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Old 12th October 2021, 10:44 AM   #1
HansMustermann
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Black Hole Universe

So I'm thinking... (Which is already a red flag.)

Let's say a truly massive star collapses into a black hole. Way I understand it:

- time and space axis reverse roles once you're past the event horizon. Space starts to act like time, and time starts to act like space.

- Every bit of matter that ever falls inside will start at the event horizon, which is to say at t=0 at the coordinate that acts like "time" (formerly known as space)

- A LOT of matter will enter at or around x=0 on the coordinate that acts like "space" now (formerly time) as the rest of the star quickly collapses into it

- the "space" and energy input of this matter will very quickly expand originally, as more and more matter falls in, as the rest of the star quickly collapses

- everyone hits the singularity at the same "time" (formerly known as space) but infinitely spread apart on the "space" coordinate (formerly known as time)

So... doesn't this sound EXACTLY like our universe if the Big Rip theory is correct? Like, from Big Bang to Big Rip, it all seems to match our falling into the biggest black hole ever.

Or what did I get wrong?
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Old 12th October 2021, 06:42 PM   #2
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The only problem with your thoughts is that you were not the first to think of the idea.

https://en.wikipedia.org/wiki/Black_hole_cosmology
https://www.forbes.com/sites/startsw...-our-universe/
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Old 12th October 2021, 06:57 PM   #3
W.D.Clinger
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Originally Posted by HansMustermann View Post
Let's say a truly massive star collapses into a black hole. Way I understand it:

- time and space axis reverse roles once you're past the event horizon. Space starts to act like time, and time starts to act like space.
No. This is a misconception that arises when people understand the Schwarzschild metric only at the level of symbol pushing, without actually understanding the mathematics.

According to the mathematics*, the event horizon cannot belong to the Schwarzschild chart, so the chart that is spatially outside the event horizon is disconnected from the chart that is inside the event horizon. Using the Schwarzschild t coordinate to mean space inside the event horizon and the r coordinate to mean time is perverse and can lead to the other misunderstandings you listed, but the mathematics allows you to use any letter you like for any coordinate so you are free to confuse yourself as much as you want.

*It should perhaps be noted here that Misner, Thorne, and Wheeler, who wrote one of the most widely read textbooks on general relativity, explicitly declined to give a mathematical definition of the concept (spacetime manifolds) that lies at the heart of general relativity. As a result, it is impossible to learn the mathematics of general relativity from reading that textbook alone; if you want to understand the mathematics, you have to learn it somewhere else.
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Old 12th October 2021, 07:44 PM   #4
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Originally Posted by HansMustermann View Post
So I'm thinking... (Which is already a red flag.)

Let's say a truly massive star collapses into a black hole. Way I understand it:

- time and space axis reverse roles once you're past the event horizon. Space starts to act like time, and time starts to act like space.
In Schwarzschild coordinates, the radial space and time coordinates switch roles. But the event horizon is a coordinate singularity in Schwarzschild coordinates. There are other coordinates where it isn’t a singularity, and no such switch takes place. Coordinates are artificial, so placing physical significance on that switch can easily mislead you. Check out Kruskal coordinates for a cool look at black holes with coordinates that don’t flip but are continuous everywhere, including across the event horizon.

Quote:
- Every bit of matter that ever falls inside will start at the event horizon, which is to say at t=0 at the coordinate that acts like "time" (formerly known as space)
No, they all END at r=0, with r (in Schwarzschild coordinates) being time-like inside the event horizon. But positive motion through time corresponds to decreasing r.

Quote:
So... doesn't this sound EXACTLY like our universe if the Big Rip theory is correct? Like, from Big Bang to Big Rip, it all seems to match our falling into the biggest black hole ever.

Or what did I get wrong?
Yeah, you got it wrong. Perhaps the most obvious difference is isotropy. The big rip should pull everything apart in all directions. But as you approach the singularity, you get spaghettified. You get stretched in only one direction (pointing towards the singularity) but squeezed in the other two. It’s a very non-isotropic process.
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Last edited by Ziggurat; 12th October 2021 at 07:47 PM.
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Old 12th October 2021, 07:58 PM   #5
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Ok, W.D., here’s one for you. I had a thought now which may or may not make sense, let me know what you think.

We’ve got a black hole that collapsed from a star, and we are looking at it in Kruskal coordinates. The outer radius of the collapsing star forms a world line that terminates at the singularity. But now I’m looking at the line of the singularity itself, and it looks to me like it’s a superluminal world line. In Kruskal coordinates, the singularity seems to be moving from the point of formation, and moving faster than light. Now, I don’t think this violates causality, because I think the singularity doesn’t carry information. The singularity’s only property is mass, and even though that’s increasing as more stuff hits it, the information about how much mass will hit it is already out there, traveling along world lines parallel to (but inside) the event horizon at c, so that the event horizon’s arrival never tells you anything new.
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Old 12th October 2021, 11:34 PM   #6
HansMustermann
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Originally Posted by W.D.Clinger View Post
According to the mathematics*, the event horizon cannot belong to the Schwarzschild chart, so the chart that is spatially outside the event horizon is disconnected from the chart that is inside the event horizon.
That much I did get. That's why my thought process dealt only with what's on the inside of it.

Originally Posted by W.D.Clinger View Post
Using the Schwarzschild t coordinate to mean space inside the event horizon and the r coordinate to mean time is perverse and can lead to the other misunderstandings you listed, but the mathematics allows you to use any letter you like for any coordinate so you are free to confuse yourself as much as you want.
Well, being free to use any letter is kinda the whole point I was thinking about. If one coordinate acts like time, maybe it IS time for all you know once you're inside.

Well, I probably got it wrong, but it was worth asking someone who understands it better.
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Old 12th October 2021, 11:41 PM   #7
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Originally Posted by Ziggurat View Post
No, they all END at r=0, with r (in Schwarzschild coordinates) being time-like inside the event horizon. But positive motion through time corresponds to decreasing r.
Well, obviously, but like any coordinate you can put the zero and the arrow in any direction you want. In this case my idea was that I put the zero at just inside the event horizon. Like, one Planck unit inside. And the arrow in the direction where that time-like flow is positive.

As for Kruskal coordinates, I'll have to look more into them, but my thought was more along the lines of what would a guy inside that black hole perceive, if the coordinates actually switch roles. As in, if he actually uses a yard stick and a stopwatch, without taking out a calculator and doing some maths to convert to other artificial coordinates.

It's probably wrong, but I thought it worth asking anyway.
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Old 13th October 2021, 03:00 AM   #8
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Originally Posted by HansMustermann View Post
Well, obviously, but like any coordinate you can put the zero and the arrow in any direction you want. In this case my idea was that I put the zero at just inside the event horizon. Like, one Planck unit inside. And the arrow in the direction where that time-like flow is positive.

As for Kruskal coordinates, I'll have to look more into them, but my thought was more along the lines of what would a guy inside that black hole perceive, if the coordinates actually switch roles. As in, if he actually uses a yard stick and a stopwatch, without taking out a calculator and doing some maths to convert to other artificial coordinates.

It's probably wrong, but I thought it worth asking anyway.
If the black hole is large so that tidal forces are small, then he will not notice anything change as he passes through the event horizon.
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Old 13th October 2021, 03:22 AM   #9
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Well, I guess I'm thinking too much about things beyond my understanding. But you can't blame a guy for thinking about *ahem* plunging into holes
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Old 13th October 2021, 04:54 AM   #10
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Originally Posted by Ziggurat View Post
Ok, W.D., here’s one for you. I had a thought now which may or may not make sense, let me know what you think.

We’ve got a black hole that collapsed from a star, and we are looking at it in Kruskal coordinates. The outer radius of the collapsing star forms a world line that terminates at the singularity. But now I’m looking at the line of the singularity itself, and it looks to me like it’s a superluminal world line. In Kruskal coordinates, the singularity seems to be moving from the point of formation, and moving faster than light. Now, I don’t think this violates causality, because I think the singularity doesn’t carry information. The singularity’s only property is mass, and even though that’s increasing as more stuff hits it, the information about how much mass will hit it is already out there, traveling along world lines parallel to (but inside) the event horizon at c, so that the event horizon’s arrival never tells you anything new.
The singularity isn't part of the spacetime manifold. It's basically a hole or rip in the manifold, with the property that it is mathematically impossible to extend the manifold to repair that rip. Thus the singularity doesn't have a world line, and so the sentence I highlighted doesn't make sense mathematically.

The singularities are mathematical signposts that tell us Einstein's general theory of relativity doesn't make mathematical sense at the extremes, which tells us Einstein's GR is at least slightly defective. We'd like to replace GR with a theory that doesn't fail at the singularities, but to my knowledge we have no such replacement for GR at this time. Thinking as though the singularity has a world line might assist a thought process that might some day lead to a theory that replaces GR, and it might be suggestive to someone's intuition or creative fiction even in the present day, but it doesn't make sense within the mathematical confines of GR as we understand it today.

ETA: But I think your final sentence, that "the event horizon’s arrival never tells you anything new", could be rephrased in a way that makes sense mathematically and would probably be correct.

Last edited by W.D.Clinger; 13th October 2021 at 04:58 AM. Reason: added ETA
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Old 13th October 2021, 05:09 AM   #11
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Originally Posted by HansMustermann View Post
Well, being free to use any letter is kinda the whole point I was thinking about. If one coordinate acts like time, maybe it IS time for all you know once you're inside.
Yes, if a coordinate acts like time, then it is perfectly acceptable to think of it as time. Pretty much the whole point of GR is that anyone is allowed to use any chart of the spacetime manifold's atlas (and there's a mathematically well-defined complete atlas that contains all charts that are compatible with the manifold).

A spacetime manifold is a pseudo-Riemannian differentiable manifold of dimension 4 with Lorentz signature. The "Lorentz signature" part implies that, for every chart, exactly one of the chart's four coordinates can and must be regarded as time. That coordinate is uniquely determined; no one has any choice in the matter of which of the chart's four coordinates acts as time.

ETA: Which, by the way, implies an important consequence for those who choose not to require charts to have a domain that is connected in the sense of topology. The mathematics allows people to make that confusing and therefore counterproductive choice of definition, but the chart's coordinates must still have Lorentz signature. That rules out the symbol-pushing idea that the t and r coordinates of a Schwarzschild chart outside the event horizon can be identified with the t and r coordinates (respectively) inside the event horizon. Even if you choose to think of the Schwarzschild chart as being defined both inside and outside the event horizon, the mathematical definition of a spacetime manifold still prevents you from making the mistake people make when they say the t and r coordinates switch roles at the event horizon.

Last edited by W.D.Clinger; 13th October 2021 at 05:20 AM. Reason: added ETA
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Old 13th October 2021, 08:32 AM   #12
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Originally Posted by W.D.Clinger View Post
ETA: But I think your final sentence, that "the event horizon’s arrival never tells you anything new", could be rephrased in a way that makes sense mathematically and would probably be correct.
That was an error on my part. I actually meant the singularity's arrival doesn't tell you anything new. You already have knowledge of how much mass it has before it reaches you.
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Old 13th October 2021, 06:05 PM   #13
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Well, I'm still confused, so here goes...

Originally Posted by W.D.Clinger View Post
ETA: Which, by the way, implies an important consequence for those who choose not to require charts to have a domain that is connected in the sense of topology. The mathematics allows people to make that confusing and therefore counterproductive choice of definition, but the chart's coordinates must still have Lorentz signature. That rules out the symbol-pushing idea that the t and r coordinates of a Schwarzschild chart outside the event horizon can be identified with the t and r coordinates (respectively) inside the event horizon.
Actually, I was identifying outside t and r with inside r and t respectively

Originally Posted by W.D.Clinger View Post
Even if you choose to think of the Schwarzschild chart as being defined both inside and outside the event horizon, the mathematical definition of a spacetime manifold still prevents you from making the mistake people make when they say the t and r coordinates switch roles at the event horizon.
Well, here's my thinking process, though: outside of the black hole, the light cones all point along the time axis. As in, if you draw a constant r line, it will be inside the light cone in any point. Once you get inside, the light cone points at the singularity, which is to say, along the former r axis. The metric also changes sign, with the signs for t˛ and r˛ being reversed.

It seems to me like for all practical purposes, the former r does act like a time axis inside, and viceversa. Like, any calculations you could do outside with t and r, now you can do with them reversed. From the chart of a one-dimensional Alice inside, the physics are essentially no different from those of a similar one dimensional Bob outside, just Alice's time axis is pointing at the singularity.
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Old 13th October 2021, 07:20 PM   #14
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Originally Posted by HansMustermann View Post
It seems to me like for all practical purposes, the former r does act like a time axis inside, and viceversa.
Yes, it does.

The potential pit fall, though, is in assuming there's any continuity between the r inside and the r outside, and conversely the t inside and the t outside. They essentially have no relationship to each other anymore.
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Old 13th October 2021, 11:40 PM   #15
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Well, luckily my whole thought process started once everything is inside, so I THINK I'm in the clear for once. I'm sure I'll make that error some other time, though.
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Old 14th October 2021, 12:10 AM   #16
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Not sure if this question has arisen, but, will dark matter have same attributes within the black hole as baryonic matter?
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Old 14th October 2021, 12:20 AM   #17
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Well, here's another thought...

Originally Posted by Ziggurat View Post
Yeah, you got it wrong. Perhaps the most obvious difference is isotropy. The big rip should pull everything apart in all directions. But as you approach the singularity, you get spaghettified. You get stretched in only one direction (pointing towards the singularity) but squeezed in the other two. It’s a very non-isotropic process.
Does it, though?

I'm going to assume that an Alice falling inside, and quite still a long way away from the singularity, is still going to assume a constant c in her chart in all directions. After all, she's at rest in that chart.

So let's measure distances on the other two directions by bouncing a photon sideways to Bob who's also inside and falling happily by her side, and back. Essentially (we have to bounce a photon from Alice's {t1, x1, y1, z1} to Bob's {t2, x2, y2, z2} and back to Alice's new {t3, x3, y3, z3}.

That photon basically draws a triangle in 4d space. In Minkowsky space (or indeed in Alice's chart) that is a nice isosceles right triangle. 45 degrees up the diagonal and 45 degrees back to Alice's coordinates some time later.

However, now let's think in terms of a chart centered on the singularity. It seems to me like when you draw the light cones still in the same r (which now acts like time for Alice) and t (which now acts as space for Alice) centered on the singularity, they start very wide (almost 180 degrees just inside the event horizon) and eventually get very narrow. In fact almost zero degrees when you're just about to hit the singularity.

When Alice and Bob just entered the singularity, if they measure it with that photon, the distance between them is practically zero. That triangle (well, it will be curved arcs now, but let's pretend they're close enough to still kinda look like a triangle) is extremely sharp, and the photon comes back after very little r (which for Alice is time.)

When they get closer to the singularity, that triangle becomes very very flat. That photon will need longer and longer to actually come back to Alice. Ergo, Bob is actually getting farther away to her side, not coming closer.

Furthermore, past a point, the photon can never make it back to Alice. Bob has just moved outside her visible universe.

Short version: time (measured by Alice, in the former r coordinate, now t for her) being stretched, also makes any sideways distances you measure by bouncing a photon there and back be equally stretched.

Doesn't that sound kinda Big Rip like?
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Last edited by HansMustermann; 14th October 2021 at 12:53 AM.
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Old 14th October 2021, 12:29 AM   #18
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Incidentally it seems to me like the same would happen as you approach the event horizon from the outside, using the normal r and t outside this time. The light cones also get narrower and narrower as you approach the event horizon. So while a distant observer might see them getting closer as they approach the event horizon, from Alice's point of view, Bob is hitting it infinitely apart from her.

Seems to me like far from getting spagettified, you'd get ripped apart. The photons between the nuclei and electrons in your body would never reach from one to another.
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Old 14th October 2021, 01:42 AM   #19
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Originally Posted by HansMustermann View Post
Does it, though?
Yes, it does.

Quote:
That photon basically draws a triangle in 4d space. In Minkowsky space (or indeed in Alice's chart) that is a nice isosceles right triangle. 45 degrees up the diagonal and 45 degrees back to Alice's coordinates some time later.

However, now let's think in terms of a chart centered on the singularity. It seems to me like when you draw the light cones still in the same r (which now acts like time for Alice) and t (which now acts as space for Alice) centered on the singularity, they start very wide (almost 180 degrees just inside the event horizon) and eventually get very narrow. In fact almost zero degrees when you're just about to hit the singularity.
You seem to be referring to something like this:
https://i.stack.imgur.com/AiiZF.jpg

But there's a serious problem with that graph. By drawing the light cones as cones with circular ends, it implies that all the spatial coordinates are behaving similarly. And that's not the case. In particular, it doesn't actually give you an accurate depiction of what happens in theta and phi. It especially doesn't tell you what the trajectory of world lines separated by theta and phi look like.

Quote:
When Alice and Bob just entered the singularity, if they measure it with that photon, the distance between them is practically zero.
First, I presume you mean they just entered the event horizon, not the singularity. And no, the distance between them is not practically zero. Don't look at the cones to figure that out (they are deceptive), look at the metric:

https://physics.stackexchange.com/qu...-leq-r-leq-2gm

Near the event horizon (r=2M), the dt coefficient becomes vanishingly small. t is now a spatial component, which might tempt you to think this means distances are vanishingly small. But this is a mistake. If Alice and Bob entered the event horizon together, they aren't separated by t, they are separated by phi and theta. And unlike r and t, phi and theta actually map directly between the outside and inside charts. So the distance between them will still be determined by their separation in phi and theta. And those coefficients are not zero, but remain finite.

Quote:
When they get closer to the singularity, that triangle becomes very very flat. That photon will need longer and longer to actually come back to Alice. Ergo, Bob is actually getting farther away to her side, not coming closer.
As they approach r = 0, the coefficients for the t term approaches infinity, which means that yes, the distance becomes larger for objects separated in t. But remember, t is now our radius-like coordinate. Bob and Alice aren't separated by t, but by theta and phi. The front of Alice and the back of Alice are separated by t, and the front of Bob and the back of bob are separated by t. And the coefficients for theta and phi terms approach zero. The separation between them doesn't go to infinity, it goes to zero. So Bob and Alice get smooshed together, and they both get stretched front to back.

And that's spaghettification. It is very, very anisotropic.

Quote:
Doesn't that sound kinda Big Rip like?
Again, no. The rip is only in one direction (your space-like t coordinate), you get compressed in the other two (theta and phi).
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Old 14th October 2021, 05:40 AM   #20
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For reasoning about what happens in the vicinity of the event horizon and continuing inside, I strongly recommend use of Gullstrand-Painlevé coordinates.

In Gullstrand-Painlevé coordinates, the time coordinate is the proper time of an unaccelerated object free-falling toward the black hole singularity.

In Schwarzschild coordinates, the time coordinate is the proper time of an observer stationed at infinity, hence viewing anything in the vicinity of the event horizon from an infinite distance. Schwarzschild coordinates are also appropriate for observers stationed at a fixed finite distance from the event horizon, maintaining that fixed distance by an acceleration away from the black hole, which they would perceive as weight. At the event horizon, the acceleration needed to stay in place would require infinite power, which is the intuitive reason for the Schwarzschild coordinate singularity at the event horizon. Using Schwarzschild coordinates to reason about objects crossing the event horizon therefore involves reasoning about a transition from acceleration that is increasing without bound into an entirely different phase.

Mathematically, you can't use Schwarzschild coordinates to reason about something that crosses the event horizon without bringing in another chart that covers the event horizon, and then using the transition maps to translate between charts. If you're trying to use Schwarzschild coordinates inside as well as outside the event horizon, then you have to use two different transition maps, just as though you were using three different charts. It's a whole lot easier and far less confusing to use a single chart (such as Gullstrand-Painlevé) that covers the entire spacetime of interest.
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Old 14th October 2021, 11:04 AM   #21
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