of the transformation laws
(where ss = v/c)
(which were generally accepted at the time),
Einstein suggested the transformations
These are the laws which H. Ott2
proposed some years later (1963), and independently of Einstein´s results. C.
Lui emphasizes that Einstein´s second letter to Laue (1953) implies universal
scepticism with respect to the memory of the relativistic thermodynamics which
were acutalized in the following decades3.
The key problem is the comparison of the
socalled “eigen.temperature” of two bodies which are in motion relative to
each other and the heat transport between bodies with variable distance and
stressed heat conductor.
From a historical standpoint, the primary idea
in the works of K. von Mosengeil4 and M Planck5
was the measurement of the black radiation temperature of moving bodies. In this
case, we do not have the problem of heat transport between bodies. A system of
black bodies at relative rest, which is in a state of radiation equilibrium,
remains in this state if the bodies are diabatically accelerated to each other.
If, in the rest state, the
“eigen-temperature” of the bodies is T0, then the “eigen-temperature” of the moving bodies become
where vA
is the velocity of a body PA relative to the rest system6.
This axiom is a result of Kirchhoff´s theorem
and is based on the validity of Wien´s law in each reference system. Wien´s
law implies that the variable n
/ T (where n is the frequency) in the Kirchhoff-Planck radiation function is
Lorentz-invariant. The the Planck-Einstein transformation of the black-body
temperature T follows from Einstein´s theory of relativistic Doppler effect and
abberation7:
The Planck-Einstein formulation of relativistic
thermodynamics is primarly the thermodynamics of heat radiation and involves the
definition of the temperature of bodies according to their black-body radiation8.
2.
Historical remarks on Planck´s relativistic thermodynamics
Planck´s relativistic thermodynamics was
originally a thermodynamics of black-body radiation or heat radiation (Planck´s
“Hohlraum-Strahlung”). In 1906, Planck published his book on heat radiation9.
It contained a definiton of thermodynamic equilibrium according to Kirchhoff´s
law and introduced an “entropy radiation”10.
Planck´s first approach to relativistic
thermodynamics was based on the thesis of his former pupil Kurt von Mosengeil.
Max von Laue pointed out the importance of this publication11.
Von Mosengeil´s transformation of heat, energy, pressure and volume V became a
special case of Planck´s notation. If the pressure p is the Maxwellian
radiation pressure, P = 2u, and if the radiation density is u = aT4
according to the Stefan-Boltzmann law12,
Von Mosengeil´s transformation of the black radiation entropy13
(where
a is Stefan´s constant)
gives, according to the Lorentz contraction, V
– (1-ss2)0,5 V0 and with respect to Planck´s theorem of the invariance of entropy, S =
S0, the temperature transformation
.
1. C. Liu: Einstein and relativistic
thermodynamics in 1952: a historical and critical study of a strange episode in
the history of modern physics. BJHS (1992), 25, 185-206
2.
H. Ott: Lorentz-Transformationen der Waerme und der Temperature. Z. Phys. (1963), 175, 70-104
3. Cf. the discussion in R. G. Newburgh:
Relativistic thermodynamics, temperature, transformations, invariance and
measurements. Nuovo Cimente (1979) B52, 219-28
4.
K. von Mosengeil: Theorie der stationaeren Strahlung in einem gleichfoermig
bewegten Hohlraum. Ann. Phys. (1907), 22, 876-906
5.
M. Planck: Zur Dynamik bewegter Systeme. Berliner Berichte (1908), 542-70
6.
A. Einstein: Ueber das Relativitaetsprinzip und die aus demselben gezogenen
Folgerungen. Jhb. Radioakt. Elektron.
7. Ibid.
8.
H.-J. Treder: Die Strahlungstemperatur bewegter Koerper. Ann. Phy. (1977), 34,
23-9
9.
M. Planck: Theorie der Waermestrahlung, Leipzig (Bahrt), 1906
10.
Ibid.
11.
M. von Laue: Relativitaetsprinzip und Wien´sches Gesetz. Ann. Phys. (1943), 34,
220-2
12.
Von Mosengeil, op. cit. (4), 876-904
13. The Problem of von Mosengeil´s thesis was a
critical revision and correction to an (non-relativistic) paper by F. Hasenroehrl:
Zur Theorie der Strahlung bewegter Koerper. Ann.
Phys. (1904), 45, 344-70. M. Abraham: Elektromagnetische Theorie der Strahlung (§44,
Dynamik des bewegten Hohlraums), Leipzig, 1920; see also W. Pauli: Relativitaetstheorie,
Enzyklop. Math. Wiss., Leipzig, 1921, xix (cf. §49a: Schwarze Strahlung in
einem bewegte Hohlraum).