I'm trying to understand the following claim from Feynman's lectures on gravitation, section 3.1 (p.30 in my edition). He's considering how heating or cooling two clouds of gas would change their mutual gravitational attraction.

I don't understand this statement. The electric scalar potential is the time component of a four vector which gets dotted into another four vector, so that the resulting physics is invariant. I don't see anything in the expression for a Yukawa potential that I can identify with the velocity dependence he's talking about. Can someone please explain?

In tensor gravity the source of the field is T^{00}, the 00 component of a rank 2 tensor. Under a Lorentz transformation, T^{00} → γ^{2} T^{00}. But in the meanwhile, the volume element contracts, d^{3}x → (1/γ) d^{3}x. So the integrated source M = ∫ T^{00} d^{3}x → γ M grows only as one factor of γ. Thus for tensor gravity the source of the gravitational field is the total energy, or "relativistic" mass, which grows with velocity like γ.

(Be clear that we are talking about random internal velocities, whose directions average out, not an overall velocity of the object.)

A vector theory of gravity would be like electromagnetism, in which the source density is J^{0}, the 0th component of a vector. Under a Lorentz transformation, J^{0} grows like J^{0} → γ J^{0}. But this is exactly compensated by the contraction of the volume element, and so the source of the gravitational field in this case is independent of internal velocities, just like the total electric charge is.

For a scalar theory of gravity, the source density is a scalar, presumably the trace of the stress-energy tensor, and is a Lorentz invariant. But the volume contraction is still present to contribute a (1/γ), and so the effective total strength of the source will decrease with increasing internal velocities.

Thanks for your reply. How exactly is the volume contraction affecting the scalar mediated interaction? If the source of the gravitational field were some sort of scalar density to which the putative scalar graviton coupled, wouldn't the lorentz contraction of the volume increase the density?

Hmm. I think I may have been putting too much emphasis on the word "density" (thinking of it as the amount of some stuff per volume, which by definition would appear to increase to a relatively moving observer) and not enough on the word "scalar"