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Positive averaged expansion implies past geodesic incompleteness under a kinematic condition

Contribution grade

Gravitation and cosmology / inflationary spacetime boundaries · Scoped theoretical result, model or research proposal

Reference score210,000Provisional editorial estimate of this exact contribution; its grade is assessed separately.

What changed

Borde, Guth and Vilenkin derived a kinematic bound on the past length of null and noncomoving timelike geodesics in an expanding spacetime. They express a directional expansion rate as the derivative of a function of the observer’s relative motion through a congruence of freely falling test particles. If the expansion averaged along the relevant past geodesic stays positive, the available affine length or proper time is finite. The argument does not require an energy condition, spatial homogeneity or isotropy, and constrains attempts to make inflation alone a past-complete description.

Scientific scope

Borde, Guth and Vilenkin derived a kinematic bound on the past length of null and noncomoving timelike geodesics in an expanding spacetime. They express a directional expansion rate as the derivative of a function of the observer’s relative motion through a congruence of freely falling test particles. If the expansion averaged along the relevant past geodesic stays positive, the available affine length or proper time is finite. The argument does not require an energy condition, spatial homogeneity or isotropy, and constrains attempts to make inflation alone a past-complete description. A suitable timelike geodesic congruence must remain defined along the observer path, and positive past-averaged expansion is an assumption on that path. Recent expansion or positivity over one finite interval is insufficient if the average tends to zero when extended into the past. The timelike bound requires nonzero relative motion at the reference event. Geodesic incompleteness of the described region does not establish a curvature singularity, rule out extensions beyond its boundary or prove an absolute beginning of the universe. Contracting, emergent and time-reversed scenarios require their own check of the averaging condition. Later papers debate the averaging conditions and their application to complete cosmological models; a 2024 paper proposes an amendment. The scope and implications of that exchange remain disputed.

Evidence

The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. A suitable timelike geodesic congruence must remain defined along the observer path, and positive past-averaged expansion is an assumption on that path. Recent expansion or positivity over one finite interval is insufficient if the average tends to zero when extended into the past. The timelike bound requires nonzero relative motion at the reference event. Geodesic incompleteness of the described region does not establish a curvature singularity, rule out extensions beyond its boundary or prove an absolute beginning of the universe. Contracting, emergent and time-reversed scenarios require their own check of the averaging condition. Later papers debate the averaging conditions and their application to complete cosmological models; a 2024 paper proposes an amendment. The scope and implications of that exchange remain disputed.

Contributors

Provisional WRS assessment, edition 2026-10-28.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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