Mass, inertia and momentum in reativity

[Green dwarf]

I read that, as an object approaches the speed of light, it's mass doesn't increase, just its inertia. As its gravitational effect is determined by its mass, this doesn't increase either.
However, it seems that momentum adds to the curvature of space, and thus to gravity. As an obect gets faster, its momentum increases, and thus one would expect that its gravity should increase.
Perhaps another way of looking at it . . . to accelerate a body, energy has to be added and that energy curves space.

 

[Dale]

However, it seems that momentum adds to the curvature of space, and thus to gravity.

Unfortunately, it is not quite so straightforward. When you say “adds to” you are comparing two scenarios, and the details really matter.

Momentum is part of the stress energy tensor, so it does matter. But momentum can be changed simply by looking at things from a different reference frame. Doing that changes the components of the curvature, but not any of the curvature invariants.

If you are just talking about a small body of constant mass as it gets closer to the speed of light in some asymptotically inertial frame e.g. due to an external pure force, that is one where the curvature invariants won’t change but the components will.

 

[PeterDonis]

I read

Where? Please give a specific reference.

as an object approaches the speed of light, it's mass doesn't increase, just its inertia.

This depends on what you mean by "mass". The invariant mass of the object doesn't change, but its "relativistic mass" does. "Relativistic mass" is not considered a very useful concept now, but it does include the increase of inertia that you describe.

However, the more important point is that all of this depends on your frame of reference, since "approaches the speed of light" itself does. In the object's own rest frame, its speed is zero, and its inertia does not increase.

its gravitational effect is determined by its mass, this doesn't increase either.

Invariant mass contributes to the gravitational effect of the object, but it's not the only possible contribution. See further comments below.

momentum adds to the curvature of space,

As @Dale says, it's not that simple. "Momentum" is frame-dependent.

Heuristically, in the object's rest frame, if we assume that stresses inside the object are negligible, the object's invariant mass, which is equivalent to its energy in this frame, is the only contribution to gravity.

In a frame where the object is moving at close to the speed of light, again if we assume that stresses inside the object are negligible, the object's momentum does contribute to gravity--but its contribution is negative, to offset the increase in the object's energy in this frame (since its energy in this frame includes its kinetic energy as well as its rest energy). The net contribution to gravity (if we define "gravity" correctly as the invariants associated with spacetime curvature) ends up being the same. It's much the same as the way invariant mass stays the same in every frame because you subtract the contribution of momentum from the contribution of energy to get something that's invariant.

 

[Herman Trivilino]

I read that, as an object approaches the speed of light,

The speed of an object depends on how fast the observer is moving. It's not absolute.

If an object is moving at nearly the speed of light relative to you, then you are moving at nearly the speed of light relative to it.

Do you think that your mass changes because an object moves at nearly the speed of light past you?

 

[Nugatory]

I read that, as an object approaches the speed of light….

Building on what @Herman Trivilino said above: Any time you say something about the speed of an object without including the words “relative to …” you are making a mistake.