Further Results on Null and Force-free Electromagnetic Fields

TL;DR

This paper advances the understanding of null force-free electromagnetic fields by establishing existence conditions for solutions based on null geodesic congruences, with applications to black hole spacetimes.

Contribution

It proves a general existence theorem for null FFE solutions, linking shear-free geodesic congruences to the existence of specific electromagnetic field configurations.

Findings

Existence of local rotations satisfying equipartition of null mean curvature.

Shear-free null geodesic congruences guarantee solutions with arbitrary functions of three variables.

Constructed new explicit null solutions in Schwarzschild, Kerr, flat spacetime, and C-metric geometries.

Abstract

The theory of Force-Free Electrodynamics (FFE) provides a robust framework for modeling the magnetospheres of compact objects, where the electromagnetic field's energy density dominates the surrounding plasma. Central to this theory is the existence of two-dimensional integral submanifolds, or field sheets, which foliate the spacetime. While it is established that every null force-free field possesses an associated 2-D null geodesic foliation, the converse, identifying which null geodesic congruences can support a force-free solution, remains a non-trivial computational challenge. In this paper, we extend the foliation-based approach to null FFE by addressing two primary obstacles to the existence of a solution: the equipartition of null mean curvature and the involutivity of the field sheet distribution. We prove a general existence theorem demonstrating that for any given null geodesic congruence, there always exists a local rotation of a 2-D basis transverse to the geodesic congruence that satisfies the equipartition condition. Furthermore, we establish that a shear-free null geodesic congruence is sufficient to guaranty the existence of an arbitrary function of three variables such that any choice of such a function will generate a null field sheet foliation. Additionally, each unique foliation will be associated with a null force-free field that further contains an arbitrary function of two variables. These results are formally linked to the vanishing of the shear tensor, providing a coordinate-independent geometric criterion for the existence of null FFE solutions. We illustrate these theorems with explicit examples in Schwarzschild and Kerr geometries and present new, non-trivial exact null solutions in flat spacetime and for the C-metric.