Tidal Forces, the Equivalence Principle, and the Emergence of the Einstein Field Equations from Worldline Non-Injectivity in de~Sitter Spacetime

This paper derives three foundational 
results of general relativity — tidal 
forces, the equivalence principle, and 
the Einstein field equations with 
cosmological constant — from a single 
geometric principle: worldline 
non-injectivity. A timelike worldline 
with Lorentz factor above a critical 
threshold intersects constant-time 
hypersurfaces in N > 1 distinct spatial 
points, generating a multi-sheet 
structure of spacetime. The gravitational 
field is encoded in the proper-time 
distribution across sheets.

The main results are as follows. Tidal 
forces emerge from proper-time gradients 
across extended bodies: the deformation 
condition ΔL/L = −Δτ/τ produces a strain 
tensor equal to the electric part of the 
Weyl tensor (Theorem 4.1). The Einstein 
field equations are derived — not assumed 
— from the topological average of the 
most general Lorentz-scalar Lagrangian 
built from the proper-time field with at 
most two derivatives (Theorem 5.1). The 
cosmological constant emerges as 
Λ_obs = Λ_0/N_0, finite and independent 
of the UV cutoff for d = 4, via the 
explicit scaling Λ_bare ~ ε^{-2} and 
N(ε) ~ ε^{-2}: no fine-tuning, no 
anthropic argument, no supersymmetry. 
The equivalence principle is derived in 
two logically distinct steps: classically, 
as the theorem that non-injectivity is 
locally removable for any smooth worldline 
(using only differential geometry, without 
ℏ); and as a quantum correction with 
minimum scale δ_min = λ̄_C/(πc), where 
λ̄_C is the reduced Compton wavelength.

The quantum correction predicts a 
violation of the Weak Equivalence 
Principle of order Δa/g ~ 10^{-13} for 
proton-electron comparisons — not 
excluded by MICROSCOPE (10^{-15}) and 
directly testable by the STE-QUEST 
mission (target 10^{-17}). This is the 
main falsifiable prediction of the paper.

The analysis is performed in de Sitter 
spacetime, the physically correct 
background for our accelerating universe, 
extending the TPST-DGQ framework from 
Anti-de Sitter to de Sitter and showing 
that the cancellation identity 
N(ε)·ε^{d-2} = O(1) holds universally, 
independent of the sign of Λ. Newton's 
constant G is the single external input, 
fixed by the Newtonian limit, in full 
analogy with Sakharov (1967), Jacobson 
(1995), and Verlinde (2011).

The paper is fully self-contained: an 
appendix provides the derivations of 
N(ε) ~ ε^{-(d-2)}, ℏ from fold 
stability, and the sheet-dependent metric 
correction δg^{(n)}_{μν} ~ ε^{d-2} from 
the Extended Lorentz Transformations, 
identical to those in the companion 
papers of the TPST-DGQ programme.

This is the ninth paper in the TPST-DGQ 
framework, which unifies holographic 
gravity, quantum mechanics, 
thermodynamics, and electromagnetism 
under the single principle of worldline 
non-injectivity.

This manuscript is current in Official Peer Review.

Not final version.
Copyright©2026 Alex De Giuseppe.
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