Closed Timelike Curves: Exploring the Boundaries

[georgir]

[Mentor's note: split from https://www.physicsforums.com/threads/gravity-instantaneous.801033/ ]

If masses can not just appear and disappear, how do solutions with closed timelike curves work? If you have some mass happily looping around such curve, from some other point of view you have some "moments" where that mass exists (twice, even, for both branches of the CTC) and then some later moments where it does not...
I'm having trouble imagining what could be the boundary between those and how it does not form a "discontinuity"...
I guess this is not entirely theoretical question too, as I've read that virtual particle-antiparticle pairs can actually be viewed as a single particle on a CTC.

 

[PeterDonis]

If you have some mass happily looping around such curve, from some other point of view you have some "moments" where that mass exists (twice, even, for both branches of the CTC) and then some later moments where it does not...

The conservation law for the stress-energy tensor doesn't work with "moments"; it doesn't depend on how you split up spacetime into space and time, and it doesn't say that the total amount of mass in a spacelike slice must be the same for every spacelike slice. It just says that, for any small region of spacetime, the amount of stress-energy "coming in" must equal the amount of stress-energy "going out". If you pick out any small region of spacetime that contains a short segment of the CTC followed by the mass, then the mass "comes in" to that small region and "goes out" of that small region, and it's the same amount of mass both times, so the conservation law is satisfied.

 

[georgir]

This is interesting... but what defines if the mass is "coming in" or is "going out" ?
If you take a small region of spacetime around the formation of the particle pair, you could say that it is just one particle, coming in and going out... or you could say that it is two particles going out, and require having enough energy coming into that region from elsewhere to compensate.

 

[PeterDonis]

what defines if the mass is "coming in" or is "going out" ?

The mass is described by a 4-momentum vector; you just look at where the vector is pointing. On one side of a given small region of spacetime, the vector will be pointing in; on the other side, it will be pointing out. (The direction of the vector corresponds to the direction in which a clock carried by the mass is increasing its elapsed time.)