Space time reversal inside a black hole

[wasphysics]

Hi guys
I had a question. We know that roles of space and time reverses inside a black hole. In the schwarzchild black hole for example r and t are interchanged into time and space respectively. this argument is used to say that because time moves in a single direction r can only decrease inside the black hole and hence nothing escapes the horizon. But if r is supposed to represent time, in the sense that an observer located inside the black hole has a watch and this watch measures r, wouldn't that really imply that r measured by him is going to increase, because time by its very essence only goes forwards, irrespective of whether it is labelled by 'r' inside the black hole. Could someone help me with this question.

 

[bcrowell]

I think it's a little bit of an oversimplification to say that r becomes time and t becomes radius. It's just that r becomes timelike and t becomes spacelike. Local frames of reference in relativity don't have a uniquely defined time axis. Picking a time axis is like picking a state of rest, which is arbitrary.

Note also that the Schwarzschild metric is time-reversal symmetric. There is nothing about it that picks out an arrow of time.

 

[wasphysics]

sure schwarzchild metric is time reversal symmetric, but a co-ordinate becoming timelike does mean that in the particular frame of reference the co-ordinate would refer to time measured by a clock in that frame, so any chain of events 'would' have to progress in an increasing direction of time, or the time like co-ordinate. Else you are breaking the relativistic principle that each frame is equivalent to any other frame, because outside the black hole time progresses in the forward direction, so inside the black hole suddenly this aspect cannot be reversed.

Also inside the black hole a inertial reference frame has time moving in the forward direction. So any other reference frame inside the black hole has to have a forward direction of motion for the time co-ordinates in order for the physical laws to be covariant.

 

[bcrowell]

It's just that (if I understood properly) you assumed that the increasing r direction had to be the forward time direction inside the horizon. It's not, it's a backward time direction. This is all obscured by the coordinate singularity at the event horizon, which isn't physically real but makes things hard to interpret.

sure schwarzchild metric is time reversal symmetric, but a co-ordinate becoming timelike does mean that in the particular frame of reference the co-ordinate would refer to time measured by a clock in that frame, so any chain of events 'would' have to progress in an increasing direction of time, or the time like co-ordinate. Else you are breaking the relativistic principle that each frame is equivalent to any other frame, because outside the black hole time progresses in the forward direction, so inside the black hole suddenly this aspect cannot be reversed.

Also inside the black hole a inertial reference frame has time moving in the forward direction. So any other reference frame inside the black hole has to have a forward direction of motion for the time co-ordinates in order for the physical laws to be covariant.

No, different local frames of reference do not have to have their time axes oriented the same way. You can even have spacetimes that are not time-orientable, just as a Mobius strip is not space-orientable.

It may be easier to think about this in terms of geodesics. There are two types of radial geodesics in a Schwarzschild spacetime. One type approaches the horizon, passes through, continues, and hits the singularity. The other type starts from the singularity and passes outward through the horizon. The two sets of geodesics are identical time-reversed copies of each other.

 

[PAllen]

sure schwarzchild metric is time reversal symmetric, but a co-ordinate becoming timelike does mean that in the particular frame of reference the co-ordinate would refer to time measured by a clock in that frame, so any chain of events 'would' have to progress in an increasing direction of time, or the time like co-ordinate. Else you are breaking the relativistic principle that each frame is equivalent to any other frame, because outside the black hole time progresses in the forward direction, so inside the black hole suddenly this aspect cannot be reversed.

Also inside the black hole a inertial reference frame has time moving in the forward direction. So any other reference frame inside the black hole has to have a forward direction of motion for the time co-ordinates in order for the physical laws to be covariant.

If you are analyzing the infall of a particle, of course you want to keep the timelike coordinate the same direction as the worldline crosses the event horizon. (If you reversing time gives, I believe, the white hole solution where everything must leave the event horizon). If you ignore coordinate labels, a key thing that happens as you cross the horizon is that the whole outside universe is in your past (backward pointing light cone). All light you emit goes 'toward' the singularity. I think you can look at it as a reverse big bang, where you're not so much moving toward the singularity as that all of space, as you can see it, is shrinking toward the crunch in your future. Your progress in time, through shrinking space is how you get to the crunch.

 

[wasphysics]

allen,
u can only make local measurements as you fall through a black hole, so before u hit the horizon, locally all the events happening in ur frame happen in the future in the 'local' frame in which the black hole is stationary. But this does not represent 'all' the universe, but just the the 'local' universe. And this local frame can only get the information of the portion of the universe it is causally connected with, which is the universe outside the event horizon. So the infalling observer can only be informed (see) the past of the universe outside the horizon. As he enters the horizon, he is entering a portion of the universe (in the frame in which the black hole is stationary), which is causally disconnected from the portion outside the horizon. What I am saying is that the knowledge of moving towards the singularity, cannot be ascertained by plainly looking at the past of the universe outside the black hole horizon.

Bcrowel
I don't think that a time non-oriented spacetime solves the einsteins equations. The two geodesics u talked about don't represent two different local frames that can compare events in spacetime. I am talking about the order of occurence between two frames happening at the same space position in spacetime cannot plainly flip. This is because flipping of directions would imply that the system cannot have a ground state, as positive energies in one frame would appear as negative energies in another frame.
Also if the forward direction of time as measured by a clock corresponds to r decreasing, then since the clock would anyway keep moving forward after r hits zero, it implies that the clock in this frame cannot be described by the r variable and that is inconsistent, because reference frames can always be chosen 'globally' to describe all of spacetime.

 

[bcrowell]

I don't think that a time non-oriented spacetime solves the einsteins equations. The two geodesics u talked about don't represent two different local frames that can compare events in spacetime. I am talking about the order of occurence between two frames happening at the same space position in spacetime cannot plainly flip. This is because flipping of directions would imply that the system cannot have a ground state, as positive energies in one frame would appear as negative energies in another frame.

GR is a classical theory, not a quantum-mechanical one.

Also if the forward direction of time as measured by a clock corresponds to r decreasing, then since the clock would anyway keep moving forward after r hits zero, it implies that the clock in this frame cannot be described by the r variable

I don't know what you mean by this.

and that is inconsistent, because reference frames can always be chosen 'globally' to describe all of spacetime.

No, frames of reference are local in GR, not global.

 

[wasphysics]

GR may be a classical theory, but in quantum mechanics time and space are conjugate to energy and momentum, so as far as measurement issues reversal of events in time between two reference frames would correspond to energy measurements which are positive in one frame becoming negative implying presence of a vacuum in one frame and abscence in another frameFrames of reference might be local, but it should be possible to chart the whole of spacetime with a single co-ordinate system, like the r and t of the schwarzchild co-ordinates, which span over the entire spacetime in a particular frame of reference which is stationary with respect to the black hole. Comparisons between reference frames happen locally, but frames of references can be defined globally, just as an inertial reference frame can be defined globally.

In this reference frame (stationary with respect to the black hole) the clocks would always go forward in time and if the clock time corresponds to the time variable inside the black hole r (which decreases as the time on the clocks start going forward), then we have a contradiction, because the time on the clock keeps going forward, while r plainly approaches zero and does not change.

 

[PeterDonis]

If you ignore coordinate labels, a key thing that happens as you cross the horizon is that the whole outside universe is in your past (backward pointing light cone). All light you emit goes 'toward' the singularity.

I agree with the second sentence quoted, but not the first, at least not as stated; your past *light cone* will not be the entire portion of the spacetime exterior to the horizon, even at the instant when you hit the singularity. (Your past light cone may be "almost all" of the exterior spacetime if you cross the horizon very "late" in terms of, say, the ingoing Eddington-Finkelstein time coordinate; but it will never quite be all of it.)

 

[pervect]

While it is true that GR in general may not be time orientable, the Schwarzschild metric, and also the Kruskal extension, are, to the best of my knowledge, both time-orientable. (Corrections welcome).

This means - to the best of my knowledge again - that you can tell the difference between a future light cone, and a past light cone. And any timelike worldline for the future from some point P will lie within the limiting boundaries of a future-directed light cone starting at said point. So you can tell the difference between "past" and "future" on any worldline on a time-orientable manifold from said time-like light-cone, given that you have a time-orientable manifold.

What happens, I think, is that r=0 in the Schwarzschild geometry, if you include the complete geometry, i.e. the Kruskal extension, is actually two surfaces. The "upper" and "lower" patches on a Kruskal diagram, such as that shown in

http://www.jessemazer.com/images/p834Gravitation.jpg

or perhaps better explained in the section "Qualitative features of the Kruskal-Szekeres diagram in the wiki"

http://en.wikipedia.org/w/index.php?title=Kruskal–Szekeres_coordinates&oldid=420900873

One is a white hole, which things can come out of, but never fall into. Another is a black hole, which things can fall into, but never go out of.

A real physical collapse probably won't be the Schwarzschild geometry, however.

So, you always see things going in only one direction - they can come out of the white hole, or fall into the black hole. And you can reverse the defintion of time, in which case things fall into the white hole (which is now black), and come out of the black hole (which is now white).

Refering to the wiki diagram, time always moves up the page, lightcones are at 45 degree angles, things can come out of region 4 and into region 1, or move out of region 1, and into region 3.

The diagram is time symmetrical because you can flip it upside down, which interchanges regions 1 and 4

But regions 1 and 4 are distinct and different regions of space-time, and things will cross the different event horizons only in one direction - the global direction of time - which you can take to be upwards.

 

[Passionflower]

Hi guys
I had a question. We know that roles of space and time reverses inside a black hole. In the schwarzchild black hole for example r and t are interchanged into time and space respectively. this argument is used to say that because time moves in a single direction r can only decrease inside the black hole and hence nothing escapes the horizon. But if r is supposed to represent time, in the sense that an observer located inside the black hole has a watch and this watch measures r, wouldn't that really imply that r measured by him is going to increase, because time by its very essence only goes forwards, irrespective of whether it is labelled by 'r' inside the black hole. Could someone help me with this question.

An observer traveling radially towards a non rotating black hole is a given proper time removed from the singularity provided he does not accelerate (proper acceleration) After he passes the even horizon he cannot even prevent to reach the singularity by acceleration.

However his clock will always more forward and then when he reaches the singularity his proper time simply stops, as his geodesic ends. But at no time does his clock go backwards.

Wrt to r for an observer who passed the event horizon. The distance between the EH and an observer between the singularity and the event horizon is always finite, however the distance between the EH and the singularity is infinite. So if we introduce a new variable $\rho_{EH}$ which represents the distance (calculated by integration of r) from the EH and a traveler passed the EH we will notice it always increases and becomes infinite when the singularity is reached.

 

[cosmik debris]

Frames of reference might be local, but it should be possible to chart the whole of spacetime with a single co-ordinate system.

No this is not possible with a non-flat geometry. Take the surface of the Earth, you need two co-ordinate patches to cover the surface. That gives singularities at the poles and a discontinuity at the dateline. On an arbitrarily curved manifold like the Universe you can sew co-ordinate patches together to cover the manifold, but you cannot do it with one co-ordinate system. In GR the patches are locally flat lorentzian patches.

 

[pervect]

For the black hole case, you can cover the entire manifold (by which I mean the extended manifold to with the Kruskal extensions) with one set of coordinates, u and v, or u0 and u1.

The usual Schwarzschild geometry only covers half the extended manifold, though.

http://en.wikipedia.org/wiki/File:Kruksal_diagram.jpg uses u0 and u1. u0 is a time-like coordinate everywhere.

The black hole horizons are the 45 degree angled lines, which separate regions I, II, III, and IV

Region I is the black hole exterior, region II is the black hole interior, and the 45 degree line separating regions I and II is the black hole event horizon at R=2M. As time goes forwards, you can pass from region I to region II, but not backwards.

Regions III and IV are the Kruskal extensions, the "white hole". A black hole that is the result of a gravitational collapse won't necessarily have regions III and IV, but it won't be time-symmetrical, either, except in the sense that you can have an equivalent time-reversed white hole solution if you time-reverse the geometry of an actual physical collapse.

 

[Lost in Space]

If time stops at the singularity, does the leaking of Hawking radiation or quantum tunnelling originate at the singularity or is it between the event horizon and the singularity?

 

[yuiop]

Has any made the following time reversal observation before? (Please correct me if I am wrong.)

If Alice jumps into a black hole from say r=4m and Bob jumps from say r =6m after he sees Alice jump (he can't bear to be without her :!) ), then after they pass the horizon Bob will see Alice as above him and when Bob arrives at the central singularity, Alice will not have arrived there yet.

 

[pervect]

What makes you think it true?

 

[yuiop]

What makes you think it true?

Take a quick look at this sketch on a kruskal Szekeres chart. The same is true on a Schwarzschild chart.

The red line is Alice's path and the green line is Bob's path. (The light blue lines are light paths)
At event B1, Bob is ar r=2.3M approx and he sees light coming from Alice when she was at r=2.1M so at this point Alice appears to be below Bob.
At event B2, Bob is very nearly at r=0M and he sees light coming from Alice when she was at r=1.5M so inside the event horizon Bob appears to have overtaken Alice (visually) and now appears to be below Alice. It does not matter how long Bob waits after Alice jumps, he always appears (visualy from his point of view) to arrive at the cental singularity before Alice. I just thought it was an interesting optical illusion.

 

[George Jones]

At event B2, Bob is very nearly at r=0M and he sees light coming from Alice when she was at r=1.5M so inside the event horizon Bob appears to have overtaken Alice (visually) and now appears to be below Alice.

Why does this mean "below"? Inside the event horizon, negative r is a future-directed timelike coordinate. What I quote above thus means that the light is received after it is emitted, as it should be.

 

[pervect]

I would visualize "above, below, future, and past" in terms of light cones, as in the large diagram below. This is assuming you are talking about what an observer at any given point would perceive in his or her "local frame". He'd perceive things in terms of his or her light cones, which would maintain 45 degree angles on the Kruskal-Szerkes chart because that's a feature of the chart.

This diagram certainly works relativistically in flat (Minkowski) space-time, as I hope you'd agree?

I've sketched the "old" outdated, non-relativistic concept of future and past in the corner, as an afterthought.

There are two issues here:

1) Use the "new" viewpoints (large diagram) not the old (small diagram).

2) r and t interchange roles as being space-and time coordinates. So in the inside horizon, if two events happen at the same r-value and different t-values, they are separated in a spacelike manner - one is "above" and one is "below", they are separated in a space-like manner.