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The speed of light is not constant everywhere in a gravitational field. Suppose there is a region of space-time which is curved due to gravity such that the speed of light or any electromagnetic radiation is more than that of in a flat or Minkowskian space-time. What will be the laws of physics in that region of space-time?

Or, more precisely, what will be the equation for mass-energy conversation, $E=mc^2$, the equation with the $c$, speed of light in flat space-time or, $E=mv^2$, the equation with $v$, speed of light in that particular space-time?

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This question is quite broad, but I'll try my best.

When making the jump from special relativity, we have two good rules of thumb:

  1. Wherever there is a Minkowski metric in SR, put a general metric in GR.

  2. Wherever there is a partial derivative in SR, put a covariant derivative in GR.

Take, for instance, mass-energy equivalence. In SR, we have $$E^2=\mathbf{p}^2c^2+m^2c^4$$ Covariantly, we write $$\eta_{\mu\nu}p^\mu p^\nu=-m^2 c^2$$ In GR, we thus have $$g_{\mu\nu}p^\mu p^\nu=-m^2c^2$$ The speed of light does not change, but the law itself does. Now we have to worry about the components of the metric. In the rest frame of the particle $E=mc^2$ still holds, however, assuming the particle is on a geodesic. This is due to the Equivalence Principle.

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  • $\begingroup$ But I don't understand why the speed of light would not change. Because, as light moves along null geodesic, we have, $g_{ii}(dx^i)^2+g_{00}(dx^0)^2=0$ and, $dx^i/dx^0=+−(−g_{00}/g_{ii})1/2$ and the quantity here is not necessarily equal to c, in different cases of gravitational field. @0celo7 $\endgroup$
    – VacuuM
    Commented Feb 27, 2015 at 18:06
  • $\begingroup$ You have written "The speed of light does not change, but the law itself does".... I think this is incorrect, because if you write your law in a generally co-variant way then your law will never change w.r.t any frame. But speed of light may change according to the law of physics. @0celo7 $\endgroup$
    – VacuuM
    Commented Feb 27, 2015 at 18:08

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