system
Given the following post, respond as if you're commenting on Reddit:
human
Why is gravitational mass proportional to inertial mass? Do both of them grow as an object reach relativistic speeds?

If they both grow, how can we explain the strong gravitational field in the object's frame of reference?

What's the difference between them on a fundamental level?
gpt
First, it's much better to think of mass in terms of rest mass, rather than relativistic mass. Almost no physicists think in terms of relativistic mass anymore; instead, it's the formulas for momentum and energy that get modified at high speeds. So we always say that an electron has a mass of 511 keV/c^2, even when it's ultrarelativistic; it just has way way more momentum and energy than you would expect for a particle with its mass and velocity in Newtonian physics.

OK, so now that we are thinking about rest masses, why is it that gravitational (rest) mass and inertial (rest) mass are the same? Per Newtonian physics this is just an accidental fact that explains gravitation nicely. Per Einstein this is a fundamental law of nature: the Equivalence Principle, which states that all objects fall at the same rate in a gravitational field. In Einstein gravity (aka general relativity) gravity is not a real force, it's an artifact of being in an accelerating frame of reference. Gravitational forces are fundamentally the same as centrifugal forces, Coriolis forces, and the force that pushes you back in your car seat when you floor it, in that they only exist when you're using the 'wrong' frame of reference. If you construct the formulae for these 'fictitious forces' you find that they are all proportional to *inertial* mass. So Einstein explains the equality of gravitational and inertial masses by saying that gravity, too, is a fictitious force, so the mass that appears in it must be the inertial mass.

Now, your question about growing relativistic mass still stands in a modified form, because Einstein gravity is sensitive to *energy*, not just rest mass. (It's also sensitive to momentum and pressure!) And the resolution is just that just as the energy, etc of a particle depend on your frame of reference, so too does the gravitational field, and everything is just right (by construction) to make the different points of view consistent. In the rest frame of a particle, though, it has only its rest mass and its ordinary gravitational field---it doesn't "know" that it's moving relativistically relative to something else.