Relativity Theory

"Relativity theory" usually refers to Einstein's theory of relativity, which is actually two separate theories, special relativity and general relativity (though the former is a special case of the latter – hence its name!); however, Newtonian mechanics is also a relativity theory!
The "relativity" in relativity theory refers to a principle (or class of principles); namely, the principle of relativity. In a nutshell, principles of relativity are assumptions about the 'sameness' of theories in physics; for example, that the laws of nature which a theory of physics describes are the same no matter who is measuring them, nor when they are measured.
Another way of saying this is that the laws of nature have some symmetry (a technical term) … and in a very deep result (called Noether's theorem, named after Emmy Noether; it's actually her first theorem) every symmetry has a corresponding conservation law. So, for example, if the laws of nature are symmetric with respect to time (i.e. it doesn't matter when you measure them, you get the same laws) then energy is conserved ('energy', here, has a specific meaning).
In special relativity, the relativity principle is that the laws of nature are the same in all inertial frames of reference (in simple words, an inertial frame of reference is one that is not accelerating) … if you are moving, relative to me, at a constant velocity (and neither of us is experiencing any acceleration), we will find that the laws of nature are the same.
In general relativity, the laws of nature are the same, period (i.e. no matter who, when, where, or how they are determined, the laws will be the same).
Of course, there's more to both special relativity and general relativity than just statements of the relativity principles they assume … just as there's more to Newtonian mechanics that the same relativity principle which special relativity assumes (e.g. Newtonian mechanics assumes absolute time).
It's astonishing just how wide-ranging the results from these simple relativity principles are, in Einstein's two theories (think of E = mc2, for example) … and how counter-intuitive some of them are too (e.g. the twins paradox, the relativity of simultaneity).
There's lots of material on the internet on relativity theory (but do be careful, some sites are just crackpot nonsense, and some are downright anti-science); two examples of good summaries/intros: Special Relativity (from SLAC), and Relativity Tutorial (Ned Wright, UCLA).
There are lots and lots and lots of Universe Today articles which include at least a mention of relativity theory; here is a sample: Small Engine for the Big Job of Testing Theory of Relativity, and Pulsars Confirm Einstein's Theories.
Likewise, relativity theory figures prominently in many Astronomy Cast episodes; here is a sample: Interstellar Travel, and How to Be Taken Seriously By Scientists.
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