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Einstein's Transition From Light to Gravity

Spurred on by the controversy with Max Abraham, Einstein slowly began to realize that the approach to gravity was not in a variable speed of light, as in his 1912 paper, but by some tensorial field that would consist of 10 independent components and characterize the geometry of space time.

In a letter to Arnold Sommerfeld, dated 29/10/1912, Einstein admits that he has "gained great respect for mathematicians, whose more subtle parts I considered until now, in my ignorance, as pure luxury! Compared to this problem, the original theory of relativity is childish."

The original "relativity" consists in the relative motion of two particles, reference frames, etc., in relative motion to one another. As Minkowski showed, they obeyed equations which resembled a rotation in space and time through an imaginary angle (hyperbolic geometry!). He found that they could be combined into an invariant

ds^2= c^2 dt^2 - dr^2 - r^2(dS^2), (1)

where S is an element of a 2-sphere. For light, (1) vanishes, and null geodesics are followed by "particles" of light.

Notice that the relative velocity of the particles, and any reference to them has disappeared from (1) leaving the hyperbolic distance as the only invariant parameter. For non-null geodesics, it is conventional to divide (1) through by dq^2, where q is some "affine" parameter. The prescription for choosing the affine parameter is that the null vector, dx^a/dq, should preserve its length under parallel displacement.

Thus, we can consider the invariant lagrangian

L=c^2(dt/dq)^2- (dr/dq)^2- r^2(dS/dq)^2, (2)

and its variation, dL=0, as deriving the Euler-Lagrange equations of motion--which are geodesic equations. Einstein's idea was to use (2) to describe the propagation of gravitational waves--at a constant speed c--by suitably generalizing it to read

L = g_(00)(dt/dq)^2- g_(11)(dr/dq)^2- .... (3)

where the g_(ij) are the so-called gravitational potentials that Einstein needed to find by inventing his "field" equations. The generalization consisted in that they are now functions of the radial coordinate, r, but not time since the gravitational field was considered static.

Now, why should (3) describe the propagation of a gravitational field just because the gravitational potentials may depend on Newton's potential? Fermat's principle considers

integral n(r)ds = extremum (4)

where n(r) is the index of refraction of the medium through which light is passing through. If the motion can be limited to the plane theta=pi/2, Fermat's principle reduces to

integral n(r){1+(phi)'^2}^(1/2) dr = extremum. (5)

With the generalized lagrangian (5), Einstein was no longer considering the gravitational field to be represented by a variable speed of light. But, it was the speed of light---not the speed of the gravitational field! Now, exactly when did c become the speed of the propagation of gravitation?

In treating his linearized equations in the weak field limit (weak with respect to what?), and in the small velocity limit (velocities of what? there is a single mass only, and that is stationary), Einstein solved his field equations in an analogous way to the linear field equations of electromagnetism. Like electromagnetism, the gravitational interaction is not transmitted instantaneously, but there is a delay due to its finite propagation speed. Now, what was the propagation of an electromagnetic wave through a non-uniform medium due to the presence of a gravitational potential has now become the propagation of the gravitational fields themselves--and at the speed of light. So the transition was accomplished to the propagation of plane gravity waves travelling at the speed of light, c, in Minkowski flat space-time (Einstein, "Naherungsweise Integration der Feldgleichungen der Gravitation" (1916).)

Electromagnetism need two potentials: the Coulomb electrostatic potential and the vector potential, introduced by Maxwell. The latter is a function of time, and its time derivative enters into the expression of the electric field. This destroys the exactness relations that would otherwise be obeyed for the gradience of the Coulomb potential. Neither Einstein, nor any one else, has succeeded to introduced a gravitomagnetic potential--except from the analogy between the linearized field equations and Maxwell's equations. The time rate of change of the vector potential, which to Maxwell expressed the electrokinematic momentum of the electromagnetic field, is responsible for induction: A charge in motion creates an accompanying magnetic field, as a varying magnetic field in time creates a flow of electric current.

There is nothing in the theory of gravitation that would even remotely resemble a magnetic field. Adding the prefix "gravito" is pulling the wool over the uneducated, layman's eyes. Because you can linearize the Ricci tensor in 4D so that it looks like a wave equation does not mean that it has anything to do with reality. If we work from the Riemann tensor, and not its averaged Ricci tensor, we would be dealing with sectional curvatures; that is, curvatures in the different planes that can be drawn. The Schwarzschild metric has taught us that these sectional curvatures represent tidal forces in 3D--not 4D which has no meaning for tidal forces. They can be referred to the principal axes of an ellipsoid such that their sum vanishes, as it should in a state of dynamical equilibrium. Who then needs an averaged quantity like the contraction of the Riemann tensor to obtain the Ricci tensor?

The nature of tidal forces is quite distinct from the non-plastic forces that are not described by a symmetric second-rank tensor? In linear elasticity theory, the answer was found by Saint-Venant in 1864. He found the condition was given by the vanishing of the curl of the curl of the strain tensor. (In gravitation the displacement field would be replaced by the gravitational acceleration.) In 3D, this condition is simply the vanishing of the Einstein tensor,

G_(ij) =R_(ij) - (1/2)R g_(ij). (6)

where R_(ij) is the Ricci tensor, and R is the scalar curvature. It can hardly be considered a coincidence that the vanishing of (6) is the condition for the description in terms of tidal forces--entirely equivalent to the existence of a strain tensor.

All this may be very well, but general relativity operates in 4D and not 3D. Tidal forces have no meaning in 4D. Moreover, the gravitational potential is static, so what does the addition of a time dimension do? One can always take time "slices" as they do in numerical relativity, but who needs the slicing since the Newtonian gravitational potential is static? And what would allow it to propagate like an electromagnetic field? Without the phenomenon of induction, where one transverse field's variation in time causes a curling of the other transverse field in space, both in the direction normal to the propagation of the wave.

Observe that the equivalence with the Saint-Venant operator and the Einstein tensor can only occur to first-order in the perturbation of the "gravitational" tensor about the flat metric. In other words, the Saint-Venant operator is the first variation of the metric tensor of the Einstein tensor evaluated in flat space (i.e., the Minkowski metric without the time dimension, or flat Euclidean space).

A tidal force description is one in which when the perturbation is removed, the system relaxes to its unperturbed state. Rather, one in which the perturbation is permanent, or plastic, means that the Einstein tensor does not vanish and requires a "non-compatibility" condition. This arises in the presence of angular momentum, which together with the gravitational force, would lead to closed (or nearly closed) Keplerian trajectories.

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