The Cosmological Constant by Emad Mostaque
Einstein called relativity a theory of principle, one where the principle itself constrains the physics, the way the axiom determines the mathematics. In 1917, he broke his own rule. He added a term to his field equations by hand, the cosmological constant Λ, because he needed the universe to sit still. A static universe that didn't collapse under its own gravity required something pushing back, and Λ was the push.
He didn't need to add it.
It was already there in his own theory, the only undetermined number the framework leaves open. The universe turned out to be expanding. Einstein withdrew the constant and reportedly called it his greatest blunder. But he was asking exactly the right question, and his own principle answers it.
One Number
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In 1998, two teams discovered that the expansion of the universe is accelerating. Riess, Schmidt and Perlmutter won the Nobel Prize for the discovery. Billions of dollars in telescopes have since been trained on a single number. That number is Λ.
In the 109 years since Einstein first wrote it down, nobody has derived its sign from first principles. This paper does. It was derived with the help of Logos, our first-principles reasoning system.
Self-Contained
In the same year Einstein introduced Λ, Kretschmann told him his reading of the relativity principle was empty. Any physical theory can be written in a form that looks the same in every coordinate system. You just dress up the equations. A criterion that everything passes distinguishes nothing. Kretschmann was right about the vacuous reading.
But Einstein was reaching for something deeper. The framework should determine the physics. No boundary conditions at infinity, no structure put in by hand. In the 1917 paper he wrote a sentence the rest of this story returns to.
"We finally infer that boundary conditions in spatial infinity fall away altogether, because the universal continuum in respect of its spatial dimensions is to be viewed as a self-contained continuum of finite spatial volume."
Self-contained. That is the precise reading.
If the algebra of spacetime symmetries has a sector it cannot see from within, whatever occupies that sector is smuggled-in scaffolding. Einstein kept finding hidden assumptions in his own foundations and throwing them out, absolute simultaneity in 1905, the preferred coordinate system in 1915.
The postulate, read precisely, requires two things of any framework. The algebra must be able to see all of its own operations from within, and the spacetime it produces must determine its own physics without requiring information from outside.
The Algebra Chooses
One Postulate showed that Einstein's relativity principle forces Lorentzian kinematics with a finite invariant speed. That fixes how observers moving at different velocities relate to each other, through three rotations and three boosts.
But observers can also be shifted in space and time. The laws of physics shouldn't depend on where you are or when you are, so the framework needs four more operations called translations, one for each direction of spacetime. How translations interact with the existing operations is already determined. The only question left is how translations interact with each other.
The answer is forced. The interaction must respect the Lorentz structure and be internally consistent, meaning the rules can't contradict themselves no matter how you combine them. All of this leaves just a single undetermined number. That number is Λ, the cosmological constant.
Every previous treatment, from Bacry and Lévy-Leblond in 1968 onward, has concluded that the algebra cannot choose. You have to go outside and measure, point telescopes at distant supernovae, build billion-dollar surveys of the sky.
What settles it is the Killing form, computed entirely from the algebra's own structure. For each generator it returns a value that tells you whether the algebra can see that operation or is blind to it. Apply it to time translation and the answer is 6Λ. The sign of that number decides everything.
Λ = 0. Spacetime is flat and infinite, the setting of special relativity as usually taught. But the Killing form returns zero on every translation. The algebra goes blind.
Every translation can be rescaled by any positive number without changing the algebraic relations, and the algebra cannot tell the difference. The shape of spacetime is fixed but not the scale. How big is a metre? The algebra cannot say. A framework that needs an external ruler is not self-contained.
Λ < 0. Spacetime curves the other way, and there is a boundary at spatial infinity. The algebra has no blindness here, but it has two problems.
It classifies time translation the same way it classifies a spatial rotation, so that "later" and "earlier" become like "clockwise" and "anticlockwise." The same algebraic rules can be realised in different ways, one in which time genuinely loops back on itself and one in which it doesn't. Same rules, different physics. The framework cannot decide between them.
The boundary at spatial infinity is timelike, which means information can flow in from the edge. The physics requires boundary conditions that the framework does not generate. Different boundary conditions, different physics, same algebra. This is precisely what Einstein was guarding against in 1917 when he introduced Λ in the first place.
Λ > 0. Spacetime has positive curvature. Space is finite but has no edge. Walk far enough in any direction and you return to where you started.
Everything works. Time extends indefinitely, never looping, never ambiguous, and every way of realising the algebraic rules gives the same physics. The algebra sees all of its generators and has no blind spots.
The algebra does something none of the others can. It builds a spacetime. Starting from a single origin, the symmetry operations sweep out a four-dimensional space of events, and the Killing form supplies the metric, fixing both the Lorentzian signature and the curvature scale. Nothing is assumed. The framework provides its own ruler, its own clock, its own geometry.
The result is de Sitter spacetime. Specify the state of the universe on a single spatial slice and the evolution is determined everywhere, past and future, with no boundary conditions, no information from an edge, no external input of any kind.
Positive
Einstein wanted the framework to determine the physics. It does. The sign of Λ was never free. It was fixed from the beginning.
The cosmological constant must be positive. The vacuum must be de Sitter. The universe must expand.