In general relativity, the presence of gravity warps space-time yet clearly me accelerating will not warp space-time. It is hints at in one of the comments of this answer that acceleration bends world lines but does not actually cause the warping of space time. Is this right? If it is then why could we not have acceleration warping and gravity bending world lines. If it is not why not?
[Physics] Why does gravity warp space time but not acceleration
accelerationcurvaturegeneral-relativitygravityspacetime
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Luboš's answer is of course perfectly correct. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve).
Image a 2-sphere (a surface of a ball). If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. Imagine a second ant and suppose he'll start to walk from the same point as the first ant and at the same speed but into a different direction. He'll also produce circle and the two circles will cross at two points (you can imagine those circles as meridians and the crossing points as a north resp. south poles).
Now, from the ants' perspective who aren't aware that they are living in a curved space, this will seem that there is a force between them because their distance will be changing in time non-linearly (think about those meridians again). This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces). You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. E.g. for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. to the precession of the perihelion of the Mercury).
So much for the explanation of how curved space-time (discussion above was just about space; if you introduce special relativity into the picture, you'll get also new effects of mixing of space and time as usual). But how does the space-time know it should be curved in the first place? Well, it's because it obeys Einstein's equations (why does it obey these equations is a separate question though). These equations describe precisely how matter affects space-time. They are of course compatible with Newtonian gravity in low-velocity, small-mass regime, so e.g. for a Sun you'll obtain that trampoline curvature and the planets (which will also produce little dents, catching moons, for example; but forget about those for a moment because they are not that important for the movement of the planet around the Sun) will follow straight lines, moving in ellipses (again, almost ellipses).
If we consider the thought experiment where we take the classic 2 dimensional plane curved in a graphic representation of the curvature of space-time, copy it and arrange the copy so the lowest points of the gravity wells are aligned. These copies can be arranged any way you like, as long as the low point, or gravity wells are aligned, then the imagery still works. Then we can see what i think you are asking about. With just a single image, it appears space-time can be stretched and pulled 'downward.' But in 3 dimensions, the second image, or many others, are also possible, implying that space is being curved down and up in the same place.
And so, i assume, your question. My thought was, where did the space go?
The other things that i consider are distance and time. Using the earth for example, over time the earth moves around the sun. In the summer the earth is curving different space-time than it does in the winter. As the earth leaves a place in space, space returns to the shape it was before the earth was present, so it seems that space-time was 'compressed' and then returned to it's original shape.
I find it interesting that space can be expanded, inflated or curved but not compressed. The confusion comes from the comparison of space to a fluid, and fluids cannot generally be compressed. Keep in mind that my first example, the rubber tarp example, and comparing space to a fluid all give us a idea of what space is like and what it is doing, but none of them are perfect.
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Although "warping" is probably not an academically well-recognized term, I feel safe to assert that warping means curvature of the spacetime. Curvature of the spacetime is the property of the spacetime being of such a nature that two infinitesimally separated geodesics (paths for which the vector generated by the parallel transport (along the path) of the tangent vector to the path at one point is also the tangent vector to the path at the resulting point) that are parallel at a point do not remain parallel in the infinitesimal neighborhood of that point. As you can easily see, the definition of the curvature is manifestly coordinate invariant. Thus, a frame being accelerated wrt the previous frame does not make the spacetime curved if it were not curved in the original frame. So, the reason acceleration doesn't warp spacetime is because the acceleration of a frame wrt the other has nothing to do with the coordinate invariant properties of the spacetime, rather it is just the relation between two frames.
Now, in some sense, according to the Equivalence principle, gravity and acceleration are the same things. In this context, gravity also doesn't always curve the spacetime. For example, if a box, in the deep empty space, is being pulled with some acceleration with respect to the local inertial frame then there would be gravity in the frame attached to the box but the spacetime is still flat.
But, there is some distinction between this kind of gravity and a rather "genuine" gravity, for example, the kind of gravity produced by the Earth or some other matter-energy distribution. This kind of gravity, called "true gravity" by Weinberg, is defined as the sort of gravity that makes the path followed by two neighboring free particles in the spacetime (who are initially going parallel to each other in their spacetime trajectories) converge. By the Equivalence principle, the space-time trajectories of the particles are precisely the geodesic curves in the spacetime. Thus, it follows that true gravity represents nothing but the curvature of spacetime. Notice that such true gravity can also be present in the perfect vacuum without any source whatsoever, e.g., in the gravitational waves. So, coming back to your question, I would say the curvature of the spacetime is what is represented by "true gravity" rather than saying that gravity causes spacetime curvature. One might be tempted to say, based on the Einstein field equations, that the energy-momentum tensor $T_{\mu\nu}$ causes the curvature of spacetime but as I said, the curvature can be present even without $T_{\mu\nu}$. So, you can not call it the exclusive cause of the spacetime curvature.