Conjugate special relativity on The Broken-Born program

Special relativity: the template

Mon, 01 Jan 0001 00:00:00 +0000

Special relativity is Einstein’s 1905 theory of how measurements of length and time compare between observers moving at constant velocity relative to one another, built on one speed that all of them measure alike, the speed of light c. Its machinery (boosts, the light cone, the relativity of simultaneity, time dilation, length contraction and the mass shell) is textbook physics. The program uses it as a template: the paper #13 copies each piece into the momentum polarization, the program’s mirror description in which momentum plays the role of position, with force in place of velocity and a maximum force c̃ in place of c. This page states the template and lists the copies; #13 tags most of its copies as derived, but no review round or script has checked them, and the checkpoint #11 does not mention them.

Maximum force and maximal acceleration in the literature

Mon, 01 Jan 0001 00:00:00 +0000

The maximum-force conjecture is a proposal within general relativity that no force can exceed c⁴/4G, a combination of the speed of light c and Newton’s constant G; Gibbons named it the maximum tension principle, and Schiller took a principle of maximum force as the starting point for deriving general relativity. An older and separate proposal, Caianiello’s, is a maximal acceleration obtained from a metric on phase space. Both are proposals rather than settled results, and #13, the program’s paper on conjugate special relativity, notes that the maximum-force conjecture is contested. They matter to the program because #13’s maximum force, c̃ = c⁴/G, has the same form up to a factor of 4, and #13 offers a reason for such a limit. This page covers the literature; the Planck units it is written in are on their own page, and #13’s own claim is on the maximum force page.

Two causal cones

Mon, 01 Jan 0001 00:00:00 +0000

The two causal cones are the light cones of position space and of momentum space. In the program’s broken vacuum the metric on phase space has one block for position and the inverse of that block for momentum, and each block defines its own cone of allowed motion: the familiar cone limits speed, |dx/dt| ≤ c, and its twin limits the rate of change of momentum, |dp/dt| ≤ c̃. A rate of change of momentum is a force, so the twin of the speed of light is a maximum force. This is the starting point of #13, the program’s paper on the special relativity of the momentum polarization. It is alive but unreviewed: #13 tags its starting point as derived, no script checks the cones, and the checkpoint #11 does not mention them. #13 also uses two different versions of the momentum cone in different sections, an inconsistency that is open.

The maximum force c̃ = c⁴/G

Mon, 01 Jan 0001 00:00:00 +0000

The maximum force is the limit that the momentum-space light cone places on how fast momentum can change, as the ordinary light cone limits speed to c: since a rate of change of momentum is a force, the limit is a largest possible force, c̃ = ℏc/λ², where λ is a length that the paper #13 calls the Born length. Setting the Born length equal to the Planck length gives c̃ = c⁴/G, about 1.2 × 10⁴⁴ newtons, which #13 describes as “the scale black holes reach when they collide”. The general-relativity literature already contains a conjectured maximum force, c⁴/4G; #13’s value is larger by a factor of 4, and the factor is open. The result is alive but unreviewed: no review round has examined it, no script checks the number, and the checkpoint #11 does not mention it.

Force boosts and force composition

Mon, 01 Jan 0001 00:00:00 +0000

A force boost is the momentum-space counterpart of a Lorentz boost: where Einstein’s boost relates observers moving at constant velocity, the force boost of #13 relates an observer under no force to one under a constant force, with the maximum force c̃ in the role of the speed of light. The same algebra then gives forces that combine like relativistic velocities, so that no composition of forces passes c̃, and observers under different forces who disagree about which events are simultaneous when the events differ in momentum. At what #13 calls the strongest sustained macroscopic force in nature, on a kilogram at a neutron-star surface, the correction is of order 10⁻⁶⁵, which #13 gives as the reason force looks exactly additive; only merging black holes approach the limit. The results are alive but unreviewed: #13 tags them as derived, but no review round or script has checked them, the checkpoint #11 does not mention them, and no observable signature in merger data has been derived.

Force dilation

Mon, 01 Jan 0001 00:00:00 +0000

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.

The invariant-length shell and Planck closure

Mon, 01 Jan 0001 00:00:00 +0000

The invariant-length shell is the momentum polarization’s counterpart of the mass shell. Every observer assigns an ordinary particle the same mass; #13, the paper on conjugate special relativity, proposes that every observer of the mirror polarization assigns a conjugate particle the same size, an irreducible length λ_χ, and obtains this by applying Born’s exchange of position and momentum to the mass shell. Two consequences are drawn. A length times the maximum force is an energy, and at the Planck length it equals the Planck energy exactly, a result #13 calls closure at the self-dual point. And a fixed size in position fixes a width in momentum, as a short pulse fixes a broad spectrum; #13 identifies that width with the momentum scale of the dark-matter velocity gate, which makes the knee in dark-matter self-interaction a dual-relativistic invariant. The shell 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. Its description of dark-matter quanta by a length has not been reconciled with the checkpoint’s ordinary treatment of them, which governs by the wiki’s order of precedence.

Dual inertia: free fall as rest

Mon, 01 Jan 0001 00:00:00 +0000

Dual inertia is #13’s account of rest and uniform motion in the momentum polarization, the mirror description in which momentum plays the part that position plays for ordinary matter. An ordinary inertial body keeps its velocity constant. In #13’s dual kinematics the rate of motion through momentum space, dp/dt, is a force, and it takes the place of velocity, so the dual-inertial motions are those of constant force: zero force, which in general relativity is free fall, and a constant nonzero force, which is a uniformly accelerated (Rindler) frame. In #13’s words, “An astronaut coasting in orbit is, to the momentum polarization, at rest.” The identification of dual-inertial frames with free fall is on #13’s list of fixed results; the dual second law built on it is marked conjecture-tier, the dual first law carries no tag, and their formalization waits on dynamics that have not been written. Like the rest of #13, the material is unreviewed and unchecked by script, and the checkpoint #11 does not mention it.

The equivalence principle as dual relativity

Mon, 01 Jan 0001 00:00:00 +0000

This is #13’s claim that the equivalence principle of general relativity, which puts all freely falling frames on an equal footing, is the principle of relativity of the momentum polarization, the program’s mirror description in which momentum plays the part of position. In #13’s dual kinematics, rest means feeling no force, which is free fall; so the dual statement that physics is the same in all force-free frames is the equivalence principle, reached from kinematics rather than from gravitation, and #13 reads Einstein’s two relativity principles as the two polarizations of one statement. The same paper lists what the two polarizations share, the time coordinate and the massless graviton, so that both fall alike, and what the breaking of the symmetry makes different: our light cone is held in place by massless gravity, the dual cone’s dynamical part by the massive field X, so dual effects are short-ranged. The claim is alive but unreviewed: it is tagged as a structural derivation by #13 itself, no review round or script has checked it, and the checkpoint #11 does not mention it. The popular article #12 states it as settled, which is not its status.

Open: the two dual cones

Mon, 01 Jan 0001 00:00:00 +0000

This open problem is an inconsistency inside #13, the paper on the special relativity of the momentum polarization: two of its sections use different momentum-space light cones. Section 1 builds the dual cone in time and momentum, with time shared between the two polarizations and the maximum force c̃ as its slope. Section 6 instead calls light’s ordinary energy–momentum relation, E = c|p|, the dual null cone; that cone lives in energy and momentum, has slope c, and, in the analysis of the v3 worklog, belongs to a different construction, the four-dimensional Born map that also exchanges time and energy and has not been built. Section 6’s claim that photon-dispersion measurements test both cones at once depends on the second reading. Paul raised the problem on 2026-10-19, while the withdrawn soliton manuscript was being revised, with the instruction to fix the two cones and settle which construction governs before any paper claims more. It is on neither #11’s list of open problems nor the current queue, and no document since has taken it up.

Open: the four-dimensional Born map and dual dynamics

Mon, 01 Jan 0001 00:00:00 +0000

The four-dimensional Born map is the version of Born’s reciprocity transformation that exchanges time with energy as well as position with momentum. #13, the paper on the special relativity of the momentum polarization, names it as the fully covariant construction in which its kinematics belongs, and it has not been built. #13 itself works in a 3+1 presentation, in which space and momentum trade places while time is shared by both polarizations. The map is also a prerequisite for the dual dynamics, the field equations of the momentum polarization (“their Maxwell, their field theory”), which do not exist: only the dual kinematics has been written, and the dual second law that #13 sketches is marked conjecture-tier until it is. Both items are open in #13 §7, and neither is on the checkpoint #11’s list of open problems or on the current queue.

Momentum-space black holes (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

Momentum-space black holes are a conjecture from the program’s original claude.ai conversation. Read in the momentum description, the program’s mirror description in which momentum plays the part of position, a black hole’s horizon would be the surface at which hovering in place requires the maximum force of conjugate special relativity; and there may be mirror objects, regions trapped in momentum rather than in space, which the handoff document calls velocity traps. The handoff records the idea among the results that exist only in the chat, at conjecture tier, together with “dual Hawking quanta” that would be huge in position space. The popular article #12 repeats part of it and adds that such a mirror object would look like compact dark matter. No document derives any of it, it appears in neither the checkpoint #11 nor the paper on conjugate special relativity #13, and it is not one of the program’s dark-matter candidates. Its status is conjecture (chat-only).