general relativity - Does a moving object curve space-time as its velocity increases?

We always hear how gravity bends space-time; why shouldn't velocity?

Consider a spaceship traveling through space at a reasonable fraction of the speed of light. If this spaceship, according to special relativity, gains mass as a factor of y as it approaches c, then its gravitational field should increase in strength as well. Hence, space-time should warp.

Note: Changes in space-time, gravity and mass should only be measureable by an outside observer with a different velocity. Those inside of the ship moving with it would not be able to measure the change in these properties.

Answer

Start with the gravitational field of the Sun. We are effectively stationary with respect to the Sun, because our relative speed is much less than $c$, and the Sun is rotating at well below relativistic speeds so we expect its gravitational field to be well described by the Schwarzschild metric. And indeed this is true: Newton's law of gravity is the non-relativistic limit of the Schwarzschild metric.

The metric tensor is invariant with respect to coordinate transformations, so if we take some observer moving at near light speed they would also find the gravity round the Sun to be described by the Schwarzschild metric. It will not look the same in the observer's coordinates, that is the individual components $g_{ij}$ will be different, but it will be the same tensor. Since in the observer's frame they are stationary and the Sun is moving, the conclusion is that velocity does not change the spacetime curvature.

Incidentally, this is why a fast moving object does not turn into a black hole.