On the Relativity of Temperature

G.R.Dixon, 11/10/2007

The Lorentz transformation makes statements of the following type: If u_{x} is the measured x-component of velocity in inertial frame K, where the measurement is made using a grid and synchronized clocks at rest in K, then u_{x}’=(u_{x}-V)/(1-u_{x}V/c^{2}) will be the measured velocity in K’, where the measurement is made using a grid and clocks at rest in K’. (V is the speed of K’ relative to K.) Implicit in such transformations is the contention of inertial observers in the respective frames that the clocks in the other frame run slowly, etc. In brief, each observer does not believe that measurements in the other frame reflect "true" relative velocities, etc. For example, an observer in K contends that the "true" value of u_{x}’ is

. (1)

In the K observer’s opinion, the K’ observer *measures* the more complicated value cited above because the clocks in K’ run slowly, etc.

By way of an example, let us consider the speed of a gas molecule in a container that is at rest in K. (I.e., the gas center of mass is at rest in K.) The molecule has measured velocity components u_{x(meas)}, u_{y(meas)} and u_{z(meas)} in K. The measured components in K’ will then be

, (2a)

, (2b)

. (2c)

What does the K’ observer contend the "true" components of velocity are *relative to K*? The Galilean transformation provides the answer:

, (3a)

, (3b)

. (3c)

Consider first the case for u_{y(true)}:

. (4)

If V>>u_{x(meas)}, then we see that, as V approaches c, u_{y(true)} approaches zero. In the opinion of K’, the particle is practically at rest in the __+__y-direction when V~c. Similar remarks apply to u_{z(true)}.

What about u_{x(true)}? According to K’,

(5)

.

In this case K’ contends that the molecule is practically at rest in the __+__x-direction when V~c.

In general when V~c, K’ contends that a gas molecule is "actually" practically at rest relative to K:

. (6)

It follows that <u_{(true)}>, the "actual" *average* speed of gas molecules relative to the gas center of mass is (in the K’ observer’s opinion) ~0. And of course <(u_{(true)})^{2}>~0.

Let us say that the gas temperature is monotonically related to <(u_{(true)})^{2}>. The K observer may *measure* <(u_{(meas)})^{2}> to be much greater than 0, and hence conclude that T_{(meas)}>>0. But according to K’, <(u_{(true)})^{2}>~0, and thus T_{(true)}~0. In brief, K’ contends that the gas temperature approaches zero as the speed of its center of mass, relative to K’, approaches c.

Such considerations are of course symmetric. If the center of mass of a volume of gas is at rest in K’, then K’ may measure T’>>0. But in the opinion of K, <(u’_{(true)})^{2}>~0 when V~c, and hence T’_{(true)}~0. In general when V~c each inertial observer contends that the molecules, of a gas whose center of mass is at rest in the other frame, are "really" practically at rest relative to the center of mass. Like length, the rate of clocks, etc., temperature is a relative matter.