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I get the idea that mass/energy distorts spacetime, justified by thought experiments with moving objects and photons bouncing between mirrors. I see how the distortion in spacetime produces the effects we call gravity. But where does inertia come in ? All mass/energy has inertia, but I don't see that inertia (or resistance to acceleration) comes naturally out of the arguments about spacetime and gravity. Why is inertial mass equal to gravitational mass, such that all free falling objects accelerate at the same rate in a gravitational field ?

I am really not qualified to come up with ideas at this level, but I thought about the problem and came up with this: An object comprising mass/energy creates a distortion in spacetime around it, its gravitational well, which extends to infinity. If the object is pushed so that it begins to move, the gravitational well around it will move too, so as to stay centered on the mass. However we know that the effect on the gravitational well cannot propagate faster than light, so some of it will remain as it was until a ripple in spacetime reaches it. Could inertia be the result of an object trying to climb out of its own 'lagged' gravitational well ? Climbing out of a graviational well requires input of energy, which requires work to be done on the object which could explain inertial forces...