Is Gravity Incompatible with Geodesic Motion?

Accelerating masses are repudiated to be the sources of gravitational waves just like accelerating charges in an antenna produce electromagnetic waves with the exception that the former needs a quadrupole, because masses have no charge, while the dipoles will suffice for the latter. Now, it is a matter of fact that all particles, whether they be 'test' particles or real ones, follow geodesics. Geodesics are paths of constant velocity so how can general relativity predict the existence of gravitational waves if masses don't accelerate?

Let us return to our discussion of the Weber force. The first two decades of the nineteenth century saw a profound interest in trying to understand the interaction of magnets with current loops. Notable among the experimentalists were Biot and Savart. But, it was Ampere who had the idea of replacing a magnet by a current, and began to experiment on the force between one current and another. That is, he hypothesized that magnetism may be the result of electrical currents; a magnetic 'molecule' in a current loop, which Gauss would latter refer to as a "galvano-electric orbit" in his 1832 paper on magnetism.

Ampere's force between any two current elements, $ids$ and $i'ds'$, had to be a function of the current strengths, $i$ and $i'$, the distance of their separation, $r$, and the area enclosed by the loop, $dsds'$. However, Ampere's force was still be be a force of "action-at-a-distance", like Coulomb's and Newton's force which preceded it. However, it would be able to account for all phenomena between current elements in motion save that of Faraday's law of induction which is related to accelerating charges. 

It is commonly believed that Weber's force rectified this situation by including the acceleration of moving charges. It consisted of three terms: the static Coulomb force, a term involving the radial velocities of the moving charges, and finally the relative acceleration between the charges. The latter was able to account for induction. Implicit in this is the transfer from current elements $ids$ and $i'ds'$ to charges $e$ and $e'$. The time increment $dt$ was introduced by setting $ids=edt/c^2$, which required the intervention of a new constant, $c$, in order to amalgam a completely static force with a moving one. Weber understood his constant as the "relative velocity which electrical masses $e$ and $e'$ have, and must retain, if they are not to act on each other at all." In 1855, Weber and Kohlrausch set out to determine the numerical value of this constant.

So action at a distance was replaced by the continuous propagation of a force and the only distinction between $ids$ and $i'ds'$ was in the difference in charges $e$ and $e'$. This, according to Maxwell's derivation of Weber's law was the result of Fencher's hypothesis that positive and negative charges travel at the same speeds but in opposite directions. This was not the first time that a flawed hypothesis was to lead to a correct result, but, it avoided the possibility of taking retarded motion into consideration which Lienard and Wiechert would do in the last decade of the nineteenth century. This seemingly minor point has been completely overlooked in the literature.

So, it really makes no difference in writing down the force law, whether it be Ampere's or Weber's. So let us consider the former. Ampere's experiments clearly led him to believe that there will be an attraction or repulsion between the current elements that depended on the direction of flow. From his first experiment he surmised that the current elements will attract or repel each other depending on their relative orientations. In other words, how does the force differ, say, when the current elements are placed longitudinally, or parallel to the line connecting them, than when they are parallel to each other, normal to the line connecting them? 

The answer to this question was found in Ampere's second experiment. From this experiment Ampere was able to conclude that "an infinitely small portion of electrical current exerts no action on another infinitely small portion of current situated in a plane which passes through its midpoint, and which is perpendicular to its direction." So the force depends on the relative orientation of the current elements. This led Ampere to write down a law of force in the form

$$\frac{ii'ds\cdot ds'}{r^n}\Phi(\theta,\theta'),$$

where $\Phi$ represents an unknown function of the angles, $\theta$ and $\theta'$, which the current elements  make with the line $r$ joining them. Undoubtedly, Ampere was led to the determination of the exponent $n$ by drawing an analogy between Coulomb's potential and that of Newton. For, in matter of fact, his law must reduced to that of Coulomb when the relative velocities tended to zero.

Ampere represented the parallel components of the current elements by $ids\sin\theta$ and $i'ds'\sin\theta'$, while $ids\cos\theta$ and $i'ds'\cos\theta'$ were the longitudinal components. According to the results of his second equilibrium experiment, there would be no product of longitudinal and parallel components, like $ids\sin\theta\cdot i'ds'\cos\theta'$, since they do not interact with each other. However, "longitudinal" and "parallel" are relative because the current elements were not constrained to lie in the same plane. Although his second equilibrium experiment assumed planarity, any non-coplanar element can always be decomposed into two components, one lying in the plane and the other normal to the plane. The fact that the normal component would not contribute to the force allows the assumption of planarity to hold sway.

Ampere's law would thus consist of a combination of parallel,

$$\frac{ids\cdot i'ds'}{r^2}\sin\theta\sin\theta',$$

and longitudinal,

$$\frac{ids\cdot i'ds'}{r^2}\cos\theta\cos\theta'$$

interactions. However, there is no reason why these force components should enter the force with equal weights. This was a consequence of Ampere's first equilibrium experiment. So Ampere introduced the constant $k$, which represented the ratio of the force between the current elements in the longitudinal position to those which are parallel. If the parallel force is taken as unity, then Ampere wrote

$$\frac{ids\cdot i'ds'}{r^2}\left(\sin\theta\sin\theta'+k\cos\theta\cos\theta'\right).$$

An additional equilibrium experiment, recorded in 1823, allowed Ampere to fix $k=-1/2$; that is, the longitudinal force is only half as great as the parallel component. The negative sign meant that the difference in angles, $\varepsilon=\theta'-\theta$ would play a role, and this was later associated with the phenomenon of induction, since it would result from the extension of the directions of the two current elements.

When Weber tried to convert Ampere's law into a dynamical one, he was unsure about the longitudinal force, $k\cos\theta\cos\theta'$, which he was inclined to drop. It was Gauss who dissuaded him from doing so, because, if Ampere's force was to stand or drop, it must do so in its entirety. And that is what Weber did---accept Ampere's law in its entirety.

We have seen how the angle $\eta$ between the two planes in which the  current element lied in, in the force law,

$$\frac{ids\cdot i'ds'}{r^2}\left(\sin\theta\sin\theta'\cos\eta+k\cos\theta\cos\theta'\right),$$

could safely be put to zero without jeopardizing the generality of the law. Maxwell also saw fit to exclude the angular dependencies in order to come out with a propagation equation for waves that travelled at a speed $c/\surd 2$. It is incomprehensible that such a phenomenon did not pass through the mind of Ampere, seeing that his colleague Fresnel was sharing his Paris apartment at the same time that he was performing his experiments. And then there was the nagging question of where should gravity fit in seeing that both force laws would start out with an inverse square law. 

Even before Weber began his investigations, Mossotti thought of a way of including gravitation under the banner of electrodynamics. If the universe consisted of equal amounts of positive and negative charges then unlike charges would be attracted by the same force that like charges would be repelled. Mossotti questioned this hypothesis and considered the possibility that the attractive force between unlike charges should slightly exceed the repulsive force between like charges. Weber later gave credence to such a hypothesis, and in a posthumously published note queried whether it would be physically possible to measure such a small difference.

Others were thinking of applying Weber's force law to phenomena that seemed to defy Newton's law of action-at-a-distance. Seegers in 1864 proposed to analyze the recently discovered anomaly of the advance of the perihelion of Mercury in terms of Weber's law. Both Scheibner and Tisserand later evaluated the anomaly using Weber's law. 

Although this was the first time that a critical speed was introduced, and corrections introduced of the order $1/c^2$, there was no rationale for taking over Ampere's, or Weber's, law lock stock and barrel. In particular, why should the parallel components be weighed twice as important as the longitudinal components? We are not treating charged particles that can show both repulsion as well as attraction. Without any reason for considering such a distinction we can set Ampere's constant, $k=-1$, based on the premise of "insufficient reason" to do so otherwise. This choice implies equal magnitudes to current elements that are longitudinal and parallel, but with a force of the opposite sign!

Weber's theorem that the electrostatic force must be reduced when the particles are in relative motion, then when they are at rest, led him to append a term involving the radial velocity, $\dot{r}$, onto Coulomb's law. Since his theorem applies both to the cases where the particles are approaching each other and receding from each other, this term must be at least quadratic in the radial velocities since the force would be impervious to the direction of flow, i.e.,

$$\frac{ee'}{r^2}\left(1+\kappa\dot{r}^2\right),$$

where $\kappa$ is a constant, which should turn out negative since motion decreases the strength of the Coulomb force.

Attraction between the particles would make them speed up when approaching each other, making $\dot{r}$ positive, and when receding from each other this velocity will become negative. At the point where the current elements are opposite one another, the relative velocity vanishes, but not their relative acceleration. This led to the introduction of a term of relative acceleration in Weber's law. However, Weber also notice that since particles in parallel current elements do not travel along the same line, the acceleration should be decreased by the distance of their separation and stand in the ratio to the square of the velocity as, viz.,

$$\ddot{r}=-\frac{\kappa}{r}\dot{r}^2.$$

But $\kappa$ must be proportional to the negative the negative of Ampere’s constant. If we call the constant proportionality, $1/c^2$, then Weber’s force can be written as

$$\frac{ee’}{r^2}\left\{1+\frac{1}{c^2}\left(r\ddot{r}-\dot{r}^2\right)\right\},$$

the terms in the round parentheses being Ampere’s law for $k=-1$.

The smallness of the factor $1/c^2$ led Weber to believe that it is possible “to grasp why the electrodynamic effect of electrical masses$\dots$ compared with electrostatic$\ldots$ always seems infinitesimally small, so that in general the former only remains significant, when as in galvanic currents, the electrostatic forces completely cancel each other in virtue of neutralization of the positive and negative electricity.”

The same will not be true when the “electrical masses” are replaced by uncharged masses, so that the most we can expect is that there will be small corrections to Newton’s law like that of the perihelion shift of Mercury.

Thus, the force law for gravity between masses $m$ and $m’$ becomes

$$\frac{Gmm’}{r^2}\left\{1+\frac{1}{c^2}\left(r\ddot{r}-\dot{r}^2\right)\right\}.$$

The Ampere term can now be written as a complete differential

$$\frac{d}{ds’}\left(\frac{1}{r}\frac{dr}{ds}\right),$$

with $k=-1$ instead of his original expression,

$$\frac{1}{\surd 2}\frac{d^2\surd r}{ds'ds}.$$

Our new expression the same equation as that of an optical ray with index of refraction, $n=1/r$. The claim is that these rays follow geodesics. According to Stokes’ theorem,

$$\oint \vec{G}\cdot d\vec{s}’=\int\int\nabla\wedge \vec{G} dsds’=0,$$

if $G$ is a total derivative like the expression above. Such behavior we should also expect of gravity. Simply said: Gravity has no curl.

In contrast, the most that could be expected from Ampere’s law with $k=-1/2$ would be the component of the force perpendicular to the plane containing the line connecting the two elements and the direction of the other current element to vanish.

In time, Weber’s correspondence to Ampere’s force is

$$\frac{d}{dt}\left(\frac{\dot{r}}{r}\right).$$

Its vanishing,

$$\ddot{r}-\frac{\dot{r}^2}{r}=0,$$

 corresponds to geodesic motion as we shall now show. Weber’s analog of Ampere’s force will vanish when $\dot{r}=\mbox{const.}\times r$. This implies that the index of refraction will be proportional to $1/r$. It will prove more convenient to consider motion in the plane, and restrict the index of refraction to a function of a single parameter, say $y$, the distance from the x-axis, which if we consider the unevenly heated plane as in the Poincare half-plane model of hyperbolic geometry, represents the axis of infinite cold. The index of refraction will be a decreasing function of height indicating ascending temperatures.

The null cones will be given by

$$c^2dt^2=n^2\left(dx^2+dy^2\right),$$

where the index of refraction is $n=1/y$. The metric will be diagonal with components, $1/y^2$. The equations of the geodesics are

$$\ddot{x}-\frac{2}{y}\dot{x}\dot{y}=0,$$

and

$$\ddot{y}-\frac{1}{y}(\dot{y}\dot{y}-\dot{x}\dot{x})=0.$$

These equations are precisely the geodesic equations

$$\ddot{z}^m+\Gamma^{m}_{ij}\dot{z}^i\dot{z}^j=0,$$

where

$$\Gamma_{ij}^m=\frac{1}{2}g^{mn}\left(g_{in,j}+g_{jn},i-g_{ij,n}\right),$$

and the commas denote differentiation with respect to the argument. In the present case all the Christoffel symbols are equal to $-1/y$. Consequently, the presence of an Amperian force indicates a deviation from geodesic motion, or constant velocity. General relativity, by its very nature, is incapable of treating such motion.