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## Main Question or Discussion Point

I have always thought that GR was identical to classical physics in the Newtonian limit. In the Newtonian limit one can define an energy density for a gravitational field parallel to the methods used to derive the energy density of a static electric field, resulting in a formula

energy density = |g|^2/8*pi*G

where g is Gm/r^2 outside a spherical mass distribution.

Shouldn't this energy contribute to the space-time curvature, or is this another way to interpret the space time curvature?

A classical integration of the mass, including the mass of the gravitational field above, blows up at a radius of rs/4, where rs is the Swartzchild radius. Is it a coincidence that the isotropic coordinates often used also have a singularity at r=rs/4?

I am currently looking for a consistent way to resolve the fact that my classical derivation does not account for time dilation as the field strength increases. My intuition tells me that the length the object travels is contracted by a net 3/4*rs by the gravitational field, but can't show this yet.

Since there is no preferred reference frame, is it valid to simply pick a convenient one? When I pick one at infinity co-moving with the central mass it appears to me that space is flowing towards the central mass with time.

Is my intuition correct that the proper time is covariant, but once I pick a frame my time is contravariant? Is space time-curvature in a covariant description the same as space flowing with time in a contravariant description?

I have a very limited knowledge of tensors (which might make these questions trivial) so a solid resource on this would be greatly appreciated.

energy density = |g|^2/8*pi*G

where g is Gm/r^2 outside a spherical mass distribution.

Shouldn't this energy contribute to the space-time curvature, or is this another way to interpret the space time curvature?

A classical integration of the mass, including the mass of the gravitational field above, blows up at a radius of rs/4, where rs is the Swartzchild radius. Is it a coincidence that the isotropic coordinates often used also have a singularity at r=rs/4?

I am currently looking for a consistent way to resolve the fact that my classical derivation does not account for time dilation as the field strength increases. My intuition tells me that the length the object travels is contracted by a net 3/4*rs by the gravitational field, but can't show this yet.

Since there is no preferred reference frame, is it valid to simply pick a convenient one? When I pick one at infinity co-moving with the central mass it appears to me that space is flowing towards the central mass with time.

Is my intuition correct that the proper time is covariant, but once I pick a frame my time is contravariant? Is space time-curvature in a covariant description the same as space flowing with time in a contravariant description?

I have a very limited knowledge of tensors (which might make these questions trivial) so a solid resource on this would be greatly appreciated.