# What is a closed timelike curve? [closed]

I was wondering if anyone could explain to me what a closed timelike curve is?

From Wikipedia:

"In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime that is "closed", returning to its starting point. "

What exactly is a lorentzian manifold ?

## closed as unclear what you're asking by Kyle Kanos, stafusa, StephenG, John Rennie, YashasDec 21 '17 at 7:46

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• The resident expert (Slereah) says: "He just gave the definition" and "What more does he want" – Ryan Unger Dec 20 '17 at 0:19
• Sorry for some reason my question didn't go up...I've added it now...what is a lorentzian manifold? – RelativelyUNcertain Dec 20 '17 at 2:50
• Suppose someone doesn't know what the words timelike, worldline, or Lorentzian mean. – Mitchell Porter Dec 20 '17 at 3:00
• No, I'm quite a layman my apologies if my questions sound silly. – RelativelyUNcertain Dec 20 '17 at 5:49

A Lorentzian manifold is a manifold on which a "Lorentzian" metric is defined. A Lorentzian metric is a Riemannian metric with the positive definiteness requirement (that for a metric $g$, $g(u,u)\gt 0~\forall ~u\neq 0$) removed and instead of having (in 4-D) signature $(+,+,+,+)$, it has signature $(-,+,+,+)$ or $(+,-,-,-)$ depending on what sign convention you prefer. For the remainder of this post, I will use the $(-,+,+,+)$ convention.
On a Lorentzian manifold, a vector is classified as time-like (using our sign convention) if it's norm ($||u||\equiv g(u,u)$) is negative, space-like if its norm is positive, and null if its norm is zero. A time-like curve is a curve on the manifold whose tangent vector is everywhere time-like (and therefore a possible world line for a material particle). A closed time-like curve is then a curve on the manifold which is time-like and passes through the same point on the manifold more than once. This means that a material object traveling along this CTC can return to exactly the same event (perhaps multiple times) all the while increasing its affine parameter (or proper time). In effect, this material object is "time traveling" to its past (hence why CTCs are always mentioned in the context of time travel).
• Who writes $\|u \|$ for $g(u,u)$? – Ryan Unger Dec 20 '17 at 4:04