# Chronology protection for non-geodesic CTCs and imprisoned curves

As far as I can make out, the quantum part of the Chronology Protection Conjecture hinges on the fact that in curved space, in the semiclassical approximation, the stress energy tensor contains a term

$$\lim_{x\rightarrow y}\sum_{\gamma \neq \gamma_0} \frac{\sqrt{\Delta_\gamma (x,y)}}{\sigma_\gamma(x,y)^2} t_{\mu\nu}$$

With $\Delta$ the van Vleck determinant, $\sigma$ the geodetic interval, $\gamma$ every geodesic linking $x$ to $y$ (the geodesic $\gamma_0$ is removed for renormalization) and $t_{\mu\nu}$ is a tensor made of the metric and tangents of the geodesic. The CPC states that CTCs will cause this term to be badly divergent because in the $x = y$ limit, only closed timelike curve will contribute to timelike geodesics, causing a rather badly divergent term in $\sigma^{-4}$.

But does this apply in some form to spacetimes where non-geodesics are CTC, but no geodesics are? And what about the case of spacetimes with imprisoned causal curves, where causal curves can remain trapped in a compact set, but never form any CTCs? I'm not quite sure you could have compactly generated regions with imprisoned causal curves, as the only example I can find is from Hawking Ellis with

$$ds^2 = (\cosh t - 1)^2 (dt^2 - dy^2) + dy dt - dz^2$$

With the identifications $(t,y,z) = (t,y,z+1)$ and $(t,y,z) = (t,y+1, z+a)$, $a$ an irrational number. In such a configuration you could still have a geodesic being arbitrarily close to itself in the past.