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- What happens to invariant mass of an object when it gets closer or further from a gravitational body?

In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1).

[tex] m^2 = E^2 - \vec{p}^2 [/tex]

This can also be written with the Minkowski metric as:

[tex] m^2 = \eta_{\mu\nu} p^\mu p^\nu [/tex]

More generally, if there is a different metric (for example Schwartzchild), you would write it as:

[tex] m^2 = g_{\mu\nu} p^\mu p^\nu [/tex]

Now the question is, if invariant mass does not change from one metric to the other, you get the equation:

[tex] 0 = (g_{\mu\nu} - \eta_{\mu\nu})p^\mu p^\nu [/tex]

This seems to give unphysical results.

I solved for a photon in the Schwartzchild metric, and the only physical solution available is if the Schwartzchild radius is 0. So this seems to imply that invariant mass (or lack thereof) is not invariant under gravitational fields.

Any help here would be much appreciated. Thank you.

[tex] m^2 = E^2 - \vec{p}^2 [/tex]

This can also be written with the Minkowski metric as:

[tex] m^2 = \eta_{\mu\nu} p^\mu p^\nu [/tex]

More generally, if there is a different metric (for example Schwartzchild), you would write it as:

[tex] m^2 = g_{\mu\nu} p^\mu p^\nu [/tex]

Now the question is, if invariant mass does not change from one metric to the other, you get the equation:

[tex] 0 = (g_{\mu\nu} - \eta_{\mu\nu})p^\mu p^\nu [/tex]

This seems to give unphysical results.

I solved for a photon in the Schwartzchild metric, and the only physical solution available is if the Schwartzchild radius is 0. So this seems to imply that invariant mass (or lack thereof) is not invariant under gravitational fields.

Any help here would be much appreciated. Thank you.