Closed Timelike Curves in ADM Spacetime

We study closed timelike curves (CTCs) in a cylindrically symmetric, stationary, axisymmetric spacetime within the Arnowitt--Deser--Misner (ADM) \( 3{+}1 \) decomposition, coupled to Einstein--Dirac--Maxwell (EDM) fields. The central observation of this work is that CTC formation is \emph{not} automatically incompatible with positive ADM energy. Beginning from the ADM Hamiltonian constraint, we prove rigorously that the ADM energy \( E \geq 0 \)for hyperbolic shift vectors of the form \( N_i = \partial_i \gamma \), yielding \( N_i = \partial_i \gamma \). The cylindrical metric admits CTCs when the coefficient \( R(r) < 0 \), a condition we translate into an explicit inequality on the determinant of the \( (t,\varphi) \) block of the metric. For the Einstein--Dirac--Maxwell system we compute the conserved Noether charge \( \QN \) and derive the critical ADM energy\( \Ecrit(\alpha, \QN) \) above which CTCs form in a tubular neighborhood of the Tipler cylinder. The quantum backreaction on the metric, encoded in the renormalized stress tensor \( \Tren \), produces a vacuum polarization (VP) distortion \[ \omega g^{\mathrm{VP}}_{\mu\nu} \;\sim\; \frac{\lPl^2}{D\,\Delta t}\sum_{n=1}^{\infty}\!\left(\frac{\delta}{2D}\right)^{\!n-1}, \] which converges absolutely for \( \delta < 2D \) to the closed-form value \( \lPl^2/(D\,\Delta t) \cdot 2D/(2D-\delta) \). We show this backreaction is consistent with, but does not by itself prove, Hawking's chronology protection conjecture. The Weak Energy Condition is satisfied globally by the ADM energy bound; local WEC violation is nevertheless required in a neighborhood of any Cauchy horizon where CTCs first form, in agreement with the Average Null Energy Condition analysis of Tipler-type spacetimes.