[raattgift]

Ok, what I take from your comment is that you're identifying General Relativity with certain solutions of GR's Einstein Field Equations. That's like deciding that algebra is just a handful of popular equations.

(Aside, I have run out of time for an editing pass on this comment, so hopefully I didn't leave in ridiculous typos or whatever).

Firstly, let's restrict ourselves to General Relativity as a physical theory. That means we don't have arbitrarily many dimensions with wild metric signatures. We have three spatial and one timelike dimension, so take a metric signature of (+,-,-,-) or (+,+,+,-) which are equivalent but end up with things being written down in different form. That's as opposed to (+,+,+,+,+,+) or (+,+,-,-), all of which can be studied using Einstein's mathematics. Indeed, it's popular particularly among quantum gravity people to work with fewer dimensions (+,+,-) or (+,-) or to go from a Lorentzian (and thus semi-Riemannian) manifold (+,+,+,-) to a Euclidean (Riemannian) one (+,+,+,+). "Quantum gravity people" here include Hawking and 't Hooft. (duck duck go or wikipedia search "Metric signature" for more)

General Relativity as a physical theory of gravitation in our universe 3-spatial-dimension-and-one-time-dimension admits all sorts of really weird spacetimes which are wildly wildy unlike anything in our universe. (In fact relativists have over the decades invented energy conditions in an attempt to remove a few "wildly"s from consideration as possible physical systems: if your spacetime doesn't fulfil some energy conditions everywhere in it, your spacetime is probably not a good match to systems in our universe. For instance, one energy condition requires that energy-density is nowhere negative; another requires that energy is nowhere observed to flow faster than c.) (search term here is "energy condition").

There are a few books worth of exact, analytical solutions to the Einstein Field Equations (those equations define a whole spacetime), some of which resemble astrophysical systems. One such 700-page non-exhaustive book: <https://www.cambridge.org/core/books/exact-solutions-of-eins...>. However there are many many more approximate solutions which have no closed form solution: that's the realm of numerical relativity.

The Schwarzschild black hole spacetime is an example of an exact analytical solution. But there is no superposition of such spacetimes available in General Relativity, so two Schwarzschild black holes in the same universe is not just some linear combination -- instead, we have approximate solutions which can only be solved numerically. That's just with the external properties of black holes: their horizon structures, at least when studying the gravitational waves such systems emit in their last bunch of orbits before merger. It doesn't matter what's inside the horizons for that.

Likewise, where isolated astrophysical black holes give us data is outside the horizon -- what's inside is pretty much irrelevant.

So, the exterior part of an exact solution like Kerr-Newman, with some small perturbations, is at least soluble such that the perturbed KN is an excellent approximation of astrophysical observations of black holes. However we have no observations of the interior part of any black hole, so no way of knowing if Kerr-Newman's interior is wildly unphysical!

(In fact Roy Kerr has said from time to time that because of the presence of matter inside a Kerr BH, the interior part of the solution he arrived at for spinning black holes is probably wrong, even though the exterior part is a remarkably good basis for modelling astrophysical black holes. An example is in his excellent 2016 talk which you can find on youtube at <https://youtu.be/nypav68tq8Q?t=2880> immediately after the 48 minute mark of the video and again at 49:20, however he develops that theme and repeats that point across much of the talk.)

The repair of an unphysical black hole interior might be to stitch together (using a thin-shell method like Darmois-Israel) the physically useful external Kerr solution with something very different inside the horizon but which is more physical. That is not the same, at all, as declaring General Relativity wrong!

Why would one do this? As Kerr implies, there are several invariants ("symmetries") of black hole solutions which are broken by the presence of matter on either side of the horizon. Is the inside part of the Kerr solution fragile to perturbations by matter? That's a work in progress. The outside is pretty clearly stable to such perturbations, mathematically: if you throw in a blob of gas, or star, or shine a very bright light at it, or throw some gravitational waves in its direction, the outside departs from Kerr for a time but soon enough returns to being very well modelled by a Kerr solution with different mass and/or spin. The stability of the inside is not settled. So maybe matter's presence forces a departure from the interior Kerr solution to something else inside, with the Kerr solution remaining in place outside.

Stitching together metrics is something we do all the time. One puts a collapsing patch of spacetime representing a galaxy cluster (in which things tend to move towards the centre, which is where one finds ginormous elliptical galaxies), or a black hole, into a an expanding cosmology (where one finds galaxy clusters flying away from one another) using methods like this. It's all GR, it's just not one single metric line element everywhere.

Of course, there might not be a useful set of metrics that covers the whole of a black hole spacetime, because some region (e.g. near the singularity) demands a theory that differs from General Relativity. For example, the coupling between gravitation and matter might "de-universalize" near the singularity, with some matter moving differently (in particular not falling inwards) compared to matter that moves on GR's trajectories, perhaps because they couple to an auxiliary gravitational field that is so weak away from the centres of black holes that it's never noticed. (This is one approach taken by people who attempt to build relativistic MOND: when gravity is at its weakest, one gets a strengthening of an auxiliary field). [auxiliary here means in addition to the metric tensor, e.g. a vector field on the left hand (curvature) side of the Einstein Field Equations].

However there is no reason from astronomers to prefer alternative theories over General Relativity. The presence of singularities inside black holes might depend on our present toolkit of exact analytical black hole solutions, which are almost all matter-free (vacuum solutions).

Down that line of thinking is confrontation with the work of Penrose and others that shows that singularities are found pretty generically in 3+1d General Relativity. And that's a topic for another time.

[scotty79]

What reason do we have to believe there's anything inside event horizons? Since the matter in dense enough state experiences near infinite time dilation relative to us couldn't it be true that real black holes currently are just very close to becoming theoretical black holes, but will actually never get there? Never as measured by our clocks.

[raattgift]

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).

[scotty79]

> 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.

I appreciate why we do it. It's better to have a mathematical model of something that doesn't exist but it's close enough to the real thing than not to have any model at all. What I'm objecting to is the claim that spherical cow in vacuum is the reality not just a lame appoximation of the actual state of affairs. Because if you claim that then generations of bright young people imagine that cows probably travel by rolling around.

> There's nothing special about our clocks.

Yes, there is. They are ours. The events that are not on them are physically outside of our scope of knowablility.

> The hyperbolicity comes from the geometry of the spacetime

I don't mind hyperbolicity. I mind linearizing it with the use of Kruskal coordinates and such just because we are curious what might happen after our clock runs for infinite time.