Einstein did not think of it as a sheet of rubber 'bending'. It's commonly used as a 2 dimensional analogue to the way gravity is space's 'metric' but I find it better to think of it as a 'Jello' that will adapt relative motion/mass/gravity/'energy'. And if you look out on the universe a dark night every star you see will have a (different) relative motion, relative you observing. That means that the universe you see is both time dilated and Lorentz contracted, which may seem weird :) but it's all a result of the invariant radiation describing what you look at.
That's also why you can't really define a time dilation, other than the twin experiment, where the twin has to come home first to 'prove it'. But it is real, both the Lorentz contraction and the time dilation, and they must be different, relative different stars.
But what communicates it all is that same 'propagating' light, coming at you at the same invariant 'speed', and that light paints the 'whole image' you obtain from observing that night sky. Think of that the next time you look up :)
