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

$$\frac{d}{d\lambda}\left(A\frac{dt}{d\lambda}\right)=0,$$

which leads to the first integral

$$A\frac{dt}{d\lambda}=\mbox{const.}$$

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

$$\mathcal{L}=\frac{1}{2}\left(A\dot{r}^2+r^2\dot{\varphi}^2-B\dot{t}^2\right),$$

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 fransform from velocity to momentum in mechanics. Consider the function,

$$\Psi(\dot{t})=\dot{r}\frac{\partial\mathcal{L}}{\partial\dot{r}}-\mathcal{L}(\dot{r}).$$

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

$$\Psi(\dot{t})=\frac{\dot{t}^2}{2A}.$$

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

$$\mathcal{L}=\frac{1}{2}\left(A\dot{r}^2+r^2\dot{\varphi}^2-\frac{\dot{t}^2}{A}\right),$$

proving that 

$$A=1/B.$$

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

$$\ddot{r}-r\dot{\varphi}^2=g\left(\frac{\dot{r}^2-c^2}{1-\alpha/r}-2v_S^2\right),$$

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

$$\mathcal{L}=\frac{1}{2}\left(A\dot{r}^2-\frac{L^2}{r^2}-\frac{\dot{t}^2}{A}\right).$$

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

$$\frac{2Gm}{c^2}\left(\frac{\ddot{r}}{r}-\frac{\dot{r}^2}{r^2}\right)=\frac{L^2}{r^3}-\frac{Gm}{r^2},$$

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

$$\frac{L^2}{r^3}=\frac{Gm}{r^2},$$

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

$$-A^{\prime}\dot{t}^2/2A^2=-A^{\prime}c^2/2=Gm/r^2$$

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.]