Re: 4D conservation of marbles, energy, work, heat

[John Denker <jsd@MONMOUTH.COM>]

At 11:44 PM 9/3/99 -0500, JACK L. URETSKY wrote:
>        I think that the vanishing of a 4-divergence is to be interpreted
> as the (in poetic terms) statement that the outward "flow" of the 3-vector
> is equal to the time rate of change of the time component.

OK so far.  That statement, if correctly interpreted, is true.

> I cannot make
> sense of the statement (in this context) that "each component" of the
> 4-vector "is conserved".

The trick here is that "the" four-vector in this sentence is almost 100%
unrelated to "the" three-vector in the previous sentence.  (Multiple layers
of meaning are often desirable in poetry, but usually not in physics :-)

We must not tangle up the scalar/vector/tensor properties of the thing
being conserved with the vector properties of the derivatives used to
express the conservation law.

In particular, consider a truly scalar quantity such as the number of
marbles.  Under many circumstances, the number of marbles is conserved.
The rate of change of marble-density is equal to the divergence of the
marble-flux field.  Now consider colored marbles.  The red ones and the
green ones are separately conserved.  Each component of the color "vector"
is separately conserved.  The number of conserved quantities has nothing to
do with the dimensionality of the space in which the marbles move.

-----------

In cases of confusion, it often helps to use Gauss's law to switch from the
derivative formulation to the integral formulation.  So let's do that now.

Also, it may help to keep in mind that there is a choice here:  We can take
the 4 dimensional viewpoint, or we can take 3+1 dimensional viewpoint.

In the 3D viewpoint, the Gaussian pillbox under observation has a 3D volume
and is bounded by a 2D surface.  We can ask about
  * the amount of X that crossed the surface during a particular time
period, and
  * the change in X within the volume during that time period

In the 4D viewpoint, the region under observation has a 4D hypervolume and
is bounded by a 3D hypersurface.  It might be a hypercube having faces on
its +-X sides, its +-Y sides, its +-Z sides, and its +-T sides.  We can ask
about
  * the amount of X crossing the hypersurface.

In particular, consider a non-conserved quantity such as the number of
bacteria.
  Case 1) A bacterial world-line enters the pillbox by flowing in across
the -X face thereof.  The bacterium undergoes mitosis.  Then *two*
bacterial world-lines exit the pillbox by flowing out the +X face.

       . . . . . . . . . . .
    -X .                   . +X
       .       _____________________
  ____________/            .
       .      \_____________________
       .                   .
       . . . . . . . . . . .

In the 3+1 viewpoint, this would be expressed in just the same way, by
surface-crossing terms in the conservation statement.

  Case 2) A bacterial world-line enters the pillbox by crossing the -T face
thereof.  The bacterium undergoes mitosis.  Then *two* bacterial
world-lines exit the pillbox by crossing the +T face.

            |    |
            |    |  +T
       . . .|. . | . . . . .
       .    |    |         .
       .    |    |         .
       .    |    |         .
       .     \  /          .
       .      \/           .
       .      |            .
       .      |            .
       . . . .|. . . . . . .
              |     -T
              |

In the 3+1 viewpoint, this would be expressed rather differently, by a
delta N / delta T term in the conservation statement.

So we see that to convert the conservation statement from the 3+1D
viewpoint to the 4D viewpoint, the delta N / delta T term is replaced by
surface-crossing terms on the +-T faces.  That's basically all there is to
it.  Simple.  Elegant.  Independent of *what* is being conserved.