Physics
Quotient structure and on-generator diagnostics of vacuum traveling-wave time machines
Xavier's vacuum traveling-wave metric admits a switchable quotient, but the closed null generator carries the same epsilon-to-the-minus-two and epsilon-to-the-minus-four singularity as Misner. A sector-level structure and discriminator paper.
Jonathan Schemoul
Xavier’s metric admits a switchable quotient in the periodic train family, while the paper’s own sech-squared and exponential examples are helical and not switchable. This paper classifies the quotient legality, computes the monodromy and the deck image sum, and tests whether the closed null generator is a core-specific discriminator.
The correct pure z-shift classification is one family: a train modulated by a pump, h = exp(2 a t) p(z) q(z - 2t) with p and q L-periodic. Its special cases are the trains (boost zero) and the pumps (z-independent, Misner pile-up). The earlier two-family dichotomy missed the boosted train family. The train’s affine circuit length is exactly constant over 24 circuits; the pump shrinks geometrically. The full deck image sum with Hadamard and Van Vleck weights converges at fixed transverse distance but diverges as epsilon to the minus two for the epsilon-to-the-minus-one weight and epsilon to the minus four for the epsilon-to-the-minus-two weight. That is the singularity of the closed null generator, not a core-specific discriminator.
The off-N horizon stress is computed at one point with the full four-dimensional deck-connecting boundary-value problem. The near-axis scaling is S proportional to C over x squared, the horizon limit is order one, and the coefficient is C1 about 52.4 at cutoff n less than or equal to 12, or 50.71 at n less than or equal to 6. The tail is estimated but not bounded. Classical stability is neutral in the analysed TT, profile, f-modulus and vector-truncation sectors; the h_xx minus h_yy polarization is expelled at k_perp = 0; the remaining sectors are open.
Key results
- Quotient classification: train modulated by a pump, with trains and pumps as special cases.
- 24-circuit monodromy: train constant, pump geometric.
- Deck image sum: epsilon-to-the-minus-two and epsilon-to-the-minus-four generator singularity.
- Off-N stress finite at one point, C1 in the 50.7 to 52.4 range with an estimated tail.
- Stability neutral in the analysed sectors.
Status and limits
This is a 24-page content-complete sector analysis. The full 10-component and longitudinal-constraint sectors, the O(A rho k_perp) problem, quotient Cauchy, nonlinear stability, the C1 tail and Maslov dressing, and a full off-N scan remain open. It has not been submitted or peer reviewed.