Does kinetic energy warp spacetime? My interpretation of GR leads me to think that energy (namely kinetic) also adds to the curvature of space-time. Which, has raised a thought experiment. If a $10000$ kg ship closely passed a $1$ kg glass ball at $0.8c$ relative to the glass ball, would the glass ball be moved in the direction of the ship for the tiny fraction of a second that its passing by, more so than if the ship popped in and out of existence at rest for the same time period relative the glass ball?
 A: The answer is yes, and this is actually an important point when considering extremely high-energy physics. As a result of this effect, gravity is the dominant interaction at sufficiently high energies, at least if we can trust the most straightforward extrapolation of the current foundations of physics. This is highlighted in 't Hooft (1987), "Graviton dominance in ultra-high-energy scattering," Physics Letters B 198: 61-63. The abstract says:

The scattering process of two pointlike particles at CM [center-of-mass] energies in the order of Planck units or beyond, is very well calculable using known laws of physics, because graviton exchange dominates over all other interaction processes. At energies much higher than the Planck mass black hole production sets in, accompanied by coherent emission of real gravitons.

A: Kinetic energy is part of the time-time component of the stress energy tensor, so by the Einstein field equations it does influence the curvature. However, the relationship is too complicated to justify a straightforward assertion that it adds to the curvature. 
First, the curvature is a rank 4 tensor, not a scalar. So it has many independent components and increasing the KE may impact many of those components, often in opposite directions. So while it certainly changes the curvature what would it even mean to simply “add to the curvature”?
Second, an increase in the KE is always accompanied by an change in momentum also. The momentum will alter one or more of the time-space components of the stress energy tensor. Sometimes the momentum changes will roughly cancel the energy-based curvature changes, leading to minimal overall change in curvature.
