Re: Newtonian gravitational field energy

[John Denker <jsd@MONMOUTH.COM>]

At 03:55 PM 8/6/99 -0700, Leigh Palmer asked for comments on what appears
to be a non-standard derivation of a standard result, namely the energy
density of a static gravitational field.

0) As preface, let me say what is good about this effort:  it is a solid
attempt to play by the rules of physics.  It does not speak about angels or
high priests or Weltanshauungs.  Instead it speaks using equations.  I
respect that.  When equations go awry, there are nice straightfoward ways
to discuss and remedy the situation.

So here we go.

1) First of all, the attempted derivation does not actually get the
standard result -- it is off by a minus sign!  After doing work on the
gravitating system, there should be *more* energy in the gravitational
field, yet the calculation purports to show there is less.  Ooooops.

Therefore we must look at the derivation to see what went wrong.

2) Physically speaking, the problem is that in a real parallel-plate
system, the fringing fields are very important.  By uncritically
considering an infinite system, these important terms have been pushed out
of sight.  The result is a system that alas does not conserve energy,
locally or even globally.

3) Mathematically speaking, it appears that the main problem is a well
known one involving an improper interchange of limits and integration, so
let's begin by discussing limits in general.

3.1) It is risky to talk about infinite quantities or infinite systems.
When in doubt, you should consider a system with a large but finite size S,
and then take the limit as S gets exceedingly large.

3.2) Similar remarks apply to infintesimal quantities, where T is never
zero but gets exceedingly small.

3.3) In general, one cannot freely interchange the order of limits.  That is,
       lim(S->inf) lim(T->0) f(S,T)
  does not generally equal
       lim(T->0) lim(S->inf) f(S,T)

3.4) An integral (even an integral over a finite volume) is itself a limit;
 recall that we need the limit as the typical term in the Riemann sum gets
very small.

4) Specifically in this case, a correct derivation might consider two
parallel disks of large but finite size S, parallel to the XY plane, and
separated by an adjustable distance D in the Z direction.  As we increase
D, there will be a decrease in the energy between the plates, but an
increase in the energy in places farther out on the +Z and -Z axes.  Surely
even without working out the details, everyone believes that this increase
MUST be taken into account.

Therefore the place where the attempted derivation is most clearly
incorrect is this sentence:

> Thus the only difference we have to consider is what occurs between the
> plates:

That sentence implicitly subtracts two infinite quantities, which is not
kosher.  To say the same thing another way, that sentence implicitly argues
that the integral (of the field-energy-differences far above and below the
plates) is zero because the integrand is zero ... but the integrand is only
zero in the limit as S -> inf, and you are not allowed to bring that limit
inside the integral.

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

Anyway, it was a respectable effort.  I hope people will learn a little
physics and even a little math by seeing what works and what doesn't.