Stationary electric charges, electric charges in motion (electric currents), static (unchanging) electric and magnetic fields, and how those fields and charges move, accelerate, and change in response to one another. Now, using the laws of electromagnetism alone, we can set up a physically relevant system: that of a low-mass, negatively charged particle orbiting a high-mass, positively charged one. This was the original model of the Rutherford atom, and it came along with a big, existential crisis. As the negative charge moves through space, it experiences a changing electric field, and accelerates as a result. But when a charged particle accelerates, it has to radiate power away, and the only way to do so is through electromagnetic radiation-light. This has two effects that are calculable within the framework of classical electrodynamics. The first effect is that the negative charge will spiral into the nucleus, as if you're radiating power away, you have to get that energy from somewhere, and the only place to take it from is the kinetic energy of the particle in motion. If you lose that kinetic energy, you inevitably will spiral towards the central, attracting object.
The second effect that you can calculate is what's going on with the emitted radiation. There are two constants of nature that show up in Maxwell's equations: ε0 - the permittivity of free space, which is the fundamental constant describing the electric force between two electric charges in a vacuum. μ0-, the permeability of free space, which you can think of as the constant that defines the magnetic force produced by two parallel conducting wires in a vacuum with a constant current running through them. When you calculate the properties of the electromagnetic radiation produced, it behaves as a wave whose propagation speed equals to the light speed. In electromagnetism, even if the details are quite the exercise to work out, the overall effect is straightforward. Moving electric charges that experience a changing external electromagnetic field will emit radiation, and that radiation both carries energy away and itself moves at a specific propagation speed: the speed of light. This is a classical effect, which can be derived with no references to quantum physics at all.
Now, General Relativity is also a classical theory of gravity, with no references to quantum effects at all. In fact, we can imagine a system very analogous to the one we set up in electromagnetism: a mass in motion, orbiting around another mass. The moving mass will experience a changing external gravitational field (i.e., it will experience a change in spatial curvature) which causes it to emit radiation that carries energy away. This is the conceptual origin of gravitational radiation, or gravitational waves. But why as one would be inclined to ask do these gravitational waves have to travel at the speed of light? Why does the speed of gravity, which you might imagine could take on any value at all, have to exactly equal the speed of light? And, perhaps most importantly, how do we know? Imagine what might happen if you were to suddenly pull the ultimate cosmic magic trick, and made the Sun simply disappear. If you did this, you wouldn't see the skies go dark for 8 minutes and 20 seconds, which is the amount of time it takes light to travel the 150 million km from the Sun to Earth. But gravitation doesn't necessarily need to be the same way. It's possible, as Newton's theory predicted, that the gravitational force would be an instantaneous phenomenon, felt by all objects with mass in the Universe across the vast cosmic distances all at once.
What would happen under this hypothetical scenario? If the Sun were to somehow disappear at one particular instant, would the Earth fly off in a straight line immediately? Or would the Earth continue to move in its elliptical orbit for another 8 minutes and 20 seconds, only deviating once that changing gravitational signal, propagating at the speed of light, reached our world? If you ask General Relativity, the answer is much closer to the latter, because it isn't mass that determines gravitation, but rather the curvature of space, which is determined by the sum of all the matter and energy in it. If you were to take the Sun away, space would go from being curved to being flat, but only in the location where the Sun physically was. The effect of that transition would then propagate radially outwards, sending very large ripples gravitational waves propagating through the Universe like ripples in a 3D pond. In the context of relativity, whether that's Special Relativity (in flat space) or General Relativity (in any generalized space), the speed of anything in motion is determined by the same things: its energy, momentum, and rest mass. Gravitational waves, like any form of radiation, have zero rest mass and yet have finite energies and momenta, meaning that they have no option: they must always move at the speed of light.
This has a few fascinating consequences. Any observer in any inertial (non-accelerating) reference frame would see gravitational waves moving at exactly the speed of light. Different observers would see gravitational waves redshifiing and blueshifting due to all the effects such as source/observer motion, gravitational redshift/blueshift, and the expansion of the Universe that electromagnetic waves also experience. The Earth, therefore, is not gravitationally attracted to where the Sun is right now, but rather where the Sun was 8 minutes and 20 seconds ago. This last statement, about the Earth being attracted to the Sun's position from 8 minutes and 20 seconds ago, was a truly revolutionary difference between Newton' theory of gravity and Einstein's General Relativity. The reason it's revolutionary is for this simple fact: if gravity simply attracted the planets to the Sun's prior location at the speed of light, the planets' predicted locations would mismatch severely with where they actually were observed to be. It's a stroke of brilliance to realize that Newton's laws require an instantaneous speed of gravity to such precision that if that were the only constraint, the speed of gravity must have been more than 20 billion times faster than the speed of light!
But in General Relativity, there's another effect: the orbiting planet is in motion as it moves around the Sun. When a planet moves, you can think of it riding over a gravitational ripple, coming down in a different location from where it went up. In General Relativity, as opposed to Newton's gravity, there are two big differences that are important. Sure, any two objects will exert a gravitational influence on the other, by either curving space or exerting a long-range force. But in General Relativity, these two extra pieces are at play: each objects velocity affects how it experiences gravity, and so do the changes that occur in gravitational fields. The finite speed of gravity causes a change in the gravitational field that departs significantly from Newton's predictions, and so do the effects of velocity-dependent interactions. Amazingly, these two effects cancel almost exactly. It's the tiny inexactness of this cancellation that allowed us to first test whether Newton's "infinite speed" or Einstein's "speed of gravity equals the speed of light" model matched the physics of our Universe. To test out what the speed of gravity is, observationally, we'd want a system where the curvature of space is large, where gravitational fields are strong, and where there's lots of acceleration taking place.
Ideally, we'd choose a system with a large, massive object moving with a changing velocity through a changing gravitational field. In other words, we'd want a system with a close pair of orbiting, observable, high-mass objects in a tiny region of space. Nature is cooperative with this, as binary neutron star and binary black hole systems both exist. In fact, any system with a neutron star has the ability to be measured extraordinarily precisely if one serendipitous thing occurs: if our perspective is exactly aligned with the radiation emitted from the pole of a neutron star. If the path of this radiation intersects us, we can observe a pulse every time the neutron star rotates. The predictions from Einstein's theory of gravity are incredibly sensitive to the speed of light, so much so even from the very first binary pulsar system discovered in the 1980s, PSR 1913+16 (or the Hulse-Taylor binary), we have constrained the speed of gravity to be equal to tie speed of light with a measurement error of only 0.2%! That's an indirect measurement, of course. We performed a second type of indirect measurement in 2002, when a chance coincidence lined up die Earth, Jupiter, and a very strong radio quasar (QSO J0842+1835) all along the same line-of-sight. As Jupiter moved between Earth and the quasar, the gravitational bending of Jupiter allowed us to indirectly measure the speed of gravity.
The results were definitive: they absolutely ruled out an infinite speed for the propagation of gravitational effects. Through these observations alone, scientists determined that the speed of gravity was between 2.55 x 10^8 m/s and 3.81 x 10^8 m/s, completely consistent with Einstein's predictions of 299,792,458 m/s. But the greatest confirmation that the speed of gravity equals the speed of light comes from the 2017 observation of a kilonova: the inspiral and merger of two neutron stars. A spectacular example of multi-messenger astronomy, a gravitational wave signal arrived first, recorded in both the LIGO and Virgo detectors. Then, 1.7 seconds later, the first electromagnetic (light) signal arrived: the high-energy gamma rays from the explosive cataclysm. Of course, we think that these two speeds are exactly identical. The speed of gravity should equal the speed of light so long as both gravitational waves and photons have no rest mass associated with them. The 1.7 second delay is very likely explained by the fact that gravitational waves pass through matter unperturbed, while light interacts electromagnetically, potentially slowing it down as it passes through the medium of space by just the smallest amount.
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