What ‘Something is conserved’ means in curved spacetime?

[Dmitry67]

In flat spacetime what we say that something (energy, information, charge, whatever) is conserved we take some region of space at moment t1, check the amount of that something, then we count the amount of the same thing at t2. What is ‘at moment t1’? It means that we cut spacetime using 2 spacelike surfaces (t1 and t2), and separate some region of space using closed 3d timelike surface.

There are some specific requirements for t1 and t2 surfaces; the improper choice of the shape of these surfaces leads to non-conservation; for example, energy is not conserved in cosmology, because cosmological time t is bent surface.

My question is, what is a general definition for ‘is conserved’ in curved spacetimes? For example, in Closed Time-like Loops? Is charge ‘conserved’ in such metrics? Say, charge enters the proximity or Kerr singularity, orbits 3 times around the ring, and escapes to CTL-free region. Local observer can see 3 'copies' of the same charge at the same time. How is it interpreted?

 

[nismaratwork]

In flat spacetime what we say that something (energy, information, charge, whatever) is conserved we take some region of space at moment t1, check the amount of that something, then we count the amount of the same thing at t2. What is ‘at moment t1’? It means that we cut spacetime using 2 spacelike surfaces (t1 and t2), and separate some region of space using closed 3d timelike surface.

There are some specific requirements for t1 and t2 surfaces; the improper choice of the shape of these surfaces leads to non-conservation; for example, energy is not conserved in cosmology, because cosmological time t is bent surface.

My question is, what is a general definition for ‘is conserved’ in curved spacetimes? For example, in Closed Time-like Loops? Is charge ‘conserved’ in such metrics? Say, charge enters the proximity or Kerr singularity, orbits 3 times around the ring, and escapes to CTL-free region. Local observer can see 3 'copies' of the same charge at the same time. How is it interpreted?

One interpretation is that the charge never leaves the CTC, and this leads to the self-destruction of time machines hypothesis of Thorne and Hawking. In that case, I don't think there is any conservation, nor does conservation even apply to that kind of geometry. You can have an arbitrary number of "copies" and energy densities. To see 3 copies as you say, would violate a lot. How the coordinates are chosen in general is just according to the rules which apply; I don't believe there is a special property to them.

This is a little tangential: http://arxiv.org/abs/hep-th/0407110

and an argument against CTCs via Brane Cosmology in general: http://arxiv.org/abs/hep-th/0211097

I found a few that are relevant, but they are all pay-to-play via Springerlink or the like.

 

[Dmitry67]

CTC does not mean that you are locked there.
On spaceship, you can fly into it, wave thru the illuminator to your copies, and then leave that area.

There are 2 cases: worldline is a loop (so particle never leaves that area) or it is a spiral: it enters from the outside, make N loops, then exits.

 

[nismaratwork]

CTC does not mean that you are locked there.
On spaceship, you can fly into it, wave thru the illuminator to your copies, and then leave that area.

There are 2 cases: worldline is a loop (so particle never leaves that area) or it is a spiral: it enters from the outside, make N loops, then exits.

I know, but why would you expect conservation of anything, or maintaining causality within something that violates them? Your example presumes a geometry that doesn't exist in this universe, so who knows what laws of conservation or not are requires for a spacetime in which you can have these CTCs? In the meantime, the notion of potentially arbitrary energy densities as a result of infinite or N transits is a decent conjecture to show that we don't and can't deal with them in our universe, at this time.

This to me, is a bit like asking about Bernoulli with the presupposition that we can fly by thinking really hard about it. What does conservation or causality matter in a situation where neither applies?

 

[Dmitry67]

Wait... What do you mean by 'geometry that doesn't exist in this universe'? It routinely exists inside the rotating black holes (based on the observating all BH in galaxies are 0.99 of the extreme). I even believe in naked Kerr singularities, so I hope we will find naked Kerr rings and CTC there. My hope is justified because I don't see anything which can stop super-extreme BH from forming if we add momentum to almost extreme BH.

 

[nismaratwork]

Wait... What do you mean by 'geometry that doesn't exist in this universe'? It routinely exists inside the rotating black holes (based on the observating all BH in galaxies are 0.99 of the extreme). I even believe in naked Kerr singularities, so I hope we will find naked Kerr rings and CTC there. My hope is justified because I don't see anything which can stop super-extreme BH from forming if we add momentum to almost extreme BH.

I personally do not believe that naked singularities exist in nature, but this is an unsettled question AFAIK. If they do, then you could be right, but I work with the standard assumption that naked singularities can be written out on paper, but not in nature. I haven't read anything that shows rotation beyond the limits of an event horizon, and without that I can't imagine a CTC in this universe, at this time, or any time in the past.

For Black Holes that are in an event horizon, does any kind of conservation even apply? That is as good a different universe.

 

[Dmitry67]

Why? These areas are part of your future lightcone, so theoretically you can jump into big rotating BH and see everything with your own eyes. So even if naked singularitites don't exist, I don't see how it magically settles the problem.

 

[nismaratwork]

It's just that once you're inside the event horizon, physics as we know it doesn't apply. What does conservation mean when GR has already broken down? I have no idea, and don't believe that an answer to that question exists.

 

[Dmitry67]

What? Do you deny the solutions for BH which don't have singularities at the event horizon, for example, Eddington–Finkelstein solution? This is something new... or may be something very old: before 1958 there was a (wrong) concept of 'frozen star' before the full solution was discovered.

 

[nismaratwork]

What? Do you deny the solutions for BH which don't have singularities at the event horizon, for example, Eddington–Finkelstein solution? This is something new... or may be something very old: before 1958 there was a (wrong) concept of 'frozen star' before the full solution was discovered.

No, I don't deny the solutions, but I don't know if they exist outside of the paper they are written on. I believe in ERBs on paper too, but I don't know if they really exist in nature.

 

[bcrowell]

My question is, what is a general definition for ‘is conserved’ in curved spacetimes? For example, in Closed Time-like Loops? Is charge ‘conserved’ in such metrics? Say, charge enters the proximity or Kerr singularity, orbits 3 times around the ring, and escapes to CTL-free region. Local observer can see 3 'copies' of the same charge at the same time. How is it interpreted?

There is no general way to define conservation laws in GR. You don't even have to have CTCs for this to become an issue. Here is a discussion: http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.5 MTW has a treatment that covers some of the same ideas.

 

[Tomsk]

What's wrong with just working with the differential version of a conservation law, e.g. $\nabla_\mu T^{\mu\nu} = 0$? Or, if you have to integrate, why not integrate over time as well, since it should be on a equal footing with space, and then look at properties of that quantity - e.g. whether it's invariant?

 

[DrGreg]

Dmitry67, are you aware of the Usenet Physics FAQ "Is Energy Conserved in General Relativity?"[/color]?

 

[Dmitry67]

Dmitry67, are you aware of the Usenet Physics FAQ "Is Energy Conserved in General Relativity?"[/color]?

Yes, but my question is more general.
Forget about the energy, think about the charge.