Gödel’s completion of the theory of gravitationFFP14, Marseille, July 18th 2014

Eric Audureaueric.audureau@univ-amu.fr

CEPERC, UMR 7359-CNRSAix-Marseille-Université

29, avenue Robert SchumanF13621 Aix-en-Provence Cedex 1

“I have often pondered why Einstein took pleasure in his conversations with me, and I believe one of the causes is to be found in the fact that I frequently was of the contrary opinion and made no secret of it”

Gödel, September 5, 1955

The study of Gödel’s contributions to RT give us a way to recover a part of the lost conversations between the two great Philosopher-Scientists

Gödel’s education and interest in physics

After his graduation from the Gymnasium, Gödel began to study theoretical physics in 1924. In particular, he attends Thirring lectures on RT. His Nachlass shows that he had extensive readings in physics (theoretical and experimental), up to 1927.

Gödel always felt concerned by physics. In the early 30’, he was looking for integrals for Einstein’s field equations. When he was in Princeton, he used to attend lectures on physics.

One of the interest of Gödel’s work on RT, and this is not the least, is to give us a detailed illustration of what he meant when he said that his interest in precision led him from theoretical physics to mathematics and mathematical logic.

How Gödel came to publish on RT

Gödel to Schilpp (25-7-46):“…I am not in position to write an article of any considerable length for the Einstein volume. I could, however, if you wish, write about 3 pages under the title: “Some remarks about the relation between the theory of relativity and Kant (…), where I would treat this question in a positive sense pointing out the similarities, which seems to me incomparably more important than the differences. I am doubtful, however, if such a contribution would be of any use, for, since this question touches the very essence of “relativity”, namely the role of the observer, I presumed it will be treated in one or more of the longer articles already.”

Remark: to claim that there are “similarities between RT and Kant” is a first opposition to Einstein.

Gödel to his mother (7-11-47) :“… I was asked to write a paper for a volume on the philosophical meaning of Einstein and his theory; of course I could not very well refuse. I am also not sorry that I have accepted and chosen this theme [the relation of Kant to relativity theory], because the problem has always interested me and its fundamental investigations has in addition led to purely mathematical results…” (Wang 87, 38)

Why the “fundamental investigations” of his problem led Gödel to purely mathematical results?

Gödel first aim, in the Schilpp paper, was to show that “One of the most interesting aspects of relativity theory for the philosophical-minded consists in the fact that it gave new and surprising insights into the nature of time, of that mysterious and seemingly self-contradictory being which, on the other hand, seems to form the basis of the world and our own existence.”

Since, says Gödel, the discovery of the relativity of simultaneity is the very starting point of SR there can be no linear ordering of events but just a partial ordering of them. Hence, there can be no “layers of “now” which come into existence successively”. “Each observers has his own set of “nows”, and none of these various systems of layers can claim the prerogative of representing the objective [one]”.

Now, the notion of change depends of the notion of an objective lapse of time, which is an interval between two objective “nows”. Hence the notion of change is deprived of objective reality: it is an appearance or an illusion.

In GR, the same situation as in SR prevails. They are still different systems of “nows” associated to each observer. But with RC the situation changes: the local times of each (physically representative) observer fit together into one world-time. So, one could conclude that the idea of an absolute time lapsing objectively is justified.

Therefore, to make his point, Gödel had to show that, in general, in cosmological solutions of the equations of the gravitational field, the local times of each observer cannot fit together into one world-time.These were the mathematical investigations to which Gödel was led. They are partly reported in the two papers sketchy papers:

- “An example of a new type of cosmological solutions of Einstein’s field equations of gravitation” (1949)-“Rotating universes in general relativity theory” (1950)

It has been repeatedly said that the universe described in Gödel’s 1st paper was deprived of physical meaning, and that in his 2nd paper he proposed more reasonable models of the Universe. Why?

Meaning of the 2nd paper (“Rotating Universes”)In these universes, the motion of the cosmic fluid is described by three different non vanishing magnitudes:- a scalar of expansion- a vorticity vector- a shear tensor

These rotating solutions have either a contraction or an expansion or both at the same time, namely a contraction in one direction of space and an expansion in another direction.

It was really a new type of solutions of Einstein’s equations since in all the preceding ones, the so-called F-L-R-W models, vorticity and shear vanish. Thus, the F-L-R-W models are just a small and restricted class of the possible solutions.

In the Universe of Gödel 1st paper, there is neither expansion, nor shear. Furthermore, there exist closed time-like lines of matter. Gödel, in a lecture given in May 1949 at the IAS (posthumously published in 1996) mentions these two features (lack of expansion and travels into the past) as “shortcomings”. Then, why he published it?

“My own work on the theory of relativity relates to the pure theory of gravitation published in 1916 which, I believe, was left, not only by Einstein himself but also by the whole generation of contemporary physicists, in its state of a torso, physically, mathematically, and with respect to its application in cosmology.”September, 7, 1955

Gödel’s 2nd paper bears upon application of GR to cosmology.

Gödel’s 1st paper is just “an example” of “new type of cosmological solutions of Einstein’s equations”. It describes the pure theory of gravitation. Here, Gödel shows that Einstein’s law of gravitation can be deduced from a general condition of isometry on an homogeneous open space-time.

The relationship between the two papers can be described by an analogy:

1st paper is to 2nd paper what Mechanics is to Hydromechanics.

But, above all, it is a direct answer to Enstein’s attempt to complete the pure theory of gravitation published in 1916

The unfinished state of the 1916 theory according to Einstein

1) It is local. Namely, to treat the problem of planets Einstein applied the equations of GR in the flat space-time of SR. It is a kind of mixture, a piece of GR immersed in the manifold of SR.

2) SR space-time shares a feature with Newton space. One can, in theses framework, pictures the material world of celestial bodies as a floating island in a void space. This picture is not very satisfying for the mind but, above all, it raises a physical difficulty.

3) A part of radiation of the celestial bodies that are on the border of the “island” should get lost in the infinite void and even some of the celestial bodies could escape from the island. Thus this universe would become impoverished. In other words, there is no way to substitute a potential function (Poisson‘s equation) to Newtonian distant action, no way to define the potential at the boundaries of the island.

4) Inertia is influenced (beeinflusst) but not determined (bedingt) by matter. (be called “Mach’s Principle” in 1918)

Einstein’s completion of the 1916 theory: the cosmology of 1917

To get rid of all these puzzling questions which, of course, are linked together, Einstein proposed his famous cosmological model which has the following features

1) suppression of the boundaries; the World is a sphere.

2) addition of an ad hoc repulsive force

3) not generally covariant: introduction of an absolute time

Einstein did not want so much to give a System of the World, as we are told in histories of cosmology . His aim was to avoid the preceding difficulties (potential at the boundary) and to complete the theory of pure gravitation by explaining the concept of inertia.

Preliminary condition to Gödel’s criticisms of Einstein’s completion of the theory of gravitation

Theorem: Gödel’s stationary stationary universe and “Einstein’s static universe are the only spatially homogeneous cosmological solutions with non-vanishing density of matter and equidistant world lines of matter”

This shows that their comparison is justified.

Gödel’s criticism of Einstein’s completion “All cosmological solutions with non-vanishing density of matter known at present have the common property that, in a certain sense, they contain an “absolute” time coordinate, owing to the fact that there is a one parametric system of three-spaces everywhere orthogonal on the world lines of matter. It is easily seen that the non-existence of such a system of three-spaces is equivalent with a rotation of matter relative to the compass of inertia. ”

(1) a rotation of matter relative to the compass of inertia = negation of Mach’s Principle

(2) non-existence of such a system of three-spaces = non-existence of an “absolute” time coordinate

Einstein’s 1917 potential is not a potential. His universe was unstable. (The ad hoc repulsive force is proportional to distance; gravity is inversely proportional to the square of distance)

For Einstein, it was crucial that GR gives Newton’s law of gravitation as a limiting case.

Gödel’s reasoningSince Einstein’s universe has equidistant lines of matter, it can be considered as a rigid body (the simplest object of physics).

Gödel obtains his universe in making Einstein’s universe rotates like a rigid body. (Here is the relevance of the aforementioned theorem).

The classical and the relativistic expressions of angular momentum are identical. The value of angular momentum increases when the angular velocity increases. Gravity forces can be assimilated to the binding forces that hold together the particles of the rigid body. Thus gravity compensates centrifugal forces exerted by rotation. We have Newton’s law of gravitation for a given value of the rotation and Einstein’s law for an other (greater) value of rotation.

Gödel’ solution gives a potential

Hence, in one move (rotation), Gödel shown to his friend that it was possible to find a potential in a Space-Time which shares the main temporal property of SR, provided that one gets rid of the materialist ideas embedded in Mach’s Principle.

Far to be deprived of physical meaning, Gödel’s solution put an end to the difficulties of the theory of gravitation.

He did so on relying on a primitive of Newton’s Principia: there exists an absolute motion, namely rotation (of the universe).

Here is probably the most direct opposition to Einstein’s views

Two remarks to complete and to conclude with the genesis of Gödel’ works on RT, :

1) Before Schilpp asked him a paper for the Einsein volume, Gödel wrote in an unpublished booknotes (Max-Phil X): Bem<erkung> (Phys<ik>): Zwei Auffassungen der vierdimensionalen Welt. Entweder 1. als etwas starr Existierendes <oder> 2. mit einer dreidimensionalen Ebene, die sich darin „bewegt“ (oder überhaupt nur dreidimensional). One can easily recognizes in these two conceptions of the 4-dimensional World precisely the universes described in his two published papers. But if Gödel had not published these papers, the remark above would have remained incomprehensible for ever. 2) It is well known that Gödel was sceptic about the possibility of unified fields theories and critic about the received conception of Quantum Mechanics. The Phil-Max notebooks contains several remarks on QM, as enigmatic as the one above. It is to be regretted that no occasion led him to publish on QM.