[raattgift]

First off, let's assume General Relativity (4d time-orientable manifold) is true. That means among other things that when I write "black hole" I mean an approximate solution which is a small perturbation of an exact solution like the McVittie metric, or even Kerr-Newman or Schwarzschild (which lack a cosmology part, but can be stitched into an expanding cosmology à la Darmois-Israel junction). Black hole solutions have inextensible curves or at least the distinction between curves vanishes at a "singularity". Curves are trajectories through spacetime that may in some places be accelerated and in some places may be geodesic. Physical objects bind to everywhere-non-spacelike curves and we further say these curves are everywhere future-pointing (no backwards time travel allowed). The "small" perturbations (e.g. adding outside matter) does not change any of these key features.

In that setting we could specify all the matter everywhere and solve the various partial differential equations and have an exact calculation for a particular spacetime. However anyone who has ever tried to do this by hand -- that includes many many grad students over the decades -- knows this is infeasible for complicated spacetimes. One can use automation with perturbation theory (e.g. https://bhptoolkit.org/ ) but then one discovers that perturbation theory breaks down deep inside black holes.

So the practical problem is that we want to do numerical relativity (NR) within black holes, and unfortunately singularities turn out to be numerically intractable with current methods.

How NR works, very roughly (and quite differently from your second-last paragraph) is that we slice up a model spacetime (with boundary and initial conditions) into 3 spatial + 1 time dimension, "foliating" on a time axis chosen (from the infinite possibilities) for pragmatic reasons. The axis is global for the whole spacetime. We have to note here that the relativity principle is that nobody's time axis is special, which means there is no right choice and no wrong choice here. However different choices come with different trade-offs, including in how straightforward it is to interconvert a system of coordinates adapted to our chosen time axis and those adapted to any other useful time axis.

We then canonicalize curves through this foliation into per-spatial-slice quantities reflecting position and momentum; we also make arbitrary choices about what represents "empty" spacetime so we can choose a shift vector (capturing how spatial coordinates differ from one slice to its neighbours) and a lapse function (capturing how coordinate time (e.g. proper-time of a massive particle) evolves from one slice to another). This gives us constraints (how quantities on a particular spatial slice relate to one another) and evolutions (how these quantities change from one slice to its infinitesimally future successor, and what was in its infinitesimally past predecessor).

There is no local "causality" clock ticking, there are only quantities on each whole-universe-spatial-slice and an ordering of slices. Suitable causal conditions -- notably globally hyperbolic solutions to a number of partial differential equations -- let us fill in the whole spacetime from infinite past to infinite future, and should be a formalism (the Initial-Value Formulation) of the full solution in general relativity up to numerical errors.

This is the most common approach to doing General Relativity on computers when perturbation theory breaks down (as it does deep inside black holes and in some cases where there are unusual gravitational-wave/gravitational-wave interactions and where matter waves strongly interact near one of these extreme events (this includes lensing close to a black hole)).

A concrete example of this is in SXS (Simulating eXtreme Spacetimes) numerical relativity kit, which you can begin reading about at https://www.black-holes.org/the-science/numerical-relativity...

It's not practical (and is likely highly error-prone) to use computers this way to calculate very close to a singularity, so various methods are used to "ignore" it, containing the inextensible curves inside a tiny region which we hope can be computationally smooth on the region's surface. This isn't totally new -- Gauss's gravity works that way too. But this raises the question: do we lose effects at the apparent black hole horizon (a "surface" we can obtain by doing local measurements, unlike the event horizon) when we blur the singularity inside the BH? And how do we calculate complete evaporation when relying on techniques like this? These and many related questions are active fields of study in NR.

A reminder: this is General Relativity, therefore the singularity is taken to be physically present. There is nothing that blocks the singularity from happening without adjusting the behaviour of stress-energy to introduce or substitute negative energy deep inside the black hole as matter moves inwards from one slice to the next (and there's no evidence that real matter does this), or substituting a global solution to the Einstein Field Equations which manifestly is not a black hole spacetime (even on the "initial" slice).

Physically, then, there is the question of whether matter somehow blocks the formation of singularities. Your "frozen star" idea is a very very very rough way of thinking about that question. There are many ideas in that space, and it's safe enough to generate those (although it's hard to keep them self-consistent) because there is no real hope for experimental verification of any of them in a human lifetime. However, there was in recent decades hope for exactly this sort of resolution when ideas in the particle physics space like (relatively) low-energy supersymmetry had not largely been killed off in contact with evidence from particle colliders like the LHC.

One can also find in the literature examples of "frozen star" ideas meaning that one doesn't use a black hole spacetime at all, for whatever reason. That raises lots of questions about why there are objects in our sky that radiate really really similarly to black hole spacetimes. "Frozen star" (of this kind) simulations tend to produce clearly wrong results far from the surface of the black hole.

> So what's the sense about talking about events that happen outside our universe?

Sure, this is the intuition between the puncture and excision approaches to the deep regions of black holes in numerical relativity. As long as whatever falls in also stays in, the approach is good. But what stops black holes from completely evaporating? In that case, what the hell is supposed to come out of the singularity / deep region / puncture region / excised region?

Sadly the lifetimes -- from our point of view -- of even low-mass primordial black holes or young stellar black holes is more than enough for anything crossing the apparent horizon (like primordial radiation, cosmic microwaves, distant starlight, and so on) to hit the singularity. This problem is even worse as we increase the black hole mass. There is no support in General Relativity for matter as we know it to "freeze" long enough in a black hole spacetime. The infalling matter doesn't care what we see happen from our perspective. We are allowed to experience optical illusions, or be misled by poor choices of systems of coordinates.

> our time reference

turns out to be a poor choice of time axis for many astrophysical events. One can choose more suitable systems of coordinates for events "over there" (or for a global foliation) and then do careful coordinate transforms from those coordinates to coordinates more in line with our day-to-day experiences.

For example one might attach Fermi normal coordinates to an infalling particle approaching an astrophysical black hole, and do a series of coordinate transformations to "cosmic time", from which we can do further transformations to TDB/TCB/TAI or whatever we want. The particle's collision with the singularity will be in our past. The flashes we detect from dust clouds and in our galaxy's central parsec or flashes from tidal disruption events in other galaxies are messages from matter which is already "in" their respective singularities.

[scotty79]

> There is nothing that blocks the singularity from happening

Apart from the small issue of infinite time needing to pass on the outside of the blackhole, everywhere in the universe before a single bit of matter can pass event horizon. And we are outside and infinite time hasn't passed yet and wont ever pass.

[raattgift]

> everywhere in the universe

No, that's not correct. It only has to happen at one place in the spacetime, namely at the horizon itself. General Relativity is a theory of point-coincidences, after all.

Additionally there are ordinary observers (cosmic microwave background radiation, in particular) on hyperbolic trajectories grazing the infall point; there are observers ultraboosted towards the black hole; and there are observers orbiting other black holes whose proper time is less tilted with respect to a free-falling infaller than your proper time is.

The key point is that one obtains the free-faller's geodesic by solving the EFEs in the block spacetime, and one notes that some of that geodesic is outside the black hole, some of it is inside the black hole, and that portion inside the black hole at no point in the future exits the black hole (barring complete evaporation that allows the infaller's worldline to be extended outside the black hole, in which case there isn't an event horizon but there's still an apparent horizon and probably other trapping surface structure).

[scotty79]

> It only has to happen at one place in the spacetime, namely at the horizon itself.

In theory yes. In practice if you can't point to a single moment and place in the concievable past and future history of human race that is simultaneous (in any practical sense) with this point in spacetime on the event horizon then from practical perspective it will never happen.

> one obtains the free-faller's geodesic by solving the EFEs in the block spacetime, and one notes that some of that geodesic is outside the black hole, some of it is inside the black hole, and that portion inside the black hole at no point in the future exits the black hole

This isn't that much different than plotting geodesic of a rock thrown in Earth's gravity and noticing that at some point it crosses the surface of the Earth and goes under. Except in GR the math itself provides the reason why the part below event horizon should be outside of our consideration.

We just chose to disreagrd it by carefully picking coordinate system so that infinities don't spoil our fun and we can seemlessly cross from physics to philosophy.

[raattgift]

The practical sense is the binary-merger waveforms detected at LIGO, Virgo, and Kagra, which were emitted in the detectors' (and humanity's) past lightcones. We don't even have to consider "the ... future history of human race"; the gravitational-wave ringdown is from after the two horizons touched (and in particular the intermediate stage of the ringdown encodes the crossing of light from the light or photon ring structure around the not-yet-merged black holes into the final merged object, and the recapture of quite a lot of gravitational radiation scattered in the strong-gravity zone around the not-quite-in-contact binary).

> Except in GR the math itself provides the reason why the part below the event horizon should be outside of our consideration

No it doesn't, and this has been known about since the 1920s (and reasonably understood since 1939 and well-understood since the early 1970s), and you can find out about this in any GR textbook, for example §12.2 of Wald, and problems 1 and 4 at the end of chapter 12. You should try solving those two exercises like any grad student (being sure to read problem 4 and to think about why it's asking for an upper limit) before making the sort of wrong claim about "the math itself" you made above.

A couple of things to understand: (1) in general black hole solutions do not superpose linearly [this is the thrust of problem 4]; (2) there are multiple lines of evidence supporting black hole mergers; (3) gravitational radiation is generic to any quasicircular orbit (not just black holes), with the emitted frequency and amplitude inversely proportional to the orbital diameter; (4) frequencies and amplitudes for some detections (e.g. GW150914) are too high for solid self-gravitating bodies anywhere in the sky.

You can't put a rock in a 100-millisecond orbit around the Earth; LIGO has ample data in the ~ 30 - 7000 Hz range. The gravitational wave ringdown of a neutron-star/neutron-star, neutron-star/black-hole, or black-hole/black-hole merger is totally unlike what one gets from a rock hitting the Earth.

The curves of rocks thrown in Earth's gravity don't become incomplete just because they transition from free-fall (geodesic motion) to accelerated (while resting on the surface). One can literally lift bits of meteorite out of craters. There's enough cosmic dust <https://en.wikipedia.org/wiki/Cosmic_dust> around launch sites that it's inevitable that some bits of meteorite have been taken back to space and are now in free-fall somewhere in or near the solar system (or as another possible example, some of the nickel or iridium in Voyager 1 components, if the ore was mined in Sudbury, Canada).

The curve of every part of a rock chucked through a black hole horizon ends inside the horizon. That curve-confinement is the characteristic feature of a black hole in General Relativity: no trapping surface, no black hole (Wald again, proposition 12.2.3). Where there is a trapping surface,no known high-energy behaviour of matter avoids collapsing gravitationally into a configuration in which every (non-spacelike) curve of the matter inside the horizon ends up becoming future-inextendible (that's what the singularity does: any curve touching it cannot be extended further within the horizon).

> coordinate system so that infinities don't spoil our fun

Curvature scalars don't diverge just below the horizon of even a stellar-mass black hole. Coordinate divergence is not the same as curvature divergence, and different systems of coordinates on a single arrangement of mass can diverge at arbitrary radial distances, including very very large ones.

> physics to philosophy

"Does the ringdown overtone of a merger of two black holes encode any of the accretion history of each black hole?" is a physical rather than philosophical question and is an avenue for testing solutions of the Einstein Field Equations (including approximate ones solved numerically) against astrophysical data. See for example https://www.ru.nl/en/research/research-news/the-observation-...

[scotty79]

> for example §12.2 of Wald, and problems 1 and 4 at the end of chapter 12

Literally the first paragraph of problem 1 states my exact objection.

"Since an observer outside a black hole does not lie within the causal future of the black hole, such an observal literally cannot "see" the black hole. As is apparent from Figure 6. 1 1 , an observer looking at a region where gravitational collapse has occurred would, in principle, see the collapsing matter at a stage where it is just outside the black hole."

What I can't understand is how people can smoothly ignore it and calculate something that is going to happen after infinite time passes on Earth. Because those things will literally never exist, because there's simply not enough time anywhere in the universe outside of any event horizon. For the points on event horizon there's no causality line that crosses our future and there's no simultaneity line (however you define it) that crosses our future. It just won't happen, ever, in any practical sense.

> The practical sense is the binary-merger waveforms detected at LIGO, Virgo, and Kagra, which were emitted in the detectors' (and humanity's) past lightcones.

Aren't those waveforms just in agreement of theoretical prediction of what happens as the matter approaches event horizon? Because we can't have any information about it crossing it. I don't have a problem with that part. I have a problem with the part that's theorized to happen after those signals get stretched, red-shifted and stopped by gravitational dilation.

> The curve of every part of a rock chucked through a black hole horizon ends inside the horizon.

Sure at t(Earth)=infintiy.

> Coordinate divergence is not the same as curvature divergence, and different systems of coordinates on a single arrangement of mass can diverge at arbitrary radial distances, including very very large ones.

Curvature is not the only real thing. Time, which is a coordinate is also a very real thing and when it diverges to infinity at any finite point for any observer, we can't just sweep it under the rug with a coordinate change.

> "Does the ringdown overtone of a merger of two black holes encode any of the accretion history of each black hole?" is a physical rather than philosophical question and is an avenue for testing solutions of the Einstein Field Equations (including approximate ones solved numerically) against astrophysical data.

I have absolutely no problem with that. What I have problem with is postulating that anything happens (from our point of view) in that observed region of space after the ringdown overtone of a merger happens. In my opinion when the signals get too faint and redshifted for us too observe nothing ever happens there, because time there physically nearly stops for us.

[lordfrito]

I very much appreciate your response... you clearly work in this field, and as I said I'm a total layman. So my musings could be utter nonsense.

That said... (bear with me)

> As long as whatever falls in also stays in, the approach is good. But what stops black holes from completely evaporating? In that case, what the hell is supposed to come out of the singularity / deep region / puncture region / excised region?

This is the rub isn't it?

First off, there's a lot of math, and the implications of said math (and GR model), which I'm absolutely not qualified to talk about (and you are). GR makes great (and testable) predictions on the outside of the horizon -- so we have immense confidence in the model, at least as far as we can observe and test.

Singularities in GR present all sorts of problems and paradoxes to us outside of the event horizon (here in the "testable" part of the universe).

I accept that that if we jumped into and found we could probe the interior behind the event horizon of, say, a sufficiently supermassive black hole where tidal forces are small enough we can continue to measure for a little while, then many of the things GR predicts should/would continue to be true. However that's a load bearing if, because it's untestable to an outside whether or not the observer jumping in actually made it inside... GR says that from the jumper's perspective and in their proper time they will cross over, but GR also breaks down in there. We have many reasons to believe the GR model is correct on the outside, but what confidence do we have that a "broken down" GR can be trusted to make predictions on the inside? Maybe things continue as expected, but maybe things don't?

The issue I have is that, from the outside of an event horizon (the rest of the universe), observers can't put together a coherent (and testable) timeline of events for an object falling in that includes a singularity. My understanding is that, due to infinite red-shift, it takes "till the end of time" for any outside observer to see an object cross the horizon. For these observers, the singularity creation event of course take place after the horizon crossing event. If the observer continues to wait for a very long time, they eventually observe the black hole evaporating. This evaporation must occur before the observer sees the object actually cross the horizon, and therefore must occur before the singularity can be said to form. So for all observers on the outside of the horizon, there is never a singularity that exists, or ever exists/existed, or can be interacted with, for all time.

Knowing GR breaks down, who's to say what form the inside of a black hole takes, if there is an "inside" to a black hole at all? The star could collapse into a new form of degenerate matter.. causality could stop... the immense gravity could create a degenerate or infinitely expanding space time where nothing can interact with anything, etc... the point is it's unknowable, and the breakdown of math suggests we're missing something. So we should be skeptical of anything the GR model predicts that's happening on the inside.

So, what I know is that:

   1) I don't have confidence GR predictions of what happens on the inside of the horizon, not just because they're essentially untestable, but especially in light of the fact the GR model breaks down

   2) I do trust the series of GR events that occurs on the outside of the horizon, in the testable part of the universe, which says a singularity makes no sense.

So I have confidence in the belief that in our universe, until the end of time, singularities can't form. Whatever happens "for real" can likely only be explained by a theory that unifies GR and quantum mechanics.

I suspect that whatever the truth behind it is, the answer would tell us a lot about entanglement/entropy, the "speed of causality", and the computational limits of the universe.

I'm a layman, I'm not just not right, I'm not even wrong on most things. Feel free to shoot all of this down!

[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.