Strong gravitational force induced by static electromagnetic fields? This 2004 preprint by B V Ivanov claims that static electromagnetic fields can create strong gravitational fields. He claims a capacitor charged to 100 kV loses 1% of its weight and a capacitor charged to 6 MV would levitate. This has long been claimed as the origin of the Biefeld-Brown effect as opposed to the conventional explanation of ionic wind thrust. Ivanov seems to use a sophisticated general relativity argument to make his claims but he hasn’t managed to publish the result.
Abstract

It is argued that static electric or magnetic fields induce
Weyl-Majumdar-Papapetrou solutions for the metric of spacetime. Their
gravitational acceleration includes a term many orders of magnitude
stronger than usual perturbative terms. It gives rise to a number of
effects, which can be detected experimentally. Four electrostatic and
four magnetostatic examples of physical set-ups with simple symmetries
are proposed. The different ways in which mass sources enter and
complicate the pure electromagnetic picture are described.

He claims that the Weyl-Majumdar-Papapetrou fields imply that the gravitational force on a test particle $g_\mu$ is given by
$$g_\mu=c^2f^{-1}\Big(\frac{B}{2}\sqrt{\frac{\kappa}{8\pi}}\bar{\phi}_\mu+\frac{\kappa}{8\pi}\bar{\phi}\bar{\phi}_\mu\Big),$$
where $\bar{\phi}$ is the electrostatic potential, $E_\mu=-\bar{\phi}_\mu$ is the electric field and $f=1+B\phi+\phi^2$ is a solution of the so-called Weyl field (see the section II Root Gravity).
He says that the coefficient of the first linear term is $10^{23}$ times bigger than the coefficient of the second quadratic term. The quadratic latter term is familiar from standard perturbation theory. This accounts for the anomalously large gravitational effect of the electrostatic field in this case.
Does his argument make sense?
 A: This is a good illustration of what happens when one forget that
Correlation  does not imply causation.
Yes, the author of the cited paper correctly describes a family of solutions of Einstein field equations with electrostatic fields. Within this family gravitational force (in the static frame) does behave according to the equation reproduced by OP. However, the conclusion drawn from that equation, that electric field causes this gravitational force is wrong. Rather both gravitational force and electrostatic field share the same cause: distribution of charged matter with a specific constant charge-to-mass ratio. But it is the “mass part” of this matter distribution that is mainly responsible for the gravitational field (long range electrostatic field contributes what amounts to a tiny correction under conditions of a reasonable experiment). So all claims made by the author (and by other authors relying on this paper) “derived” from the behavior of this solution family extrapolating the effects on situations without the constraints on  charge and mass densities are unfounded.
Note that Majumdar–Papapetrou family has a lot of solutions with interesting properties such as multicenter black hole solutions and almost black holes. In all such solutions with strongly curved spacetimes variation of the electrostatic potential between the asymptotic region and strongly curved region is of order of Planck voltage which is unity in GR units or $c^2/\sqrt{4\pi\epsilon_0 G}\approx 1.04 \times 10^{27} \text{ V}$ in SI units (see this PSE answer for the discussion of Planck voltage). So if we consider MP solutions with potential  variations much smaller than $10^{27}\text{ V}$ these solutions would be almost flat and since they are also static, to analyze them one would only need the Newtonian gravity together with equations of electrostatic. So this is what we'll do.
Let us assume that we have a system of charges at rest with constant charge to mass ratio given by (in SI units):
$$
\frac{q}m =\sqrt{4πϵ_0 G }\approx 8.62\times10^{-11}~\text{C}\cdot\text{kg}^{-1}. \tag{*}
$$
This corresponds to about one elementary charge for about $10^{18}$ nucleon masses. It is easy to see that in this system the force of gravitational attraction acting on each charge is precisely compensated by the force of Coulomb repulsion:
$$
\mathbf{F_\text{g}}=-\sum_i\frac{G\, m m_i \,\mathbf{r}_i }{r_i^3}=- \sum_i\frac1{4\pi\epsilon_0}\frac{q q_i \,\mathbf{r}_i }{r_i^3} = - \mathbf{F}_\text{e}.
$$
So this system of charges initially at rest would remain at rest indefinitely. And Newtonian gravitational and electrostatic potentials are related via:
$$ 
\phi_\text{g}=-\sqrt{4πϵ_0 G} |\phi_\text{e}|.
$$
To reiterate, while this last equation has $\sqrt{G}$ and electrostatic potential on the rhs, it does not mean that electric field causes this gravitational field.
Now, imagine the system of MP charges is placed in external (Newtonian) gravitational field. If this external field is approximately homogeneous on a  scale of MP subsystem then it would move as a single body possessing mass and charge (satisfying $(*)$) and would also generally undergo deformations by tidal forces. But there would not be any unusual “antigravity” (or other paradoxical) effects. And if the MP subsystem is mechanically connected to some structure neutralizing its charge (such as a plate of capacitor) there would by no unusual gravitational of electromagnetic effects.
