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All other forces have some kind of charge system, electromagnetism has positive and negative, weak force has hypercharges, and strong force has colors. Why doesn't gravity have anything like these?

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It does! The 'charge' of gravity is 'mass'. One aspect of the equivalence principle says that the gravitational-charge "mass" is the same as the inertial "mass". Why that seems to be the case is unclear. It's also unclear why there is no (or why we haven't observed any) 'negatively'-charged mass, but there is no known theoretical reason why this must be the case.

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  • $\begingroup$ Doesn't the charge of mass come from the interaction of the Subatomic parts? Maybe gravity causes charge but does charge come directly from it? $\endgroup$ – Bill Alsept Dec 15 '15 at 18:33
  • $\begingroup$ In order for massless spin-2 particles (gravitons) to scatter non-trivially then all the charges must be identical. This is the theoretical reason why gravity couples universally. There is a nice derivation and discussion of this fact at the end of chapter 9 in Schwartz's QFT. $\endgroup$ – Evan Rule Dec 15 '15 at 18:43
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    $\begingroup$ Hmmm ... in the structure of general relativity the equivalence of inertial and gravitation mass is explained: it follows from the "force" of gravity being an inertial pseudo-force in just the same way that the Centrifugal and Coriolis forces are proportional to (inertia) mass. I'm also told that the tensor nature of GR implies that a field theory built on it would be strictly attractive, but I don't understand that statement myself. $\endgroup$ – dmckee Dec 15 '15 at 18:45

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