The Need for a Road Map in General Relativity

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.

Weber taught us that when static and motional forces are combined, it is necessary to introduce a limiting velocity. In fact, in 1856 Weber and Kohlrausch determined its value. According to Weber, $c$ is "that relative velocity which electrical masses $e$ and $e'$ have and must retain, if they are not to act on each other at all." His fundamental law was

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

where it turned out that $c'$ exceeded Maxwell's constant, which is the inverse of the square root of the product of the dielectric  and magnetic permeabilities of the vacuum, by $\surd 2$,  We may see how this arises by considering the Lagrangian,

$$\mathcal{L}=-r(c\dot{t})^2+\frac{\dot{r}^2+r^2\dot{\varphi}^2}{r},$$

in the plane $\vartheta=\pi/2$. The EL equations,

$$\frac{d}{d\tau}\frac{\partial\mathcal{L}}{\partial\dot{x}}-\frac{\partial\mathcal{L}}{\partial x}=0,$$

are explicitly given by

$$2\frac{d}{d\tau}\left(\frac{\dot{r}}{r}\right)+\frac{\dot{r}^2}{r^2}-\dot{\varphi}^2+(c\dot{t})^2=0,$$

$$\frac{d}{d\tau}r\dot{\varphi}=0,$$

and

$$\frac{d}{d\tau}c^2r\dot{t}=0.$$

The second and third equations lead to the first integrals, $r\dot{\varphi}=v=\mbox{const.}$, and $c^2r\dot{t}=H=\mbox{const.}$. Whereas $v$ has the physical significance of a velocity, $H$ doesn't have one so we can set it equal to unity. That is to say, $\dot{t}$ does not have any physical significance regarding a 'coordinate' time, nor does $\tau$ refer to 'proper' time. It is put there only to provide a limiting velocity, as Weber very well knew. 

For upon introducing the first integrals into the radial equation there results

$$\frac{1}{c^2-v^2}\left(2\frac{\ddot{r}}{r}-\frac{\dot{r}^2}{r^2}\right)+\frac{1}{r^2}=0.$$

This is the condition that the Weber force vanish with the only difference that Weber's constant, $c'$, is now given by $\surd(c^2-v^2)$, which explicitly states that $c$ is a limiting velocity. It also points to the fact that motion causes a contraction in the radial direction, which predates relativity by almost a half a century.

In the previous blog we noticed a difference between the way general relativity treats the advance of the perihelion as opposed to the deflection of light by a massive body.  The latter case could be derived from the same procedure as above adopting the Lagrangian

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

which is obtained by dividing the metric

$$-c^2d\tau^2=-(1-\mathcal{R}/r)c^2dt^2+\frac{dr^2}{1-\mathcal{R}/r}+r^2d\varphi^2,$$

through by $(1-\mathcal{R}/r)^{-1}$, $d\tau^2$, and  considering the Lagrangian to be half the right-hand side. Even if  subtracted a constant from the Lagrangian it would have no effect on the EL equations. Yet, if we were to solve

$$-c^2=-(1-\mathcal{R}/r)c^2\dot{t}^2+\frac{\dot{r}^2}{1-\mathcal{R}/r}+r^2\dot{\varphi}^2,$$

obtained by dividing through by $d\tau^2$, and solving the equation with respect to the first integrals,

$$(1-\mathcal{R}/r)\dot{t}=c/v=\mbox{const.},\hspace{20pt}\mbox{and}\hspace{20pt}r^2\dot{\varphi}=L=\mbox{const.},$$

the result is entirely different. In fact, the left-hand side is necessary for introducing the Newtonian potential, and the fact that there is a limiting velocity. This apparently gives some credence in treating two times: one coordinate and the other a proper time. However, appearance can be deceiving.

Solving the metric with the aid of the first integrals, and introducing $u=1/r$ gives the equation,

$$\varphi=\varphi_0+\int^u_{u_0}\frac{du}{\surd\left(W+\mathcal{R}u/L^2-u^2+\mathcal{R}u^2\right)}.$$

After multiplying through by $1-\mathcal{R}/r$ in the metric, the constant term, which represented the square of the increment in proper time, introduces a $-1$ in the first term in the term

$$W=\frac{(c/v)^2-1}{L^2},$$

and the linear term $\mathcal{R}u=2GM/rc^2$. Closed orbits having negative total energy require $W<0$, which seemingly violates the fact that $c$ is a limiting velocity. The linear term is essentially twice the Newtonian potential which wouldn't be there if we did not consider the "proper" time. Why should "proper" time be necessary to introduce a static potential?

Be this as it may, we continue to follow conventional "reasoning".  Quoting from the classic text, Adler, Bazin, and Schiffer,  Introduction to General Relativity, "Unfortunately even though this [the above expression for $\varphi$] is an exact solution to the problem, it expresses the angle $\varphi$ as an integral of $u=1/r\ldots$ To make the problem more transparent differentiate with respect to $\varphi$ so that it looks like a classical  Kepler problem."

The question is why bother when we could have begun directly with the Lagrangian,

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

where the last term is $c^2d\tau^2$, after having divided through by $d\tau^2$? It is a constant, and as such, it will not influence the EL equations of motion. The EL equation for the radial coordinate is easily found to be

$$(1-\mathcal{R}/r)\ddot{r}-g\frac{\dot{r}^2}{c^2}=\frac{L^2}{r^3}(1-\mathcal{R}/r)^2-g(c/v)^2,$$

where $g=GM/r^2$. It doesn't at all look like the equation obtained by setting the difference in the metric equal to zero. Moreover, the fact that $\dot{r}/\dot{\varphi}=dr/d\varphi$ has been used in the derivation. This contradicts the assumption that the "field of a static body cannot depend on time which is ensured by $R_{01}=-\dot{A}/rA=0$, which implies $A=A(r)$", where $R_{01}$ is a component of the Ricci metric and $A$ a coefficient in the metric. 

The metric coefficients have to be a function of time if $r$ is. Moreover, the need to consider proper time, or for that matter time at all, for introducing a static potential is absurd. The need for introducing time at all is analogous to the way Weber combined Ampere's force with Coulomb's. Time itself does not come into play except for determining the rate of change of the radial coordinate connecting the two current elements.  Hence, in order to get the  sought after results in general relativity, you have to follow a road map!