A bit of everything

# Maccone's magnetic wormhole

Due to their peculiarities, texts on wormholes can pop up in rather mysterious places outside of dry theoretical physics papers, from NASA's Frontiers of Propulsion Science to weird government reports to UFO enthusiasts, all with varying degrees of rigor. While physics papers are easy enough to find electronically, as we've had the use of arxiv for about thirty years, and generally a better access to the older papers as well, those other papers can be somewhat more challenging to find.

Somewhat on the more reasonable side of that scale is the Journal of the British Interplanetary Society. Here we find two papers which are sometimes referred to elsewhere : "Interstellar Travel through Magnetic Wormholes" by Claudio Maccone (vol. 48, pp. 453-458, 1995) and its rebuttal, "“Magnetic Wormholes” and the Levi-Civita Solution to the Einstein Equation" (vol. 50, pp. 155-157, 1997). Neither of these are available electronically, not even on request, nor can they be purchased. But if you ask nicely enough and leave some donation to the society of the appropriate value, someone may photocopy it for you.

I thought that therefore it may be a good idea to put this up here for posterity, in case someone needs to know what the magnetic wormhole is and why it does not work.

First, we need to look at the Levi-Civita spacetime. While indeed discovered by Levi-Civita, in some untranslated Italian paper (recently translated here), it is more commonly known under its modern name of the Bertotti-Robinson spacetime, as it was later on studied by Bertotti and Robinson's A solution of the Einstein-Maxwell equations (Bull. Acad. Polon. Sci. Math. Astron. Phys. 7, 351), itself apparently not available online. It's not the most popular of metrics to study, although it is discussed in Pauli's Theory of relativity (p. 171-172) and of course, Stephani et al.'s Exact Solutions of the Einstein Field Equations (section 12.3).

The Bertotti-Robinson spacetime's basic idea is quite simple, it is a spacetime with a constant electromagnetic field upon it. What it means for the EM field to be constant isn't quite trivial since the EM tensor components will depend on the coordinates, but we'll see what it means later on. As can be guessed from the definition, the Bertotti-Robinson spacetime is an electrovacuum solution, which solves the Einstein-Maxwell equations

$$R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = (F_{\mu\sigma} {F_{\nu}}^\sigma - \frac{1}{4} g_{\mu\nu} F_{\rho\sigma} F^{\rho\sigma})$$

The stress-energy tensor can be shown to be traceless, as the electromagnetic tensor itself is, leading to $R = 0$, so that

$$R_{\mu\nu} = (F_{\mu\sigma} {F_{\nu}}^\sigma - \frac{1}{4} g_{\mu\nu} F_{\rho\sigma} F^{\rho\sigma})$$

$$x$$ ...

There are two invariants of the EM field we can consider to define it as constant,

\begin{eqnarray} I &=& F_{\mu\nu} F^{\mu\nu}\\ J &=& \varepsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma} \end{eqnarray}

As these are scalars, they can be used to define some notion of constant field independently of the coordinates. In flat space they roughly correspond to the quantities $-(\| \vec{E} \|^2 - \| \vec{B} \|^2)$ and $-(\vec{E}\cdot\vec{B})$ (up to some factors), and we therefore say that if $J = 0$ and $I \neq 0$, then either $I < 0$ and the field is electrostatic, or $I > 0$ and the field is magnetostatic (as in, some observers will measure such a field).

...

There are many forms of the Bertotti-Robinson metric

$$ds^2 = \frac{1}{\lambda^2 r^2} (-dt^2 + dr^2 + r^2 d\theta^2 + r^2 \sin^2(\theta) d\phi^2)$$ $$ds^2 = - \lambda^2 (\rho^2 + z^2) dt^2 + \frac{1}{\lambda^2 (\rho^2 + z^2)} (d\rho^2 + dz^2 + \rho^2 d\theta^2)$$
Posted on 2019-07-05 11:27:59