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

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

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

\begin{equation} ds^2 = (\cosh t - 1)^2 (dt^2 - dy^2) + dy dt - dz^2 \end{equation}

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.