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

Thank you Igor for your illuminating discussion. I have some questions and
comments pertaining to the following excerpted postions:

<snip>

> 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.

<snip>

> 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.

These observations are very pertinent to how one interpret the results in
the draft paper I posted at:
http://home.nycap.rr.com/jry/Papers/Uncertainty%20and%20GR.pdf. I suspect
that your reaction in general will be the same, namely "What you [do] 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," especially since that is exactly what I am dealing with
there.

What we learn from general relativity is that we must seek to describe
events which have physical significance, and, since coordinate systems have
no physical significance, what we describe physically cannot depend on our
coordinates.

Yet, what I think you are saying is that when we integrate a tensor density
and end up with an integrated form that has one free index, "the result of
integration of their components [is not] in general coordinate independent."

So, here is the crux of the question I think I am exploring: Does this tell
us anything more, other than about the inability to have different observers
attribute the same objective meaning to their own independently-conducted
integrations of a tensor density? In particular, is there a *physical*
manifestation of this dilemma? Is this a problem with coordinates and
mathematics only, or is this an indication of natural, physical phenomenon?
Does this problem that Igor walks through above show up in observable
physics? Is there some *physically meaningful* statement that characterizes
the degree to which the integrated tensor densities with a free index *fail
to be objective from one observer to the next*?

I suspect this is more than a problem with coordinates and integrating
tensor densities objectively, and I suspect the observable physics it shows
up in is the uncertainty principle which gives us, not an objective
observation of energy and momentum over finite regions of spacetime, but a
statement that in general one cannot make a coordinate-invariant statement
about the results of integrating energy and momentum over a finite region of
spacetime, and a statement about the degree to which this integration is
imprecisely objective.

In the draft paper referenced above, following integration of a tensor
density, we end up with an integrated quantity with a free, four-component
index, and that free index seems to amenable to a connection with "delta E
delta t" and "delta P_k delta x_k", k=x,y,z.

In other words, is the uncertainty principle the *physical* manifestation of
the problems with integrated tensor densities and different frames of
reference / choice of coordinates being discussed above?

If so, then Heisenberg uncertainty may be the direct *physical* consequence
of the *mathematical / coordinate* problem of integrating tensor densities
in general relativity.

I think I have pointed out some mathematical connections which are
indicative in this direction. I am walking through the wilderness of trying
to find the right language and perspective to talk in "objective" terms
about a situation where different observers cannot, even in principle, make
"objective" statements about their integration of tensor densities.

Help?

Thanks.

Jay.





















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