Weber versus Weber: Why the Electrodynamic-Gravitational Analogy Breaks Down

We read from Forward, in "Guidelines to Antigravity", that (Joseph) Weber used Einstein's theory to determine the acceleration of a small mass $m$ by a big mass $M$. The resulting force  was found by Weber to be given by

$$ \vec{F}/m=-\frac{GM}{r^2}\left(1+\frac{\vec{v}\cdot\hat{r}}{c}\right)\hat{r}+\frac{4GM}{c^2r}\left(\vec{a}+\frac{\vec{v}\cdot\hat{r}}{c}\vec{a}+\frac{\vec{a}\cdot\hat{r}}{c}\vec{v}\right),$$

From Ampere to Post-Newton?

Ampere's generalization to Coulomb's force of attraction between bodies which depended not only on the velocities of charged particles but also on their angles of interaction was a natural for attempts to generalize Newton's law of attraction between masses in motion. The synthesis between Ampere and Coulomb was accomplished by Weber which caught the eye of astronomers seeking a way to incorporate planetary motion into Newton's law of attraction.

Turning the Tables: Forces Determine Gaussian Curvature


It is commonly believed that for a rotationally symmetric metric with a conformal factor $\varphi$, all sectional curvatures lie between $-\varphi^{\prime\prime}/\varphi$ and $(1-\varphi^{\prime 2})/\varphi^2$, where the prime stands for the derivative with respect to the independent variable, like time or the radial coordinate. This supposedly encompasses all sectional curvatures ranging from radial to tangential curvatures, respectively.

Is General Relativity the Final Word?

Early on in his quest for a description of gravity that would go beyond Newton's law governing the attraction of bodies, Einstein realized that any such theory must  include:the properties that all bodies possess inertia, and matter has inertia only in the presence of other massive bodies. In essence, Einstein was trying to incorporate Mach's principle, whereby the rotation of the heavens determines the inertia of all bodies on earth, into his "general" theory. 

Why Ricci and not Ampere?

The mechanical prescriptions of general relativity tells us to add up Gaussian curvatures to form the Ricci tensor. If the "universe" is empty set the Ricci components equal to zero and determine the metric coefficients. If not zero, set them equal to the corresponding eigenvalues of the energy-stress tensor. If a perfect fluid is chosen, the latter contains mass density and hydrostatic pressure ensuring that the universe is homogeneous and isotropic.

Newton versus Volta or Gravitational versus Electromagnetic Waves

In order to combine Coulomb's law with Ampere's law to get Weber's law, the latter had to electrostatic units into electromagnetic ones. The factor of proportionality is c, the speed of light. If we want to carry over the analogy to gravitation, there must be something analogous to electromagnetic charge corresponding to dynamical mass in contrast to statical mass that we find in Newton's law of gravitational attraction.

Is there a Gravitational Effect Which is Analogous to Electrodynamic Induction?

This is the title taken from a little known (perhaps intentionally!) of a 1912 article of Einstein to show that the mass of a system is modified by the presence of other matter by an amount $GMm/Rc_0^2$, where $m$ is within a hollow sphere of mass $M$ of radius $R$, and $c_0$ is the speed of light at infinity. This is the amount of increase in mass independently of whether we are considering $m$ or $M$. Instead of the electrodynamic repulsion between two circuits, embodies in Ampere's law, Einstein always comes out with an attraction between the masses. 

What Does Electrodynamics Teach Us About Curvature?

Einstein made a big story out of Ricci's tensor and its dubious relation to a matter stress tensor. In the very few analytical models that are solvable, the force does not correspond to a linear combination of sectional curvatures. But general relativity does not define forces, but it does define pressures, and the pressure is the force per unit area.

Let's Talk Curvature

If we agree that the sectional curvatures of a surface have something to do with the forces that perhaps create them, then it is manifestly obvious that the average curvature afforded by the Ricci tensor has nothing to do with the forces themselves. For it would be equivalent to the condition of mechanical equilibrium whereby the sum of the forces  vanish in a state of mechanical equilibrium.