Chapter II

Three Options.
One Survivor.

Once you accept Einstein’s two postulates, the algebra of spacetime turns out to have exactly three possible forms — one for each possible sign of the cosmological constant. The algebra itself decides which one is allowed.

Three spacetime curvature diagrams — 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 postulate doesn’t provide.

B(P_μ, P_ν) = −6Λ η_μν < 0

The Killing formThe Killing FormA mathematical tool invented by Wilhelm Killing in 1888 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 — the algebra sees the boundary.

✕ Violates Postulate 1. 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 Postulate 2 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.

✕ Violates Postulate 2. 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 radius L = 1/√Λ is set by the algebra itself — 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. Both postulates satisfied.

B(P_μ, P_ν) = −6Λ η_μν > 0

L = 1/√Λ

The Killing formThe Killing FormA mathematical tool invented by Wilhelm Killing in 1888 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.

✓ Both Postulates Satisfied

The Verdict Table

A Victorian-style assessment of 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\|	—	1/√Λ
Metric determined?	Yes	No	Yes
Globally hyperbolic?	No	Yes	Yes
Boundary conditions needed?	Yes	No (but scale undefined)	No
Postulate 1 satisfied?	No	Yes	Yes
Postulate 2 satisfied?	Yes	No	Yes
Both postulates satisfied?	No	No	YES

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 radius 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.

“The withdrawal was the blunder. Not the introduction.”

Three Options.One Survivor.

What happens

What happens

What happens

The Verdict Table

The Universe That Cannot Help But Grow

Three Options.
One Survivor.