Why the curvature of spacetime is related to momentum?

[Brucezhou]

Well, I'm totally in a mess now

 

[Dale]

It must be for a relativistic theory of gravity. We know that gravity is related to mass, and mass in one frame is mass and momentum in another frame. So any relativistic theory of gravity must be related to momentum.

 

[Psychosmurf]

The Einstein tensor, which describes the curvature of spacetime, is a rank 2 tensor, so the stress-energy tensor that appears in the field equations (and describes the distribution of mass and energy throughout spacetime) must also be a rank-2 tensor. It just so happens that the "time-space" components of this tensor look a lot like what we normally think of as momentum in Newtonian mechanics.

It should also be noted that this momentum is not about how fast a particle travels through space (with respect to some observer), but rather, it is momentum density through a volume element in spacetime.

 

[WannabeNewton]

It should also be noted that this momentum is not about how fast a particle travels through space (with respect to some observer), but rather, it is momentum density through a volume element in spacetime.

Slight correction here but it's through a volume element in space. For example in flat space-time if we have a distribution with energy-momentum $T^{\mu\nu}$ and define a slicing of space-time relative to a family of inertial observers with global inertial frame $(t,\vec{x})$ based on their global simultaneity slices $\Sigma_{t}$ (which, as per standard simultaneity, is $t = \text{const.}$) then $P^{i}(t) = \int _{\Sigma_t}T^{0i}(t,\vec{x})d^{3}x$ is the total momentum of the distribution. Similarly the total angular momentum of the source is $S^{i}(t) = \sum _{j,k}\epsilon^{ijk}\int _{\Sigma_t}x^{j}T^{0k}(t,\vec{x})d^{3}x$.

 

[dauto]

Not just momentum but also energy density, energy flux, pressure, and mechanical stress. In relativity all those things are different components of a single entity called "The tensor of density and flux of energy and momentum" - the "energy tensor" for short.