Does kinetic energy of an object curve spacetime? Based on general relativity mass and energy distribution curves spacetime. Thus, if an object with 1kg rest mass moves with constant speed and has a speed very close to speed of light, then it has an enormous kinetic energy which should curve spacetime much more than a 1kg body at rest would. On the other hand, an observer moving with the same speed as of the 1 kg body do not see any kinetic energy and should not see any spacetime curvature.
This seems paradoxical! How to resolve this paradox?
 A: Yes and no.
The kinetic energy counts, but so does the momentum, and they do so in such a way that they tend to cancel. The rest mass is constant, so the left hand side of $(mc^2)^2=E^2-p^2c^2$ is also constant, and so any increase in momentum has to be cancelled by a corresponding increase in energy - called the kinetic energy. In general relativity, the source of gravity is not just the mass (=energy), but the energy-momentum-stress tensor, so the momentum contributes too. If you simply change reference frame, you get the same very small curvature, simply 'rotated' in spacetime.
However, if you have a relativistic particle confined to a box, bouncing backwards and forwards between the ends, then the momentum of box and particle going in both directions at once cancels out, and the kinetic energy does indeed contribute to the mass and hence curvature of spacetime.
A: You are only a step in figuring out the answer.
Both Energy and Momentum cause curvature.
The curvature caused by energy is attractive. But the curvature caused by momentum is repulsive.
There is an analogy between gravity and electromagnetism, called gravitoelectromagnetism. Similar to how the electric force (caused by charge density) is repulsive for two like charges, the magnetic force (caused by a moving charge) is attractive.
This video I found might be helpful:
https://www.youtube.com/watch?v=rKFzV8sVDsA

For the advanced viewer who got stuck, a typical Lorentz transformation (Assuming the curvature is weak, or the observer is far away) would transform the coordinates of the stress-energy tensor the same way as how the Ricci tensor would transform. Both tensors are invariant, but their components transform. So both observers, the stationary and the comoving see the same curvature, but they express it in their respective coordinates differently because they express the energy tensor differently.

Also, I would like to add, that the analogy I stated before is enough because you are looking at the momentum energy tensor and not the curvature. I urge you to look up the equations for gravitomagnetism and use a Minkowski spacetime diagram, and see that the gravitational force is equal regardless of the frame.
