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From: ethan@utastro.UUCP (Ethan Vishniac)
Newsgroups: net.physics
Subject: Re: A question about mass and energy
Message-ID: <365@utastro.UUCP>
Date: Tue, 16-Jul-85 20:21:45 EDT
Article-I.D.: utastro.365
Posted: Tue Jul 16 20:21:45 1985
Date-Received: Sat, 20-Jul-85 05:08:14 EDT
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Organization: U. Texas, Astronomy, Austin, TX
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>     The problem is gravitation.  All other forms of energy have (at least)
> one thing in common:  in general relativity, they are source terms on
> the right hand side of Einstein's field equation, G = 8piT.  That is, they
> curve space, and cause a relative geodesic deviation of nearby world lines
> that pass through that region of space.  No such thing is true of any
> definable "local gravitational energy" because of the equivalence principal.
> The equivalence principal implies that for every neighborhood in space-time
> there is a coordinate frame in which all local gravitational fields 
> disappear.  (I.e., all Christoffel symbols vanish.)  Thus any attempt
> to define a local gravitational energy (or, more precisely, an energy
> momentum tensor) will fail.
>      Note, however, that gravitation does contribute to the energy;  
> the energy of the solar system is less than the energy that the system
> would have if its parts were at infinite separation.  That is undeniable.
> What is deniable is the localizability of gravitational energy.  
> Gravitational energy is a global effect, caused by global curvature.
>      This difficulty is why it took seventy years after the discovery of
> GR to prove that the total energy in a region surrounded by asymptotically
> flat space is positive, because, by choosing an appropriate coordinate
> system, you can make the energy at any point whatever you want.
> 
> Most of this argument was take from Misner, Thorne, and Wheeler, 
> "Gravitation", p466ff.

There is, however, at least one *nonlocal* rigorous definition of energy
due to Penrose which allows us to calculate the energy contained inside
any volume on an arbitrarily chosen hyperspatial plane.  The definition is
analogous to Gauss's law for electromagnetism in that it uses only the value
of the metric (and various derivatives) on the surface of the volume.
The main drawback to this method is that "calculable" is a theoretical not
a practical, description of the definition.  In practice only a few simple
situations give calculable answers.

-- 

"Don't argue with a fool.      Ethan Vishniac
 Borrow his money."            {charm,ut-sally,ut-ngp,noao}!utastro!ethan
                               Department of Astronomy
                               University of Texas