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Old 4th December 2018, 07:15 AM   #1
HansMustermann
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Black Hole Episode 4: A new hope ;)

Ok, so last time I learned that inside a black hole time and space switch positions. Which makes sense in 2d, which is to say in t and r, far as my limited understading goes.

But it just occured to me to try that in THREE space dimensions, and, well, I kinda went cross eyed trying to figure how to swap 1 coordinate with 3 coordinates. Not even Indiana Jones had to pull THAT kinda sleight of hand

Or basically, if I stick to polar coordinates, so I can swap t and r, then what happens to the two angular coordinates? Is there some formula for those? I would expect so.

So, err, can anyone explain to me how that works, please?
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Old 4th December 2018, 07:21 AM   #2
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You learned that time and space switch positions? Nobody knows what's inside a black hole or even if there is an inside. One theory has it that black holes are 2d, just a surface on which information collects.
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Old 4th December 2018, 07:40 AM   #3
HansMustermann
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Well, that's what folks here told me, at any rate.
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Old 4th December 2018, 09:23 AM   #4
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Originally Posted by HansMustermann View Post
Well, that's what folks here told me, at any rate.
Ummmm, ...

I seriously doubt that you were actually told such a thing by anyone on this Forum.
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Old 4th December 2018, 10:56 AM   #5
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Originally Posted by baron View Post
You learned that time and space switch positions? Nobody knows what's inside a black hole or even if there is an inside. One theory has it that black holes are 2d, just a surface on which information collects.
Originally Posted by HansMustermann View Post
Well, that's what folks here told me, at any rate.
Originally Posted by Crossbow View Post
Ummmm, ...

I seriously doubt that you were actually told such a thing by anyone on this Forum.
Well I am pretty sure someone here with the relevant knowledge of the Swarzschild metric agrees with what Hans learned. From wikipedia:

Originally Posted by Wikipedia
The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although it has some rather bizarre properties. For r < rs the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike.
https://en.wikipedia.org/wiki/Schwarzschild_metric
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Old 4th December 2018, 11:49 AM   #6
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And a question that has been answered in this link: https://physics.stackexchange.com/qu...e-a-black-hole

However, if you are asking non-mathematically, no; space is space and time is time. The math and behavior of those descriptors seem to behave differently inside the EH.
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Old 4th December 2018, 12:28 PM   #7
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Originally Posted by HansMustermann View Post
Well, that's what folks here told me, at any rate.
Someone is playing loose with the idea of "direction", but what they're probably talking about is what happens with Schwarzchild coordinates.

When you're very far away from your black hole (or actually any spherically symmetric mass), Schwarzchild coordinates look like a polar version of flat space. You've got your radius r, your two angles theta and phi, and your time coordinate t, all nicely labeled in a way that makes intuitive sense. r, theta, and phi are all spatial coordinates, t is a time coordinate, and everyone is happy.

Once you're inside the event horizon, though, r is no longer a space-like coordinate, it's a time-like coordinate. Conversely, t is no longer a time-like coordinate, it's a space-like coordinate. One of the consequences of this is that we can see escape is impossible and hitting the event horizon is inevitable: you can only move one way through time, so if r is time-like, then it must decrease continuously for anything that has fallen inside.

But it's possible to take these coordinates a little too literally. Space and time don't actually switch places. What has happened is that the Schwarzchild metric has a coordinate singularity (not a physical one) at the event horizon. This means that the coordinates inside don't match up with the coordinates outside. It may look continuous because the same metric equation is used for both inside and outside, but r and t don't mean the same thing anymore, and they won't change smoothly for a world line crossing the event horizon either.

This is one of the reasons it's often better to use other coordinate systems to describe what's going on inside a black hole. For examply, there's Kruskal–Szekeres coordinates, which replace t and r with T and X, which are related to the former by some equations given at the link (theta and phi remain the same). These coordinates take some getting used to, they're not very intuitive, and they're not well suited for doing stuff far away from a black hole. But they have three very significant advantages here. First, the T coordinate is always time-like and the X coordinate is always space-like, so there's no swapping involved. Second, the event horizon is not a coordinate singularity in Kruskal-Szekeres coordinates. This means that the coordinates inside the black hole smoothly match the coordinates outside the event horizon, and trajectories of objects passing through the event horizon do not have discontinuities in T or X. Third, light cones are always drawn at 45 degree angles on an X-T graph, no matter where you are on the graph. This makes it easy to see what world lines are possible.

In Kruskal coordinates, time and space don't switch places. But something weird still happens: the singularity at the center of the black hole becomes a space-like surface (ie, the line is more horizontal than vertical). This means that once you're inside the event horizon, all trajectories within your light cone will intersect with the singularity. The singularity is not separated from you by distance, but by time. It is your inevitable future, with no possible escape.
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Old 4th December 2018, 12:52 PM   #8
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Thanks much 'Ziggurat' for that excellent explanation as it is a big help.

Also, I do hope that 'HansMustermann' heeded your answer as well.
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Old 4th December 2018, 03:10 PM   #9
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Originally Posted by HansMustermann View Post
Well, that's what folks here told me, at any rate.
As I recall, only one person told you that.

You were also informed (by a different person) that what you were being encouraged to think is a really bad way to think about it.

Furthermore (and I don't recall whether you were told this earlier), some authors require charts to be connected. The reason some definitions of differentiable manifolds require charts to be connected is that it eliminates the confusing pathology one person was encouraging you to embrace.
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Old 4th December 2018, 04:45 PM   #10
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Ok, maybe I've played fast and loose with the terminology. I hope I can be at least partially excused on account of, well, not knowing what I'm talking about

But, ok, let's say we had t and r outside, and we have w and z inside. And they're just space-like and time-like.

I'll stick to Schwarzschild coordinates, because, well, unless I'm mistaken, that's what the guy falling into the black hole can measure with his yardstick and chronometer. Which is really what's confuzzling me, rather than how you calculate stuff.

Anyway, as long as it's two dimensions, I can dig that. My question is basically what to do with four dimensions. Are there now three time-like coordinates and one space-like one?
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Old 4th December 2018, 06:12 PM   #11
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Originally Posted by HansMustermann View Post
Ok, maybe I've played fast and loose with the terminology. I hope I can be at least partially excused on account of, well, not knowing what I'm talking about

But, ok, let's say we had t and r outside, and we have w and z inside. And they're just space-like and time-like.

I'll stick to Schwarzschild coordinates, because, well, unless I'm mistaken, that's what the guy falling into the black hole can measure with his yardstick and chronometer. Which is really what's confuzzling me, rather than how you calculate stuff.

Anyway, as long as it's two dimensions, I can dig that. My question is basically what to do with four dimensions. Are there now three time-like coordinates and one space-like one?
There are always 3 space and 1 time coordinate. The 2 angular coordinates are spatial on both sides of the event horizon. And the falling astronaut's ruler and stop watch never measure Schwarzchild coordinates except far from the black hole.
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Old 4th December 2018, 08:09 PM   #12
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Ahh, thanks. Yeah, the two angular coordinates were confuzzling me. Now I think it's clear.
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Old 4th December 2018, 11:36 PM   #13
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Originally Posted by baron View Post
You learned that time and space switch positions? Nobody knows what's inside a black hole or even if there is an inside. One theory has it that black holes are 2d, just a surface on which information collects.
https://en.wikipedia.org/wiki/Holographic_principle

Even if the holographic principle applies, there is still an interior to the black hole, it's the "projection" of the information on the surface.

Under the holographic principle the information content of any space is contained on it's boundary. In that case the interior of a black hole is just as real as the exterior.

Although if you accept the idea of a black hole "firewall" then it becomes a little less clear what the interior represents.
https://en.wikipedia.org/wiki/Firewall_(physics)
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Old 5th December 2018, 02:56 AM   #14
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Originally Posted by Roboramma View Post
https://en.wikipedia.org/wiki/Holographic_principle

Even if the holographic principle applies, there is still an interior to the black hole, it's the "projection" of the information on the surface.

Under the holographic principle the information content of any space is contained on it's boundary. In that case the interior of a black hole is just as real as the exterior.
But there is no interior. That's the point, it's like a hologram where all the information is contained on the 2d surface and the 3d projection is an illusion. In a holographic universe there is no third dimension, all that exists is the surface.
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Old 5th December 2018, 06:51 AM   #15
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PBS Spacetime: How Time becomes Space like inside a Black Hole.
https://youtu.be/KePNhUJ2reI
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Old 5th December 2018, 07:44 AM   #16
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TBH, and this is probably Dunning-Kruger speaking, but I never understood why it has to be either-or. Why not both?

Way I see it, from the point of view of an observer that's safely outside the black hole, there's nothing in it. Not only everything that ever fell into it never crossed the event horizon, but even anything that was already inside that radius when the black hole formed appears to be on its outside now. Basically everything is smeared on the surface of the black hole, and none of it is inside.

That's your firewall right there, right?

From the point of view of someone that fell inside, though, it does have an interior and they crossed the event horizon without noticing anything particularly different at that point.

I don't really understand why some people have to decide why it's one or the other. Seems to me like it's both, just in different frames or charts, innit?
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Old 5th December 2018, 07:54 AM   #17
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Originally Posted by baron View Post
But there is no interior. That's the point, it's like a hologram where all the information is contained on the 2d surface and the 3d projection is an illusion. In a holographic universe there is no third dimension, all that exists is the surface.
I think most proponents of the holographic universe, and by that I mean the physicists, would be really surprised to hear that there is no interior. Especially Susskind, whose whole position was to oppose the idea of a missing chunk of space behind an event horizon all along.

But anyway, from what I understand it, it's more complex and, well, HOLOGRAPHIC than just everything being on the surface of the universe or of a black hole. In fact, it applies to any piece of space you can imagine. Your room for example. Or your computer's case. There's a fundamental limit to how many bits of information you can fit in that space, and that limit is its outer surface in plack lengths squared.

I'm not sure it means that the interior space doesn't exist, and I'm getting the impression that Susskind didn't think so either, at the very least. It is more of an issue of the stuff (or more accurately, information) IN that space, than meaning that space itself doesn't exist in your room.
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Old 5th December 2018, 08:42 AM   #18
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Originally Posted by HansMustermann View Post
I think most proponents of the holographic universe, and by that I mean the physicists, would be really surprised to hear that there is no interior. Especially Susskind, whose whole position was to oppose the idea of a missing chunk of space behind an event horizon all along.

But anyway, from what I understand it, it's more complex and, well, HOLOGRAPHIC than just everything being on the surface of the universe or of a black hole. In fact, it applies to any piece of space you can imagine. Your room for example. Or your computer's case. There's a fundamental limit to how many bits of information you can fit in that space, and that limit is its outer surface in plack lengths squared.

I'm not sure it means that the interior space doesn't exist, and I'm getting the impression that Susskind didn't think so either, at the very least. It is more of an issue of the stuff (or more accurately, information) IN that space, than meaning that space itself doesn't exist in your room.
I don't think so, but I'm willing to be corrected. The ubiquitous location of a hologram on its surface (i.e. you can break a piece off and the hologram still exist, albeit diminished in clarity) is only an attribute of the hologram. The hologram itself - the information - only exists in 2D.
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Old 5th December 2018, 10:13 AM   #19
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Originally Posted by baron View Post
I don't think so, but I'm willing to be corrected. The ubiquitous location of a hologram on its surface (i.e. you can break a piece off and the hologram still exist, albeit diminished in clarity) is only an attribute of the hologram. The hologram itself - the information - only exists in 2D.

I don't think the holographic principle states no interior to a BH, but that physics happening in a 3D universe is identical/analogous to physics happening on a 2D surface. It is thought it might apply to our universe because the entropy of a BH scales with the surface area of it's event horison and not it's volume.
If true it would mean not only BHs are 2D but the whole damn thing, us included.
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Old 5th December 2018, 01:38 PM   #20
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As I was saying, though, that applies to ANY space, although it's less interesting as an upper limit for anything else than a sphere. For the same volume, say, a cube has more outer surface than a sphere, so being limited to the surface area is less of a limitation.

But basically it also puts an upper limit on how many bits of information one can store in a room, or for that matter a silicon chip, to its outer surface. Which incidentally puts the kibosh to the "simulation inside simulation inside simulation" argument some people have been proposing.

But that's basically IMHO also the argument against the whole 'inside space is missing' idea. There's nothing more special about the outer surface of the whole universe than to the outer surface of my room. The fact that we can define such 3d portions of space anywhere and expect the same consistent behaviour, is inherently requiring a 3d space to draw that curve in.
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Old 5th December 2018, 01:45 PM   #21
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Originally Posted by HansMustermann View Post
There's nothing more special about the outer surface of the whole universe than to the outer surface of my room.
If I understand it correctly then your room has no outer space. The 2d surface is the cosmological horizon. Your room is encoded onto this surface. There is no spoon room.
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Old 5th December 2018, 01:54 PM   #22
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Yet I can define another surface INSIDE my room, say the surface of my computer, which has the same properties. Or imagine me holding a baketball. Inside my room. The basketball has the same property: the upper limit for the number of bits that can exist inside it, is limited by the outer surface of the ball, in plank lengths squared.

Hell, I can define a center, say, where my index finger is now, and imagine a thousand concentric spheres centered on that point, in 1mm radius increments. Spheres inside spheres inside spheres. And each of them has that property.

In what do I define that inner 3d surface, if not 3d space?

As I was saying, the issue isn't actually lacking 3d space, it's all the bits of information inside (from electrons, photons, etc) that are limited by the surface.

Edit: and to make it clear, there isn't just ONE surface that would be THE hologram. All those concentric spheres, my room, the outer shell of the whole house, the limits of the atmosphere, etc, ALL have the same property. It's not just ONE surface that has that property, and which we could say it's THE hologram for everything. Every region of space you could define, its outer surface is just as good a hologram as the any other outer surface. Basically don't let yourself be fooled by the terminology. It's really NOTHING like a hologram you could make with lasers and film.
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Old 5th December 2018, 02:01 PM   #23
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Originally Posted by HansMustermann View Post
Yet I can define another surface INSIDE my room, say the surface of my computer, which has the same properties. Or imagine me holding a baketball. Inside my room. The basketball has the same property: the upper limit for the number of bits that can exist inside it, is limited by the outer surface of the ball, in plank lengths squared.

Hell, I can define a center, say, where my index finger is now, and imagine a thousand concentric spheres centered on that point, in 1mm radius increments. Spheres inside spheres inside spheres. And each of them has that property.

In what do I define that inner 3d surface, if not 3d space?

As I was saying, the issue isn't actually lacking 3d space, it's all the bits of information inside (from electrons, photons, etc) that are limited by the surface.
It's not a case of there being a 3d space with limited particles inside. Because the number of particles can be defined in relation to the 2d surface it is postulated that the 3d space does not exist and is an illusion arising from the 2d information. Hence 'hologram'. If you have a hologram of your bedroom (for some reason) it makes no sense to ask about the surface that bounds it because the bedroom doesn't exist in 3d space and is defined only by information on the flat surface.
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Old 5th December 2018, 02:05 PM   #24
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Yes, but see the edit above. I can define a surface inside that hologram of a room, and that itself acts as a hologram too.
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Old 5th December 2018, 02:47 PM   #25
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Originally Posted by HansMustermann View Post
Yes, but see the edit above. I can define a surface inside that hologram of a room, and that itself acts as a hologram too.
You can't, because a hologram has no 'inside'. The hologram is a flat surface, it exists nowhere else.
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Old 5th December 2018, 02:59 PM   #26
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Yes you can, and exactly that's the difference between the holographic universe and a hologram you do with lasers and a piece of film. You seem to be thinking in terms of the latter. In the holographic universe, yes you can define a surface inside another surface, and it will be just as much a hologram.

Basically don't read too much into what label someone slapped on it. Last time I made that mistake, well, let's just say the sign on the door said "women", but there was just a toilet inside
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Old 5th December 2018, 03:04 PM   #27
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Originally Posted by HansMustermann View Post
Yes you can, and exactly that's the difference between the holographic universe and a hologram you do with lasers and a piece of film. You seem to be thinking in terms of the latter. In the holographic universe, yes you can define a surface inside another surface, and it will be just as much a hologram.

Basically don't read too much into what label someone slapped on it. Last time I made that mistake, well, let's just say the sign on the door said "women", but there was just a toilet inside
At least there was an inside. I had to piss on the boundary wall.
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