Re: Covariant Volume Integration and Stokes' / Gauss' Theorem in Curved Spacetime

Jay R. Yablon wrote:
> Hello to everyone:
>
> There was some discussion with DRL in an earlier SPR post regarding
> volume integration in curved spacetime and what is and is not possible.
> I have brushed up on this a bit with some off-line advice from Igor K.
> as well. Thank you to both of these folks.
>
> In a document at:
>
> http://home.nycap.rr.com/jry/Papers/Volume%20Integration.pdf

> I'd appreciate if some of the sharp folks here might take a look and
> make sure I did this right, before I do try to use the formalism here
> for some new things.

First, I have to say that I find your notation somewhat confusing. What
do you denote by the symbol dx^u? Is it a 1-form? Is it a component of
a 1-form? (That is a kind of object one would expect to find when
dealing with integration on manifolds). Or do you treat them more as
formal symbols, like in elementary calculus, marking the integration
variable? Be careful, it is very easy for notation to lead you into a
trap.

What you is, as far as I can tell, mathematically valid. However, you
get into trouble as soon as you try to assign physical interpretation
to any integrated quantity with a free index, like for instance what
you call "the integral form of Maxwell's equations". You can always
perform the integrals that you write down and get some numbers out.
However, you will not be able to compare your results with a friend or
a collegue. That is because the other person can do the exact same
computation but with a different choice of coordinate system (say a
non-inertial coordinate system) and get different numbers. In other
words, your results depend on the choice of coordinates on space time.
But one of the wisdoms that everyone should learn from GR is that any
such results have no physical significance by themselves.

Lets review Stoke's theorem and see exactly why that happens. To
perform an integral over a p-dimensional piece of the manifold V, you
need degree-p differential form, say w. In other words, we can write

<V,w> = integral over V of w

and obtain a number that is independent of any particular
coordinatization of V. There are two natural operations that we can
perform. We can take the boundary of V, call it @V, which is a
(p-1)-dimensional piece of the manifold. And we can take an exterior
derivative of a differential form, and obtain a differential form of
1-higher degree. Actually, lets suppose that there is a degree-(p-1)
differential form u such that du = w. Note that u is just of the right
degree to be integrated over the boundary @V. Lets put all of this
together in the statement of Stoke's theorem:

<V,w> = <V,du> = <@V,u>.

It's as simple as that. In other words, we can think of the operations
(exterior derivative) d and (boundary operator) @ being adjoints with
respect to the bilinear pairing

<(piece of manifold),(differential form)>.

Incidentally, what I'm calling "piece of a manifold" is formally called
a "chain". Chains can be simply thought of as domains of integration.
We can take linear combinations of them with real coefficients:

int_(aV_1 + bV_2) = a*int_(V_1) + b*int_(V_2).

Incidentally, that is also why the above pairing < , > is bilinear.

In short, now you have two pieces of information. (1) Integration over
a chain can only be done with respect to a differential form of the
right degree. If you don't have such a differential form, you don't
have an integral. (2) You have a formulation of Stoke's theorem in this
formalism, which you can (if you wish) translate to the classic
component notation for tensors.

Now, lets see what happens when we try to deal with pseudotensors. In
GR, we not only have a manifold, but also a metric. Whenever we have a
metric, we can define a canonical degree-n differential form (if we are
in n-dimensional space-time), the volume form vol. In any given
coordinate
system, its components are proportional to the antisymmetric
Levi-Civita symbol, with the constant of proportionality being
sqrt(-det|g|). If we have a scalar function f, we can construct the new
n-form f*vol. This n-form can now be integrated over space-time, and
that's how must integrals in GR are constructed. It so happens, that if
f is the divergence of some vector field v, f = div v. Then we can find
an (n-1)-form u such that du = (div v)*vol = f*vol. That's great, it
means we can use the above formulation of Stoke's theorem to generalize
the classic divergence theorem to curved space-times.

What about tensor densities? Well, in a given coordinate system, the
components of a tensor T can be considered as scalars. If you've had a
look at MTW's book on Gravitation, you should know that T acts as a
linear black box that eats vector fields and 1-forms and spits out a
scalar. In the case of a given coordinate system, we can use this
property on the vector fields parallel to the coordinate axes and
1-forms taken as differentials of the coordinate functions to extract
what we usually mean by the components of the tensor T. In other words,
given a coordinate system, you can extract some scalars from the tensor
T and you can integrate them over space time, just like in the previous
paragraph. But I hope you've noticed by now that the numerical results
that you get depend intrinsically on the coordinate system that you
chose (or equivalently on the vectorfields and 1-forms that you chose
to plug into your tensor to get its components).

Coming back to the divergence theorem. Say you have a degree-2
contravariant tensor (like the stress energy tensor), and you take its
divergence. You still have one free index left. In any coordinate
system, you can extract its components. However, the scalar components
will not in general be themselves divergences of some vector field.
Unless, that is, the coordinate 1-forms you used to extract these
components are covariantly constant (their covariant derivatives
vanish). This does happen for inertial coordinates in Minkowski space,
but not in general. The conclusion is that, in general, there is no
divergence theorem for tensors, nor is the result of integration of
their components in general coordinate independent.

Hope this helps.

Igor

.

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