system
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human
What is it about GR that causes the subtle changes in the Perihelion Precession of Mercury?

I create and present Planetarium shows at the local college. I've done show on Special and General Relativity, so I have a solid understanding of these effects, but the mechanism by which GR affects the orbit of nearby celstial bodies still eludes me.

What I understand: Mercury's orbit varies ever so slightly from the orbit calculated with Newtonian physics. Apparently this variation is due to relativistic effects since it orbits so closely to the Sun (this is about as far as most internet sources will delve into it). In the last day or two, there are reports that the closely orbiting stars around Sgr A\* (the supermassive black hole at the center of the Milky Way) also display variations in \*their\* orbits which confirm some effect of GR. Are these the same effects, just at different scales? Are their orbits slowing during their closest approach as compared to an external observer? What is it \*actually\* that's going on that modifies these orbits?

Thanks in advance!
gpt
So, in Newtonian gravity, the potential from a star is proportional to 1/r

In General Relativity, it turns out that this is just an approximation that's only accurate when momentum is low and distance is large. The fact that *momentum* matters is a clue here. In General Relativity, gravity isn't only caused by the rest mass of the objects - the flow of energy and momentum actually matters too.

This extra term is proportional to the sum of the masses and the square of angular momentum, divided by the speed of light squared, divided by distance *cubed*. This is technically another approximation, but it's a better one.

Dividing by the speed of light tells us that this is going to be a small number most of the time. Dividing by distance *cubed* means it's going to be particularly small if you're not really close. So "real" gravity has this small extra bit that's so small that it's hard to measure except in odd circumstances.

The measurable bit is the *precession* of the orbit. For a 1/r potential, you get "conic section" orbits - every orbit is an ellipse or circle, a hyperbolic orbit, or a parabolic orbit. For elliptical and circular orbits, they always trace out *exactly* the same track. This is something unusual about 1/r potentials, and isn't a universal thing with orbits in general.

However, if your potential isn't exactly 1/r, then your elliptical orbits aren't exactly the same every time. The orbit *precesses*, and rotates slowly around the Sun in a spirograph kind of pattern. This can happen classically, because the influence of Jupiter etc means that the potential isn't perfectly 1/r, and planets are actually the main effect on the orbit of Mercury. However, the planets don't actually add up to quite enough, and you need to take into account this 1/r^3 potential from General Relativity as well to make the orbits work.

So, we're not talking about stars getting so close to a supermassive black hole that they experience Interstellar level time dilation - you have to be right on top of the black hole for that to matter. The more common effect is that gravity is *a little bit stronger* in General Relativity than you'd expect to find classically, and the potential has a slightly different shape so it's not something you can explain by just correcting the mass of the Sun or black hole.