Is Gravitation a Longitudinal or Transverse Force--Or Both?

The idea behind Ampere's investigations was that charges in motion exert forces on each other and the force is directed along the line connecting the charges or the current elements which he dealt with. Historically, Ampere's work was anteceded by the experimentation of Biot-Savart who studied the interaction of currents and magnetics. It was Ampere's idea that a magnetic could be modeled as a small current loop. It is commonly knowledge that the two laws are equivalent when one of the current loops is closed. 

As common as knowledge is, this is wrong. A force which acts in the direction of the line connecting two current elements cannot be likened to the force normal to the current element and the line connecting the elements. Ampere's force is a longitudinal law, while Biot and Savart defined the magnetic field which is normal to the direction of flow of electric charges. In fact, the discovery of cathode rays led to the demise of Ampere's force since longitudinal currents were deemed disruptive to the flow of conduction currents. As a matter of fact, Ampere's law is related to the displacement current, but that's another story. 

About two decades later, Grassmann, unhappy with the angle dependencies in Ampere's law, developed his own law According to Grassmann's view, a pair of current elements do not attract or repel each there; each experience a force that is perpendicular to itself which is caused by the other circuit element.  The force on a current element, due to the other element, lies in a plane containing the direction of the current element and the line connecting the two elements. And because of Ampere's rule, the interaction of current elements can always be reduced to a two dimensional analysis in which all components normal to the plane produce no forces on the element in question. Therefore the interacting components of the Grassmann force all lie in the same plane.

So Grassmann's force, which was later to become the magnetic component of the Lorentz force abandons the description of the mutual attraction or repulsion of current elements, and violates Newton's third which states that for every action there is an equal and opposite reaction. This sounded the death knell of classical physics. And if special theory is an outgrowth of Lorentz's force, and general relativity an outgrowth of special relativity then gravity does not describe the attraction of masses at all.

In fact, general relativity is as close to Newton's law of gravitation as Grassmann's force is to Neumann's force. Grassmann's force is the triple vector product

$$\vec{F}_G=\frac{1}{r^3}\left(i'd\vec{s}'\wedge(id\vec{s}\wedge\vec{r}\right),$$

acting on $ids$ for current elements, $id\vec{s}$ and $i'd\vec{s}'$, and $\vec{r}$ is the radial vector that connects them. There are two components to Grassmann's force,

$$\vec{F}_G=\frac{ii'}{r^2}\left(d\vec{s}\cos\theta'-\hat{r}\cos\epsilon\right),$$

where $\theta'$ is the angle that element $id\vec{s'}$ makes with the unit normal, $\hat{r}$, along the line connecting them. The angle $\epsilon$ is the angle formed from the continuation of the lines directed along the two current elements. It is the second term that is related to the Neumann force

$$F_N=\frac{ids\cdot i'ds'}{r^2}\cos\epsilon.$$

Whittaker in his A History of the Theories of Aether and Electricity  shows that the other term in Grassmann's force vanishes when summed over a close circuit. If this term is related to relativistic effects then what Whittaker showed was that all relativistic effects vanish upon closing the circuit. This is independent of the magnitude of the relative velocities.

What we shall now show is that Ampere's law gives the exact perihelion shift for Mercury if the ratio of electromagnetic to electrostatic units is reduced by a factor of 3. Now Ampere's law is a longitudinal, ponderomotive force, and not an electromotive force,  The adjective 'longitudinal'  is superfluous since all ponderomotive forces lie along the line connecting the masses.  

When Einstein determined the advance of the perihelion of Mercury, it was well known the non-Newtonian advance of the orbital perihelion per revolution was $6\pi M/ac^2$, where $M$ is the mass of the sun, $a$ the semi-latus rectum of the orbit, and $c$ a propagation constant that relates the Newtonian potential to one that is a function of the velocity of the motion of the planet. So the challenge was to find a potential which gives this value.

According to A. A. Vankov (which we wholeheartedly agree) in his manuscript entitled "General relativity problem of Mercury’s perihelion advance revisited",
 

"... Mercury’s relativistic effect has never been directly observed and even not evaluated from circumstantial astronomical evidence. The matter is that the GR theory, at least as it given in literature, does not provide a clue about distinguishing between the classical drag along with the equinoxes precession, on the one hand, and relativistic effect, on the other hand. There is no other way but look for an admissible anomaly gap to be filled with the predetermined perihelion advance of 43′′ per century as tight as possible, no matter of what kind the effect is. Such “a gap fitting” cannot be termed “the confirmation of the GR prediction”. Another issue is a statistical meaning of the gap fitting. There is no single publication devoted to the treatment of observations of Mercury perihelion advance; the claimed numbers are stated in different works on empirical data not treated in rigorous terms of statistical theory. A bad fitting practice and the precision concept abuse should be noticed. At the same time, the usage of standard “precise” initial conditions in ephemerides calculations makes the results stable what creates an illusion of their high-precision, while their real precision remains unknown."

"Gap-fitting" is an academic exercise so that if one knew the potential that is responsible for the effect one might try to rationalize it physically. A high school teacher named Paul Gerber did just this, and anteceded Einstein by about two decades. The potential he proposed was

$$V(r,\dot{r})=-\frac{GM}{r}\left(1-\frac{\dot{r}}{c}\right)^{-2}.$$

The second factor, appended on to Newton's law has led to some heated debates so as not to detract from Einstein's result. Many an author has given short shrift to it, as the statement "what Gerber brings forth as physical considerations appears unintelligible" made by the staunch Einstein supporter, von Laue. If the second factor were reduced to $-1$, one might argue that it was due to taking into account retarded time. But the additional factor escapes all rationale. 

It will become apparent that Gerber\s potential is identical to Weber's,

$$V_W=-\frac{GM}{r}\left(1-\frac{1}{2}\frac{\dot{r}^2}{c^2}\right),$$

at least to order $1/c^2$.

The force that is to be equated with the radial acceleration $\ddot{r}-r\omega^2$, is given by the Lagrangian term

$$F=\frac{d}{dt}\frac{\partial V}{\partial\dot{r}}-\frac{\partial V}{\partial r}$$

$$=-\frac{GM}{r^2}\left(1-\frac{\dot{r}}{c}\right)^{-4}\left\{\frac{6r\ddot{r}}{c^2}-\frac{2\dot{r}}{c}\left(1-\frac{\dot{r}}{c}\right)+\left(1-\frac{\dot{r}}{c}\right)^2\right\}.$$

When this is expanded in powers of $\dot{r}/c$, there results

$$F=-\frac{GM}{r^2}\left(1-3\frac{\dot{r}^2}{c^2}+6\frac{r\ddot{r}}{c^2}-8\frac{\dot{r}^3}{c^3}+24\frac{r\dot{r}\ddot{r}}{c^3}-\cdots\right).$$

This represents the force per unit mass that acts in he direction of the central gravitating mass $M$ and is equal to the radial acceleration, as Ampere-Weber would assume. In fact, if we define $\tilde{c}^2=c^2/6$, and retain terms up to $1/c^2$, we get

$$F_A=-\frac{GM}{r^2}\left(1-\frac{1}{2}\left(\frac{\dot{r}^2}{\tilde{c}^2}\right)^2+\frac{r\ddot{r}}{\tilde{c}^2}\right),$$

which is exactly Weber's law with the Newtonian force replacing the Coulomb one where $\tilde{c}$ is the number of units of statical mass (electricity) which are transmitted by a unit material (electric) current in a unit of time, to borrow Maxwell's definition of $c$. It presupposes some unit of mass just light in Le Sage's (1784) explanation of gravity in terms of a tiny, rapidly moving particles in space. These particles would travel at the speed $\tilde{c}$=c/\surd 6$, or around 17/% the speed of light.

There is no tangential acceleration for the vanishing of $r\dot{\omega}+2\dot{r}\omega$ leads to the conservation of angular momentum, $L$. Calling $u=1/r$, Einstein got the trajectory of the motion as

$$\frac{d^2}{d\varphi^2}+u=\frac{GM}{L^2}+3\mathcal{R}u^2,$$

whereas Gerber got

$$\frac{d^2u}{d\varphi^2}+u=\frac{GM}{L^2}-\frac{1}{2}\frac{GM}{\tilde{c}^2}\left(\frac{du}{d\varphi}\right)^2-\frac{GM}{\tilde{c}^2}\frac{d^2u}{d\varphi^2}.$$

Any correction to the equation for the orbit,

$$\frac{d^2u}{d\varphi^2}+u=\frac{GM}{L^2},$$

will lead to an 'opening' of the orbit and some type of precession. The second term on the right of Gerber's equation does not result in a first-order correction, and it is by providence that the last terms in the general relativity equation give the same result for the  precession where the speed is $c/\surd 6$. But, speed of what? The nonlinear term in Gerber's equation does not enter the first-order correction to the orbit so it is impossible to distinguish between $k=-1/2$ and $k=-1$, where $k$ is the ratio of the longitudinal to the parallel  force, taking the parallel force as unity. These are two extremes orientations of the current elements that always interact along the line connecting them. There are no tangential components to the acceleration.

What Gerber says, in essence, is that gravity is far-action (according to Newton) and push (according to Le Sage). Only be a quirk in the numerical equivalence of the first-order correction term can we say that gravity is due to warped space-time (according to Einstein). Even if the latter would be true, it gives us no understanding of the mechanism that causes the perturbation in the orbit, which should also hold for the other planets, and not Mercury alone.