by raattgift | 2025-06-01 15:36:21+00:00
Solutions of the Einstein Field Equation give rise to the geodesic equation for each such solution. In an exact, analytical solution like Schwarzschild or Kerr, every geodesic is solved for the whole spacetime. The spacetime is a block universe, and one can pick out individual geodesic worldlines as a sort of thread that runs from the infinite past to the infinite future. The most interesting worldlines are lightlike and timelike geodesics; the latter are those which a physical observer (with mass) would follow in eternal free-fall. Timelike geodesics have the property that the time dimension is longest dimension of their thread-like presence in the block spacetime.

So far there are no coordinates in place at all. There's just talk of a block fully described by a system of differential equations. We can then proceed to apply coordinates to the block, and can use any coordinates that we might want. No choice of coordinates can change the geodesics through the block, only the way in which they talk about it. For instance, let's look at using different 2-d coordinates on the a restaurant where you and a friend are looking at each other across a dinner table. One could say you are north of the friend looking south, or your friend is in front of you looking back at you, or you are in front of your friend and looking back at him. Someone else in the restaurant could say that you are to the left of your friend, or that you are slightly northeast and your friend slightly southeast; a different person elsewhere in the restuarant could describe it exactly opposite: you are to the right, your friend to the left. And so on. But you aren't actually moving around the table or restaurant: the configuration of you and your friend isn't changed as we apply different sets of coordinates. And we can move the origin of Cartesian or polar or whatever coordinates anywhere in the restaurant, so you're at x=0,y=0 and your friend is displaced in the y direction. Or you're bothed displaced in x and y from the x=0,y=0 preferred by an onlooking diner at a different table.

In a black hole spacetime, there are geodesics which are interior to a set of horizons: once they cross such a horizon they stay crossed. It does not matter at all what the coordinates are that are used to describe the horizons or the geodesics.

> relative to us

This is just you applying your coordinates to a block.

Doing so doesn't change the geodesics which cross a horizon in one direction only: their eternal past might be "free" but their eternal future is within the horizon.

> time dilation relative to us

Again, this is applying "our" coordinates. However even those might differ: you are probably thinking in terms of the standard Schwarzschild coordinates or something close to them, whereas I might reach for Kruskal coordinates instead, which absorb timing differences. Just like someone in the restaurant might prefer north-south/east-west coordinates instead of a personal left-right/foreground-background system.

> theoretical black holes

Well, yes, we assume General Relativity is sufficiently correct that astrophysical candidates are compared to black hole solutions in General Relativity rather than something different in a different theory.

We also make some simplifying assumptions which are known to be nonphysical but which represent very minor perturbations of black hole solutions to the Einstein Field Equations. Otherwise we would have no hope of calculating anything, and could not even approximate astrophysical candidates' behaviour.

> Never as measured by our clocks

There's nothing special about our clocks. That seems to be the problem you are wrestling with.

You could get a hyperbolic clock on your smartwatch that slows down as you age, and ultimately becomes so slow that a tick on your wristwatch is more than long enough for a free-falling body to cross a black hole horizon. The physics doesn't change; the free-falling body's worldline in the block universe has a portion outside the horizon and a larger portion inside the horizon. You're just using different algorithms on your wristwatch to apply timelike coordinates to a part of that free-falling body's worldline.

Because one can use any sort of coordinates on a block universe with a set of solved-by-the-equations worldlines (or at least timelike and lightlike geodesics reasonably near a subsystem of interest) one can make poor choices about what set of coordinates to apply to a particular subsystem of the block universe. "Our [linear] clocks" is a poor choice for describing systems with strong gravitation, speeds comparable to c, strong acceleration, or any combination of those, because there is always a nontrivial element of hyperbolicity in such systems' worldlines. The hyperbolicity comes from the geometry of the spacetime, and is always there in any Lorentzian spacetime (meaning it has 3 spacelike and 1 timelike dimension where the latter is related to the former in line elements by the constant c and a change of sign).