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

Hi John Denker-
You wrote, in part:
************************************************************************
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".
***************************
I do not understand you, unless I confused the issue by neglecting
to write: "of a 4-vector" in the initial statement that should have read
"I think that the vanishing of a 4-divergence of a 4-vector is to be..."

Then you added:
*********************************************************************
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 :-)
**********************
But this statement cannot be true because I am referring to the
3-vector (meaning spacelike 3-vector) part of some arbitrary 4-vector that
has vanishing divergence.

Then you added:
******************************************************************
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.
***************************
Yes, that was implicit in my remark.

******************************************************************************
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
*********************************************************
Yes, I've taken the 3-d viewpoint. But Gauss's law makes no reference
to an "X". It says that the time-derivative of the time-component of the 4-vector,
integrated over a 3-volume, is equal to the integrated outward flow of the 3-vector
through the boundary of that same volume ("integrated outward flow" is poetry for
Surface integral over the boundary surface of the dot prduct n.V, where n is the outward
normal to a point on the boundary and V is the value of the 3-vector at that point).
How do we get to an "X"? We get that from the physics associated with
L-transformations. If it so happens that there is a L-frame in which the 4-vector
has only a time component, then we can identify the time component with the
density of some quantity X. In any other L-frame, then, the space-vector part
represents the flux of that same X.
All of the above was implicit in my posting. I did not include it because
it was not relevant to my main point:
Vanishing 4-divergence of a 4-vector does not mean that each of the 4
components of the 4-vector is separately conserved. If anything in your posting
contradicts my point then I totally misssed it.
Regards,
Jack



"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography