# Wormholes and Closed Time-like Curves

I've been doing a presentation on Wormholes for my Science class, and I've been wondering about the difference of Closed Time-Like Curves and Wormholes, as most articles I read can't truly explain the difference, even most search engines i've tried can't explain the difference. I came to this site to ask this simple question. I'm not exactly great at formulas and stuff like that, but I can understand some simple things. If anyone can explain the answer well, that would be great. Thanks!

Since you have linked to Wikipedia for both CTC and Wormhole I'll try not to repeat what they say and address the distinction you are seeking.

A CTC is a curve along which something e.g. an observer may travel, starting at time $t_0$, always moving "forward" in time from its own perspective but nonetheless finding itself back where it started at some positive Proper Time later, but at coordinate time $t_0$ again.

On the other hand, a wormhole is a topological feature of spacetime; topology is the mathematical study of the way things are "connected up". The classic example is the comparison of a donut and a sphere: if you draw a circle on the surface of a sphere you can shrink that circle down to a point, but on the surface of a donut, if that circle passes through the hole it cannot be shrunk down to a point. The donut has a hole in it and the sphere doesn't.

A wormhole is a topological feature of spacetime that works in just the same way: you can go through the wormhole and back to where you came from but that path has a certain minimum length, whereas if you go around in a circle without going through the wormhole there is no such minimum length.

It is claimed that wormholes can be used to create closed timelike curves, and this is a) widely accepted in the literature and b) a problem because CTCs create real problems for physics. (The fact that you need "exotic" matter to keep the wormhole open isn't a theoretical problem, it's just a practical problem).

The "go back in time and kill your grandfather paradox" (due originally to Schachner in 1956, Amazing Stories sci-fi if I recall correctly) isn't a real paradox at all strangely. If you assume, which you must if causality is to make any sense at all, that if things can be consistent they must be consistent, it can't happen: the existence of a CTC just changes the conditions under which the laws of physics are satisfied, i.e. to be consistent the setup (the so called "initial condition") has to ensure that even if you do travel backwards in time, any attempt to kill your grandfather will fail... because it did fail. Physicists have constructed so called "billiard ball" time travel scenarios in which a ball collides with itself - and they are indeed perfectly consistent!

I also suspect (personal thought) that if one takes quantum mechanics into account too, the temporal interference effect on the wave function will also ensure consistency because impossible outcomes will cancel out and consistency will reinforce, just as light interferes in making the bright and dark lines of the Double Slit experiment

But, the point here is that if a wormhole could be used to create a CTC, that CTC passes through the wormhole, which is why the two things are often spoken of together and quite easy to confuse.

However, you do not need to have unusual topology to have a CTC: Kurt Godel's (and van Stockum's, and various others') spacetimes do not have wormhole-like features, but because of the way the coordinates twist around you can go around in time circles (what you need are so called-cross-terms between space and time in the spacetime metric). Einstein wasn't very happy with Godel when he discovered that possibility!

Technically speaking it's the nature of the spacetime metric that accounts for this, but you can think of the time axis as just being bent around.

So, to conclude with a summary distinction: a CTC is a curve that makes a circle in time, a wormhole is a topological feature of space (ignoring the time part) that has circular paths which cannot be shrunk down to nothing (and which can become CTCs under appropriate circumstances).

• I'm curious: what exactly does a time travel billiard ball scenario look like if you were to actually render it into a physically accurate movie? Is there any such video out there (i.e. a simulation showing a particle interacting with its past self via, say, a suitably crafted wormhole, with accurate trajectories and all)? – The_Sympathizer Sep 10 '18 at 3:18
• A quick google around the title "Billiard balls in wormhole spacetimes with closed timelike curves: classical theory" (Phys Rev D 44, 1991) doesn't show any video, but that paper with some illustration is at authors.library.caltech.edu/6469/1/ECHprd91.pdf – Julian Moore Sep 10 '18 at 5:51
• @The_Sympathizer See above – Julian Moore Sep 10 '18 at 6:04
• @Julian Moore, so any wormhole that is used for time travel to the past will necessarily create a CTC, yes? – Satyajit Sen Mar 12 at 6:52
• @SatyajitSen - sorry for the delay. Time travel and CTC are synonymous/interchangeable here; the wormhole does not "create" the CTC but it does allow it to exist (there are other ways, incldding cosmic strings, rotating black holes, rotating universe etc.) – Julian Moore Mar 27 at 18:53