Monopole and dipole layers in curved spacetimes: Formalism and examples

The discontinuities of electromagnetic test fields generated by general layers of electric and magnetic monopoles and dipoles are investigated in general curved spacetimes. The equivalence of electric currents and magnetic dipoles is discussed. The results are used to describe exact “Schwarzschild disk” solutions endowed with such sources. The resulting distributions of charge and dipole densities on the disks are corroborated using the membrane paradigm.

Figure 1

Equivalent points and associated 3-volumes. The timelike coordinate $x^0$ is suppressed. (For notation see the text.)

Figure 2

The motion of two associated infinitesimal charges. The spacelike coordinate $x^2$ is suppressed. (For notation see the text.)

Figure 3

The time component of the surface current $s_{(e\mathcal{M}o)}^a$ (i.e., the charge density) endowed with electric charges for the parameters $r_{0+}=5.1M$, ${\theta }_{0+}=0.7\pi$, $z_0=1.7M$.

Figure 4

The time component of the surface current $s_{(e\mathcal{D}i)}^a$ of the disk endowed with electric dipoles for the parameters $r_{0+}=6.5M$, ${\theta }_{0+}=0.48\pi$, $z_0=1.7M$.