# Geodesics and CTC

What is the exact explanation for which CTC (closed timelike curves) are not geodesics? I have already looked up in many papers but none provides the exact reason.

• I'm fairly sure there is no rule forbidding geodesics from being CTCs or vice versa. I'm fairly certain Hawking considered closed null geodesics in his work on the chronology protection conjecture. Did you have some specific situation in mind? For example in the Gödel universe there are CTCs but no closed geodesics. Nov 13 '16 at 20:13

Closed timelike curves can be geodesics. In some spacetimes it's even possible that every geodesic is a closed timelike curve. What I think you're refering to is the chronology protection theorem, which states that closed causal geodesics will cause a divergence in the energy of the vacuum.

A rather suggestive form for the propagator of quantum field is the Hadamard form

$$G(x,y) = \hbar \sum_\gamma \frac{\Delta_\gamma (x,y)^{\frac{1}{2}}}{4\pi^2} [\frac{1}{\sigma_\gamma (x,y)} + v_\gamma (x,y) \ln |\sigma_\gamma (x,y)| + \omega_\gamma(x,y)]$$

Which is a sum over all geodesics linking the points $x$ and $y$ of those various functions, for which $\Delta$ is the van Vleck determinant, and $\sigma$ is the geodetic interval. The other functions depend on the exact differential operator you're considering.

The stress energy tensor depends on that quantity, in the coincidence limit of the points $x$ and $y$. This quantity is obviously divergent but can be simply enough renormalized by removing the divergent quantities of the geodesic linking the two points.

With closed causal geodesics, though, you will still have divergent parts that is not renormalized. It's hard to prove in the general case (that's why it is still a conjecture), but it is true for a few well known examples such as wormholes and Misner space.

From a physical perspective, this can be explained the following way :

Closed causal curves are pretty bad in field theories. There are many configurations in which a field, when confronted with a closed causal curve, will just circle round and round, getting blue shifted with each cycle, and just diverging to hell. This isn't necessarily too terrible, because this may only happen for some configurations of the field.

Unlike a classical theory, the quantum vacuum will run over every momentum, which guarantees that some of them will get blueshifted. As once approaches the Cauchy horizon, the null curves will get closer and closer to self-intersection, the blueshift caused will increase until, theoretically, becoming infinite at the horizon.

So for a spacetime with closed timelike curve to not blow up in such a manner, it is supposed to be necessary to avoid any closed causal geodesics.

Spacetimes without closed causal geodesics but still closed causal curves (those aren't too difficult to cook up, even if compactly generated) should be fine, although note that this analysis is only valid for free fields. I'm guessing that in general closed timelike curves will have pretty bad consequences on quantum fields.

• And in general, do curves and geodesics refer to the same thing? Both describe a path in a curved spacetime, don't they? Dec 10 '16 at 10:47
• No, a geodesic is a curve that follows the geodesic equation, $u^\mu \nabla_\mu u^\nu = 0$, with $u$ the tangent vector. Dec 10 '16 at 11:03
• @Timetraveler Both describes paths in a curved spacetime but the (timelike) geodesics are those special paths who extremizes the proper time. Mathematically, geodesics follow the geodesic equation as Slereah mentioned. Jun 23 '17 at 10:58