I honestly thought that both theories of relativity were theories of gravitation. Special and general relativity, the latter being the generalized version of the latter. That makes sense, right?
Then I started actually digging into special relativity. I studied the original annus mirabilis paper, entitled “On the Electrodynamics of Moving Bodies”. Yes, the title was the first clue, although I didn’t see it at the time. The second clue lied in a conspicuous absence: neither gravity nor gravitation is mentioned in the paper. Not even once. No, instead, Einstein spends 5 sections on kinematics (which is basically about geometrically deducing the value of some parameters of the mechanical system from others given as initial conditions); then he applies his kinematics to electromagnetism.
This is when I finally realized that special relativity had nothing to do with gravity. General relativity, that’s a theory of gravity, one you can compare with Newtonian Mechanics in a variety of ways. But special relativity is not a theory in this sense; that’s the wrong type signature. No, special relativity looks far more like a meta conservation law.
Because instead of conserving a quantity, special relativity assert that the laws of physics themselves are invariant by specific transformations called Lorentz transformations. This is stated by Einstein itself in his popular account of his theories (Relativity: The Special and General Theory, Chapter 14):
Once in possession of the Lorentz transformation, however, we can combine this with the principle of relativity, and sum up the theory thus:
[...] Or in brief: General laws of nature are co-variant with respect to Lorentz transformations.
This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found which did not satisfy this condition, then at least one of the two fundamental assumptions of the theory would have been disproved.
Although the transformations had been discovered independently earlier (by Lorentz notably), Einstein rederived this constraint on laws of nature from his two principles:
That the laws of physics are invariant by uniform (so not accelerated) motion of the reference frame (technically that they’re the same in all inertial reference frames)
That the constancy of the speed of light in the void is such a law of physics
In classical mechanics, 2 contradicted 1 because the addition of velocities meant that light coming out of a moving source would have to move faster or slower than the constant speed of light. But since Einstein assumed both principles as strong phenomenological laws, he looked for a way to reconcile them. And Lorentz transformations give just that: they transform coordinates from a stationary reference frame to a uniformly moving one while maintaining the constancy of the speed of light in the new frame. Which they do by contracting the lengths and dilating time, two ideas familiar to anyone who has read pop physics books.
But more than the precise mechanics of the Lorentz transformation, what I find fascinating here is that what I thought was an imperfect theory of gravitation actually proved to be an (imperfect) conservation law, a constraint on the laws of nature.
What can you do with such a constraint? I see two main applications, both of which were pursued by Einstein himself and other physicists at the time.
Test on existing theories
As explained in the paper, Maxwell’s theory of electromagnetism passed this test: it was already a relativistic theory. Classical mechanics on the other hand contradicted it directly (because of the addition of velocity breaking the constancy of the speed of light). This spurred on physicists to look for a relativistic theory of gravitation, which would eventually lead to general relativity.
Tool for constructing/reconstructing theories
Einstein ends section 8 of his paper by this paragraph:
All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of coordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.
This demonstrates the constructive and productive power of the constraint. It doesn’t only tell you when something is wrong; it also lets you deduce new laws and constraints, here transforming the moving frame into a stationary frame by the Lorentz transformation and then applying the laws for the static case.
Similarly, not only Einstein’s paper but multiple other sources highlight how special relativity lets you simplify Maxwell’s theory, by removing the need for additional special electromotive forces that depend on motion, and instead just deducing the electrodynamics of moving bodies (ahah, that’s why the paper is called like that!) from stationary electromagnetism.
In the end, what impresses me the most is how imbalanced the trade is. How little gets in, and how much gets out. By simply accepting two phenomenological laws, Einstein derived an incredibly fecund constraint on the laws of physics, one that clarified previous theories and led him to one of the most impressive achievements in all of science, general relativity.
Seriously, try reading the first 5 sections of the paper. It’s incredibly simple, understandable with minimal physics, mostly basic geometry. And yet it unfolds the world.
Nitpick: The usual physical term for this kind of thing would be "symmetry", not "conservation law".
(Though by Noether's theorem, symmetries imply conservation laws.)
I agree with the impression that symmetries seem super overpowered for inference. One of the favorite tricks for causal inference I've learned is that you can easily uniquely pin down causality from correlation if you've got symmetries across different contexts with sufficiently different distributions.