Gerber's theory of gravity

Long time I considered Gerber's work as frivolous... until I read that the light deflection in the gravitational field in the framework of his theory would be larger by half than that of Einstein. (Roseveare, N. T: Mercury's perihelion from Leverrier to Einstein. Oxford: University Press 1982). This has me thinking, I already knew that in the immediate vicinity of the sun, the observed bending of light rays is much larger than predicted by the theory of relativity. Schmeidler in 1984 calculated the average additional of 0.3" or + 17% and in 1956 Michailov reported that in Russia in 1937 the light deflection of 2.74" or +57% was measured.

Paul Gerber recognized two factors that can influence the Newtonian law of gravity:

„First, the potential must indeed begin to form at a distanceof the masses, whereis positive whenincreases and negative when it decreases, to achieve stationary size in the inverse proportion of, because it would otherwise not be possible to see how this relationship could be fulfilled when the masses are at rest. But the effect is not immediately imposed on, because the conditional process starts from the attracting mass and needs time in order to proceed to the attracted mass. Of course, the same kind of progression also occurs from the attracted to the attracting mass... The outbound potential at the distanceof the attracting mass manifests itself therefore inonly at a later time, after the distance has become equal to.

Second, the potential as a long-distance effect would indeed appear immediately in its full amount; however, time and space are to be taken into account in the assumed way, so a certain duration of time is also necessary, so that, once arriving atthis mass influences, i.e. causes the corresponding state of movement of... If the masses are at rest, the movement of the potential tohappens at its own pace; then the value transferred onis defined by the inverse proportion to the distances. When the masses are hastening towards each other, the time of the transfer is reduced, and thus also the value of the potential transferred in proportion of the speed of the potential itself to the sum of this speed and the speed of the mass, since the potential has this overall speed in relation to.“ Paul Gerber, Die räumliche und zeitliche Ausbreitung der Gravitation. Zeitschrift für Mathematik und Physik. 43, 1898, S. 93-104, Translation of Hadley Jones.

As I am reading, I think I understand the reasoning of Paul Gerber.

According to Oppenheim, the idea for the first factor was originally the idea of ​​Neumann: the potential of the mutual attraction of two masses requires some time to get from one mass to another mass. This correction factor is determined by

( - radial velocity.)

The second factor is however the invention of Gerber:  the duration of the impact of gravitational potential. According to Gerber, the gravitational interaction has the constant speed (of light) only relative to the mass, from which it emanates. This leads to greater "exposure time" during the removal of the masses of each other and vice versa. For better understanding let's look at the picture below.

How much time needs a change of field from the massto run by the masswith diameter ?

At rest:

When running togetherness (left):

When running away (right):

Obviously the same holds for field change of the massby masse . With the consideration that the radial velocity is positive with the increasing radius, we get the expression for the second factor:

Together, these two factors make the potential:

Gerber's derivation of the formula for the perihelion shift requires neither relativity nor relativistic space -time continuum, but a little care. I would like to suggest the main lines.

Gerber's gravitational potential can be represented through binomial series:

Because here other than the function ofstill exists the dependence of, Gerber used general Lagrangian equation of motion to find the acceleration:

After some calculations Gerber could get to the formula for elliptical planetary orbit:

Here arereduced mass,angular momentum, M and N Integration constants and

Now is first printed through the ellipse parameters. Figure below shows the names of the orbital parameters, which Paul Gerber used.

The comparison with the general form of the orbit equation

and corresponding calculations provide

To find the change of, so the perihelion shift, it could be necessaryto identify from the left side:

are large and small half-axes. It follows

Finally, Gerber finds:

The integration provides the perihelion shift per revolution:

This formula helped Gerber in 1898, the perihelion shifts of the planet to calculate. The result was consistent with the observations. 18 years later appeared the same formula in General relativity.

Walter Orlov, 2011

Paul Gerber