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A Proposed Relativistic, Thermodynamic Four-VectorIt 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
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 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 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 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 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, 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, As, in
the thermodynamic case, the observer in K' required a thermal gradient,
if it is seen approaching by the observer in K', and
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. 17Consider
the case of a unit "train" approaching the observer at velocity, v. Its
Lorentz contracted length must be
1/
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/[ v/[
A similar argument may be made for the receding train, with the result that the observed length is now {(1
- v/c)/(1 + v/c)}1/2 < 1/
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