Weber's gravitational law for zero acceleration In: http://www.ifi.unicamp.br/~assis/J-Advanced-Phys-V5-p176-179(2016).pdf equation (5) Weber's gravitational law is given by (assuming zero acceleration):
$F = - G m_1 m_2 \frac{\hat{r}_{12}}{r_{12}^2}(1 - \frac{6}{c^2} \frac{v^2}{2} )$
Now assuming that $m_1, m_2 > 0$ and that in this case the force is attractive, we get $F \le 0$ from which it follows that $(1 - \frac{6}{c^2} \frac{v^2}{2} ) \ge 0$ and from this it follows that $v \le \frac{c}{\sqrt{3}} \equiv 0.577 c$.
However in experiments with electrons velocities greater than this are observed. So where is the flaw in the argument?
 A: Tajmar & Assis (2016)'s equation (5) uses $\dot{r}_{12}$, not $v$.
$$v \not\equiv \dot{r}_{12}.$$
$v$ is an absolute quantity (speed with respect to a third point). $\dot{r}_{12}$ is a relational quantity (the change in the distance between 1 and 2), or as Tajmar & Assis (2016) put it:

$\dot{r}_{12}=dr_{12}/{dt}$ is the relative radial velocity between them

See Relational Mechanics Appendix A (pp. 493-8) for the difference between relational and nonrelational quantities.
A: This expression for the force is only applicable when the speed of the particles $v$ is much smaller than $c$. This technique called post-Newtonian expansion assumes that the speed of particles satisfies condition $v\ll c$ which means that the ratio $\frac {v^2}{ c^2}$ is a small parameter and expands expressions  with respect to it, and usually keeping only lowest corrections to the Newtonian case (which is zeroth order of approximation). As speeds of particles increase, higher-order terms, like those proportional to $\frac{v^4}{c^4}$ become comparable with those of lower order and have to be included to achieve the same degree of precision. As the speed becomes comparable with $c$ such approximation breaks down and fuller theory must be applied to achive meaningful results.
The full theory that describes gravitation for relativistic objects is general relativity. For situations, where one of the gravitating objects is much heavier than the other and when we can neglect effects of gravitational radiation, the motion of lighter particle is described by a geodesics in Schwartzschild metric. These curves are well studied for all values of speed of relativistic objects and do not include "antigravity-like" effects. 
