General relativity generalizes the line element

$$-c^2d\tau^2=-Bc^2dt^2+Adr^2+r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2),$$

In our last blog we noticed (at least) two fallacies in the derivation of the equations of motion of the gravitational field equations: 

I. The constraint $A\dot{t}$, where $A$ is any metric coefficient depending on $r$, and $\dot{t}$ is the derivative of time (only one time!) with respect to an affine parameter, $lambda$, cannot be introduced into the variational equations more than once, and

II. Since $t$ is a cyclic (ignorable, kinosthenic, or whatever you want to call it) variable the Lagrange equation reads

There has been much discussion in the literature of anti-gravity machines based on gravitational repulsion that would be the source for propulsion of payloads in the foreseeable future. The possibility of gravitation repulsion comes from an analysis of the Schwarzschild metric by Droste and Hilbert which predict that "a large mass moving faster that $c/\surd 3$ could serve as a driver to accelerate a much smaller mass from rest to a good fraction of the speed of light" (Felber: http://arxiv.org/abs/0910/1084).

We have seen that the same formulas may have entirely different meanings and the same same physical phenomena may be explained in many ways. Two such examples are the general relativistic calculation of the deflection of light and the advance of the perihelion of Mercury. Both these phenomena were known before the advent of general relativity, and even the numerical gap in the advance of the perihelion was known, but not to the degree it is known today.

In our last blog we proposed a metric,

$$-c^2d\tau^2=-rc^2dt^2+\frac{1}{2r}\left(dr^2+r^2d\varphi^2\right),$$

in the plane, $\vartheta=\pi/2$, for the Weber force,

$$F_W=\mbox{const.}\times\left(\ddot{r}-\frac{\dot{r}^2}{2r^2}+\frac{c^2H^2}{r^2}\right),$$

whose vanishing coincides with the Euler-Lagrange equations, or, equivalently, the geodesics. 

The Lagrangian describing this geometry is

We have seen, in the last blog, that there is a rather strict analogy between the Schwarzschild metric and Gerber's potential, both of which give the same expression to the perihelion shift of Mercury to order $1/c^2$. It has long been known that to this order the force derived from Gerber's potential is the same as the Weber force. There is a slight difference, however.

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. 

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?

It has often been claimed that Ampere's force is equivalent to Grassmann's force, which is a precursor of the Lorentz force, when the force is integrated round a circuit, in which the current element forms a part, vanishes. This is taken to mean that a force exerted by a complex circuit element is a right angles to the element. As such it was taken to be equivalent to the earlier discovered Biot-Savart law. The Biot-Savart law,

$$\vec{B}=\frac{\mu}{4\pi r^2}(\vec{v}\wedge\hat{r}),$$

The critical density, as we have seen, is determined by introducing the Hubble parameter, $H$, defined from the velocity distance relation, $v=HR$. Rather than being a 'receding speed', $v$, and $R$ the distance the galaxy and the observer, the former appears as a revolving speed, so that $H$ is an angular speed of rotation, and the latter the radius measuring the distance between the galaxy and the cosmic center.