Is spacetime wholly a mathematical construct and not a real thing?
TL;DR This is a complicated question and anyone who tells you a definitive answer one way or another is either a philosopher or is trying to sell you something. I justify arguments either way below, and conclude with the AdS/CFT correspondence, in which two theories on two vastly different spacetime manifolds are in fact equivalent physically.
First, let’s clear things up:
Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'.
No. This is simply how easy-to-digest PBS documentaries and popular science books explain the idea of spacetime. In reality, it is a (pseudo-Riemannian) manifold, meaning that it locally (for small enough observers) looks like regular flat spacetime that we are used to dealing with in special relativity. The main difference here, is that for larger observers, the geometry may start to look a bit foreign/strange when compared to the “flat” case (for instance, one might find a triangle whose angles don’t add up to 180 degrees). These are just geometrical properties of the world in which the observer lives, and it happens that the strange geometrical measurements happen to coincide with areas of concentrated mass/energy. This effect of wonky geometry, coupled with the fact that observers naturally follow the path of least spacetime “distance” (proper time) account for what we’re used to calling gravity.
In addition to this, if it's a mathematical thing, then what does its distortion/bending mean?
Again, this is a PBS documentary image that anyone wishing to actually understand physics needs to abandon. Spacetime doesn’t “warp,” “bend,” “stretch,” “distort,” or any other words popular science books care to tell you. What these terms are really referring to is the geometrical properties of different parts of spacetime being different than in the special spacetime of special relativity. In particular, they refer to the geometrical notion of curvature, which is simply a value measurable by local observers which is zero for flat spacetime and nonzero for others, but has nothing to do with stretching, pulling, distorting, or what-have-you.
Finally, let’s get to the meat of the question:
Is spacetime real, or is it a mathematical construct?
Short answer: Yes to both.
Spacetime is, from a mathematical viewpoint, a manifold, which is a set of points equipped with a certain structure (being locally flat). Physically, each point corresponds to an event (a place for something to happen, a time when it happens), and local flatness simply means that small enough observers can find a reference frame in which they would locally feel like they are in flat spacetime (this is Einstein’s equivalence principle).
Mathematically, spacetime has a little more structure. It has a metric tensor, which is the fundamental geometric variable in relativity, and physically corresponds to being able to measure distances between nearby “events” and angles between nearby “lines.” These both certainly seem physical.
As you can see, each mathematical property of spacetime manifests itself to the observer as a physically measurable property of the world. In this sense, spacetime is very physical. However, one could argue the other way as well.
I really like the way that Terry Gannon puts it in his book “Moonshine Beyond the Monster.”
...we access space-time only indirectly, via the functions (‘quantum fields’) living on it. (Gannon, 117)
And this is the sense in which spacetime is just a mathematical tool. We never interact with “spacetime.” What we interact with are the functions whose domains are the abstract manifold we call spacetime when describing them (gravitational fields, electromagnetic fields, etc.), and to make any measurement about spacetime occurs only indirectly through the measurements of these fields. Even something as simple as measuring distance requires a ruler, which can only be read through electromagnetic interaction (light).
The truth is, this is a completely and utterly complicated question which we may never know the answer to. Instead, I leave you with this:
Holographic theories (AdS/CFT and variations thereupon) suggest that a gravitational system (spacetime + curvature) and a certain non-gravitational quantum theory of fields (spacetime + fields + no curvature) in one dimension fewer have exactly the same physics. That is, no measurement could meaningfully tell you which system you’re in, because, physically, they are the same system.
So where did the extra dimension of spacetime come from, and where did all the curvature come from? If spacetime is real, then why can I describe two identical theories on two very different spacetime manifolds?
As a final thought:
Physics does not aim to find “truth”(so much is the subject of philosophy or metaphysics). Physics seeks only to find models of reality which are useful in predicting the outcomes of experiments or processes. Thus, physics can say nothing of the “reality” of spacetime, so long as there are two different theories on two different spacetimes which give the same result of every possible experiment.
Gravitational waves carry energy, momentum, and angular momentum from one point to another. In my book, this makes spacetime just as real as an electromagnetic field, which also transports these quantities. You feel electromagnetic energy whenever you walk outside on a sunny day. If gravity weren’t so weak, you would feel the energy in gravitational waves.
Most physicists consider these fields just as real as matter, which in modern physics is explained using yet more fields!
spacetime is three dimensions of space and one of time. ... my question is, whether spacetime is real or is just a mathematical construct
Here is different from there and now is different from then whether you do the math or not. A mathematical construct cannot be the difference between a near miss and a catastrophic collision.