Math proof draws new boundaries around black hole formation

There's a nice graph in the midde of this[2] page that shows the difference between the proper time and the aparent time observed by the outside observer. At r = 2m you can see the aparent time goes to infinity and the quickly back again.

[1]: https://physics.stackexchange.com/questions/718222/proper-ti...

[2]: https://www.mathpages.com/rr/s6-04/6-04.htm

Below I'm going to ignore angular momentum; black hole spin changes the details but not the central thrust of my comment.

You could think about it this way: Schwarzschild is an eternal vacuum solution (to the Einstein Field Equations of General Relativity (EFEs)). Like other solutions to the EFEs, it tends to be investigated by tracking the behaviour of "test particles". Test particles don't change the EFEs themselves -- they don't have mass, they don't have any non-gravitational interactions at all, they're not physical, they're just a tool used to trace out the geometry of the spacetime. Throwing in a test particle doesn't change the eternal nature of the Schwarzschild black hole -- it always has the same mass, and the Schwarzschild radius is totally determined by that mass (so the horizon is always of constant size).

That is, the test particle is not a perturbation of the Schwarzschild black hole.

A significant mass would perturb the Schwarzschild metric though.

Unlike test particles, that mass enters into the EFEs. The relevance is that the Schwarzschild radius is totally determined by the central mass. When the significant additional mass (the perturbing infaller) is far away from the central mass, the geometry (described by the EFE's metric tensor) still looks a lot like Schwarzchild. However, as the perturbing mass approaches and falls in, things depart from Schwarzschild for a bit, then returns to Schwarzschild but with an increased central mass.

The infaller ultimately ends up at the singularity, leaving M_before < M_after, so the horizon must grow in proportion to the mass that fell in because the Schwarzschild r_s = 2GM/c^2, where M is the central mass. Distant viewers can measure the central mass in several ways; it's observable.

Since throwing actual mass (rather than test particles) into Schwarzschild changes the size (and, briefly, shape) of the horizon, you could consider it as if the object you see dimming and moving verrrry slowly doesn't just come to a halt at a constant horizon: instead, the horizon "reaches out" and snatches the infaller just outside the M_before horizon.

The worry that nothing actually falls in arises if one insists on keeping M constant (M_before == M_after) even as one has mass outside the centre of the black hole. Constant M is easier to work with mathematically, which leads to things like test particles or Hawking negative energy quanta, and so on. Extrapolating from those convenience uses tends to lead to confusions like "nothing initially outside can actually end up inside", which is just wrong.

Finally, using perturbation methods, a compact infaller like a neutron star would raise a significant bump on the horizon, distorting it slightly from ~spherical. That distortion vanishes in a short time (even for an outside observer, who can also detect gravitational waves), and the post-infall result is a bigger spherical horizon. We have observed several neutron star-black hole mergers. The result, generically, is a more massive black hole (and a lot of gravitational radiation).

tl;dr: that the body is all attached for some time after crossing the horizon is [a] likely and [b] sufficent to justify the notion of stimulation in the feet being processed in the brain, when falling feet-first within a black hole.

Firstly and basically the whole of [b], I'm just gonna assume that nerve impulses are the result of chemicals spat within neurons and between them, all very slowly compared to the speed of light, and all quite local (one chemical change or protein-conformation change at a time, step by step). I don't know the full details, but the relevant thing is that the speeds are slow, and that's the basis for the question ("how does something not faster than light rise within a black hole?").

Now on to what happens to a self-connected body inside a black hole, i.e., a justification for [a].

My nth iteration of an answer to this (abandoned a few because they got too technical or were unsatisfying; really fatigue drove me to a decision to submit as-is or to just walk away from an attempt at a good answer) is that the "rules" of a black hole are that your body's centre of mass cannot climb relative to the centre of mass of the black hole. More particularly, acceleration of the body's centre of mass is a vector (i.e., with magnitude, direction and sense) which is constrained to have the sense be at least slightly inwards at every point inside the horizon.

However, there is no rule about where exactly your centre of mass is found relative to the bits of you stuck to your skeleton and encased in your skin. Equivalently, the sense constraint does not apply to very small parts of your body so long as those small parts do not break the rule that applies to the bulk's centre of mass. (This is another way of stating the universality of free fall version of the weak equilvalence principle, where the vacuum worldline is that of the bulk, the microscopic components of the bulk (atoms in haemoglobin in blood in the aorta etc) are obviously not in vacuum themselves.)

Internal adhesive forces (and other contact/surface forces) win for a while in a battle to keep your extremities from breaking off and taking their own free-falling trajectories, and so (if you are falling feet-first) you could think of your head supporting the weight of your body. As you fall inwards, your feet will feel heavier and heavier and ultimately you become what medical people like to call "disarticulated".

This is a result of three rules:

* A spinning spherical mass becomes oblate. Earth is very slightly narrower between the poles than between two opposite points on the Equator.

* Objects falling onto a spinning spherical mass become prolate, that is longer in a vertical direction and narrower in the directions perpendicular to that.

* Objects dropped above different points on a spinning spherical mass will follow converging trajectories. Handy little diagram: <https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...> If we put the stick figure at the equator on the tropic of Cancer instead, the fat arrows pointing towards the centre still converge, just with a smaller angle.

These are all taken to exxtreem in a spinning black hole. The hole itself is slightly oblate. This induces different notions of "straight down" that get weirder as one gets deeper within. The vertical stretch and inward squashed induced on an infaller becomes strong as one gets close to the black hole's centre. And how much inducement there is for parts of a body to find their own notion of "straight down" (the gravitational "pseudo force") becomes stronger than the inducement on them to stay attached to the rest of the body.

If you are falling feet-first, the skin cells on your shoulders want to take their own notion of "straight to the centre", but they are dragged onto your centre of mass's "straight to the centre" because of the forces that stick your skin cells to the layers underneath. Your feet want to fall faster than your hips, but your connective tissue and skeleton keep them from doing that. Your head wants to lag behind, but it too is prevented. Until you split apart (e.g. your femoral artery ruptures, your spine separates, whatever) this structural integrity supports nervous impulses and blood rising from your feet towards your brain. You can still think on your way down.

You could consider this as your internal forces changing the local spacetime geometry to allow nerve impulses and blood to rise upwards; or alternatively you could think of it as your upper bits suspending your lower bits at a higher gravitational potential. The real physics is a complex set of tensors, and with enough initial data you could grind out a full picture with mathematical rigour. You'd then cut down the tensorful field equations into vectors (which way do your various bits get accelerated) by choosing a suitable set of coordinates; probably you'd want to use ones where your centre of mass is always at the origin, or where each of your bits and pieces have their own set of coordinates ("Fermi coordinates") and you do some coarse-graining like averaging. Either way, your body-bits' mutual accelerations overwhelm the tidal forces (and the tidal forces are themselves vectors which we extract from the Weyl tidal tensor in applying the same suitable coordinates).

In support of this line of thinking about how you can still think when you're in a black hole is the "no-drama" conjecture. The radius of curvature just outside the horizon of a supermassive black hole is much greater than that at Earth. Our bodies tend not to be disassembled by Earth's gravitation (in the "tidal" sense; that's that sense which is captured in the three rules above -- a Nasa-produced animation exaggerating the prolation induced by the moon on the Earth's oceans <https://www.youtube.com/watch?v=l37ofe9haMU>), so it would be surprising if they were disassembled by the gentler gravitation just outside a big black hole.

You should just free-fall through a big black hole's horizon without feeling much in the "tidal" sense (your view of distant stars would give things away though, even before you are at the point of no return). For a supermassive black hole you should continue not to notice much for some time (minutes on your wristwatch). Eventually (possibly an hour or so on your wristwatch) you notice and it hurts and your head probably pops off ~minutes later, and your remains "spaghettify".

Unfortunately we don't have enough evidence to select an low-curvature-radius-but-still-near-horizon effective theory, so there are certainly plausible descriptions very different from mine above.

For example, certain physicists reject the no-drama conjecture quite strongly, often because their preferred theories of quantum fields (or strings) predict that the horizon is a tangible surface which supports very high energy radiation. Black hole horizons in general relativity do nothing of the sort; they're just a notion of a point of no return[*], and any particle crossing that point falls inwards, be they super high energy or super low energy or something in between. (The curvature singularity isn't a surface that supports anything either; infallers' remains don't really collide with it or pile on top of it, or rather we can't say one way or another what the case is because the standard equations become divergent. They do not diverge at the horizon though.) In some of those theories you are fried by high intensity gamma radiation before you get to the horizon; in others there is a "firewall" just inside that blows you apart with ~Planck-length radiation. If those are right, there's drama at or very near the horizon, and nothing will be able to spaghettified because it will be disintegrated by radiation.

[*] yes yes, but you (complaining expert) don't know which horizon I mean, and there are many to choose from (Visser, <https://arxiv.org/abs/1407.7295v2>, 2nd paragraph)

Now, one of the things one can think about carefully is the dynamics of extended bodies in general curved spacetime. Extended body dynamics is already a problem in gravity-free special relativity (e.g. Bell's spaceship paradox). I'd treat it slightly differently when gravity is important enough. The centre of momentum of a human obeys the universality of free fall; a quick bungee jump proves this pretty well. However, the internal components stuck to the skeleton do not individually free-fall (otherwise the end of the falling part of a bungee jump would go pretty badly) because they experience internal contact forces (cf. body forces like Newton's gravity).

An infalling human is not a cloud of fine dust, or a big gassy star: there are intermolecular forces in play even in the wet flowy bits of the anatomy, and while far from outright rigidity, a puny human's microscopic bits zip around on accelerated curves rather than the free-falling geodesics of a pre-dustified (atomized? turned into a fine spray with particle sizes much much smaller than the size of a red blood cell) human. It's also why you aren't a microscopically thin puddle on the floor right now, and indeed why there is a floor to be on (or "mountains" on the surface of a neutron star or why a neutron star/BH (NS-BH) collision like GW200115 is different from a BH-BH collision, especially with a stiff equation of state).

The mechanisms of support against freefall for every microscopic part of the anatomy in turn must be encoded in the stress-energy tensor rather than some notion of ultraspecific background curvature; consequently local T_munu != 0 suddenly becomes quasilocally relevant since it determines the curvature. For test particles, we don't care, we keep T_munu = 0. For puny humans the temptation is to do the same, but realistically we circle back to a statement like "nobody's developed a justifiable intuition for this" (although you can certainly look at a multiplicity of numerical results).

Inside general black holes (following Lluis Bel) the gradient of the electrogravitic tensor is probably the killer; that's what captures the stretching up-to-down (cf the rope in the Bell's spaceship) and what also encodes inwards squeezing in the perpendicular directions. Essentially, you are liable to be resemble a radially-squeezed tube of toothpaste with the flowier bits spraying out the bottom when there is a local failure of your tissues' binding energies to decouple tissue-components from converging geodesics (roughly, the contact forces that slosh momentum into the spatial diagonal of the stress-energy tensor eventually fail to prevent it sloshing right back out) or kinda like if a bit of gas near the inner surface of a balloon were to become sharp as you really stretch the filled baloon along one axis (i.e., making it prolate)). One might then spend remaining moments of consciousness wondering if there are conditions in which free-falling distant starlight (null geodesics[*]) passing through the bloody mist (now on timelike geodesics[*]) could produce a rainbow for some intact massive object further below.

In short, "think about it carefully" is the sort of thing one might do for a PhD thesis (demonstrating that you can actually do research) rather than a brief hackernews comment (which rarely demonstrates any such thing).

[*] downside, interior solution of the geodesic equation depends on the entire history of the BH. And using Kerr as a starting point is fraught. David Madore (an engaging mathematician) put it pretty well when he wrote this years ago <http://www.madore.org/~david/math/kerr.html#course.real_blac...>. There are pretty visualizations there, but if you get extended-body/black-hole-interior trajectory intuitions out of them, send a postcard back to this thread.

Oh, in addition to his agreeable couple paragraphs of objectsions, the space outside an astrophysical black hole isn't empty (otherwise how is there anyone to jump in?), and the spacetime there isn't even asymptotically flat. So now, for a black hole huge enough not to destroy you while you're still outside it, you get to think about: coupling with the host galaxy (does the shell theorem have an analogy there?); the cluster, whose metric may be quite inhomogeneous; and the cosmological constant; and justify whether those contributions are marginal or irrelevant in grinding out accessible trajectories. Maybe start with Schwarzschild-de Sitter as an approximation of the galaxy cluster all smooshed into a ginormous black hole in the far future, and take Schwarzschild->Vaidya (relic cmb) and then to spinning Vaiyda-de Sitter, extra credit for contrasting with AdS solutions since those thermalize but have off the shelf computer program implementations. Along the way you can re-calculate what happens to a human being dropped towards each iteration's central mass. Extra extra credit for considering binary black holes at various mass ratios: can we go beyond a gross qualititative guess like, "infaller dies faster in smaller of the pair"?