Before you can ask what the cosmological constant is — or what its sign must be — you need to understand the rules Einstein set for himself. He didn’t build general relativity by guessing. He started with two inviolable principles and derived everything else from them.
The paper by Emad Mostaque shows that these same two principles — written in 1905 and 1907 — already contained the answer about Λ. Not as a consequence you had to look for. As a logical necessity you couldn’t escape.
The Relativity Principle
“The laws of physics take the same form in all inertial frames and at all spacetime points.”
In plain English:
No special place. No special time. No preferred frame of reference. The rules apply equally everywhere and for everyone.
What it prohibits:
Any spacetime that requires information flowing in from a special boundary — a “wall” at the edge of the universe — would violate this principle. The laws would not be the same near the wall as they are in the interior.
The Equivalence Principle
“Gravity is geometry. The metric must emerge from the framework — not be assumed independently.”
In plain English:
The “ruler” of spacetime — the thing that tells you how far apart two events are — must come out of the physics itself. You’re not allowed to declare it from outside.
What it prohibits:
Any spacetime whose algebra can determine shape but not scale— that needs an external number plugged in to set the ruler — violates this principle. The metric must be self-determined.
The Key Implication
Both postulates are prohibitions: they forbid external structure. Any theory that needs data from outside itself to work violates them. This is the test that the cosmological constant must pass. The paper shows that Λ > 0 passes both tests — and that Λ ≤ 0 fails at least one.
Albert Einstein
“The laws of physics take the same form in all inertial frames and at all spacetime points.”— Albert Einstein, Postulate 1, 1905
“Gravity is geometry. The metric must emerge from the framework, not be assumed independently.”— Albert Einstein, Postulate 2, 1907 (paraphrase)
This is the precise condition that eliminates Λ = 0: when Λ is zero, the algebra cannot determine the scale of the metric from within the framework. The ruler must be supplied from outside — exactly what this principle forbids.
These two rules narrow the possible universes to exactly three. In the next chapter, we meet all three — and watch two of them fail.
Three Universes →