Very massive objects cause the so called 'frame dragging' that can increase the speed of a beam of light to a total aggregate speed faster than the speed of light in normal circumstances so my question is: can a very massive fast object drag the'frame' along its trajectory so a theoretical space ship, chasing it, would have its speed increased due to this possible linear 'frame dragging'?
I don't see why the linear effect shouldn't be there. After all, you can envision the rotation as a following up of infinite linear motions. At any instant, a body in circular motion is composed of bodies with linear motion. If you envision two massive plates moving parallel through space. The plates are kept at rest wrt each other. Between the plates, there is no gravitation due to mass, but you will be dragged along in a direction parallel to the plates if you find yourself in between them (or outside them but in between you'll experience no "normal" gravity). You will end up with the same velocity as with which the plates are moving. This goes to show that space is connected to matter and not to mass, which is just a property of matter. If only mass were the cause of curvature, then no framedragging was present.