At 08:18 PM 9/5/99 -0500, JACK L. URETSKY clued me in that while his
question asked about one thing, my attempted answer talked about another,
and I did not sufficiently clarify the distinction.
Executive summary:
Jack is right, the equation that says
four-divergence of [energy,momentum] = 0
does NOT correspond to the standard 3D energy-conservation law or
momentum-conservation law. The equation is not wrong -- it just expresses
something else.
First I will explain what the foregoing equation means, and then present
*another* equation that expresses conservation of [energy,momentum].
We start by considering some particles (which we can call snarkons) as they
move through spacetime, and in particular as they move through some
Gaussian pillbox that we are watching.
If the snarkons have no velocity in the lab frame, we can describe this in
4D or 3D language:
-- 4D statement: Snarkons enter the pillbox through the t0 face and exit
through the t1 face.
-- 3D statement: d/dt (# of snarkons in pillbox) = 0
If OTOH the snarkons are moving, we can say
-- 4D statement: Some snarkons enter the pillbox through the x0 face and
some exit through the x1 face.
-- 3D statement: There is a flux of snarkons entering and leaving the
pillbox.
We can mathematicize all this by defining a snarkon [density, flux] vector,
which we can call S: The zeroth component of S is the number-density
(snarkons per unit volume), and the other three components are the
number-flux (snarkons per unit area per unit time). The law of
conservation of snarkons can be written as
del_mu S^mu = 0 (4D language)
or
d/dt(S_0) + del_i S^i = 0 (3D language)
Futhermore, we can multiply this S vector by the rest-energy of a snarkon
to get an [energy, momentum] vector, which we can call Q. The law of
conservation of snarkon rest-energy can be written
del_mu Q^mu = 0
and by now you can figure out the 3D version for yourself.
This is the law that Jack was asking about. It's an interesting and useful
law.....
BUT (!!!) this is *NOT* the law of conservation of old-fashioned energy or
momentum in the lab frame. Snarkons are conserved, and their rest-energy
is conserved, but the real laws of conservation of energy and momentum say
much more than that.
===============================
To model conservation of [energy,momentum], suppose that rather than having
a few isolated, independent snarkons moving around, we have a *fluid* of
snarkons. Some of them bump into their neighbors, transferring energy and
momentum. By this process, energy and/or momentum can be transferred into
our pillbox *without* any noticeable number of snarkons entering or
leaving. We believe that the law of conservation of energy and momentum
exists in its own right, independent of the law of conservation of snarkons.
In order to write down this fancy law, we need a tensor, which we can call
T (with components T^mu^nu). I know tensors scare some folks, but sorry,
that's the way it goes sometimes.
The zero,zero component of this tensor represents the flow of energy in the
time direction. Physically, this represents energy that flows into the
pillbox through the t0 face and out through the t1 face. The one,two
component of this tensor represents the flow (flowing in the X direction)
of the Y-component of momentum. A physical example of such a flow occurs
due to viscosity, when there is a shear layer. Imagine a shear layer
perpendicular to the X direction, and on one side of the layer the fluid is
moving in the Y direction while on the other side it is stationary.
In terms of this tensor, the genuine law of conservation of energy and
momentum can be written
del_mu T^mu^nu = 0
where nu is an unbound variable, meaning that the foregoing equation states
that each of the four components (nu=0123) are separately conserved.