Many fast moving objects exist in the space.

If there is very fast moving big object, like planet, and not rotate, are the object's front side gravity and rear side one same?

We assume speed of the object is 1/5 of the light speed.

Edited February 8, 201114 yr by alpha2cen

Do we mean accelerating? If the body is accelerating due to the gravitational field of some other object then describing the interaction of these two fields will in general be difficult. The field equations are non-linear.

 

If we have a non-accelerating planet the standard thing to do would be to pick the frame in which the object is stationary. You put the planet at the centre and then worry about describing the vacuum around it. Your situation would reduce to the Schwarzschild metric.

I mean steady state velocity. And the speed is 1/5 of the light speed.

I mean steady state velocity. And the speed is 1/5 of the light speed.

 

Relative to what?

 

You want to consider what the gravitational field would look like to a distant observer comoving at 1/5 c with respect to the planet/star?

 

I think this would still correspond to the Schwarzschild solution, up on a suitable change of coordinates. Have a look for Gullstrand–Painlevé coordinates, these may be of interest.