Rotating Cosmic-String Worldtube (Jump-Rope Mode): Closed Timelike Curves from a Spinning Troposkein, Energy Conditions, Einstein–Cartan Extension, and Chronology Protection

We analyse a finite spinning cosmic string in the jump-rope (troposkein) configuration as a source of closed timelike curves (CTCs) in General Relativity and Einstein–Cartan (EC) theory. The troposkein is the equilibrium surface of revolution swept by a flexible string spinning at constant angular velocity Ωrot between two fixed endpoints (“handles”); it is neither straight nor cylindrical, but bows outward from the rotation axis, reaching maximum radius Rtrop at the midplane z = 0 and returning to r = 0 at the endpoints z = ±L/2. In the weak-field, fast-rotation limit the linearized metric admits the form ds2 = −dt2 − 2ω(r, z) dt dϕ + (r2 − ω2) dϕ2 + dr2 + dz2, where ω(r, z) ∼ ω0 Ψ(z) (r/Rtrop)2 near the axis, Ψ(z) encodes the troposkein profile with Ψ(0) = 1 and Ψ(±L/2) = 0. The r2 scaling follows from the dipole character of the gravitomagnetic field of a finite loop source. We prove: (i) CTCs exist in an annular region r− c (z) Rtrop; this is resolved by a three-layer exact construction (Section III): (L1) the truncated Lanczos–Van Stockum rotating dust interior is an exact nonlinear solution — no approximation used — with r2 frame-dragging, CTCs sourced by positive-density ordinary matter, and a compact cylindrical chronology horizon; (L2) the troposkein z-profile continuously deforms this exact solution, preserving CTCs and closing the horizon into a compact surface of revolution; (L3) the linearized metric is the weak-field limit of (L1), analytically tractable for computing CJR. The jump-rope troposkein provides a finite, single-source CTC construction with a naturally compact horizon and an exact nonlinear existence proof from ordinary matter (via the truncated Lanczos–Van Stockum construction, Layer L1). Whether it survives quantum back-reaction depends on CJR (Claim 1), which has not been computed.