Flaw when considering gravitation from Newton versus GRT point of view

Question

Stage: Two massive objects are moving slowly with respect to each other and to some galactic background. For example Sun and Earth. We apply Newtonian gravitation concepts and the force felt by the Earth is proportional to the square of the distance between Sun and Earth. Of course, we are making some approximations here: The distance between Sun and Earth varies over time - so which distance should we use when? Newton assumes that the gravitational effect is immediate and travels at infinite speed, but the error we make with this remains small.

Change of perspective: Now let us put on general relativistic (GRT) glasses to correct for this negligence and get more precise results.

What would now play the role of "distance" in the GRT point of view? As the Newtonian result is a good low speed low energy approximation there should be some kind of physical concept in the GRT perspective which approximates the role of distance.

This concept certainly is not (Newtonian, spatial) distance, as in GRT spatial distance is not invariant and depends on the choice of the coordinate system. So, well, then let us use space-time distance, which is invariant. To calculate space-time distance we need to choose two events. Let us pick location Earth and as time 1. 1. 2022 12:00 for the first event. What would be the appropriate second event? Well, assuming gravity travels at the speed of light, let us connect this earth event with an event located on the Sun which we reach by following the backward light-cone from our earth event. That should provide the correct position/distance of the sun for that event. Following that light-like path gives us a space-time distance of...zero. Hm. Not what we wanted.

The problem seems obvious: We were measuring distance in a kind of light-cone coordinate system for which the Newtonian law of gravitation, of course, does not hold.

My Question: What is the physical concept which tells us something about the strength of the gravitational effect the Sun has on the Earth in the GRT perspective?

I am interested in a real (as in: independent of the choice of the coordinate system) and physical (as opposed to a purely mathematical concept, eg. the Ricci tensor) concept which gives a rough idea of the gravitational effect. I want to understand the physical mechanism of gravitation and my mental model obviously leads me astray.

Answer 1

What would now play the role of "distance" in the GRT point of view?

There is nothing that plays that role in GR.

This is analogous to Coulomb’s law and Maxwell’s equations. Coulomb’s law is an approximation to Maxwell’s equations, valid in electrostatics, and indeed Coulomb’s law can be derived from Maxwell’s equations with that assumption. However, there is no corresponding $r$ anywhere in Maxwell’s equations.

Like Maxwell’s equations, the Einstein Field Equations are a set of differential equations. They describe how the field varies locally. In the appropriate circumstance you can produce the $r$ dependence found in the simplified law, but that $r$ dependence does not appear in the general law and it is only meaningful in the specific cases where the simplified law applies.

For gravity, Newton’s law of gravitation is the simplified scenario, the equivalent of Coulomb’s law. It is a limiting case of the Einstein Field Equations, and in that limit the $r$ is the same as the Newtonian $r$. Outside of that limit, in general scenarios, that concept isn’t important in GR, just the local changes in the fields.

Answer 2

The tensors that occur in General Relativity are what you are looking for (in particular, the Ricci tensor, but also the Riemann tensor, and the metric, and even the Christoffel symbols, which are not really tensors).

In GR, gravity is an effect due to the curvature of spacetime. This curvature is quantified in a coordinate-independent manner by the metric and its derivatives. If you want to know how the Earth will move close to a given event (assuming the Earth is there), all you gotta do is evaluate all of these tensors in the proximity of that event and solve the geodesic equation.

Distance is no longer something that explicit, because now gravity is not due to a force, but due to spacetime itself. The Sun bends spacetime and the Earth (which we are here approximating as a really light object) moves on this curved spacetime. This curvature leads to the effects which we observe as gravity.

If you are looking for a scalar that quantifies the strength of gravity in each point, you won't find it unless you start contracting the metric or the Riemann tensor with all vectors of some basis to get the components in an invariant way (for example, $g_{ab}\xi^{a}\xi^{b}$ is coordinate-independent). The curvature is characterized by tensors and gravity is characterized by curvature.

There are, of course, some numbers which can hint on one or other aspect. The Ricci scalar can tell you how intensely bodies are being contracted, or the Kretschmann scalar can tell you how intense are the tidal effects. In the end of the day, you can't reduce everything down to a single number and you'll end up working with something that is no better than working with the tensors we started with (but perhaps it will be more complicated to compute with).