How to Derive the Correct Equations of Motion for the Electro-and Gravito-Fields

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


which leads to the first integral


All the literature uses the above condition, but, with $\dot{t}=\mbox{const.}$, meaning that $A\neq A(\lambda)$. However, since $A=A(r)$, this implies that $r\neq r(\lambda)$, which clearly contradicts the equations of motion. Moreover, since $r=r(\lambda)$, it is also a function of $t$ because of the constraint, $A\dot{t}=\mbox{const}$. Finally, this violates the so-called condition of a static field in that the components of the Ricci tensor no longer vanish; in particular, $R_{01}=-\frac{\dot{A}}{Ar}$, and, consequently, all the rest.

We also used the condition that the metric coefficients, $A$ and $B$, satisfy $AB=1$, in the Lagrangian


in the simple case of motion in the plane, $\vartheta=\pi/2$. 

Now, the last term in the above expression for the Lagrangian can be looked upon as a constraint on the motion. That term is the Legendre transform of the first term, considered as a kinetic energy, This is the same way as we transform from velocity to momentum in mechanics.

Consider the function,


Now, define $\dot{t}=A\dot{r}$, and introducing this gives


When this is subtracted from the Lagrangian, $\mathcal{L}=(1/2)(A\dot{r}^2+r^2\dot{\varphi}^2)$, we get


proving that 


Finally, when the constraint $\dot{t}/A=\dot{r}$ is introduced into the variational equation, we have to choose that speed $\dot{r}$ which is the maximum speed, namely $c$, or in Weber's case $\surd 2 c$.

For $A=1/r$ we get the Weber field; for $A=1/(1+\mathcal{R}/r)$, we get the Schwarzschild field, where $\mathcal{R}$ is the Schwarzschild radius, and for $A=1+\mathcal{R}/r$ we get the Gerber field, which he used to determine the advance of the perihelion of Mercury. In the case of the Schwarzschild field, the equation of motion is


where in the planar case $v_S=r^2\dot{\varphi}^2.$  This shows definitively that there is no gravitational repulsion if the second postulate of the special theory holds.

We can also treat $\varphi$ as a kinosthenic, (or ignorable) variable which can be eliminated in favor of the angular momentum, $L$, which is defined by $L=r^2\dot{\varphi}=\partial\Phi/\partial\dot{\varphi}$. Subtracting off $\Phi=L^2/2r^2$ from the Lagrangian gives


Choosing $A=\mathcal{R}/r$, the radial equation is


after introducing $\dot{t}=Ac=2\frac{Gm}{c^2r}c$. For transverse motion ($\dot{r}=0$), we get the condition,


or, equivalently, $r\dot{\varphi}^2=Gm/r^2$ for orbital motion. This clearly shows that the gravitational potential which was introduced through the term


has nothing at all to do with time. The factor of $2$ in the definition of the Schwarzschild radius $\mathcal{R}$, is required for the variational principle, and not the asymptotic time limit since there is no time involved. In this sense, the Newtonian potential is a completely static field which does not enter into the coefficient of the $dt^2$ in the metric. For if clocks slow down in a gravitational field, they will also do so for the electric field for, in fact, the gravitational scalar potential was introduced by the analogy with the electric field [Moller, The Theory of Relativity, p. 246.]