The Transformation of Temperature

G.R.Dixon, 11/05/2007

Instructive Link: AIP, Physics News Update, Relativistic Thermodynamics

In this article we address the following question: If T is the temperature of an ideal gas in the container’s inertial rest frame, K, then what will T’ be according to inertial frame K’, where K’ moves in the positive x-direction of K at speed u? We begin by stating the combined laws of Boyle/Mariotte and Gay-Lussac/Charles:

. (1)

Here P is the gas pressure, V is its volume, m is the number of moles and R is the gas constant. The number of molecules is

, (2)

where N_{o} is Avogadro’s number. Assuming

, (3)

we have

. (4)

Pressure is force per unit area. Let us say that our container is a cube with side A parallel to the yz-plane and side B parallel to the xz-plane. Let F_{x(A)} be the gas force on side A. According to the Lorentz transformation F_{x}’=F_{x}. Since the area of side A is the same in K and K’, we have

. (5)

But F_{y}’=g^{-1}F_{y} in the case of side B. On the other hand, in K’ the area of side B is (owing to length contraction) only g^{-1} its value in K. Thus here again we have

. (6)

In general, P’=P.

What about volume? In K’ the "cube’s" volume is *not* equal to the volume in K. Owing to length contraction,

. (7)

Putting it all together, we find that

, (8)

and thus

. (9)

Viewed from frame K’, the "moving" gas is colder than the "resting" gas is in K.

It seems plausible to generalize this result, and to state that moving systems are colder than resting systems. Note in Eq. 9 that, as u approaches c, T’ approaches absolute zero.

It is entertaining to think about space travelers, moving relative to us earth-bound observers at a speed close to c. They and their ship are of course length-contracted into wafers. And all on-board processes are slowed down from our perspective. Furthermore, they are practically "frozen stiff!" An on-board thermometer might indicate a comfortable room-temperature reading. But we would point out that, like on-board measuring rods and clocks, the on-board thermometer does not provide "correct" information. We would contend that the on-board temperature is "really" near absolute zero.

In theory we should be able to verify this by measuring the temperature in the spaceship using a thermometer at rest relative to ourselves. In practice such a procedure may be impractical.