Why Gravitational Repulsion Doesn't Exist

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).

One thing that gravitational repulsion would do for sure is to make Newton turn over in his grave. The possibility of gravitational repulsion is the result of misconceived use of  eliminating a cycle, or ignorable, variable, and then reintroducing it into the Euler-Lagrange equations of motion. There is nothing special about the time variable we use to clock time, or the angle variable we use to measure angles. Both are "kinosthenic", in the terminology of J J Thomson, who thought of the possibility of developing a mechanics without forces  by assuming the presence of such coordinates. They are also referred to as speed coordinates which can be eliminated.

The morale of the story is you can't use a constraint twice in the same variational principle. A good example can be found in Thermodynamics of Irreversible Processes, p. 100. What was a minimum to start with then becomes a maximum, and vice-versa. A double dose of a constraint led Onsager to conclude that heat flow can only lower, or cause no change at all, the entropy production. We would expect just the opposite to happen, and so, too, with the variational principles of GR.

Consider the metric,

$$-d\tau^2=-Bdt^2+Adr^2+r^2d\varphi^2,$$

in the plane $\vartheta=\pi/2$. $\tau$ is referred to as 'local' or 'proper' time, and $t$, coordinate time, the time that would be measured in the laboratory. If we divide through by $d\tau^2$ and consider the Lagrangian,

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

where the dot denotes differentiation with respect to $t$, and $A$ and $B$ are functions of $r$. It makes not difference if we exchange $t$ for $\tau$ because $t$, or $\tau$ are ignorable coordinates like $\varphi$; their differentials appear but not the variables themselves. The Euler-Lagrange equations are

$$\frac{d}{d\tau}\left(A\dot{r}\right)-\frac{1}{2}A^{\prime}\dot{r}^2-r\dot{\varphi}^2+B^{\prime}\dot{t}^2=0,$$

$$\frac{d}{d\tau}=r^2\dot{\varphi}=0,$$

$$\frac{d}{d\tau}B\dot{t}=0.$$

The last two equations are examples of ignorable, or cyclic, coordinates. They can be eliminated by the first integrals

$$r^2\dot{\varphi}=L \hspace{20pt}\mbox{and} \hspace{20pt}B\dot{t}=c.$$

This is a way of introducing the constants, $L$ and $c$, which are angular momentum and the speed of light, respectively. 

Introducing them into the radial equation of motion leads to

$$\ddot{r}+\frac{A^{\prime}}{2A}\dot{r}-\frac{L^2}{Ar^3}+\frac{B^{\prime}}{2AB^2}c^2=0.$$

So far, so good. Now comes the error. There must be something very special about $t$---which we just eliminated---to want to reinsert it again! In the words of McGruder [Phys, Rev. D 25 (1982 3191] "[t[o obtain the Schwarzschild radial acceleration as a function of the Schwarzschild radial velocity $\dot{r}=dr/dt$, and the transverse velocity $v_S=(\dot{r}^2\dot{\vartheta}^2+r^2\sin^2\vartheta\dot{\varphi}^2)^{1/2}$ we first insert [$\dot{t}=c/B$] into the identities

$$\frac{dx^i}{d\tau}=\frac{dx^i}{dt}\frac{dt}{d\tau}, \hspace{10pt} i=1,2,3,"$$

where McGruder uses the parameter $p$ for our $\tau$. He then obtains

$$a_s\equiv\ddot{r}-\frac{r\dot{\vartheta}^2}{A}-\frac{r\sin^2\vartheta\dot{\varphi}^2}{A}=\left[\frac{B^{\prime}}{B}-\frac{A^{\prime}}{2A}\right]\dot{r}^2-\frac{B^{\prime}c^2}{2A}.$$

There is absolutely no reason to favor $p$, $\tau$, $t$, etc., but what is important is the introduction of $c^2$, as Weber fully recognized.

For it is the first term on the right-hand side that shouldn't be there, for what should have been $\dot{r}^2-c^2<0$ has now become

$$a_S=\frac{g}{c^2}\left\{\frac{3\dot{r}^2}{1-\alpha/r}-2v^2_S-c^2(1-\alpha/r)\right\},$$

where $g=GM/r^2$, and $\alpha=2GM/c^2$. This leads McGruder to the conclusion that: "In Newton's theory the acceleration of gravity is always negative, indicating gravitational attraction. In Einstein's theory, however, [the above equation] shows that for

$$\dot{r}^2>\frac{1}{3}c^2(1-\alpha/r)+\frac{2}{3}c^2(1-\alpha/r),$$

we have $a_S>0, implying gravitational acceleration." This violates the second postulate of special relativity, allowing the radial velocity to become even infinite.

There is nothing scared about the time $t$, and introducing the first integral, $B\dot{t}=c$ twice has led to the error resulting in the above inequality. 

We can show this explicitly by deriving Einstein's equation for the deflection of light which involves setting the Lagrangian 

$$A\dot{r}^2+r^2\dot{\varphi}^2-A^{-1}\dot{t}^2=0,$$

where we set $A=B^{-1}$. Eliminating $\dot{t}$ through the constraint, $A\dot{t}=c$, results in

$$A\dot{r}^2+r^2\dot{\varphi}^2-Ac^2=0.$$

It doesn't make one iota of a difference what the dot stands for because on dividing through by $\dot{\varphi}^2$ there results

$$\left(\frac{dr}{d\varphi}\right)^2=\frac{c^2}{L^2}r^4-\frac{r^2}{A},$$

where we eliminated $\dot{\varphi}$ by the expression for the conserved angular momentum. Notice that if we had used

$$\frac{d}{dp}r^2\frac{d\varphi}{dp}=0,$$

we would have come out with

$$\frac{r^2\dot{\varphi}}{1-\alpha/r}=L,$$

which is no longer the conservation of angular momentum! True, the second term in the denominator is small, but to neglect it like Moller did would be to avoid using the transform from $p$ to $t$ which introduces the extraneous term, $B^{\prime}\dot{r}^2/B$, in the equation of motion for the radial coordinate given above.

Having set $dr/dp/d\varphi/dp=dr/d\varphi$, and all traces of the affine parameter $p$ have completely disappeared leaving the equation for the trajectory. Had we used the above transform from $p$ to $t$, given by Gruder, who referenced it from Weinberg's Gravitation and Cosmology, we would be led to an erroneous result.  Differentiating the equation for $dr/d\varphi$ by $\varphi$ and transform to the variable, $u=1/r$, give

$$\frac{d^2u}{d\varphi^2}+u=3gu^2,$$

which is exactly the same equation that GR found for the deflection of light by a massive body. 

Parenthetically, we might add that to get the shift in the perihelion of Mercury, all we have to do is to set the Lagrangian to a constant, say $-1$, instead of zero. This has falsely led credence to the distinction between proper and coordinates times. Although this introduces the constant term $\alpha/L^2$,  we can't believe that proper time has led to the introduction of a static potential. It has the effect of adding a constant to the Lagrangian  which does not affect  the Euler-Lagrange equations.

Weber never had any trouble in defining the time variable, he transformed the currents into charges, where the velocity time charge $ve=c\times ids$, $c$ times the current element, $ids$. The $c$ so introduced was defined as the number of statical units of electricity which are transmitted by a unit electric current in a unit of time. The transformation from $s$ to $t$ was

$$\frac{dr}{dt}=v\frac{dr}{ds},$$

but no one ever stopped to ask which $t$? It was merely a deux ex machina for introducing $\surd 2 c$, which was Weber's constant.

Hence, there is no such thing as gravitation repulsion and the second postulate of the special theory is upheld.