Does a body curve spacetime at higher velocities?

Question

Suppose we have two objects where the distance over time decreases. Now, as I understand it, general relativity says that we can observe the Universe from the perspective of both objects an get a equal valid view, i.e. it's squally valid to say that object $A$ is in rest while object $B$ is the one moving as to say that $B$ is at rest and $A$ is moving.

Now, GR also tells us that the relativistic mass increases by the gamma factor $\gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}}$. As the mass increases, so must the gravitational force and thus the curvature of spacetime.

But, now GR states that the curvature of spacetime is not relative, but absolute ($G_{\mu v} = R_{\mu v} -1 \frac{1}{2} g_{\mu v} R$). How is this consistent with claiming that both $A$ and $B$:s perspectives are equally valid?

possible duplicate of reletivistic mass as seen by different observers — John Rennie, Dec 11 '14 at 06:51
See also Does a moving object curve space-time as its velocity increases? though the answers to this don't much more than say the curvature is an invariant, as you've already pointed out. — John Rennie, Dec 11 '14 at 06:55
I don't think the "relativistic mass" should be regarded as a true mass in any sense. It's simply the time component of the 4-momentum vector (divided by $c^2$). At relativistic speeds, that value is pretty meaningless without the other three components. At a guess, the relevant quantity for "amount of curvature" is likely to be the invariant magnitude of the 4-momentum...which is just the rest-mass. — Nathan Reed, Dec 11 '14 at 08:34
Also note that the values of $G_{\mu\nu}$ and $T_{\mu\nu}$ change in different reference frames too. They are no more or less "absolute" than the 4-momentum is. — Nathan Reed, Dec 11 '14 at 08:36

Does a body curve spacetime at higher velocities?

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