How do you find the circumference of a closed timelike curve?
You may know how curves in spacetime can be classified as timelike, spacelike or lightlike (null). You may also know how some timelike curves can be closed timelike curves. Well, I was studying some general relativity (self study, not classroom, I was not even a physics major in college). Specifically, I have been looking into closed timelike curves.
Long story short, I learned how you can parameterize a curve in spacetime(we will call our curve x(s) which of course has the parameter s), and how you can apply certain formulas involving the metric tensor in order to find out whether the curve is spacelike, timelike or lightlike, and what the proper time or arc length between two events is along a curve.
However, I also learned that your parameterized curve's components have to have derivatives that are always positive for your curve to be a valid curve. Having said that, there is no way for the curve to mathematically appear to loop back on itself and return to the event from whence it began in order to create a closed timelike curve. In other words, if your curve is something like x(s) = [10s, s, s, s] (which has derivatives that are always increasing), then no two values of s will ever lead to the same event in spacetime (at least if you just go by the numbers that you plug in).
Instead of having the values in the curve determine whether or not the curve is a CTC, you have to look at the geometry of the spacetime itself to determine whether or not your curve loops back to the same event in spacetime. Basically, the spacetime geometry has to be something like a circle or a sphere (a geometry on which going forward along a curve will eventually take you back to the same point). In other words, the spacetime geometry has to kind of be like the unit circle. On the unit circle, assuming that you have coordinates (r, theta), the point (r, 0) is exactly the same point as (r, 2pi) (and of course on the actual unit circle r = 1). You could say that the unit circle has a "period" or rather a circumference of 2pi.
What I would like to know is, how do you find the "period" or "circumference" of a closed timelike curve in a spacetime such as the Kerr metric or the Godel metric? An even better question is, how do you deduce that the geometry of a curve in spacetime is actually circular or spherical or something closed (because not every timelike curve is a closed timelike curve even in spacetimes that allow for closed timelike curves)? Would it have something to do with the Ricci curvature tensor or the Riemann tensor?
Long story short, I learned how you can parameterize a curve in spacetime(we will call our curve x(s) which of course has the parameter s), and how you can apply certain formulas involving the metric tensor in order to find out whether the curve is spacelike, timelike or lightlike, and what the proper time or arc length between two events is along a curve.
However, I also learned that your parameterized curve's components have to have derivatives that are always positive for your curve to be a valid curve. Having said that, there is no way for the curve to mathematically appear to loop back on itself and return to the event from whence it began in order to create a closed timelike curve. In other words, if your curve is something like x(s) = [10s, s, s, s] (which has derivatives that are always increasing), then no two values of s will ever lead to the same event in spacetime (at least if you just go by the numbers that you plug in).
Instead of having the values in the curve determine whether or not the curve is a CTC, you have to look at the geometry of the spacetime itself to determine whether or not your curve loops back to the same event in spacetime. Basically, the spacetime geometry has to be something like a circle or a sphere (a geometry on which going forward along a curve will eventually take you back to the same point). In other words, the spacetime geometry has to kind of be like the unit circle. On the unit circle, assuming that you have coordinates (r, theta), the point (r, 0) is exactly the same point as (r, 2pi) (and of course on the actual unit circle r = 1). You could say that the unit circle has a "period" or rather a circumference of 2pi.
What I would like to know is, how do you find the "period" or "circumference" of a closed timelike curve in a spacetime such as the Kerr metric or the Godel metric? An even better question is, how do you deduce that the geometry of a curve in spacetime is actually circular or spherical or something closed (because not every timelike curve is a closed timelike curve even in spacetimes that allow for closed timelike curves)? Would it have something to do with the Ricci curvature tensor or the Riemann tensor?
