How does this cause a point particle to accelerate towards the sun? Must be something about the gradient, but how does the gradient of time cause you to curve towards the sun?
That's a great question. The answer is, the stuff you are reading in this thread is not right (you figured it out). The real version of the story is, there is this thing called the "Christoffel symbol," which tells you where, at every point in space, you would end up if you went in a certain direction, including which way you would be facing if you went that way. It relates three vectors: your direction of motion, the direction you are currently facing, and the delta to your direction of facing that would result from taking that direction of motion.
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.
For a nice introduction to relativity, look at The Einstein Theory of Relativity: A Trip to the Fourth Dimension by Lillian R. Lieber. $15 on Amazon. Written in 1945 and still quite good.
No, but you can talk about changes in perturbations of fields over time in QFT (which has its own representational issues). A particle is a useful metaphor.
Their point in this case is that a wavefunction is spread out over space, which would cause it to be subject to a local clock gradient in curved spacetime. If you wanted to use particles, you'd need to use a Feynman-style "integrate over all possibilities" approach, which would again be subject to a clock rate gradient over space.
The mathematics of this is a bit too complex to reproduce in a comment here, but in, say, the Earth's gravitational field, taking this effect into account (approximating GR as a field of locally varying clocks, then allowing, e.g., an electron's wavefunction to evolve on that spacetime) would reproduce gravitational acceleration / free fall towards the Earth.
Said differently: this is precisely the kind of nuanced scenario where getting sloppy with metaphors gets you into trouble very quickly. Quantum mechanics in curved spacetime is not to be dabbled with lightly.
I think it takes a very narrow understanding of physics discourse to call the particle a terrible metaphor. What's next, entirely discarding the teaching of newtonian physics in high school?
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.