Simultaneity of events inside a singularity

[Alyx.Vox]

Hello,

I was wondering if the following idea had been proposed in one form or another within
a scientific community. Provided that the space within the singularity is dimensionless,
that being 0 in length on all the axes (is it?) the following is in order.
From this follows that the Planck's Length is zero, therefore the Planck's Time is zero as well.
All that means that if whatever happens, if it happens at all inside singularity must happen
at the exact same instance as all the other events happening inside the same singularity (and every other singularity as well).

Now, don't be too quick on throwing rotten tomatoes at me. My expertise lies within the realms of computer science.
I am not educated into physics formally if you don't count high-school and my general interests (I might get a degree later on, however unlikely that seems to be).
My thought rests on one assumption only: the lack of dimensions inside a singularity and I'm wondering if I'm wrong about that.
If that assumption is correct the train of thought should be correct as well.

Thank you.

 

[PeterDonis]

Provided that the space within the singularity is dimensionless,
that being 0 in length on all the axes (is it?)

Classically, yes. But not when quantum effects are taken into account. See below.

From this follows that the Planck's Length is zero, therefore the Planck's Time is zero as well.

No, because the Planck length and Planck time are quantum concepts. We do not have a theory of quantum gravity yet, but there is pretty general agreement that whatever that theory turns out to be, it will involve something different happening at singularities; instead of going down to zero size on all dimensions, some different physics will come into play at length and time scales on the order of the Planck length and Planck time. So what we now call a "singularity" would not have zero size when quantum effects are taken into account.

 

[arindamsinha]

...whatever happens, if it happens at all inside singularity...

The main problem is talking about an 'inside' of a singularity, which as you mentioned is dimensionless. I assume you didn't mean event horizon.

If we can't talk about an inside of a singularity, any physics discussion there would be speculative, since we have no theory to guide us.

 

[bcrowell]

Provided that the space within the singularity is dimensionless,
that being 0 in length on all the axes (is it?) .

GR doesn't describe the singularity as part of spacetime at all. The answer might be different in a theory of quantum gravity, but we don't have a theory of quantum gravity.

 

[atyy]

As others have explained, the question doesn't make sense within the theory of general relativity. However, there are a couple of related questions that the OP may be interested in, eg. the classification of singularities as spacelike or timelike, and the BKL conjecture for spacelike singularities.

"1. The first part of the conjecture states that spatial points decouple as one goes to a spacelike singularity in the sense that the evolution can be described by a collection of systems of ordinary differential equations with respect to time, one such system at each spatial point. (“A spacelike singularity is local.”)

2. The second part of the conjecture states that the system of ordinary differential equations with respect to time describing the asymptotic dynamics at any given spatial point can be asymptotically replaced by the billiard equations. If the matter content is such that the billiard table has infinite volume, the asymptotic behavior at each point is given by a (generalized) Kasner solution (“Kasner-like spacelike singularities”). If, on the other hand, the matter content is such that the billiard table has finite volume, the asymptotic behavior at each point is a chaotic, infinite, oscillatory succession of Kasner epochs." http://relativity.livingreviews.org/Articles/lrr-2008-1/

 

[Bill_K]

Provided that the space within the singularity is dimensionless, that being 0 in length on all the axes (is it?)

No, it's not. We can't talk about the singularity itself, but we can describe the surrounding spacetime, and it's totally different from what you may imagine. When you hear "r = 0" you naturally think, "aha, point!" but it isn't. Inside the event horizon, the metric components change sign and r and t switch roles. r becomes a timelike coordinate and t becomes a spacelike coordinate. And so in a sense, the singularity r = 0 is indeed "all at the same time". But since t can be anything, it's not "all in the same place"!

To lessen the confusion, let me use different names and call r = T and t = Z. The Schwarzschild metric is then

ds² = (1 - 2m/T)dZ² - (1 - 2m/T)^-1 dT² - T²d²Ω

where d²Ω is solid angle. T²d²Ω is still the metric on a sphere, but the radius is now T. As T goes to zero the sphere shrinks to zero circumference and zero radius. Rearranging things slightly,

ds² = (2m/T - 1)^-1 dT² - (2m/T - 1)dZ² - T²d²Ω

And approximating T ≈ 0,

ds² = (T/2m) dT² - (2m/T) dZ² - T²d²Ω

What is happening? Inside the event horizon the spacetime is no longer time independent. Rather it has a translational symmetry. In fact the spatial part of the metric has cylindrical symmetry. It's a world in which the radius is collapsing, but the distance along the axis Z is stretching. Particles that came in at different times t will hit the singularity at different places Z.

 

[Alyx.Vox]

A lot of different explanations were given, but I think this one...

...

... is the most interesting one to me. Darn that human intuition! :D When I think about time-space I think about two different entities, when in fact they're different sides of the same coin. My question was about the time-part of space-time indeed, not space-part.

Now, the question is: if they all happen at the same time, but not at the same place... Then where do they happen? Multi-universes? Other dimensions?

I have very little to no knowledge of quantum mechanics, so that part of the discussion is over for me, at least for the time being, but please be free to try and elaborate it to me if you feel the need and I'll try to understand them on my part (hmm, I thought nobody understood quantum mechanics :P ).

Oh, yes, one more question concerning Planck's Length. In the context of the singularity "prior" to Big Bang, could it be that Planck's Length was initially set just as any other natural constant and that it "was" therefore undefined "before" the BB?

 

[Sayajin]

A lot of different explanations were given, but I think this one...

... is the most interesting one to me. Darn that human intuition! :D When I think about time-space I think about two different entities, when in fact they're different sides of the same coin. My question was about the time-part of space-time indeed, not space-part.

Now, the question is: if they all happen at the same time, but not at the same place... Then where do they happen? Multi-universes? Other dimensions?

I have very little to no knowledge of quantum mechanics, so that part of the discussion is over for me, at least for the time being, but please be free to try and elaborate it to me if you feel the need and I'll try to understand them on my part (hmm, I thought nobody understood quantum mechanics :P ).

Oh, yes, one more question concerning Planck's Length. In the context of the singularity "prior" to Big Bang, could it be that Planck's Length was initially set just as any other natural constant and that it "was" therefore undefined "before" the BB?

All this things are very speculative because we don't have any theories that can describe this. When you try to describe this infinite small and dense object all the theories break down(GR and QM). The fact that we are dealing with infinities is telling us that something is wrong.

The Singularity and the Big Bang Bang are the biggest unsolved problems in modern physics. The closest thing to the Big Bang that we can observe is about 300000 years after the big bang (the cosmic microwave background radiation). We only know that the universe was expanding from some point but we don't know what was the big bang what caused it, how it hapen, was there space and time, were the laws of physics valid and so on.

In the beginning physicist were thinking that the BB was some event from which everything including space time and the laws of physics appeared from nothing and it was the beginning of everything. However this was only an assumption and nowadays many theories as Inflation, String/M-Theory discribe the big bang as an event happening in much larger Universe and possibly only one of many such events. Nothing of this is proven or well known it can turn out to be anything.

BTW you can try to search in youtube for Leonard Susskind's Lectures(General/Special Relativity,Quantum Mechanics, Classical Mechanics, Cosmology, Particle Physics). They are very very good source of real physics unlike the most things that are on the internet today.
I am just like you Computer Scientist interested in physics so in my spare time I watch them sometimes. You can find them interesting too. The best thing is that they are made for people who know math and want to see the real physics but at the same time don't have too much time to spend.(They tell you stuff really fast)

 

[PeterDonis]

if they all happen at the same time, but not at the same place... Then where do they happen? Multi-universes? Other dimensions?

As bcrowell pointed out, the singularity itself (r = 0) is not part of spacetime, but events with r coordinates arbitrarily close to zero are; and for any r > 0 there are an infinite number of such events that all "happen at the same time". In fact, the coordinate "r" *is* that time. That's why some people prefer to rename the "r" coordinate to something else, like "T", inside the event horizon (as Bill_K did), since in that region it is timelike instead of spacelike.

As for "where" all those events happen, they happen in spacetime. There's no need for multi-universes or other dimensions. Inside the event horizon, "space" is infinite, so there's plenty of room for an infinite number of events that all happen "at the same time".

(A technical point: all of these statements are really dependent on adopting a particular set of coordinates to describe spacetime inside the event horizon. There are other coordinates that have different properties: for example, there are coordinates in which "r" is still spacelike inside the horizon, and "space" is not infinite inside the horizon, and events with the same "r" coordinate don't all happen "at the same time". So things are actually even weirder than we've been describing. )

 

[Alyx.Vox]

...

I knew that much, except for the lectures of Susskind. Thanks, I'll check them out.

...

I wasn't talking about the event horizon. I was talking about the singularity, i.e. the center of the black hole. I asked about a multiverse and higher dimensions, exactly because the gravity at that particular point is enormous (an understatement), enough to rip a hole in time-space. Well, as others have mentioned, everything apart from string-theory and quantum mechanics is pure speculation in this context.

 

[PeterDonis]

I wasn't talking about the event horizon. I was talking about the singularity, i.e. the center of the black hole.

Yes, I understand; I was pointing out that, first, the singularity itself isn't actually part of the spacetime (as bcrowell said), so we can only talk about getting closer and closer to the singularity (i.e., "r" values closer and closer to zero, but still greater than zero); and second, the "weirdness" about "r" being a time coordinate instead of a space coordinate, and there being an infinite number of "spatial points" at a given "time", applies everywhere inside the event horizon, not just very close to the singularity.

I asked about a multiverse and higher dimensions, exactly because the gravity at that particular point is enormous (an understatement), enough to rip a hole in time-space.

Classically, this isn't true; spacetime curvature (which is the correct term for what you're calling "gravity" here) is finite anywhere short of the singularity, no matter how close you get to it, and any finite amount of spacetime curvature is allowed, and won't do any "damage" to spacetime itself, so to speak.

Quantum mechanically, as has already been said, we don't have a good theory, and we can really only speculate about what happens to spacetime once curvature gets large enough (roughly, when the radius of curvature of spacetime gets small enough to be of the same order as the Planck length), or whether "spacetime" is even the right description at that scale, as opposed to something completely different.

 

[Alyx.Vox]

...the "weirdness" about "r" being a time coordinate instead of a space coordinate, and there being an infinite number of "spatial points" at a given "time", applies everywhere inside the event horizon, not just very close to the singularity.

Hmm, so why is it constrained to within the event-horizon and not outside it (anywhere else in the universe)? The event horizon is just a point starting at which even light can't escape the black hole, it's not all too special otherwise in terms of bending the laws of nature or is it?

 

[PAllen]

Hmm, so why is it constrained to within the event-horizon and not outside it (anywhere else in the universe)? The event horizon is just a point starting at which even light can't escape the black hole, it's not all too special otherwise in terms of bending the laws of nature or is it?

The core reason is simply that it is impossible to maintain a constant distance (defined as staying on a circle of constant circumference, to avoid measuring distance to a singularity) from the singularity: even light shined away from the singularity gets steadily closer to it and soon hits the singularity. Thus any spacetime path of constant r is spacelike path (it must pull ahead of outward directed light).

Outside the event horizon, it is possible to maintain a constant r, and not proceed toward the singularity. The force required to hold constant r approaches infinity as r approaches the horizon (from outside; from inside, even infinite force would do no good).

 

[PeterDonis]

Hmm, so why is it constrained to within the event-horizon and not outside it (anywhere else in the universe)?

I'm not sure what you mean; what do you think is "constrained"?

Perhaps this will clarify somewhat:

(1) Outside the event horizon, a curve of constant r (and constant theta, phi--but I'll assume throughout this post that the angular coordinates are constant) is timelike: that means it can be thought of as describing "a point in space" existing for an infinite time (from t = minus infinity to t = infinity). "Space" outside the horizon extends out to r = infinity, so "space" is also infinite outside the horizon; we can think of a curve of constant t as describing a "space" extending from r = 2M (the horizon) to r = infinity.

(2) The event horizon itself is a curve of constant r = 2M (where M is the mass of the hole and G = c = 1) which is null: it describes an outgoing light ray.

(3) Inside the event horizon, a curve of constant r is spacelike: that means it can be thought of as describing an "instant of time" that extends infinitely in space (from t = minus infinity to t = infinity). If we want to emphasize this, we rename the coordinates inside the horizon, r -> T and t -> R, and we say that a curve of constant T describes an instant of time that extends infinitely in space from R = minus infinity to R = infinity.

The event horizon is just a point starting at which even light can't escape the black hole, it's not all too special otherwise in terms of bending the laws of nature or is it?

The laws of nature are the same at or inside the event horizon as they are anywhere else.

 

[Alyx.Vox]

Well all I learned from this is that I'm quite a dummy when it comes to anything outside of classical mechanics in the context of physics. All this is starting to make my brain hurt...