Does its gravity field compress when an object travels at almost the speed of light? Since gravity propagates at the speed of light, what happens to an object's field of gravity if the object travels very near to the speed of light (like 99.99999%)? Does it stop propagating forward, and only propagates to the sides and behind? And is this also what happens to the tiny gravity of a photon? And wouldn't this mean that the gravity of near-lightspeed objects and photons does not fully (but only partially) join the cumulative and continuous effect of gravitation on a cosmic scale, since no gravity is projected ahead of them...? And finally, are there any popular science books that go into these things: fundamental properties of objects travelling very close to lightspeed, and the specific gravitational properties of photons? Thank you.
 A: to your question, let's take the case of particles with rest mass first.
Let's take a neutrino, and see what happens to the gravitational field it (its stress-energy) creates and how it propagates to its sides, behind it and forward.
Well, actually to the sides and behind was not your question, but as per SR, massless particles (photons) seem to travel at c from all inertial reference frames, regardless of the speed of the frame.
In this case, thought the neutrino travels at almost light speed, it still creates a gravitational field around it, and the gravitational field as you say propagates with speed c.
From the inertial frame of reference of the neutrino, it will seem as the gravitons (or the gravitational field propagates with speed c) forward. 
So although the neutrino itself travels near speed c, it will see the gravitational field in front of it propagate with speed c. 
This is like to say that since gravitons are massless, and travel at speed c, SR rules apply to them the same way as to photons.
So gravitons will seem (or the gravitation effects) to propagate at speed c from all inertial reference frames.
A: It is important to note that in special relativity, all objects that travel at the speed do so in every inertial reference frame, including other inertial frames that are themselves traveling at the speed of light. The photon "observes" gravitons radiating away from it (if indeed that picture is accurate) at the speed of light, because in it's reference frame it is at rest. Likewise the gravitons that affects it coming from other objects, are "seen" to approach it at the speed of light. And so it's response to gravity in its reference frame is identical to an object experience gravity at rest. It is only when observing the photon interacting with the graviton from an outside reference frame that any peculiarity is noticed.
