In discussing relativity with a
(somewhat mathematical) friend
Nope, he's not mathematical.
the other day, I ran into a problem showing why special and/or
general relativity could be considered as exact descriptions of
reality rather than just approximations that are working okay so far.
Both special and general relativity are approximations. Why? They are non-quantised theory. There is a special relativity-quantum mechanics merger called Quantum Field Theory, though, but not yet a universally accepted general relativity-quantum mechanical merger (well, actually there is, semi-classical gravity, but it isn't complete, because gravity is still considered classical, there. String theory happens to solve the problem, along with some other such QG theories, most of which have been proven wrong)
Here's his argument:
You can't argue without mathematics involvedd.
Special relativity applies to a situation where there is no curvature
Yes, this is right.
Since both matter and energy cause curvature,
Or simply, the stress-energy-momentum tensor, is a much better way to say "both mass (not matter) and momentum and pressure and shear stress".
the only situation where special relativity would apply would be one
which has no matter and energy,
Yes, but only if you want an exact result (actually, it is still not exact, because it isn't QUANTISED)
and therefore one which is not achievable in any experiment.
How much does a coconut curve spacetime? Not much. Definitely not enough to have any significant (observable) deviation from SR.
And even as a thought experiment, a clock has to have mass and energy
and so there can be no exact predictions in such a theory.
A clock is lighter than a coconut (unless it is a grandfather clock, but even that is much lighter than a coconut tree, and how much does a coconut tree curve spacetime?.
He went on to say that therefore the general theory of relativity
rests on a very shaky foundation.
You eat a coconut from a coconut tree, and deduce that it is spoilt, unlike what your ancestor (Newton) told you. Also, its flesh is green in colour, you deduce.Then, you eat all the coconuts from the tree and deduce that all are spoilt, not all of them are green. Many are actually very very slightly bluish green (small moons), some are very slightly bluish green (small planets), some are slightly bluish green (planets like the earth), some are bluish green (Stars like the sun), and some are blue (black holes). But you weren't wrong that some coconuts (no-SEM-tensor reigons of spacetime) are actually green in colour.
Ok, let me do a better, more relevant example. Special Relativity is fine as long as gravity is weak. For example, if you are suspended in some reigon of spacetime, then, you are lighter than a coconut tree and you curve spacetime very little, and nothing with a large SEM-tensor is there near your so special relativity can analyse your motion very well. But if you are suspended near the sun, special relativity can't be used, or you will fall into the sun! Call Kerr and Newmann to tell you what to do!
This is as compared, for example, to thermodynamics which rests
comfortably on statistical mechanics and quantum mechanics.
Since when did CLASSICAL thermodynamics rest on quantum mechanics? And general relativity improves special relativity, not rest on it. Whereas a lot (but not ALL) of thermodynamic actually uses results from statistical mechanics. I think you're discussing physics with a "non-mainstream" (which is a polite word for saying...) friend?