What is the reference coordinate system for "curvature of space-time"? Physicists describe gravity waves as "a ripple in the curvature of space-time".   As a mathematical concept, isn't curvature of a coordinate system only meaningful relative to a coordinate system exhibiting a different curvature?  
Alternatively, what is the context for curvature of a coordinate system? I.e., "Space-time may be curved - but with respect to what?"
related question here 
 A: To summarize: The curvature tensor of a (pseudo)Riemannian manifold $(M,g)$ is independent of the choice of (local) coordinates on $(M,g)$. 
This also applies to the claim that the curvature is defined once two coordinate systems are given. Actually, in modern treatments one defines  the curvature tensor without using any coordinates whatsoever. Thus, you should not talk about "curvature of a coordinate system" but, rather, of $(M,g)$ itself. As of gravity waves being "ripples" in something, this sounds more like a metaphor rather than a mathematical statement. My favorite treatment of Riemannian geometry is do Carmo's "Riemannian geometry". Reading the first 4 chapters of the book will help to clear many misconceptions. One thing you will realize after you are done with Chapter 4 is that a tensor $T$ on a differentiable manifold is not some expression with several upper and lower indices (which is an impression one typically gets from reading physics literature), but $T$ becomes such after  you introduce local coordinates (one local coordinate system at a point suffices)! This is analogous to the fact that (in the classical mechanics) the velocity vector of a point-object exists independently of whether or not somebody introduced coordinates in the space; the velocity vector becomes an expression of the form $(v_1,v_2,v_3)$ only after coordinates are introduced. 
