# In the context of General Relativity, does having gravitational mass imply resistance to acceleration?

My question is essentially this one, but since that one never got a satisfactory answer, and it's of such fundamental concern, I'd like to try asking it in a different way.

The question is whether general relativity offers a reason why gravitational and inertial mass are equal. But this is not fully answered by the law of geodesic motion; that merely removes gravity from the list of forces, and changes the meaning of "acceleration" to "deviation from a geodesic path". So the question becomes: how does having gravitational mass cause a body to resist deviating from its own geodesic path, when a non-gravitational (eg. EM) field is present?

Meanwhile, the field equation associates "gravitational mass" with positive timelike Ricci curvature (actually, positive Ricci curvature in all spacetime directions, if that matters).

And finally, since we are working in the confines of GR, we must represent the "non-gravitational external field" as yet another region of non-zero Ricci curvature. If we're talking EM, its form would thus be given by the electromagnetic stress-energy tensor. This does not capture the full character of the EM field, but it's the best GR by itself can offer. So finally we arrive at the fully-GR version of the question:

Does the presence of higher positive Ricci curvature along a worldline imply that worldline will deviate less in the presence of a given external background of non-zero Ricci curvature?

You asked a couple of different questions here, but the one that's in the title and introduced earlier is this:

The question is whether general relativity offers a reason why gravitational and inertial mass are equal.

The answer to this might depend a bit on what you mean by a reason. From a mathematical point of view, this is a statement of the equivalence principle, which is often taken as one of the postulates of general relativity. As such it is not "explained" but an assumption. From a more physical point of view, the equivalence principle is justified by various thought experiments that suggest it should not be possible to tell the difference between an accelerating frame and gravitation (e.g. the various versions of an experimenter "dropping" something in an elevator).

The rest of your text seems to be trying to work it the other way around, starting with postulates about how differential geometry maps to physics and inferring the equivalence principle. You can find texts that make different choices about what's assumed and what's derived, but I have trouble following your particular chain of reasoning as it starts from more abstract mathematics and tries to back-fit physics to it. (At least as I understand what you wrote.)

• Here's how I see it: the Einstein equivalence principle says that gravitational motion is the same thing as inertial motion (they are both geodesics). But it doesn't say why, for example, two objects that differ only in their mass, in a given EM field, will accelerate in accordance with their masses. That is not part of the EEP, and not the job of GR. However, since GR gives us new representations of both mass and external field -- in terms of curvature -- maybe from these new representations we can get some insight into the latter question... Oct 5 '21 at 23:37
• The "equivalence" in the equivalence principle is exactly the equivalence between inertial and gravitational mass. Your comment is mixing up different things. en.wikipedia.org/wiki/… Oct 6 '21 at 2:22

By gravitational mass of an object, I understand for example that when a scale in the surface of the earth shows a force $$F$$, its mass is $$m = \frac{F}{g}$$

If the same object is in the ground of a rocket in the outer space with an acceleration $$g$$, as measured by an inertial comoving frame, a scale in the rocket will also shows a force $$F$$. And in this case, the conclusion is that the object requires the same force to get the acceleration $$g$$. So, its mass is also its resistance to acceleration.

Basically, GR explains the equivalence of gravitational mass and inertial mass by doing away with gravitational mass. It does not attempt to explain inertia, although Einstein did establish that all forms of energy have inertia (in his paper "Does the inertia of a body depend on its energy content?", which established the famous formula for the inertia of energy, $$m = \frac{E}{c^2}$$). The stress-energy tensor which determines the curvature of spacetime is dominated by energy under ordinary conditions, and hence we see that gravity is mostly caused by energy, which has inertial mass.