Re: temperature of flowing fluid

["John S. Denker" <jsd@MONMOUTH.COM>]

Bob Sciamanda wrote:
...
> In Moller's text  "The Theory of Relativity"  (Par 78, Eq 124), it is
> asserted that under a Lorentz boost
> T=To*SQRT(1-u^2/c^2).

1) OK, so to first order in u, everybody agrees that
that T=To -- which I think answers the question that
was originally asked.

2) If in the ?good? old phys-l tradition we want to
pursue this into the relativistic regime, we have some
interesting physics obscured by terminological inconsistency.

First, the not-very-interesting terminological issue:

By way of analogy, suppose we have an electron whizzing
around in a syncrotron, and somebody asks "what is the
mass of that electron?"  In modern usage, the standard
answer is that "mass" refers to the invariant mass in
the particle's own frame.  We know that it takes a lot
more force to turn this electron than it would to turn
a nonrelativistic particle of the same (rest) mass, but
we deal with that in the equation of motion, not in the
definition of mass.

But the terminology was not always so.  Reputable guys
like Einstein used a "mass" that grew in proportion to
gamma.

I'm not an authority on such things, but I believe something
similar is going on with temperature.  For instance, MTW
page 558 defines T to be the temperature of the fluid in
its own frame.  When we consider equilibrium between two
non-comoving fluids with the same (rest) T, things will get
tricky, but I believe the modern convention is to put the
trickery into the equations, not into the definition of T.

But certainly it was not always so.  Reputable guys like
Planck and Einstein and Tolman and Moller used a frame-
dependent notion of T.

=============

OK, now for some physics:

Imagine two flat streams of gas moving past
each other but not touching:

            ---------------- ==>
        <== ----------------

Now, suppose that they are in thermal equilibrium.  We
need to think about what that means.

In equilibrium, the (rest) temperature of the two frames
is equal.  That is, thermometers attached to stream A
read the same as thermometers attached to stream B, when
things are in equilibrium.  This should be obvious by
symmetry.

But suppose the A-team doesn't want to measure the
temperature of stream B using B's thermometers.  Suppose
they use an optical pyrometer instead.  What will they
see? For one thing, they will impute to B a larger density
(large compared to A's density) because of the Lorentz
contraction.  So if that were all there were to it, stream
B would appear brighter;  there would be more black-body
radiation arriving from B than is leaving from A.  But
there's another part to the story:  The radiating atoms in
B are using time-dilated clocks (or so the A-team says).  So
it appears to A that each radiator doesn't radiate as often
as it should.  This corresponds to black-body radiation at
a lower temperature.  Bottom line:  when you track down all
the details, A imputes to B a larger density but a lower
temperature.  The energy flow is perfectly balanced in equilibrium,
which is the defining property of equilibrium.

There are about ten other ways of reaching the same conclusion.
The conclusion is not sensitive to the type of measuring-
device used.  But if you use anything other than light to
carry the energy from one place to the other, you need to
think pretty carefully about what happens to that thing when
it gets accelerated or decelerated.

==========

Beware:  The sign of the effect is the opposite of what most
people would guess.  If I increase the speed of something
relative to my reference frame, its apparent temperature
decreases (while its proper (rest) temperature remains the
same).  Most people would argue that the overall energy is
increasing, so the thermal energy must increase in proportion.
Well, that's not how it works.  Remember that thermal energy
has to do with the random _spread_ in energy.  When I increase
the speed of something, the center of the distribution is
moved to higher energy, but the amount of spread decreases.