Do the Earth and I apply the same gravitational force on each other in GR?

Question

Our high-school teacher told us that the Earth pulls us with some force $F$ and we pull the Earth with the same force $F$. Within Newtonian physics this is true because of [Newton's 3rd law](https://en.wikipedia.org/wiki/Newton's_laws_of_motion), but let’s consider [Einsteinian gravity](https://en.wikipedia.org/wiki/General_relativity). My mass is small; so I don’t warp space-time much. But Earth’s large mass warps space-time to a far greater extent. So do I pull Earth with the same force it pulls me? If yes, how?

Answer 1

Newton's 3rd law is a consequence of the conservation of momentum. So the question is whether the conservation of momentum is obeyed in General Relativity. For this we have to go deeper: The conservation of momentum is guaranteed by Noether's theorem if the laws of motion (read Lagrangian) of GR is invariant under translations. Therefore, if the Lagrangian of GR is translation invariant, GR would obey Newton's 3rd law. Now, the Lagrangian of GR depends on the metric tensor which is not always translation invariant, e.g. Schwarzschild metric. So, if the metric tensor is translation invariant, GR would obey Newton's 3rd law.

Logic flow:

Translation invariance of metric tensor => Translation invariance of GR Lagrangian => Conservation of momentum => Newton's 3rd law

Short Answer:

Newton's 3rd law is only obeyed in GR if the metric tensor does not change under translations in space.