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A Proposed Relativistic, Thermodynamic Four-Vector
It is customary to begin a discussion, in special relativity, by first choosing two Cartesian frames of reference, K and K', which are oriented with their three spatial axes coinciding. These frames are then put into relative, uniform translation, with velocity, v, in the x direction. For two such frames of reference in relative motion, with K' moving in the positive x direction relative to K, the Lorentz transformations take the familiar form:
where
= 1/(1 - v2/c2)1/2, and c = 3 X 108
m/sec is the speed of light, a constant in all frames of reference. It
is also customary to define any set of four quantities, which transform
according to eq. 1, as components of a relativistic
four-vector. Familiar examples of such four-vectors include the velocity
four-vector, the energy-momentum four-vector, and the current density
four-vector.
In the calculation that follows, it will be shown that any vector, V = (Vx, Vy, Vz), and scalar, S, which are related through a continuity equation, i.e., an equation of the form
S/in all frames of reference, transform according to eq. 1, and, therefore, comprise components of a four-vector; i.e., that V and S satisfying eq. 2 is a sufficient condition for their Lorentz-transformability. It will also be proposed that certain thermodynamic quantities, which are shown to be related through a continuity equation, be investigated as components of a relativistic, thermodynamic four-vector.
We begin by using eq. 1 to transform the differential operators
from K to K'. We first use the chain rule to write
We next use eq. 1 to show that
We finally substitute eq. 5 into eq. 4, to obtain
Let us now consider a set of quantities comprising a vector, V, and a scalar, S, in the frame of reference K, which are related through a continuity equation; i.e., for which
We substitute, into eq. 2a, the transformed operators from eq. 6
and rearrange terms
If we now require that eq. 8 have the same form in K' as in K; i.e., that eq. 8 be a continuity equation in K', then, there must be a vector, V' = (V'x', V'y', V'z'), and a scalar, S', in K', such that
and, (comparing eq. 8 with eq. 9)
Eq. 10 has a form identical to eq. 1, and we are forced to conclude that the vector, V, and the scalar, S, together form components of a relativistic four-vector. (Q. E. D.).
The above argument tells us that relativistic four-vectors may be identified from the continuity equations of physics. The remainder of this discussion is devoted to a continuity equation whose terms may not have received much attention as a four-vector; i.e., one involving thermal heat flux, and thermal energy density.
The heat conduction equation is
where q = (qx, qy, qz) is heat flux in cal/(m2 sec), k is the thermal conductivity of the medium, and T is the absolute temperature. The thermal diffusion equation is
where a is the thermal diffusivity of the medium. Taking the divergence of eq. 11, and using eq. 12, we obtain
from which, immediately, follows a continuity equation
The quantity (k/a)T is, dimensionally, a thermal energy density. Together, the quantities, qx, qy, qz, and (k/a)T, must form components of a relativistic four-vector. Thus, they must satisfy the system
To date, the author and his colleagues have not seen this particular system of equations in any of the literature on special relativity. Eq. 15 appears to provide interesting insights into the behavior of thermodynamic systems, as seen by observers in different states of relative uniform motion.
Consider
a one dimensional problem in which a long [infinite], solid rod,
a black body, is lying along the x-axis, at rest in the frame of reference
K. Let the rod have a uniform temperature, so that 
T
= 0, and, therefore, qx = 0, everywhere along the rod in K.
Equations 15 then simplify to
The first
equation in 16 suggests that an observer in a frame
of reference, K', in relative motion to the observer in K, will measure
a non-zero heat flux, qx', of magnitude v(k/a)T.
He will also observe that the thermal energy flow is in the same direction
as the motion of the rod in his frame of reference. This result might
be understood by invoking a Doppler shift in the thermal radiation emitted
by the rod. The portion of the rod which is approaching him (i.e., which
lies in the positive x' direction) will be blue shifted relative to the
portion of the rod receding from him. Since, for a black body, the peak
in the observed radiation frequency is proportional to the temperature
at which the radiation is emitted (Wein's law), the observer must conclude
that the portion of the rod ahead of him is hotter than the portion behind;
i.e., that a thermal gradient exists along the rod, driving the heat flux
which he observes.
Alternately, let two points, A and B, be marked on the rod such that the observer in K is situated midway between them. Let clocks, and thermometers, be placed at each of the points, and let the clocks be synchronized in K. Finally, let the observer in K' be seen, by the observer in K, as moving toward point B, and away from point A. At a time t0, when the observer in K' is adjacent to the observer in K, let both observers record the temperatures at A and at B by observing the thermometers placed at each of the points. They might do so by sending out light pulses to illuminate the thermometers. In each frame of reference, the light pulses must travel out to the thermometers and be reflected back in order for the thermometers to be read. Following this procedure, the observer in K measures equal temperatures. The observer in K', however, does not. The observer in K' does not see the clocks at A and B as reading the same. He sees the clock at B as reading an earlier time than the clock at A. Since the rod is radiating, it is losing thermal energy with time, and cooling. Observations made at different times must therefore record different temperatures. More specifically, for observations made at two different times, the earlier observation will record a higher temperature than the later one. Again, qualitatively at least, we must conclude that the observer in K' will record a higher temperature at B than at A.
The second equation in 16 suggests that the observer in K' will detect a lower thermal energy density, [(k/a)T]', than the observer in K. This result might be understood by invoking the time dilatation. The thermal energy density is made up of a summation of the individual random thermal oscillations of the molecules making up the rod. These motions must exhibit a distribution of frequencies which correlate with the frequencies of the observed thermal radiation. In K', these frequencies must have a lower value than in K, and hence must indicate a lower thermal energy density.
Additional
insight may be gained, at this point, by a comparison with the current
density four-vector. Let jx, jy, and jz
be components of the current density, and 
be the static charge density in some frame of reference. Then for the
frames K and K' being considered above,
As previously,
we consider a one dimensional problem in which a long [infinite]
wire is lying along the x-axis, at rest in the frame of reference K. Let
the wire have a uniform surface charge density, ,
and carry no electric current, so that jx = 0. Equations
17 then simplify to
This
case is analogous to the thermodynamic case we have been discussing. A
uniform, cylindrical electric field exists everywhere along the wire in
K. This electric field corresponds to the radiation field of the rod;
the static charge density, ,
to the thermal energy density, (k/a)T; and the non-zero current density,
jx', to the non-zero heat flux, qx', in K'. (A difference
exists between the two cases, in that the observer in K' will detect a
magnetic field in the electric case, which has no counterpart in the thermodynamic
case.) Eq. 18 tells us that, while the observer in
K detects a uniform static charge distribution everywhere along the wire,
and no electric current, the observer in K' detects a reduced static charge
distribution, and a non-zero electric current, moving in the same direction
as the motion of the wire in his frame of reference.
As, in
the thermodynamic case, the observer in K' required a thermal gradient,
T,
to drive the heat flux along the rod, so the observer in K' must infer
an electric potential gradient, -
,
along the wire to drive the electric current. This gradient arises from
the nature of the charge distribution, along the wire, as seen by the
observer in K'. Recall that the charge distribution is uniform in K. In
K', it is relatively lower seen approaching than receding. If unit charges
are placed at regular intervals along the wire in K, (i.e., at x = 0,
1, 2, etc.), and the charge density is defined as the [average]
charge per unit length of wire, then, due to the finite propagation speed
of light, and taking into account the length contraction, the separation
between the regular intervals in K will appear larger, seen approaching,
and smaller, seen receding, by the observer in K'. In fact, a unit length
in K will appear as the length
|
[(1
+ v/c)/(1 - v/c)]1/2
|
19.
|
if it is seen approaching by the observer in K', and
|
[(1
- v/c)/(1 + v/c)]1/2
|
20.
|
if it is seen receding17. The linear charge densities will appear, respectively, lower, if seen approaching, and smaller, if seen receding, in K'. The observer in K' will infer, from this difference in charge densities, the required electric potential gradient.
These results are preliminary. Perhaps they will shed some new light on the relativistic four-vector in general, and its possible application to thermodynamics in particular.
=
(1 - v2/c2)1/2 Imagine
that an observer uses a camera with a rapid shutter to photograph the
approaching train. A wave front, which left the front of the train at
the time t, and another, which left the rear of the train at a time 1/[(c-v)]
earlier, arrive simultaneously at the film plane, and are recorded. Thus,
the image of the approaching unit train appears, not contracted, but stretched
out. In fact, the train appears to have length
(c-v)]
+ 1/
= {(1 + v/c)/(1 - v/c)}1/2 > 1. A similar argument may be made for the receding train, with the result that the observed length is now
.
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