Is curved spacetime a real thing or just math? I was curious if the curving of spacetime by mass/energy was actually a real thing or is it just a mathematical construct, a way of visualizing the force of gravity and explaining it and that there is not truly a "fabric of spacetime" and this fabric doesn't actually curve. The math simply explains our observations, but isn't literally what is happening. 
I'm looking for a pure GR/SR answer to this, not asking about newer theories that build on GR. Did Einstein himself believe spacetime and its curvature was a real physical thing or did he know he was just using math and geometry to explain a phenomenon everyone can see/feel? 
 A: I think you are asking whether GR is of the same class of theories as what we had for electromagnetism: you define a gravitational field and that lives atop of a flat space, or perhaps a flat Minkowski spacetime. You are wondering if the rest of the theory can be understood in roughly this way, with gravitational waves being propagating waves in the field and so forth.
In truth, it would be hard to understand GR in this way. There are several indications why.
So the defining feature of special relativity is that when you accelerate in any direction with acceleration $a$, you see clocks ahead of you by some coordinate $z$ tick faster by a proportion $az/c^2,$ and clocks behind you tick slower by the same proportion. You can derive the general Lorentz transformation from simply this, and it has a direct physical consequence, which is that if you are accelerating, there is a surface behind you at coordinate $z=-c^2/a$ where time appears to stand still; call it a "wall of death." 
One defining feature of general relativity is that when you are in free-fall you are in a flat Minkowski spacetime, hence if you are not in free-fall you must be accelerating away from a planet and thus seeing clocks at higher elevations ticking faster than your clocks are. This is called gravitational time dilation and it is an observed consequence of general relativity. It is very hard to explain in any sort of field approach since those typically have universal time coordinates; you would have to postulate that the mechanisms in the clock depend in complicated ways on the field. Similarly, strong sources of gravity called black holes have a wall of death intrinsically when viewed from a far-off distance: clocks that fall into them appear to tick slower and slower as they fall, when seen from a far-off distance. But though the clock seems from the greater distance to stop, general relativity allows you to calculate from the in-falling clock's perspective and in that perspective nothing special happens at this surface: you just happen to pass through it after some time and then no more of your light can reach the distant observers who were looking at you. So the natural language of the theory does not permit you to say objectively that the clock "stops"; it allows you to see things also from the clock's perspective where it does not stop. This would be even harder to do on a flat background.
Finally as kind of the simplest example of this, you would have to do some very nontrivial coupling of electromagnetism to a gravitational field to get a gravitational field to bend light rays the way that lensing has been observed to do. But that seems very difficult.
A: We can observe that light rays are bent by heavy objects, despite being massless, but more importantly it is bent twice as much as Newtonian models would imply but exactly what GR predicts. Indeed, this is enough to show gravitational lensing from remote galaxies and clusters. There is a delay when photons passes close to heavy objects. There is excess perihelion advance of Mercury that fits curved space. Frame-dragging has been measured in satellite experiments. We can observe that there are things that do not emit light, yet have significant mass and fit in very well with black hole solutions of GR. Even better, gravitational waves have been observed. If nothing else GR is a very good fit to reality and makes nontrivial predictions.
Now, you can try to say that space actually is flat and there are peculiar deflections, delays and redshifts that exactly imitate a curved spacetime (including some kind of waves that manage to exactly imitate gravitational waves). That might be possible to get to work, but it breaks Occam's razor by adding a lot of epicycles to a flat spacetime compared to what is actually a fairly simple theory if one accepts curvature.
The real test that could completely invalidate flat space theories would be to observe a nontrivial topology. Unfortunately we do not have any black holes or even better wormholes to experiment on at the time being.
In the end physics can never prove what truly exists. It can just build explanations that make nontrivial predictions that can be tested. GR has done a fine job with that.
A: With all due respect, what you are looking for is a circular way of thinking:

The math simply explains our observations, but isn't literally what is happening. I'm looking for a pure GR/SR answer to this, not asking about newer theories that build on GR.

Fundamentally, physics does not answer what is really happening but rather provides frameworks that predict observations to some satisfying degree of precision: One hope is that a "final" theory would be able to provide exact predictions to what we know from experiment, but ultimately all we know is the agreement of experiments with theory.
Specifically, General Relativity (GR) is our current mathematical model that explains the perceived force of gravitation as curvature. We call this theory correct because it has been tested experimentally, such as in Gravity Probe B. Such a theory is typically interpreted as being completely correct for the purpose of further inquiry, but strictly speaking such a claim is pseudoscience as it is unfalsifiable. If you are looking for a pure GR answer then spacetime is indeed curved because that is effectively the premise.
A result of the (current) untestability of all aspects of a physical theory is explicitly seen in the newer theory of Teleparallelism (or more accurately the Teleparallel Equivalent of General Relativity, TEGR), which re-creates gravity as the torsion of spacetime. Within this framework spacetime has zero curvature. How do we know which one is "correct"? We do not and can not know because as far as I am aware there is no inconsistent prediction between the two theories.
A: Excellent question. You are on the right track to understand it. 
We use the phrase "gravity bends space" mainly because:


*

*all known particles are affected by gravity

*gravity is always attractive
Now in reality, we do not know what is bent, why it bends, or what spacetime is, or what the fabric of spacetime is, or how it bends. What we do know is, that as per our currently accepted theories, the SM, and GR, the gravitational field has an effect on spacetime so that that region of spacetime that is in the gravitational field, will have an effect on all known particles, so that all these known particles will have a altered trajectory (from 3D straight) when they interact with the gravitational field.
Now we use this phrase "gravity bends space" because all known particles will be affected and they all will bend in the same direction towards the center of mass. But in reality we do not know if anything is bent at all, or what is bent or why or how it bends. What we do know, is that all the data from the experiments fit the theory of the SM and GR.
In reality we do not know if anything is bent at all, all we do know is that the particles do have a altered trajectory.
Now to understand why we might not know if anything is bent or maybe nothing is bent, you have to understand the same way the EM force.
There are two main differences between how we use the phrase "gravity bends space" and why we do not use the phrase "EM force bends space":


*

*the EM force does not affect all known particles, just some

*the EM force might be attractive or repulsive, so the direction of bend is not trivial
Now could we say the same way that the EM force bends spacetime? No, because a neutrino or a neutron will not have an altered trajectory as it passes through the EM field. So we could say, but if spacetime would be bent by the EM force, then why does a neutrino or a neutron not have an altered trajectory at all?
So the answer to your question is, if we would discover a particle, that is not affected by the gravitational field, then we could easily say that gravity does not bend space at all, it just has an effect on space, where the gravitational field interacts with the particles so that they have an altered trajectory (relative to 3D straight).
But until then, you could have the idea of something called space to be bent by gravity, since there are no experiments to disprove it, all known particles will have a bent trajectory.
A: Spacetime is curved by gravity, this is a very useful model, but this model of curved spacetime is not compatible with quantum mechanics. However, gravity may also be represented in the form of gravitational time dilation in flat, uncurved space.
This may be shown easily for the Schwarzschild metric:
$$ ds^2 = -(1 - \frac{2GM}{c^2 r}) c^2 dt^2 + \frac{1}{1 - \frac{2GM}{c^2 r} } dr^2 + r^2 (d\Theta^2 + sin^2 \Theta d\Phi^2)$$
Gravitational time dilation from the point of view of a far-away observer is
$$ C = \sqrt{1 - \frac{2GM}{c^2 r}}$$
As you may observe, we can insert the second equation into the first one, and we get: 
$$ ds^2 = -c^2 (Cdt)^2 + {(\frac {dr}{C})}^2 + r^2 (d\Theta^2 + sin^2 \Theta d\Phi^2)$$
This is still the Schwarzschild metrics of curved spacetime. Now we compare this equation with the Minkowski metrics of flat space:
$$ ds^2 = - c^2 dt^2 + dr^2 + r^2 (d\Theta^2 + sin^2 \Theta d\Phi^2)$$
We see that both equations are only differing (twice) by the factor C which is gravitational time dilation, and we may conclude three things with respect to the Schwarzschild metric:


*

*Gravity may be perfectly and completely expressed by gravitational time dilation, both notions are equivalent

*Gravity may be expressed also in uncurved space as gravitational time dilation and

*Accordingly, instead of by spacetime curvature, the attraction force of gravity may be described as the tendency of particles to maximize their own time dilation. This model of gravity in flat spacetime does comply with quantum mechanics.
