The article says 4+ dimension black holes can exist. Just curious about an oddball idea from complete laymen perspective - could a blackhole with a time dimension/component exist then? I'm sure someone's thought about that before me :)
Came to comment about the same thing. It's funny - higher dimension black holes can confidently exist in a model, but because the model isn't tied to reality, it's not a very meaningful statement.
Time ends at black holes. They stretch out time infinitely, from an outside perspective.
I think what they meant by end is that time appears to stop at the event horizon for outside observers. As something approaches a black hole event horizon, time appears to infinitely slow, for an outside observer, such that a frozen image of the object gets stuck on the event horizon. But this image can't be seen because it gets infinitely red shifted such that no detection device can keep up with the longer and longer wavelengths.
You can measure time dilation at different distances from black hole, then extrapolate and see that time dilation goes to infinity at the event horizon.
Black holes already have a time component since they exist in a universe with 3-dimensional space and 1-dimensional time. The equations include both the spatial components and the temporal components.
One of the curious features of black holes is that they reverse the roles of time and space in a sense. In ordinary space we have freedom to move around in space, but are constrained to only move forward in time. Different reference frames may move forward at different rates relative to us, but they always move forward.
By contrast, once you pass the event horizon of a black hole, these properties swap. It's possible to "move freely" in time in the sense that you can find reference frames that appear to move backwards in time. But all these reference frames are constrained to move forward in space.
The article explicitly says that it's talking about number of spatial dimensions, it's not including time. It says the new theorem shows that black holes can form into up to 7 spatial dimensions (plus 1 for time). (This isn't to say they can't form beyond that, just this new theorem doesn't show that.)
Not really. It just appears to take an infinite amount of time for an outside observer.
For someone falling into a black hole, it takes a finite amount of proper time[1] to reach the event horizon.
For that infalling observer, the horizon is a boundry where, once beyond, the singularity is always in their finite future. No matter what you do inside, you will reach it at some point.
Yes, this is correct. If you look at the Schwarzschild solution to the Einstein field equations you'll find that there are two apparent singularities: one at the center of the black hole, and another at the event horizon. However, the apparent singularity at the event horizon is not a true singularity because you can perform a coordinate transformation in which it disappears. Mathematically, this is doing what you describe: going from the reference frame of an external observer to one who is falling into the black hole. To the outside observer you appear to take an infinite amount of time to fall in, but from the perspective of the person falling in, it happens in a finite amount of time.
The other singularity at r = 0 is different, though. It is a true singularity because there is no coordinate transformation you can make in which it disappears.
If the escape velocity from the event horizon is (defined to be) light speed, that seems to mean falling in from indefinitely far away, you would get indefinitely close to light speed (as perceived from at rest outside) at the point when you cross it,.
From the perspective of the person falling, it happens from one moment to the next. You as the person being accelerated don't notice any change in your own proper time, only distant observers will see that.
If you were to fall radially inwards towards a non-rotating black hole, that is not spiraling around it but "straight in", it would be the same as with plain Newton's law[1].
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.
Thank you for the response and links. The second link is particularly helpful. I'm surprised that Newton is equivalent here, but the reason appears simple: although the inbound observer's time slows, their velocity does not.
A distant observer is unable to perceive them as they approach the event horizon as the inbound actor grows infinitely dim thanks to time dilation.
But, if I am concerned solely with my own perspective (since there is no privileged viewpoint), is it fair to say that nothing yet has ever been observed entered a black hole?
The answer is simply that black holes grow when you throw mass at their horizons. The small region you last saw the mass (dimly and red-shifted) just outside the black hole ultimately ends up being on the inside, even from your perspective.
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).
Aside: the fact that the radial spatial dimension towards the centre of the black hole where the singularity lies becomes time inside the event horizon is why all the drivel about “what you’d see if you crossed the event horizon” is drivel. You’d die instantly since circulating your blood or sending nerve impulses would essentially become impossible because it would involve reverse time travel — i.e., blood moving toward the centre would never be able to move out again because that would involve it travelling back in time. Greg Egan mentioned this in a short story of his.
This is not true. It is possible for two observers inside the event horizon to send signals to each other. It's just that those signals cannot be sent to any radius larger than the radius they originated at.
The only danger to a human would be the tidal forces. For a supermassive black hole, the tidal forces at the event horizon are quite modest, so you really would not notice it when you passed across the event horizon. It's only when you get close to the singularity that the tidal forces tear you apart.
Thanks for asking this good (and pretty difficult) question, rather than trying to make an assertion; cf. a sibling comment of mine.
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 ...
Thanks. Unfortunately, I noticed an error I made during some rewording: "an low-curvature-radius-but-still-near-horizon effective theory". The indefinite article should be "a", and the reciprocals are wrong -- the radius of curvature near the horizon is high and the magnitude of the curvature scalar is low (it's a function on position in spacetime and goes to infinity as one approaches r=0; that behaviour of the Kretschmann scalar is used to show there's an actual curvature singularity rather than some artifact of a choice of how one chooses the "r" coordinate).
Thank you for your very comprehensive and insightful reply. As an applied mathematician specialised in entirely different fields, I must ask though: why the focus on the centre of mass of the whole infilling object? Surely one can subdivide the infalling object into a number of sub-units, each with its own centre of mass, all the way down to the subatomic level, and come to the conclusion that though each object’s centre of mass is relevant, it conflicts with the overall capacity to propagate signals across the breadth of the aggregate object? And what of the situation when the object is (from its own perspective) crossing the event horizon, so the centre of mass of the overall object is not yet within the event horizon itself? Folliwing your argument, if the centre of mass is what matters, then signals could still propagate upwards from the part that is already within the event horizon and into the parts of the object that are not yet within the event horizon, violating the whole intuition about what an event horizon actually is and what it means in a physical sense?
If you think about it carefully you’ll realise that a human being’s body would span a breadth of radii and therefore the parts closer to the singularity would not be able to send signals or matter to the parts that are further out, and that this would impede circulation and signal propagation.
Ok, on the way to thinking about it carefully I'll ask you to look at all the little pockets one can hide a puny human in on a (r, \theta, K) or plane contour graphset of the Kretschmann scalar for a large general black hole (BH). Unfortunately scalar invariants plots don't really offer anyone much intuition about the detailed fate of an astronaut jumping in, and anyway it's more fun to think about Taub-NUT BHs because nobody repeats slogans about those on hackernews.
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 po...
> By contrast, once you pass the event horizon of a black hole, these properties swap.
Sigh. No. They do not. Moreover, nothing special at all happens to you when you pass the event horizon.
In fact, if you are free-falling then you should not even be able to detect the crossing using only local experiments.
> It's possible to "move freely" in time in the sense that you can find reference frames that appear to move backwards in time.
Nope.
What happens is that your spatial directions become more and more constrained, until they collapse into a single point (the singularity). And then you'll just exist in this single point forever, according to GR.
From your viewpoint, it'll look like the singularity becomes an infinite plane that cuts off most of your field of vision. You'll be falling towards this plane, but until the last moment you'll be able to receive signals from outside of the black hole.
If someone puts a stationary clock outside the black hole's event horizon, you won't see it going faster or slower. And for any realistic black hole, your trip to the singularity will consume only a short time according to that clock.
And to add to this a singularity is predicted by GR, we don't actually know if such a thing exists and there are models where it doesn't like LQG where black holes are just a different kind of star whose collapse is opposed by a repulsive force that is predicted to exist by the uncertainty principle.
Technically, GP is right that they (mathematically) swap, but yeah, it has a meaningless physical effect (the geodesic is always smooth). It's akin to describing the rotation of a kicked ball with imaginary exponentiation and thinking something spooky is going on.
So I would agree with this for the Schwarzschild metric. The swapping of signs in the space and time components of the metric does not have any real physical consequences. But when you move to a Kerr metric it absolutely does because it permits the construction of closed timelike curves.
It doesn't happen in Gullstrand-Painleve coordinates which would be a "correct" set of coordinates for a freely falling observer as they enter into a black hole. Or the Lemaitre coordinates work too.
Passing the event horizon in Schwarzschild coordinates is meaningless, it never happens.
> From your viewpoint, it'll look like the singularity becomes an infinite plane that cuts off most of your field of vision.
You might need to elaborate because the singularity in Schwarzschild is not something you can spoke with a stick, let alone see. It is a spacelike singularity, meaning in this case that it lies in the future of any observer falling into the black hole. It is like death: It is certain you will encounter yet in the future, yet you don't see it.
I haven't read the paper, but I suspect the whole concept of a 4, 5, 6, or 7-dimensional black hole is nonphysical, but the math works out if you just close your eyes and assume such a thing can exist. The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n you have, an n-sphere of radius r has a much larger boundary for the same volume. Conversely, that means that for a given surface area, you have less volume, so, for a constant mass density, you're enclosing less "stuff" inside.
What prevents the formation of many sub-universes with different space-time configurations? We happen to live a large 3+1 one, but others may exist. Speculating further, some black holes could be bridges between e.g. a 3+1 world and a 4+3 world. The observation about 7-dimensional spaces would mean that above that level there are no such boundaries between world, they all become one universe.
I believe that when people talk about N-dimensional spaces they typically talk about spaces with a Euclidean metric unless they explicitly say otherwise. The properties of a sphere in an N-dimensional Euclidean space are not all the same as those in an N-dimensional Minkowski-like spacetime (either 6+1 or 4+3 or any combination).
> The reason I suspect n=7 is as high as it goes is because the volume to surface area ratio of an n-dimensional sphere is r/n, which means that the higher n you have, an n-sphere of radius r has a much larger boundary for the same volume.
No, I haven't read the paper in detail either, but the reason is very likely the same for why Schoen & Yau (mentioned in the article) could initially only prove their Positive-Mass Theorem up to spatial dimension n=7: In n=8 spatial dimensions minimal hypersurfaces can suddenly develop singularities[0] and you can no longer treat them as manifolds but need tools from geometric measure theory (so-called minimal currents) to describe them. This makes proofs relying on minimal-surface theory in dimensions n > 7 much, much harder (though not necessarily impossible).
[0]: The singularities, when zooming in, essentially look like the tip of the 7-dimensional Simons' cone, https://encyclopediaofmath.org/wiki/Simons_cone , making the latter the prototypical example. In fact, the existence of Simons' cone is the whole reason why (i.e. essential to the proof that) stable minimal hypersurfaces in n > 7 can have singularities but in lower dimensions they cannot and are instead smooth: There are no singular stable minimal hypercones in lower dimensions.
> The article says 4+ dimension black holes can exist.
Unfortunately, the article is not particularly precise here. The actual statement is that if you're in n < 7 spatial dimensions, then the existence theorem of the paper holds. (I.e. if conditions A, B and C are met, there will be a black hole, more specifically an apparent horizon.)
As far as we know, we are in n=3 spatial dimensions, though. (Yes, string theory claims something else but the results of the paper only hold for n non-compactified spatial dimensions, so they don't apply to string theory.)
It's not a practical possibility. The black hole wouldn't last long and would be too small to actually absorb anything. It's the equivalent of asking if a nuke would set the atmosphere on fire.
Even a "large"ish primordial black hole would probably just pass straight through the Earth without anyone noticing.
Remember that Strange matter is only dangerous assuming a specific range of values for its surface tension, otherwise it's harmless and doesn't catalyze the conversion of normal matter into Strange matter.
I'm not sure. Somewhere around 10^12 kg of initial mass would be evaporating today (1). So perhaps there is no meaningful minimum, only a minimum initial mass. If it's just about to evaporate, it could perhaps be arbitrarily small. Earth is ~10^25 for reference.
It only has the gravity of a 1 kg object and is extremely small, so it's not likely you'd be able to find an environment where mass is going to be getting jammed into it at quadrillions of tons per second even ignoring the fact that it's sort of a continuous nuclear explosion — it's radiating all its mass energy at a rate a few million times higher than the output of the sun in the ballpark of a star actively going nova.
A black hole is just a "normal" mass object. If the Moon was replaced by a black hole of the same mass, there'd effectively be no change except for it not being reflective like the Moon.
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[ 11.9 ms ] story [ 431 ms ] threadTime ends at black holes. They stretch out time infinitely, from an outside perspective.
I wouldn't think so because empiricism implies experience which implies time
One of the curious features of black holes is that they reverse the roles of time and space in a sense. In ordinary space we have freedom to move around in space, but are constrained to only move forward in time. Different reference frames may move forward at different rates relative to us, but they always move forward.
By contrast, once you pass the event horizon of a black hole, these properties swap. It's possible to "move freely" in time in the sense that you can find reference frames that appear to move backwards in time. But all these reference frames are constrained to move forward in space.
Everything past the event horizon is speculative.
For someone falling into a black hole, it takes a finite amount of proper time[1] to reach the event horizon.
For that infalling observer, the horizon is a boundry where, once beyond, the singularity is always in their finite future. No matter what you do inside, you will reach it at some point.
[1]: https://en.wikipedia.org/wiki/Proper_time
The other singularity at r = 0 is different, though. It is a true singularity because there is no coordinate transformation you can make in which it disappears.
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
A distant observer is unable to perceive them as they approach the event horizon as the inbound actor grows infinitely dim thanks to time dilation.
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).
The only danger to a human would be the tidal forces. For a supermassive black hole, the tidal forces at the event horizon are quite modest, so you really would not notice it when you passed across the event horizon. It's only when you get close to the singularity that the tidal forces tear you apart.
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 ...
The radius of curvature is 1/|K| where one chooses a curvature scalar -- Kretschmann, Gauss, others may apply -- and finds a matching "kissing circle" (osculation is kissing). Here's an example in 2d, \rho is the radius of curvature and we're asking about the radius of curvarure at P on the curve AB: <https://undergroundmathematics.org/glossary/curvature/images...> (Two other examples <https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...>, <https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...>). For a point on the surface of a shell, we'd use an osculating sphere, and so on in additional dimensions.
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 po...
Sigh. No. They do not. Moreover, nothing special at all happens to you when you pass the event horizon.
In fact, if you are free-falling then you should not even be able to detect the crossing using only local experiments.
> It's possible to "move freely" in time in the sense that you can find reference frames that appear to move backwards in time.
Nope.
What happens is that your spatial directions become more and more constrained, until they collapse into a single point (the singularity). And then you'll just exist in this single point forever, according to GR.
From your viewpoint, it'll look like the singularity becomes an infinite plane that cuts off most of your field of vision. You'll be falling towards this plane, but until the last moment you'll be able to receive signals from outside of the black hole.
If someone puts a stationary clock outside the black hole's event horizon, you won't see it going faster or slower. And for any realistic black hole, your trip to the singularity will consume only a short time according to that clock.
Technically, GP is right that they (mathematically) swap, but yeah, it has a meaningless physical effect (the geodesic is always smooth). It's akin to describing the rotation of a kicked ball with imaginary exponentiation and thinking something spooky is going on.
Passing the event horizon in Schwarzschild coordinates is meaningless, it never happens.
You might need to elaborate because the singularity in Schwarzschild is not something you can spoke with a stick, let alone see. It is a spacelike singularity, meaning in this case that it lies in the future of any observer falling into the black hole. It is like death: It is certain you will encounter yet in the future, yet you don't see it.
No, I haven't read the paper in detail either, but the reason is very likely the same for why Schoen & Yau (mentioned in the article) could initially only prove their Positive-Mass Theorem up to spatial dimension n=7: In n=8 spatial dimensions minimal hypersurfaces can suddenly develop singularities[0] and you can no longer treat them as manifolds but need tools from geometric measure theory (so-called minimal currents) to describe them. This makes proofs relying on minimal-surface theory in dimensions n > 7 much, much harder (though not necessarily impossible).
[0]: The singularities, when zooming in, essentially look like the tip of the 7-dimensional Simons' cone, https://encyclopediaofmath.org/wiki/Simons_cone , making the latter the prototypical example. In fact, the existence of Simons' cone is the whole reason why (i.e. essential to the proof that) stable minimal hypersurfaces in n > 7 can have singularities but in lower dimensions they cannot and are instead smooth: There are no singular stable minimal hypercones in lower dimensions.
Unfortunately, the article is not particularly precise here. The actual statement is that if you're in n < 7 spatial dimensions, then the existence theorem of the paper holds. (I.e. if conditions A, B and C are met, there will be a black hole, more specifically an apparent horizon.)
As far as we know, we are in n=3 spatial dimensions, though. (Yes, string theory claims something else but the results of the paper only hold for n non-compactified spatial dimensions, so they don't apply to string theory.)
If an LHC collision were to form a blackhole,
1. How long would it last
2. Could we detect it
3. How much mass would need to be collided to suck in Earth?
Even a "large"ish primordial black hole would probably just pass straight through the Earth without anyone noticing.
Strange matter on the other hand...
(1) https://en.wikipedia.org/wiki/Micro_black_hole#Expected_obse...
[1]: https://en.wikipedia.org/wiki/Hawking_radiation#:~:text=The%...
A significant fraction of the mass of the Earth. Black holes don't "suck" any harder than other objects of the same mass.