The Overlooked Principle of Einstein: Unveiling the Birth of the Cosmological Constant

[Un article de The Conversation écrit par Olivier Minazzoli – Astrophysicien, Université Côte d’Azur]

For Isaac Newton, no movement can be designed without the notion of a rigid and absolute space, serving as a reference to define the trajectory of objects. In contrast, for Christian Huygens, Gottfried Wilhelm Leibniz and, later, Ernst Mach, these are the other objects that serve as landmarks. Consequently, no movement should be able to be defined in the total absence of material: remove any material reference, and it becomes impossible to say whether an object is in motion or at rest.

Influenced by Mach, Einstein erected this point of view in principle founding of his theory of general relativity. He mentioned the idea for the first time in 1912, then described it in turn as “hypothesis”, “postulate”, before finally calling it “Mach principle” in 1918, in homage to the Austrian physicist and philosopher who had fueled his reflection.

At first glance, general relativity seems in accordance with this principle, insofar as space is no longer the rigid and absolute structure of Newton, but a dynamic object influenced by matter. Indeed, in the theory of Einstein, gravitation is not a force properly speaking, but the manifestation of the curvature of space and time, induced by matter: any source of energy or mass deforms space-time, and it is this curvature which rules the movement of bodies.

Nevertheless, Einstein was aware that, to fully satisfy this principle, it was not enough for matter to influence the geometry of space-time: it had to determine it entirely. This is what led him to modify his equations by introducing the cosmological constant – which we interpret today as the dark energy, responsible for the acceleration of the expansion of the universe.

Why Einstein introduced the cosmological constant

Two problems seemed to arise in Einstein in the initial version of his theory of general relativity, without this constant.

The first is that his equations admit emptiness solutions: they allow, in theory, the existence of a space-time even in the complete absence of matter. This directly contradicted his “Mach Principle”. He also wrote in 1918:

“In my opinion, the theory of general relativity is only satisfactory if it shows that the physical qualities of space are entirely determined by matter alone. Therefore, no space-time can exist without the material that generates it. Albert Einstein, Proceedings of the Royal Academy of Sciences of Prussia, 1918.

The second problem concerned the limits of space-time, that is to say, of the supposed structure of infinity space-time. It seemed necessary to introduce them, but they were either incompatible with its principle or led, he thought, predicting a movement of the distant stars which was not observed. It is perhaps a reinterpretation of this difficulty that the idea was born, today widespread, according to which Einstein would have introduced the cosmological constant to conform his theory to a preconceived vision of a static and therefore eternal universe.

A finished universe but without edge

As he explains in the article where he introduces the cosmological constant, this makes it possible to avoid the use of problematic limits, by authorizing a finished but without edge universe.

To imagine what a “finished but without edge” universe is, one can think of the surface of a sphere. It does not have a border – you can circulate without ever reaching the edge – and yet its surface is finished (equal to four times the ray on the square).

This is how Einstein invented the very first physical model of the universe as a whole! It is a spherical universe, in balance, with a uniform distribution of matter. If its model is that of a static universe (that is to say, which does not evolve over time), it is clear that for Einstein it is only a first approximation which makes it possible to solve the equations.

A theoretical impasse

But we know it today: the Einstein model does not correspond to our universe. What is more, it is theoretically unsatisfactory, because unstable: any gap, even minimal, to the homogeneous approximation that it uses leads to predict its destruction. However, our universe is certainly not exactly homogeneous!

Then, it is now established-since the Dutch astronomer Willem of Sitter demonstrated it-that, even with a cosmological constant, general relativity admits solutions in which space-time can, in theory, exist without any form of matter.

From Mach to dark energy: the ironic rehabilitation of a constant

The introduction of the cosmological constant therefore fails to make general relativity compatible with the principle of Mach-which was, let us remember, the initial intention of Einstein by introducing the cosmological constant in his equations. It is probably for this reason that Einstein will later qualify his introduction as “the biggest error in his life” (words relayed by George Gamow in his autobiography).

This forgotten principle thus prompted Einstein to think of the universe as a whole, giving birth to the very first mathematical solution of a universe in general relativity – and therefore to modern cosmology. By an ironic chain of scientific history, he led him to introduce a constant for a reason which, in the end, will not work – but whose adequacy to observations will be confirmed later, in 1998, by the observation of distant supernovae. This discovery will be worth in Saul Perlmutter, Brian Schmidt and Adam Riess the Nobel Prize in Physics in 2011, almost a century after the development of Einstein's theory.

The role of hypotheses in science

The capacity of the human mind to conceive of abstract principles which can reveal the structure of reality questions. How an idea a prioriborn without empirical data, can she lead to an objective law of the world?

This is simply explained simply: scientists formulate hypotheses, principles and models – some fail and fall into oblivion, others prove to be fruitful. This is what the beautiful image of Novalis thinker suggests:

“The hypotheses are nets: only those who throw them catch something.” Novalis, The disciples at know and other fragments, thoughts on philosophy, love and religionpage 68, 1802.

It was by throwing his that Einstein gave birth to modern cosmology.