Does mass really increase with speed

[PeterDonis]

But leaving free fall requires a force and I am afraid as soon as we do that your next question would be "Where does this force comes from and what about the involved energies?" Therefore I prefer a setup with gravitational interactions only.

Fine. What does that have to do with whether looking at such a scenario using coordinate acceleration is *required*, as opposed to one possible method but not the only one?

 

[DrStupid]

What does that have to do with whether looking at such a scenario using coordinate acceleration is *required*, as opposed to one possible method but not the only one?

Nothing.

 

[timmdeeg]

My problem is to understand why "active gravitational mass of a moving object" isn't a priori in contradiction with 'mass is invariant'.

Because you're using the wrong definition of "active gravitational mass"; you're plugging numbers into the Newtonian formula for gravitational "force" and trying to read off what the "active gravitational mass" is by applying F = ma, but the Newtonian formula for F is not correct; it doesn't fully describe the actual "force" exerted by a massive object.

Ok, there is no contradiction. And it is probably simply wrong to compare 'invariant mass' (a term within SR) with 'active gravitational mass' (GR).

But how about this reasoning:

A heated body has increased mass and thus an increased gravitational field due to the increased kinetic energy of the particles from which it is composed. A heated body weighs more. Now let's imagine a cold spherical mass M, which's radius exceeds 2GM very slightly. Would it form a black hole on heating (assuming the coefficient of thermal expansion low enough)?

The szenario of DrStupid is much different, but kinetic energy is involved as well.

 

[PeterDonis]

A heated body has increased mass and thus an increased gravitational field due to the increased kinetic energy of the particles from which it is composed.

Yes. But bear in mind that this assumes that the net momentum of the particles composing the mass is unchanged by the heating process (for example, it could be heated by radiation falling on it from all directions in a spherically symmetric manner). This is similar to the stipulation in DrStupid's scenario where the rockets' momenta are equal and opposite so they sum to zero.

Now let's imagine a cold spherical mass M, which's radius exceeds 2GM very slightly. Would it form a black hole on heating (assuming the coefficient of thermal expansion low enough)?

In principle, yes, you can cause an object to collapse into a black hole by heating it. Technically, the exact scenario you describe cannot be realized because it is impossible to have a body in stable equilibrium with a radius less than 9/8 of the Schwarzschild radius (i.e., 9/4 GM / c^2). So a body whose radius exceeds 2 GM / c^2 only very slightly would not be stable, it would already be collapsing into a black hole. But you could take a body that was just at the stable limit and add heat to it, and that would push it "over the edge" into collapsing (because its radius would now be slightly *less* than the minimum for stability for its new, slightly higher mass).