You also have to include the total gravitational energy or T^{ab} due to
local sources and Λg^{ab}.
The ADM Hamiltonian constraint is NH = 0 where this Hamiltonian is
determined by the traceless transverse part of the extrinsic curvature or
Gauss fundamental form. For a general spacetime manifold there is no way to
define mass-energy and for most Petrov types the mass-energy is simply no
defined. Think of a spherical space with matter throughout. There is no way
to construct a Gaussian surface with which to integrate a total mass or
energy. Also if that putative surface is embedded in mass-energy then that
surface is subject to diffeomorphisms of local curvature. Energy is then
not localizable, and in general things that we want invariant are so
independent of such diffeomorphisms.
LC
On Tuesday, September 3, 2019 at 10:00:55 PM UTC-5, Alan Grayson wrote:
>
> Just sum over the estimated total of 10^80 particles, using mc^2 by first
> estimating the average mass of those particles for the rest energy, adding
> their average potential gravitational energy and their average kinetic
> energy. Why not? AG
>
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