A cosmological model can reach a boundary without answering every question about the beginning of reality. The boundary may mark incomplete coordinates, an incomplete spacetime, or a regime where the physical approximation fails. These possibilities require different arguments.
Ask what is approaching a limit
In a power-law model a(t) = (t/t₀)p, positive p makes the scale factor approach zero as t approaches zero. Whether that signals a physical singularity depends on the full geometry and its extension, not only on the appearance of a(t).
A simple coordinate example demonstrates why. In flat spacetime, set T = t coshχ and R = ct sinhχ, with t > 0. These coordinates cover the interior of the future light cone of an event. They give the Milne metric ds² = −c²dt² + c²t²[dχ² + sinh²χ dΩ²].
The spatial scale factor ct vanishes at t = 0. Nevertheless, the spacetime is part of ordinary Minkowski geometry with zero curvature. The underlying geometry extends beyond this coordinate patch. A vanishing scale factor alone therefore cannot be a sufficient test for an unextendible physical beginning.
This example does not remove singularities from realistic cosmology. It shows why a diagnostic must distinguish an incomplete description from an obstruction belonging to the spacetime itself.
The full metric changes the answer
Set c = 1 for the following calculation and choose the curvature convention in which the FLRW scalar is R = 6[ä/a + (ȧ/a)² + k/a²]. For Milne, a = t and k = −1. The terms are 0, 1/t² and −1/t², so R = 0.
Now keep a = t but set k = 0. The scalar becomes R = 6/t² and diverges at zero. The same functional scale factor has produced a different curvature because the spatial geometry changed. The cancellation was a property of the entire Milne metric.
Even a finite scalar R is not enough to establish regularity in every spacetime. Other curvature invariants, tidal effects and extendibility must be considered. A single convenient number is a diagnostic, not a universal definition of a singularity.
What a singularity theorem establishes
The Hawking–Penrose work derives geodesic incompleteness from specified geometric, causal and energy conditions. Roughly, at least one relevant freely falling path or light ray cannot be continued indefinitely within the spacetime, even though its proper time or affine parameter has only a finite range.
That is not automatically a theorem that all density scalars become infinite, or that there was creation from absolute nothing. Nor can its assumptions simply be omitted. A proposed modification must say which condition changes and whether the resulting model is actually complete.
The Milne patch helps distinguish issues: a restricted region can contain incomplete paths even when a larger regular spacetime is available. A physical singularity analysis must examine whether such an extension is allowed and meaningful for the model under consideration.
Describe a proposed continuation operationally
A bounce, an emergent phase or a different description of the earliest regime is more than a name for what happens beyond a boundary. It must provide equations, admissible states and a mapping to the later observables that make the proposal cosmological.
For example, suppose a rule replaces a singular scale factor by a smooth minimum. One must still check matter evolution, stability and perturbations. A smooth curve for a(t) alone does not ensure finite tidal effects or a successful spectrum of later structure.
In SMM/DTT, a generative description may be proposed as more basic than continuum geometry. The crucial bridge is then the recovery of a controlled effective metric where established physics works, together with explicit predictions where that approximation fails.
The book's discussion of cosmological beginnings distinguishes an early cosmic epoch from perpetual renewal. This allows the question of ongoing generation to remain meaningful without identifying each completion with a repetition of the hot Big Bang. The distinction sharpens both the physics and the philosophical inquiry.
Sources and further reading
- S. W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology, Proceedings of the Royal Society A 314, 529–548 (1970).
- Ned Wright, cosmological coordinates in flat spacetime. The coordinate substitution and curvature comparison above are worked examples.
- Continue with limits and idealizations and cosmology and creation.