Maximum force and maximal acceleration in the literature

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.

Status: Established as literature: the proposals are published, not settled, and the maximum-force conjecture is contested — [T], with both citations marked † (from model memory, not yet checked against the papers) — #13 notes the maximum-force conjecture’s contested status [T] without citing the dispute; the † references wait on the literature battery. #13’s use of the conjecture is its own, unreviewed and not machine-checked, and #11 v2.2 does not mention it.

Why c⁴/G is a force [T]#

The speed of light c and Newton’s constant G can be combined into exactly one quantity with the dimensions of a force, c⁴/G, and exactly one with the dimensions of a power, c⁵/G [T]. A bound on force built from classical gravity alone is therefore a pure number times c⁴/G, and a bound on luminosity a pure number times c⁵/G; the proposals in the literature, and #13’s, differ in that number. Planck’s constant ℏ does not enter either combination, so a limit of this kind is a statement about classical gravity.

The maximum tension principle [1†]#

#13 §1 introduces the literature in one sentence: “A maximum force c⁴/4G was conjectured within general relativity as the “maximum tension principle” [1†], with a companion maximum luminosity c⁵/4G approached by binary black-hole mergers”. Reference [1]† gives two papers:

  • G. W. Gibbons, “The maximum tension principle in general relativity,” Found. Phys. 32, 1891 (2002), hep-th/0210109;
  • C. Schiller, “General relativity and cosmology derived from principle of maximum force,” Int. J. Theor. Phys. 44, 1629 (2005).

As the two titles say, Gibbons stated the bound as a principle within general relativity, a limit on the tension or force that general relativity allows, and Schiller reversed the direction, taking a maximum force as the postulate from which general relativity and cosmology are to be derived [T]. The companion bound c⁵/4G is a limit on power, the luminosity of any source; #13 describes it as “approached by binary black-hole mergers”, that is, by the power two black holes radiate in gravitational waves in their final orbits and merger. The same mergers are the one natural process #13 places near its own limit (Force boosts and force composition).

Contested. #13 §1 notes “the conjecture’s contested status” in the general-relativity literature and tags the remark [T]. It cites no paper on either side of the dispute, and the wiki adds none. The conjecture is therefore recorded here as a proposal, not as a result of general relativity.

Maximal acceleration [2†]#

#13 §1 adds that “a maximal acceleration from a phase-space metric is older still [2†]”. Reference [2]†: E. R. Caianiello, “Is there a maximal acceleration?” Lett. Nuovo Cim. 32, 65 (1981).

As usually summarized (†, from model memory; #13 states none of it), Caianiello joined a particle’s velocities (for a particle of given mass, equivalently its momenta) to its spacetime coordinates and gave the combined space a single line element, so that position and velocity enter one geometry, as Born’s reciprocity puts position and momentum into one phase space (Born reciprocity). Requiring that line element to remain real bounds the particle’s proper acceleration, in the way the Minkowski interval bounds speed (Special relativity: the template). The bound so obtained involves ℏ and depends on the particle’s mass; it is not a single universal force. #13 does not state how its maximum force relates to Caianiello’s acceleration.

Planck units [T]#

The Planck length l_P = √(ℏG/c³) and the Planck energy E_Planck = √(ℏc⁵/G) are built from ℏ, c and G; their ratio is c⁴/G, the Planck force, which contains no ℏ, as the section above requires of any force built from c and G [T]. #13 sets its Born length equal to the Planck length, written λ² = ℏG/c³, which turns c̃ = ℏc/λ² into c⁴/G (§1; The maximum force c̃ = c⁴/G), and obtains the Planck energy as “E = l_P · c⁴/G = √(ℏc⁵/G)” (§4; The invariant-length shell and Planck closure). The units, the Planck mass and its two conventions are on Planck units and the two Planck masses.

What #13 takes from this literature#

  • The form of the limit. #13’s value c̃ = c⁴/G has the same form as the conjecture’s c⁴/4G and is larger by a factor of 4. #13 §1: “The factor of 4 is not fixed here [O]”; §7 lists “the factor 4” among its open items. HANDOFF.md §2 says c̃ “matches Gibbons–Schiller maximum-force conjecture up to the factor 4”.
  • A reason for the limit. #13 §1: “The framework’s contribution is a reason: a maximum force is the statement that nothing outruns light in the momentum polarization.” In #13 the limit is the invariant speed of a second light cone, the one of momentum space (Two causal cones).
  • Literature tethers. #13 §7 lists “maximum-tension conjecture [1†], maximal acceleration [2†], Born’s self-dual program [3†, 4†]” as the work it ties itself to.

These uses are #13’s claims, not results of the literature. #13 was never put through a blind review round, no verify/ script checks its numbers, and #11 v2.2 mentions neither the maximum force nor the conjecture.

Verification still to do#

All four of #13’s references are marked †: cited from model memory and to be checked against the sources before any external use. HANDOFF.md §8 item 6 asks for “every † reference in #11 and #13 source-checked”; in CLAUDE.md’s current queue the † literature battery is part of item 1. #11 §14 item 10 (OP10) carries the same task for #11’s own references, “Literature battery completion; every † reference source-checked [O]” (Open: the † literature battery (OP10)). #13’s [3]† and [4]† (Born, 1938 and 1949) are #11’s [1]† and [2]†; [1] and [2], the two references on this page, appear in no other document’s reference list; HANDOFF.md §2 names the “Gibbons–Schiller maximum-force conjecture” without a citation.

Used by#

Sources#

  • papers/conjugate-special-relativity.md §1 — the conjectures as #13 cites them, their contested status [T] and #13’s reason; §7 for the literature tethers and the open factor 4; References [1]†–[2]† for the citations.
  • papers/conjugate-special-relativity.md §4 — the Planck energy as l_P · c⁴/G.
  • HANDOFF.md §2 — “up to the factor 4”; §8 item 6 for the literature battery.
  • CLAUDE.md ‘Since the handoff’ — the battery as part of item 1 of the current queue.
  • papers/broken-born-checkpoint-v2.md §14 item 10 — the battery for #11’s references (OP10).
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