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Comment by meindnoch | Hacker News Reader

I've read somewhere an article which posited that our 3D universe might be inside a 4D black hole. When you cross a black hole's event horizon, the radial coordinate becomes timelike, so you lose one degree of spatial freedom. Movement is still possible in the tangential directions however, so what you get is basically an N-1 dimensional universe. So maybe our 3D universe is actually matter that fell into a 4D black hole, and our 3D black holes contain 2D flatland universes. And of course, the outer 4D universe might be in a 5D black hole, etc.

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Yes, and then there's the parlor game of guessing what familiar property of our known universe is actually a spaghetified fourth dimension.

I guessed c once. It would be a constant. Maybe all the constants are spaghettified remains of a superior universe.

I don't think c is a good candidate, because it's not really a parameter. It's just a correction factor for our mis-judgment in picking different units for time and space.

In "natural units", we define the units so that the important conversion factors (c, G, h-bar, etc) work out to exactly 1. You can say that c is one light-year per year and then forget about it.

The true parameters of the universe are the dimensionless constants: the fine structure constant, proton-electron mass ratio, 3+1 dimensions, etc.

> I don't think c is a good candidate, because it's not really a parameter.

dont be so sure! there is no way to experimentally know if c is a parameter or not. there are consistent physics formulations which have variable, even anisotropic c. physicists dont usually explore them (e.g. tangherlini relativity) though because the math is considerably harder.

Physicists (and in particular the subset doing physical cosmology) don't usually explore parametrizations of c because they're not clearly physical (and sometimes even clearly unphysical), or alternatively don't help solve astrophysical or cosmological problems.

Relativists sometimes like to explore things that make using the Tangherlini transformations rather than the Lorentz transformations look positively benign. (To be clear, the Tangherlini synchronization system is clearly unphysical, requiring infinite speeds. His thesis also proposed using a distinguished global frame, which is not really philosophically different from how the standard cosmological frame is used, and seems OK because the distribution of stress-energy can pick out useful systems of coordinates in standard relativity. Unfortunately his method frustrates and probably outright breaks comparisons between inertial reference frames related by a boost, which the standard cosmology does not, and it's hard to see an alternative method that preserves his central ideas.)

But why even be stuck with 3+1d spacetime like Tangherlini? He was trying to do physics. But an unphysical metric signature with 47 plusses and no minuses is really cool!

In our observed universe, FAPP, c is the same everywhere after recombination, and we get that from spectral lines. You have to play really weird games to preserve the Lyman-alpha forest's apparent isotropy while introducing spacetime (or redshift-space, here) anisotropy. Things like BAOs make the problem even harder.

If we strip away all that pesky radiation and the information its structure encodes, analysing variations of c gets a lot easier. A relatively recent paper (Lewis & Barnes 2021) I enjoyed considered anisotropy in the one-way speed of light in an FLRW cosmology with zero energy density (well, really the convenient Milne model, which is also far from spatially flat). "So far, we have considered two cases, where either the speed of light is isotropic, or the extreme case where the anisotropic speed is 1/2 in one direction, and infinite in the other. The question remains whether this holds true in general case, for an arbitrary κ": https://www.cambridge.org/core/journals/publications-of-the-... (arxiv: <https://arxiv.org/abs/2012.12037>). "For more general cosmological models, where the presence of mass and energy results in curved space-time, the picture is more complicated as there is no simple mapping of the modified Lorentz transformations into the general relativistic picture. We leave this discussion for a future contribution."

Sadly there doesn't seem to be a future contribution yet, at least going by published citations (<https://scholar.google.com/scholar?cites=2012575105829699847...>). (Of those, I've put the Chamberlain paper on my to-read pile; you'd appreciate how it relates to Tangherlini, "credence is given to one-way infinite light-speed inward to each particle in direct comparison against Einstein’s isotropic (c=constant) light-speed").

Of course there's also the excellent Magueigo 2003 VSL overview <https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R...> copy <https://cds.cern.ch/record/618057/files/0305457.pdf> preprint <https://arxiv.org/abs/astro-ph/0305457>.

And even if you can make sense of an f(c) cosmology in the early visible universe, you will get to epochs before recombination and try to make sense of the later universe's chemistry, which of course relates to big bang light nucleosynthesis, baryogenesis and electroweak ssb. How do you abolish Lorentz symmetry in those epochs? Good luck!

(I mean, I think if you are doing physical cosmology you ought not to ignore gauge theory...)

my understanding is vsl physics can eliminate singularities if you have c approach zero in the presence of mass in a flat spacetime. AIUI this is equivalent to einsteinian curved spacetime except as you approach zero and deviate from linear in your c(m) formula.

you get pseudo black holes but depending on the extremeness of the deviation from linear, the difference to black holes might not be observable with current tech.

Not sure what you mean; you can't have any mass in a flat spacetime and obey the Einstein Field Equations for a Lorentzian spacetime (because T_{\mu\nu} doesn't vanish everywhere).

There are a variety of types of variable speed of light. If we foliate to 3+1 the usual picture is that c is constant on all spatial slices. Some VSL theories have the same c at all points on a given slice, but introduce a time variation of c. Other VSL theories introduce spatial variation as well (or instead). These families of theories all have significantly different equations of motion or actions from one another (cf. <https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_actio...>). There's no obvious reason why c couldn't relate in a more complicated way to the stress-energy tensor than the Einstein gravitational constant does, but there's also no obvious reason to think such an alternative theory should produce free-fall trajectories similar to those from GR.

In any case, I think you have to choose your function on c, obtain the field equations, decide which energy conditions and constraint equations you want to impose, set appropriate boundary conditions, choose a curve along which to foliate, and run with enough different initial-value surfaces (each of which must satisfy the constraints initially), that eventually an intuition develops. A Will-like parameterized post-Newtonian formalism approach would also be a good idea (<https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_f...>).

Unfortunately I'm unable to guess your choice of "c(m) formula".

I thought that's what the high dimension counts of string theories were: taking constants and turning them into dimensions.

Or is that too simplified?

Gravity, obvs.

> When you cross a black hole's event horizon, the radial coordinate becomes timelike, so you lose one degree of spatial freedom.

The second half is incorrect. Since the time coordinate becomes spacelike in turn you'll still have 3 spatial degrees of freedom. Dimensions can't just vanish if you believe that spacetime is a 4D Lorentzian manifold (as physicists do).

Moreover, the singularity is not a place you can poke with a stick, once you've entered the black hole. It lies in your future, in the same way as your death.

> The second half is incorrect. Since the time coordinate becomes spacelike in turn you'll still have 3 spatial degrees of freedom. Dimensions can't just vanish if you believe that spacetime is a 4D Lorentzian manifold (as physicists do).

Can we say that one of the spatial dimensions (the radial dimension) and the time dimension combine into a single dimension? After crossing the event horizon aren't they 1:1 correlated?

No, there's no change in dimensionality.

The swapping of timelike and radial dimensions are a "game" frequently played with families of coordinates, including Schwarzschild coordinates. One can apply any system of coordinates on a physical system without changing the behaviour of the physical system: coordinates are unphysical. Think of navigating around in a neighbourhood: you can talk about going forward a few blocks then turning left, after which you go forward two more blocks; or for the same journey, going "city north" a few blocks then going "city west" two blocks. Here assuming that (initially) "forward" is in the "city north" direction (and "city north" is not necessarily exactly magnetic north nor a section of a meridian of longitude). After the left turn, "forward" is "city west". There's an analogue to the discussion's (ab)use of Schwarzschild coordinates.

In Schwarzschild spacetime, without applying any system of coordinates, just floating in free-fall far from a black hole extremizes your travel in the timelike dimension. (You can do this at home: you stay put at some point on Earth (whether you use GPS latitude/longitude/altitude or some other system of coordinates) but your wristwatch keeps ticking). Inside the black hole horizon, just floating in free-fall extremizes your travel in the direction of the singularity. Far from the black hole, accelerating as strongly as you can in any direction takes travel from the timelike dimension and puts it into one or more spatial dimensions. In particular, you have the freedom to increase the spacetime interval between you and the singularity. Within the horizon, however strongly you accelerate the spacetime interval between you and the singularity shrinks. This behaviour seems to invite the use a different set of coordinates applied to a patch of space around an observer far from the black hole and a patch of space around an observer inside the horizon. It's some human cognition thing, and in the early 20th century it took decades to discover systems of coordinates that work for observers far from the black hole, at the horizon, and inside the horizon. And even today, most people don't seem to try to enhance physical intuitions by swapping among arbitrary systems of coordinates (including no coordinates) on a single physical system like a black hole and a pair of observers (one inside the horizon and one far outside the black hole).

The Schwarzschild black hole interior is still locally Lorentz-invariant everywhere (because the whole Schwarzschild spacetime is a Lorentzian manifold).

The various local interactions of the Standard Model will keep working inside a black hole. In a really tiny patch around every point, everything behaves as if its in Minkowski space (flat 4-d (3 spatial + 1 time) spacetime).

(That's one of the problems of quantum field theory on curved spacetime: the "focusing-pressure" [for experts: this is encoded in the Weyl curvature tensor; my "scare quotes" take a view of this in a Raychaudhuri equation way] gets so high that the unknown ultraviolet behaviour of the Standard Model (a quantum field theory) becomes relevant. The Weyl behaviour in Schwarzschild is that quasispherical objects are strongly prolated with the long axis aligned radially: a soccer ball or basketball starts looking like an American or Canadian or Rugby football ball. The radial stretching "spaghettifies" by ripping apart weaker bonds (like intermolecular ones, and molecular ones, and ionizing atoms), but the tangential squashing ("focusing") must eventually generate more nuclear interactions, probably up to quantum chromodynamics (QCD) energies possibly before the radial stretching starts generating hadronization.

How this works in the Standard Model is just unknown. However simpler "test" quantum field theories (fewer, or even no, interactions; and often no colour-confinement-like processes) raise really difficult questions.

Finally, back to the Standard Model as local theory: how does any allegedly quantum nonlocality behaviour work? Local here in the sence that states can be distinguished by local measurements alone. Related questions: can you entangle particles deep inside a black hole? If an entangled pair fall in together, how does the entanglement evolve? Or obsessing black hole information people, what if you throw in only one half of an entangled pair and locally measure the outside half? Nobody has great answers for these sorts of questions at present, and there's no near-term hope of testing any proposals in laboratories or via astrophysical observation.

I don't think the spacetime swap idea is particularly well explained though? Like although it's sort of mathematically true, my impression was that it's not like time suddenly becomes a dimension you're moving in once inside the event horizon, just that spacetime is acting so weird because there's now a deliberate direction where one did not exist before.

> I don't think the spacetime swap idea is particularly well explained though?

What exactly do you mean by "spacetime swap idea"? If you're saying the behavior of Schwarzschild coordinates at the event horizon is not well-understood, then I disagree. There is nothing particularly weird or surprising going on, there's just a trapped surface[0].

[0]: https://en.m.wikipedia.org/wiki/Trapped_surface

The problem is probably connecting an image of a tilted causal cone at p near but outside a regularized r_s in Schwarzschild spacetime with the idea that from p there is a limited number of null geodesics escaping to the asymptotically flat region, that the fraction decreases with decreasing regularized r coordinate, and that from p' at the trapping surface there are no such null geodesics, as all non-spacelike curves (accelerated or not) at any point on or interior to the trapping surface terminate at the singularity. The idea that the singularity inevitably lies in the future of any observer at p' is behind the "spacetime swap" notion.

Some of the problem is that Schwarzschild coordinates have surprises buried in them, and what \Delta r and \Delta t mean are not what most people tend to think.

Someone should do an ELI12 of Unruh's (ca. 2014) excellent (give or take varying the spelling of Martin Kruskal's surname) Schwarzschild BH global coordinates pedagogic review <http://theory.physics.ubc.ca/530-21/bh-coords2.pdf> and add in a bit on Fermi normal coordinates as a maybe-obvious not-a-chart follow-on to the commenary just above eqn (55). But on "maybe-obvious", Unruh has the choice line: "Since in a large number of cases, the single horizon coordinates were discovered long before Schild’s coordinates, this is an exercise in alternate reality – what could have so easily happened if only the generators of those coordinate systems had recognized what they had."

Hi. Sorry to bother you here. Could you point out what error am I making in my, I believe, very simple objection?

https://news.ycombinator.com/threads?id=scotty79&next=441240...

tl;dr The farrrr-from-the-horizon part of Schwarzschild spacetime is just not like our spacetime. Only near and outside the horizon (or better, in the absence of a horizon) does Schwarzschild become a decent physical approximation for anything in our universe.

Schwarzschild infinity is unphysical, while your notion of t(Earth) is physical because we can associate a worldline with the planet's centre of mass (COM), hold the COM at the spatial origin of a system of spacetime coordinates, and use whatever "timestamps" we like on the time axis. But we could decide that t(Earth)=infinity could be yesterday, or tomorrow, or a billion years ago, or a couple billion years from now; if we count of seconds before or after t(Earth)=infinity, we still have t'(Earth)=infinity, so it's not a very good choice of coordinate.

I think you have a misunderstanding that is probably beyond my ability to help you with, since we can't do interactive blackboard work in HN comments. The root of your problem seems to be mis-identifying the local time at Earth with the Schwarzschild time at infinity in the Schwarzschild solution. We aren't at infinity to any known black hole: between us and the most distant black holes we know of is expanding spacetime not found in Schwarzschild's solution; betwee us and the nearest black holes is substantially and lumpily curved spacetime and plenty of matter unlike Schwarzschild's unique pointlike mass surrounded by non-lumpy matterless vacuum; none of the astrophysical black holes are infinitely old today (whereas Schwarzchild black holes are infinitely old at every time, otherwise the spacetime would not be static); and in general exact solutions of the Einstein Field Equations -- even ones that are not eternal -- do not superpose cleanly with solutions for other black holes (and crucially there are no black hole mergers in Schwarzschild), ordinary stars, galaxies, clusters, and expanding spacetime. As an example: hover just above the apparent horizon of Sagittarius A*. Look at a stellar black hole in our galaxy. What do you make of infallers plunging towards the smaller black hole? What do you make of the evolution of mass of the stellar black hole, from your vantage point hugging an SMBH's horizon?

Short of taking a series of courses or finding an informal short-term tutor to walk you through particular things (you can find either at your local tertiary education school, like a community college or university), there are plenty of good textbooks on General Relativity. You seem to have found Wald's, which is probably the most rigorously and densely mathematical of several popular teaching choices, and it does not seem to have helped you. I'd guess you'd be better off with e.g. Carroll's Spacetime and Geometry or Wheeler's Gravity and Spacetime.

There is also the Israel-Darmois thin shell method, which is technically annoying but lets us cut the central part of an e.g. Schwarzschild solution and paste it into a cosmology populated with other such pasted-in subregions. We can then trace light rays from e.g. a quasar, across early expanding space to a SMBH or elliptical galaxy acting as a gravitational lens, and then across later expanding space to an approximation of our neighbourhood, adapting the rays at each shell boundary. Although there is very definitely a subregion of black hole solution in that kind of approach, the asymptotically flat part of Schwarzschild is cut away along with its distant infinities. One can compare this cutting and pasting to the Hill sphere of influence of Jupiter and those of its satellites, for example, if one were interested in a navigational plan like Juno's or JUICE's.

> Only near and outside the horizon (or better, in the absence of a horizon) does Schwarzschild become a decent physical approximation for anything in our universe.

I guess that's my point. Noting at or inside event horizon (of any kind, not just Schwarzschild solution) is physical. It's pure math, no matter how fun, has nothing to do with reality.

> But we could decide that t(Earth)=infinity could be yesterday, or tomorrow, or a billion years ago, or a couple billion years from now;

No, we cannot. Because as you stated t(Earth), by which I meant time as it passes on Earth, is physical... Now I think I should write t_Earth instead so it looks more like subscript not function application. So t_Earth = Infinity is the moment after all of the time already passed. After every finite moment already occurred. So t_Earth = Infinity doesn't really exist ever. It's purely mathematical concept. Abstract limit of the real thing.

> I think you have a misunderstanding that is probably beyond my ability to help you with, since we can't do interactive blackboard work in HN comments.

That's probably true. Even a blackboard wouldn't help because you seem to be interested mostly in minutia and specifics while my problem lies with general reality (pun intended) of all of this and which part is real and which is just extrapolation to times that don't exist and why physicists usually don't seem to care to differentiate between one part and the other.

> We aren't at infinity to any known black hole

I'm saying exactly opposite. The black hole (its event horizon to be precise) is at infinity (infinite time) to us. Any infalling object at Kruskal diagram crosses the line clearly labeled as t=infinity which in reality can't happen because there's simply isn't infinite amount of time in the universe.

> none of the astrophysical black holes are infinitely old today

To be honest that's another claim I just don't take on faith, especially in the light of the discovery of early developed galaxies and the fact that galaxies developing this early this fast would emit so much light that it would contribute to CMB (even up to 100% of it) which throws off all of our precise math theories of how everything started. I'm more inclined to believe that every astrophysical black hole existed at the the time when CMB was emitted (and before) and had exactly same size (of the event horizon) as it has today.

> and crucially there are no black hole mergers in Schwarzschild

Do you say that because they are mathematically impossible (which I would agree with) or just because Schwarzschild modeled just one black hole so there's nothing to merge with?

> hover just above the apparent horizon of Sagittarius A*. Look at a stellar black hole in our galaxy. What do you make of infallers plunging towards the smaller black hole?

Let's assume your trajectories are parallel to skip issues of special relativity. If he's closer to his event horizon than you are he's slowed down in time for you as he is for the rest of the universe outside (just slightly less). If he's farther away he lives at a pace accelerated relative to you, in the same manner that the outside world is accelerated for you. The specific math of how both of you tend to infinite time dilationas you approach your respective event horizons should show if relative time dilation between you tends to some ratio (or 1) or infinity. I don't know which is the case.

> There is also the Israel-Darmois thin shell method, which is technically annoying but lets us cut the central part of an e.g. Schwarzschild solution and paste it into a cosmology populated with other such pasted-in subregions.

I'm not that interested in pasting them statically far away. I'd really love to see what shape two Schwarzschild blackholes (and by that I mean their event horizons because, I don't believe anything beyond them is real to us) hitting each other could look like. Maybe (|)

Thanks for recommendations for further reading.

Sorry, I don't want to get into metaphysics.

Black hole mergers are studied using post-Newtonian methods and numerical methods because there is no general analytical approach known. SXS, Simulating eXtreme Spacetimes, and the black hole perturbation toolkit both have web presences, you could start there. There is also an academic literature on matching the waveforms in both regimes. These are checked against results from multimessenger astronomy.

> I'd really love to see what shape two Schwarzschild blackholes (and by that I mean their event horizons because, I don't believe anything beyond them is real to us) hitting each other could look like

This is well into the numerical relativity regime.

ETA: I'd pick <https://www.youtube.com/watch?v=jkpfXByQHxA> (SXS collab, "Event horizon for equal mass inspiral BBH in two coordinate systems") and the zoom-in at <https://www.youtube.com/watch?v=p4MTsCDtHMM> from a quickie cruise through some visualizations. There are links in the video description. Do beware that there are several types of horizon involved here, and they will not match your intuitions from Schwarzschild (see the point made in the zoom-in video description) which I would wager are built on the presence of a static Killing field which becomes null at the horizon, but the entire Killing field doesn't exist in these BH merger spacetimes. Roughly, though, if anything is in an orange region, it stays in an orange region. That includes a lot of gravitational radiation moving inwards in the purple region. [ETA again: the related Phys. Rev. D paper <https://arxiv.org/abs/1606.00436> has some nice details about the "duck bill" topology, too, and offers further detail on the purple region.]

https://www.youtube.com/@mpi_grav has several videos particularly in their NR playlist <https://www.youtube.com/watch?v=acHmN2MlJQQ&list=PLSYkic-Csf...>. Look for distortions in the BH horizons (whether it's an apparent horizon or some comparable surface gets into metaphysics; apparent horizons are at least locally observable), particularly the so-called "duck bill". Bear in mind the these are data visualizations principally of the waveforms, and the choices in intensities and hues are probably not going to be aligned with your intuition.

SXS has several videos too https://www.youtube.com/@SXSCollaboration - last month there was a major catalogue reorganization by the SXS collab so it may be that some internal links and semi-recent videos have issues.

And see for example https://www.black-holes.org/2024/10/02/BBH-mergers-with-spec...

Generally such simulations allow one to trace lightlike geodesics as a local probe of the lightlike horizon surfaces.

> I don't know which is the case

Exactly. That honest self-admission must be made near the start of any research programme.

> Any infalling object at Kruskal diagram crosses the line clearly labeled as t=infinity

Do check out Lemaître and Gullstrand-Painlevé coordinates for the Schwarzschild black hole.

I’ve always like to explore the idea of our universe being in a static 5th dimension where the 5th dimension represents randomness/entropy. The same way to think about exploring a 2d plane in a 3 dimensional space where the 3rd dimension is constant. We just happen to be in a random big bang in this 5th dimensional space

You can't have the curl operator in 4D.

What's in a 1D black hole?

/dev/null I presume

I'm guessing it'd look something like this on a 1-dimensional number line:

    --- >   | > >> . << < |   < ---

The dot in the middle would be the singularity, the pipes the event horizon, and the contents would be increasingly warped spacetime that may or may not exist, depending on your interpretation of things.

What? Wouldn't that mean an object's speed in some direction determines how time passes once it crossed over, and conversely, it would experience its old time dimension as spatial and be able to "move through (old) time" freely after crossing the event horizon?? My head hurts.

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