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 [see Govind Menon, Classical Quantum Gravity 38, 045011 (2021) for details], the converse—identifying which null geodesic congruences can support a force-free solution—remains a nontrivial 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 guarantee 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, nontrivial exact null solutions in flat spacetime and for the C-metric.

Close

PrevNext

View figure in article

• Classical black holes
• General relativity
• General relativity equations & solutions