Force dilation

Force dilation is the momentum-space counterpart of time dilation. In special relativity a fast-moving clock runs slow; in the dual kinematics of #13, a system under a large sustained force evolves slowly, by the same square-root formula with force in place of velocity and the maximum force c̃ in place of the speed of light. Its mirror is a contraction of the range of momenta a forced system spans, in place of the contraction of its length. #13 restricts the effect to systems of the conjugate (momentum) polarization, leaves open which physical clocks respond, and states that both effects are unmeasurably small below the forces of black-hole mergers. The result is alive but unreviewed: #13 tags it as derived, but no review round or script has checked it, and the checkpoint #11 does not mention it.

Status: Alive, unreviewed — [D — structural; which physical clocks respond is polarization-dependent, O], as #13 §3 tags it; force dilation is on #13 §7’s fixed [D] list, the contraction of momentum spread is untagged — no blind review round has examined #13, no verify/ script checks the formula or the sizes quoted, and #11 v2.2 does not mention it. The clock-response question is open and is missing from #13 §7’s open list.

The dual proper time#

#13 §3: “The dual proper time of a trajectory p(t) is”

dτ̃ = dt √(1 − (1/c̃²)|dp/dt|²) = dt √(1 − F²/c̃²)

The template is the proper time of special relativity, dτ = dt√(1 − v²/c²), the time read by a clock that moves at speed v = |dx/dt| [T]. The transcription replaces the velocity dx/dt by the rate of change of momentum dp/dt, which is the force F, and c by c̃ (The maximum force c̃ = c⁴/G). The formula has the form of the dual interval along the trajectory divided by c̃, dτ̃ = ds̃/c̃ with ds̃² = c̃²dt² − dp² (Force boosts and force composition), as the template’s proper time is ds/c; #13 writes the result without that intermediate step.

Read as #13 reads it, a system under a constant force F advances its dual proper time more slowly than the coordinate time t, by the dual Lorentz factor: dτ̃ = dt/γ̃ with γ̃ = (1 − F²/c̃²)^{−1/2}. The slowing depends only on the size of the force, |dp/dt|, and not on how the force changes, as time dilation in the template depends on speed and not on acceleration [T]. At F = 0 the two times agree; as F approaches c̃ the dual proper time stops advancing. The plain passage of §3 puts it as “hard-pushed clocks run slow — a system under enormous force ages more slowly than an unforced one, by the same square-root formula with force replacing velocity.”

What it dilates: conjugate-polarized systems#

The technical statement is narrower than the plain passage. #13 §3:

“so sustained force dilates the internal evolution of (conjugate-polarized) systems as sustained velocity dilates ours [D — structural; which physical clocks respond is polarization-dependent, O].”

The parenthesis restricts the effect to matter of the other polarization. Postulate P4 sorts matter into two kinds: “Matter comes in x-polarized (Standard Model) and p-polarized (χ) representations; both couple universally to H; gauge forces do not cross polarizations” [P] (#11 §4). Ordinary matter is x-polarized, and its clocks are slowed by velocity in the usual way. #13’s claim is that p-polarized systems, the program’s conjugate matter, are slowed by sustained force in the same way. The Abstract’s shorter phrase is “force-dilation of clocks”, and §7 lists “force dilation” among its fixed [D] results without the qualifier.

Open: which clocks respond#

The second half of #13 §3’s tag is open: “which physical clocks respond is polarization-dependent, O”. #13 does not say which physical systems would show force dilation, whether any x-polarized system would, or how the dependence on polarization would be decided. Nothing has been computed. The question is absent from the lists that would schedule it: #13 §7’s own open list (“the factor 4; the 4D covariant construction; dual dynamics; whether merger-band gravitational-wave data can bound dual-kinematic corrections”), #11 §14’s open problems, and the current queue in CLAUDE.md. The program’s other open items are collected in Open problems and the queue.

The mirror: contraction of momentum spread#

#13 §3, untagged:

“The mirror of length contraction: a forced system’s extent in momentum — its momentum spread — contracts by γ̃ in the force direction.”

In the template, a rod moving at velocity v is shorter along its motion by the factor γ than in its own rest frame, and its transverse size is unchanged [T]. The transcription replaces the rod’s extent in position by a system’s extent in momentum, the range of momenta it spans, and the velocity by the force: a system under the force F has its momentum spread along F reduced by the factor γ̃. The claim is not on §7’s fixed list and carries no tag of its own. #13 states it in one sentence and does not discuss how a momentum spread would be measured.

Size of the effects#

#13 §3 closes: “Both effects are unmeasurably small below merger-scale forces (§2).” The factor that governs both is γ̃ − 1 ≈ ½(F/c̃)², and §2 evaluates it for “the strongest sustained macroscopic force in nature — on a kilogram at a neutron-star surface, 10¹² N”, giving “ 10⁻⁶⁵” (Force boosts and force composition). The plain passage of §3 states the threshold as “Below 10⁴³ newtons the effect is utterly negligible, which is why nobody has noticed.” Only comparable-mass black-hole mergers, which #13 places at F ~ c̃, would bring the factor near 1 (Open: merger-band corrections). No verify/ script reproduces any of these figures (the iron rule).

The popular article #12 (‘A second speed of light’) says “Clocks subjected to sustained force run slow, by the same square-root formula that governs fast clocks, with force in place of velocity.” It drops both the restriction to conjugate-polarized systems and the open question of which clocks respond, and so states more than #13 does. #13 governs; #12 presents the framework as if it were confirmed and is never a source for status (The popular articles (#10, #12)).

Standing#

HANDOFF.md §2 lists “force dilation” under dual special relativity among the alive results, without a machine-check mark. #13 tags the dual proper time [D — structural] and lists force dilation among its fixed results, but no blind review round has examined #13, no verify/ script checks the formula or the numbers, and #11 v2.2 does not mention force dilation. The result is alive and unreviewed, and its physical scope, which clocks respond, is open (Conjugate special relativity).

Sources#

  • papers/conjugate-special-relativity.md §3 — the dual proper time, its scope and open clock question, momentum-spread contraction, the size of both effects; §2 for the neutron-star estimate and the merger arena; §7 for the fixed and open lists; Abstract for “force-dilation of clocks”.
  • papers/broken-born-checkpoint-v2.md §4 — postulate P4, the two polarizations of matter.
  • HANDOFF.md §2 — the alive listing.
  • popsci/universes-other-half.md ‘A second speed of light’ — the unqualified popular statement (not a status source).