# When periodic solutions are combined with timelessness, do we get closed timelike curves?

In quantum gravity, ADM wavefunctional solutions have to satisfy the Wheeler-DeWitt equation. This leads to timelessness. What happens if we have a time periodic solution? In classical general relativity, a time periodic solution just means that and no more. But when combined with a timeless wavefunctional, if the same configuration occurs twice or more often, the coefficient of the configuration component in the wavefunctional has to be exactly the same both times around, both in magnitude and in phase. That is because we have no disambiguating clock external to the timeless wavefunctional, which is not multivalued. But this is exactly the description of a closed timelike curve.

Assume space is compact so we do not have to worry about asymptotic infinity.

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If there is a time periodic solution of $\hat{\mathcal{H}}\Psi=0$, then obviously there are only a finite number of solutions throughout the whole of time. A closed timelike curve is a closed worldline, i.e., the particle whose worldline is a CTC must have a repeated 4-position. If the position $\overrightarrow{x}$ has a linear dependence on time, then your argument is true and a CTC is created. If no such linear dependence exists, then the particle may end up with the same time coordinate, but not the position - and this is not a CTC.