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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2603.14046 |
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| _version_ | 1866910052934221824 |
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| author | Menon, Govind Adhikari, Rakshak |
| author_facet | Menon, Govind Adhikari, Rakshak |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14046 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Further Results on Null and Force-free Electromagnetic Fields Menon, Govind Adhikari, Rakshak General Relativity and Quantum Cosmology Mathematical Physics 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. |
| title | Further Results on Null and Force-free Electromagnetic Fields |
| topic | General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2603.14046 |