Why did Einstein Reject his Field Equations of General Relativity in 1912 only to Come Back to them in 1915?

It has to do with Newton:

mw04660

Sir Isaac Newton, by Sir Godfrey Kneller, Bt. oil on canvas, feigned oval, 1702.

In a forward from 1931 to Newton’s book, Opticks Einstein wrote: “In one person he combined the experimenter, the theorist, the mechanic, and, not the least, the artist of exposition”.

Starting in summer 1912, Einstein undertook a long journey in the search for the correct form of the field equations for his new gravitation theory. He began collaborating with his friend from school Marcel Grossmann.

In 1912 Einstein wrote a form of the Ricci tensor in terms of the Christoffel symbols and their derivatives. This was a fully covariant Ricci tensor in a form resulting from contraction of the Riemann tensor.

Einstein then considered candidate field equations with a gravitational tensor constructed from the Ricci tensor. This gravitational tensor was called by scholars the “November tensor”. It transforms as a tensor under unimodular transformations.

Setting the November tensor equal to the stress-energy tensor, multiplied by the gravitational constant, one arrives at the field equations of Einstein’s first paper of November 4, 1915.

Einstein hoped he could extract the Newtonian limit (i.e. the Newtonian gravitational field equation, Poisson’s equation for gravity) from the November tensor. Poisson’s equation for gravity: the Newtonian gravitational field equation for gravity; the potential at distance from a central point mass (the fundamental solution) is equivalent to Newton’s law of universal gravitation. But Einstein found it difficult to recognize that the November tensor reduces to the Newtonian limit. Thus, he finally chose non-covariant field equations, the so-called Entwurf field equations.

In 1913 Einstein and Grossmann wrote a joint paper in which they established the Entwurf field equations through energy-momentum considerations instead of the November tensor; these equations could also reduce to the Newtonian limit. Unfortunately, these equations were not covariant enough to enable extending the principle of relativity for accelerated motion and did not satisfy the equivalence principle.

Towards the beginning of November Einstein was led back to his starting point, namely to the November tensor of 1912. He gradually expanded the range of the covariance of his gravitation field equations. Every week he expanded the covariance a little further until, on November 25, he reached his fully generally covariant field equations.

Einstein explained to his colleagues that he had already considered the November tensor with Grossmann three years earlier (in 1912). However, at that time he had concluded that these field equations did not lead to the Newtonian limit. Einstein now understood that this was a mistake. In retrospect, he realized that it was rather easy to find these generally covariant equations but, it was not easy at all and even extremely difficult to recognize that they are a generalization of the Newtonian Poisson’s equation and that they satisfy the conservation of energy-momentum.

In 1915 Einstein explained this point of view to David Hilbert:

Hilbert

The Collected Papers of Albert Einstein, Vol. 8.

Einstein told Hilbert that the difficulty was not in finding the November tensor.

In my paper, “Why did Einstein Reject the November Tensor in 1912-1913, only to Come Back to it in November 1915?”,  published a year ago in May 2018, in the journal Studies in History and Philosophy of Modern Physics:

An ArXiv PDF version of this paper.

studies

I ask three questions:

1) Why then did Einstein reject the November tensor in 1912-1913, only to come back to it in November 1915? Einstein explained that he found it difficult to recognize that the
November tensor reduces to the Newtonian limit.

2) Why was it hard for Einstein to recognize that the generally covariant equations are a simple and natural generalization of Newton’s law of gravitation?

3) Why did it take him three years to arrive at the realization that the November tensor is not incompatible with Newton’s law?

In my paper, I consider each of these three separate questions in turn. I examine these questions in light of conflicting answers by several historians: Michel Janssen, John Norton, Jürgen Renn, Tilman Sauer, and John Stachel.

Fast forward to the 1920s. In 1923 Élie Joseph Cartan provided a generally covariant formulation of Newtonian gravity (called the Newton-Cartan theory), though his formulation was much more complex than the generally covariant formulation of Einstein’s 1915-1916 general theory of relativity.

On a four-dimensional manifold, he introduced two geometrical objects the metric tensor and a symmetric affine connection. In general relativity, the metric tensor describes through the line element the behaviour of clocks (the chronometry) and measuring rods (the geometry) of space-time. We combine the one-dimensional chronometrical and the three-dimensional geometrical structures into the chrono-geometrical structure of space-time. In the four-dimensional version of Newtonian gravitation theory, the chronometry is represented by a scalar field on the four-dimensional manifold, called “absolute Newtonian time”, and the geometry by a degenerate metric field of rank 3. In the space-time formulation of Newton’s theory, proper time is not defined.

In Cartan’s formulation of Newtonian gravitation theory, the weak equivalence principle (Galileo’s law of free-fall) is taken as a principle: In a gravitational field, all local freely falling objects are fully equivalent. Free material particles follow the geodesic curve and a tangent vector field is parallel-transported along a geodesic curve. This is parallel transport of a tangent vector in an affine connection on a manifold.

Consider a freely falling particle moving in a gravitational field. We look first for the equation of the trajectory traced by the particle and then express this equation in geometrical language. A particle in free fall moves along a geodesic in space-time (in 3+1 space-time in Newtonian theory). The geodesic equation can be written in terms of the affine connection.

One cannot distinguish Newtonian inertial systems (material particles in free fall following geodesic lines in curved non-flat space-time) from local inertial systems (material particles in free fall following geodesic lines in flat space). We cannot, therefore, locally separate gravity from inertia. Gravity and inertia are described by a single inertio-gravitational field. We incorporate an inertio-gravitational field into the geometric formulation of Newtonian gravity theory. We cannot separate a flat affine connection of space-time from a field describing gravitation (a gravitational potential).

Cartan, therefore, found a general dynamical non-flat connection which represents both inertia and gravity. Rather than thinking of particles being attracted by forces, one thinks of them as moving along geodesic lines in curved, four-dimensional space-time. Therefore gravity is not a force anymore but is interpreted geometrically and should be considered as a curvature of space-time. The above non-flat affine connection defines the amount of curvature of geodesic lines. Cartan’s connection is still not the metric connection. The Christoffel symbols do not serve as the components of the connection (the Levi-Civita connection) and the connection is not written in terms of the metric tensor components.

Cartan introduced a Newtonian classical space-time: A four-dimensional manifold endowed with a Euclidean special metric and absolute time. Cartan then defined the total mass-momentum field associated with the matter field. To find the field equations he defined the Newtonian field equations similarly to the Einstein field equations: Field equations relate chrono-geometrical quantities and the non-flat connection to the matter distribution. Given the Newtonian Poisson equation (the mass density related to the gravitational field) the Poisson equation is now replaced with a generalized Poisson equation written in terms of the Ricci tensor and Cartan’s non-flat affine connection. The Ricci tensor is defined in terms of the Cartan connection. Finally, it turns out that the condition that space-time is flat is fulfilled.

The significance of Cartan’s formalism in general relativity is the following: Cartan’s affine connection is non-flat and cannot break up into a Newtonian flat affine connection and a gravitational potential. In a freely falling system, we cannot separate gravity from inertia. This embodies the inertio-gravitational field (equivalence principle). Accordingly, in the Newtonian limit, although we obtain Poisson’s equation and space-time is flat, Cartan’s affine connection remains non-flat, that is to say, the Ricci tensor is expressed in terms of Cartan’s non-flat connection. Thus, the four-dimensional formulation of the Newtonian Poisson field equation is a precise definition of the Newtonian limit of Einstein’s field equations. This actually finally solved Einstein’s problem of obtaining the Newtonian limit. Between 1912 and 1915 Einstein lacked Cartan’s formalism, i.e. more advanced mathematical tools and John Stachel argues that these could be responsible for inhibiting him for another few years. I extensively explain this in my second book: General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists

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