Mass as Curvature in GR: Deriving Inertia as a Force

[Crosson]

I am a student who is studying General Relativity, and I don't know enough tensor analysis to answer the following straightfoward question:

Since the stress energy tensor is just a sophisticated representation of mass, and since einstiens field equations equate this energy to a representation of curved spacetime, is it appropriate to say that mass is the curvature of spacetime?

More precisely, is it possible to derive from the field equations the existence of a "force" (spatial rate of change in energy) that is proportional to acceleration and "mass" (E/c^2) in the Newtonian limit?

Conceptually, I am asking if GR predicts inertia as a force which does work on spacetime.

If true, notice that Newtons Law (sum of F = ma) reduces to the following elegant statement:

Sum F = 0

(a force of -ma occurs when an object with gravitational mass accelerates)

 

[pervect]

It is possible to approximate gravitation as a "force", yielding the familiar Newtonian laws, only when all velocities are much lower than the speed of light, and when the distortions of space and time due to gravity are not too severe.

This approach is often discusses as the "PPN" approximation to general relativity. It's good enough for usage anywhere in the Solar system, but it will not work under conditions of extreme gravity.

The more general approach of the full theory requires that one view gravity as a curvature of space-time rather than a force. The conditions which allow one to approximate gravity as a force are the conditions where one can ignore the spatial part of the space-time curvature.


The stress energy tensor is really a lot more than a sophisticated representation of mass, as one might guess from the fact that it has 10 components at a point in space-time, while mass is a scalar quantity that has only one component. The conditions that allow one to approximate space as flat also simplify the stress-energy tensor, though - for instance, the pressure terms of the stress energy tensor contribute to gravity in the full theory, but in the PPN approximation one assumes that the contribution of pressure to gravity is negligible. Similarly, momentum contributes to the stress energy tensor, but because all velocities are much lower than 'c', these contributions can be ignored in PPN theory.