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I read Spacetime and Geometry by Sean Carroll. In p. 166 there is a comment that GR's action is nonlinear because if it is linear like the EM field, then graviton will not interact with each other, then a massive object A interacts with a massive object B and this system's inertial mass equals to mass A + mass B + interacting energy while its gravitational mass equals to mass A + mass B.

Then if we require that the gravitational mass equals to inertial mass, the action must be nonlinear. While not all nonlinear interaction term will satisfy that the gravitational mass equals to inertial mass, it is only a necessary condition. While Einstein-Hilbert action obviously satisfies this requirement.

My question is: What's all possibilities of action that can satisfy that the gravitational mass equals to inertial mass? Or Einstein-Hilbert action is only one that can satisfy this requirement?

John Rennie
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  • If I remember correctly, doesn't Carroll cover the possible actions that satisfy this? – Jim Nov 10 '14 at 14:28
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    This may be too big a question to work on stackexchange. If you were teaching a graduate course on GR, a big landmark of the course could be to prove this. –  Nov 10 '14 at 15:35
  • If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Nov 10 '14 at 19:21

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