What Are Closed Timelike Curves?

[Champion]

i was going over some relativity search online and came across what is called a closed timelike curve and that i actually allows time travel, am i right?

 

[Fredrik]

A "curve" is a function $c:A\rightarrow M$ where A is some interval of real numbers and M is the spacetime manifold. "Timelike" means that if we call the tangent vector v and the metric tensor g, then g(v,v)<0 at all points on the curve. (If we define the metric with a +--- signature instead of -+++, it's ">" instead of "<"). "Closed" means that c(t+T)=c(t) for some real numbers t and T such that both t and t+T are in A.

A timelike curve is the type of curve that can represent that path of a massive object through spacetime. If it's closed, the object will meet a younger version of itself at some point.

Edit: I should probably add that the condition g(v,v)<0 can also be written

$$g_{\mu\nu}v^\mu v^\nu<0$$

and that in a local inertial frame, this is just

$$-(v^0)^2+(v^1)^2+(v^2)^2+(v^3)^2<0$$

 

[Champion]

i was thinking it over,about the constant energy.ctc's would violate 2nd law of therodynamics,also i looked up that GR allows ctc's but QM does not.