# Where is the Charge in the Kerr-Neumann Solution?

The Kerr-Newman solution to the Einstein field equations describes the gravitational and electromagnetic fields of a spinning, charged, point mass. When the mass is smaller than the charge or the spin (in suitable geometrical units) the solution has a naked singularity in the form of a ring. This is the case considered here. It would describe the electron or the proton, say, were it not for quantum mechanics.

Does the electric charge reside on this ring, or on the disc of which the ring forms the boundary? The answer is "both", but the charge on the disc is opposite in sign to that on the ring. This is illustrated by Figures in the pdf below. Consequently, the charge seen at spatial infinity is less than the charge on the ring due to partial cancellation by the negative charge on the disc. In fact, the charge on the disc and the ring are both infinite, so their finite difference involves an infinite cancellation, reminiscent of renormalisation in quantum field theory. However, this view of things applies only if we restrict attention to the usual 'physical' region with positive radial coordinate.

The maximal analytical extension of the Kerr-Newman spacetime manifold includes a region with negative 'radial coordinate', joined to the physical region via a 'bridge' or 'wormhole' at the disc. If the whole spacetime is considered, the charge disappears. The lines of electric force are everywhere continuous. Since they neither originate nor terminate anywhere, there is no charge. Instead the electric lines of force form closed loops. Thus, electric charge may be an illusion arising from the multiply connected nature of spacetime.

Read the pdf: Where the Charge is in the Kerr-Newman solution

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