general relativity - Invariant interval and gravity

Gravity is inversely proportional to the distance between objects.

Do we use Euclidean distance or the invariant interval for that distance?

Using the invariant interval makes everything a bit more complex: An object that is one lightyear away, is also one year "younger" when we observe it. Does that temporal distance contribute, so that when we calculate trajectories and physical phenomena on a distance - they are different from what's observed locally?

Does this affect observed things such as the trajectory of two equally distant (from us) objects and rotational velocity of distant objects?

An example:

When we look towards the center of the milky way, we're looking about 27 000 years back in time; i.e. the temporal distance is 27000 years, and spatial distance is 27 kly

When we look at the center of the andromeda galaxy, it's about 25 Mly away. When we look at another object in the andromeda galaxy, which is also 25 Mly away - the temporal distance between those objects, relative to us is 0.

So; is the relative temporal distance between objects and the observer incorporated in Einsteins field equations?

Answer

Gravity (the potential that is, not the force) is only inversely proportional to actual, spatial distance in the Newtonian approximation of GR, also known as the weak field limit, i.e. for bodies (like the earth) that are very far from becoming a black hole. One obtains it e.g. in the Schwarzschild solution of GR for $\frac{r_S}{r}\ll 1$.

But, in general, gravity is not proportional to anything - the presence of matter determines the geometry of spacetime through the Einstein field equations, and "gravity" is the observation that that geometry differs - in whatever form - from the gravity-free flat space that is Minkowski space.

About the temporal distance being included or not included - the Einstein field equations are, as a tensor equation, invariant under arbitrary coordinate transformations, just as every good SR equation is invariant under Lorentz transformations. Both invariances mean that the term temporal distance (which is not invariant) cannot really play a role in these, because, from another frame, that temporal distance will be different (or non-existent, or have a different sign). You can, invariantly, talk about spacetime distances being time-like, light-like or space-like depending on their sign. But, there is nothing to include here - no matter whether SR or GR, we have given up on assigning any special treatment to time: It is just another coordinate on the spacetime manifold (well, in SR, it's the one with the odd sign in the metric, but that's really it), and our equations and techniques apply to everything on that manifold equally.