Undergrad What is the Alice/Bob thought experiment in Malament-Hogarth spacetime?

[nomadreid]

The article https://en.wikipedia.org/wiki/Malament–Hogarth_spacetime, the possibility of a space with a "worldline λ and an event p such that all events along λ are a finite interval in the past of p, but the proper time along λ is infinite" is discussed, and the suggestion is made that if you ignored Hawking radiation, the inner event horizon with the Kerr metric of a black hole would be such a space. Could someone explain why, or at least present an Alice/Bob thought experiment to illustrate, in terms that do not require much background in Kerr metrics? Attempting to understand a simplified version, in which one refers to the (outer) event horizon of a stationary black hole, only ends up with the opposite: to an outside observer, the time of an in-falling object is infinite, but the proper time of that object is finite.

 

[PeterDonis]

to an outside observer, the time of an in-falling object is infinite, but the proper time of that object is finite.

This is not the same thing as an M-H spacetime. The time "to an outside observer" is not a proper time along the infalling object's worldline, or an interval between events on that worldline. It is just a coordinate artifact.

What is happening in an M-H spacetime is much more counterintuitive, since it is not a coordinate artifact. Kerr spacetime inside the inner horizon is a M-H spacetime because it has closed timelike curves--any spacetime with CTCs is a M-H spacetime, because the CTC has a finite "length" (since it's a closed curve, just like a circle), so there is a finite interval between any two points on it, but an object can go around it an infinite number of times.

 

[nomadreid]

Enlightening explanation, PeterDonis. Thanks. If I understand the last phrase correctly, the "completed infinity" arises because the object comes back to the same point in spacetime, so an infinite amount of time of the object can pass according to its proper time even though a finite amount of time has elapsed. (Or if it came back to just an infinitesimal point in time afterwards each time, the same conclusion would apply, I suppose.)

 

[PeterDonis]

the "completed infinity" arises because the object comes back to the same point in spacetime, so an infinite amount of time of the object can pass according to its proper time even though a finite amount of time has elapsed

It's not that "a finite amount of time has elapsed"--"elapsed" is a description of proper time experienced by the observer/object. It's that the closed timelike curve, considered as a curve in spacetime (without considering how many times an object/observer traverses it), has a finite arc length.

 

[nomadreid]

Thanks, PeterDonis, for pointing out this distinction.