Chapter III

Three Options.
One Survivor.

Once you read Einstein’s founding relativity principleA PostulateA foundational assumption accepted without proof, the bedrock on which a theory is built. In this paper's framing, Einstein's founding postulate is the relativity principle: physics admits no preferred frame or external structure. Read precisely enough, that single principle rules out boundary-dependent universes and also forces the metric's scale to arise from within the framework, selecting Λ > 0. precisely enough, the algebra of spacetime turns out to have exactly three possible forms, one for each possible sign of the cosmological constant. Only one is both self-contained and internally scaled.

Three spacetime curvature diagrams showing Anti-de Sitter, Minkowski, and de Sitter

Three possible spacetime geometries. Left: walled, constrained (Λ < 0). Centre: flat, undetermined (Λ = 0). Right: open, self-contained (Λ > 0).

Λ < 0

Anti-de Sitter Space

A universe with walls

Timelike boundary walls at the edge, information flows inward

What happens

When Λ is negative, the resulting spacetime, called Anti-de Sitter space, has a boundary at the edge: a timelike wall. Information can flow in from this wall. The physics isn’t self-contained. It needs boundary conditions that the relativity principle does not provide.

B(P_μ, P_ν) = 6Λ η_μν

The Killing formThe Killing FormA mathematical diagnostic rooted in Wilhelm Killing's 1888 work and developed centrally by Élie Cartan to measure the 'size' or 'non-degeneracy' of a symmetry algebra. Applied to spacetime, it produces a tensor that either vanishes or does not. When it vanishes on the translation generator P₀, it signals a missing scale, a universe with no built-in unit of length. When it does not vanish, the universe has a natural ruler baked in. is negative on translations, so the algebra sees the boundary.

✕ Not self-contained under relativity. Eliminated.

Λ = 0

Minkowski Space

A universe without a ruler

Clean flat grid, but the scale bar is missing

What happens

When Λ is zero, the algebra can determine the shape of spacetime but not its scale; it knows what the metric looks like but not how big it is. The ruler has to be supplied from outside. That’s exactly what the strong reading forbids. The algebra goes blind on the translations.

B(P_μ, P_ν) = 6Λ η_μν = 0

When Λ = 0 the Killing form vanishes on translations. The algebra can’t see them. The scale is undetermined.

✕ Needs an external ruler. Eliminated.

Λ > 0

de Sitter Space

A universe that carries its own ruler

Open, expanding, self-contained, the curvature radius emerges from within

What happens

When Λ is positive, the algebra’s diagnostic registers a positive value on the translations. The curvature radiusCurvature Radius LA length that describes how tightly spacetime is curved. Derived directly from Λ: L = 1/√Λ. A very small Λ means a very large curvature radius, spacetime is nearly flat on human scales, which is why we don't notice it. L ≈ 9.3 × 10²⁵ m (≈ 10 billion light-years) — the built-in curvature scale of the vacuum set by Λ. For comparison, the observable universe has a radius of ≈ 47 billion light-years. L = 1/√Λ is set by the algebra itself, with no outside input needed. The universe is globally hyperbolicGlobal HyperbolicityA technical requirement for a spacetime to have consistent, deterministic physics. A globally hyperbolic spacetime is one where, given initial conditions on a single moment of time (a 'Cauchy surface'), the entire future and past of the universe can in principle be calculated without needing information from some external boundary. Our universe is believed to be globally hyperbolic. Anti-de Sitter space is not, it has a boundary at spatial infinity that can send signals back inward.: you can specify the state of the whole universe on a single time-slice and predict everything else. Self-contained. Every internal requirement satisfied.

B(P_μ, P_ν) = 6Λ η_μν

L = 1/√Λ

The Killing formThe Killing FormA mathematical diagnostic rooted in Wilhelm Killing's 1888 work and developed centrally by Élie Cartan to measure the 'size' or 'non-degeneracy' of a symmetry algebra. Applied to spacetime, it produces a tensor that either vanishes or does not. When it vanishes on the translation generator P₀, it signals a missing scale, a universe with no built-in unit of length. When it does not vanish, the universe has a natural ruler baked in. is positive on translations. The algebra knows its own scale.

✓ Survives Every Requirement

The three candidates

Property	Λ < 0	Λ = 0	Λ > 0
Killing form on P₀	Non-zero	Zero	Non-zero
Algebra type	Anti-de Sitter (AdS)	Poincaré (flat)	de Sitter (dS)
Curvature radius L	1/√\|Λ\|	None	1/√Λ
Metric determined?	No	No	Yes
Globally hyperbolic?	No	Yes	Yes
Boundary conditions needed?	Yes	Yes (external scale required)	No
Self-contained under relativity?	No	No	Yes
Built-in scale from within?	Yes	No	Yes
All requirements satisfied?	No	No	YES

“We finally infer that boundary conditions in spatial infinity fall away altogether...”

Albert Einstein, Kosmologische Betrachtungen, 1917

Victorian astronomical diagram showing an expanding universe

The Universe That Cannot Help But Grow

de Sitter space, the universe with Λ > 0, is not just “allowed” to expand. It is fundamentally, algebraically required to. The curvature radiusCurvature Radius LA length that describes how tightly spacetime is curved. Derived directly from Λ: L = 1/√Λ. A very small Λ means a very large curvature radius, spacetime is nearly flat on human scales, which is why we don't notice it. L ≈ 9.3 × 10²⁵ m (≈ 10 billion light-years) — the built-in curvature scale of the vacuum set by Λ. For comparison, the observable universe has a radius of ≈ 47 billion light-years. L = 1/√Λ is determined internally. The expansion rate H = √(Λ/3) follows. These are not facts you add to the theory. They emerge from the theory itself.

Einstein’s 1917 introduction of Λ was not a patch or a fudge. It was the only self-consistent choice.

Three Options.One Survivor.

What happens

What happens

What happens

The Universe That Cannot Help But Grow

Three Options.
One Survivor.