A strange exact solution in Einstein–Maxwell electrostatics

Abstract

The solution is static, axisymmetric, electrovac, and flat at infinity. It has an isolated source. Both the electric and gravitational fields are zero on the symmetry axis near the source. An explanation is offered.

Appendix

The solution in Sect. 2 has its origin in a paper [3] published in 1979 giving a static, axially symmetric, electrovac solution in prolate spheroidal coordinates. This is

$$\begin{aligned}&\displaystyle ds^{2} =-a^{2}U^{2}V^{2}\left[ \frac{(UV-BV-CU)^{2}}{(\eta ^{2}-\mu ^{2})^{3}} \Big (\frac{d\eta ^{2}}{\eta ^{2}-1}+\frac{d\mu ^{2}}{1-\mu ^{2}}\Big )+ \frac{(\eta ^{2}-1))(1-\mu ^{2})}{(UV-BV-CU)^{2}}d\psi ^{2}\right]&\nonumber \\&\displaystyle +\frac{(UV-BV-CU)^{2}}{U^{2}V^{2}}dt^{2},&\end{aligned}$$

(24)

$$\begin{aligned}&\displaystyle \phi =\frac{C}{V}-\frac{B}{U},&\end{aligned}$$

(25)

where

$$\begin{aligned} U=B+A\mu -\eta ,\end{aligned}$$

(26)

$$\begin{aligned} V=C+A\mu +\eta . \end{aligned}$$

(27)

In the above $\phi$ is the electrostatic potential, and $a,A,B,C$ are real constants satisfying

$$\begin{aligned} A^{2}=BC+1. \end{aligned}$$

(28)

The main interest in the solution when it was published in 1979 was that it had three free parameters, and was not a Weyl solution, that is, the gravitational and electric potentials were not functionally related. The field at infinity was studied.

Carminati [7], Alekseev [12] and later, Cabrera-Munguia et al [8], noted that the solution could usefully be expressed in bipolar spherical coordinates and, eventually, this idea was used to reformulate the solution so that it described the static field of two charged masses [9, 10]. In this reformulation the parameter $A$ is crucial in determining the distance between the two sources, and putting it equal to zero we obtained the present solution referring to one source.

With $A=0$ the solution falls into the Weyl class, and Weyl solutions had been treated in the important paper by Cooperstock and de la Cruz [4] in 1979, using cylindrical coordinates. The approach in [4] is elegant and simpler than that in prolate spheroidal coordinates, and this is what we have used in this paper.