Why masses attract each other?

Question

According to the gravitational law, every mass attracts each other. But why the masses attract each other? Why they don't repel each other?

Answer 1

Mass is gravity's equivalent of electric charge, with two obvious differences:

• Charges can be positive or negative, but masses are positive. We can discuss "what about the mass-energy stored in gravitational fields?" or any number of gotchas, but you and me and planets, as familiar examples, definitely have positive mass.
• If the dissimilarity ended there, you might expect masses to repel: after all, positive charges repel each other. Mathematically, we need to get a sign change from somewhere.

Where we get it from is a very complicated theoretical question. The first few chapters of Quantum Field Theory in a Nutshell derive the mathematics, but here's the short version:

• Interactions are due to the exchange of particles of integer spin, coupling to a conserved tensor-valued current whose rank is that spin;

• Positive attracts positive if that spin is even, or repel if it's odd, because otherwise the action we minimise wouldn't have a kinetic cost, so would be unstable;

• The spin has to be $0$, $1$ or $2$;

• Electromagnetism is due to a spin-$1$ photon, coupling to a conserved vector current, so like charges repel;

• Gravity acts on all mass-energy contributions to the rank-$2$ stress-energy tensor, which requires a spin-$2$ graviton, which implies positive masses are attractive.