The Equivalence Principle Determines the Quadratic Form in Finslerian Extensions of Gravity
Year: 2026
Type: preprint
Abstract: We prove that the equivalence principle, in its Finslerian formulation, forces the effective metric on which matter propagates to be independent of the four-velocity. When applied to Finsler functions of the form F² = H f(β²/H), where H ≡ −g_μν ẋ^μ ẋ^ν and β ≡ A_μ ẋ^μ, this condition imposes f″ = 0, and hence f(x) = 1 + cx. After a suitable renormalisation of the vector field A_μ, one obtains the unique quadratic form F² = α² + β². This result converts the quadratic Finsler function from an arbitrary choice into a necessary consequence of a fundamental physical principle. The mathematical ingredient underlying the proof is the vanishing of the Cartan tensor, which is the standard condition for a Finsler metric to reduce to a Riemannian one. What is new here is not the mathematical fact itself, but its physical interpretation: the equivalence principle provides a fundamental reason why the quadratic form is singled out. The result is established entirely within the ansatz F² = H f(β²/H), and no additional dynamical assumptions are invoked.
Author: Abdel-Sattar, Mahmoud F.
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225259546
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