Coupling Higher Form Structures of the EFT of Force Free Electrodynamics to Gravity

Abstract: We know that the charged Reissner Nordstr$ö$m black hole metric is obtained from the Einstein Hilbert gravitational action. This action has the kinetic term $F² = (da)<sup>2$. Motivated by the higher-form symmetry structure of the EFT of Force Free Electrodynamics, we replace the Maxwell field-strength contribution in the Einstein Hilbert action by the gauge-invariant combination $(b-da)<sup>2$, where $a_μ$ is the worldsheet gauge field and $b_{μν}$ is a background two-form field. This ensures that the new action has a higher form symmetry $b \rightarrow b+dΛ, a\rightarrow a+Λ$. Here, unlike in qed, $Λ$ may be any one form (not necessarily a differential one form $\partial_μφ$). The higher form symmetry here is one with the conserved current being a two form and the charge integrated on surfaces. Intuitively, it is the number/current of vector field lines that is conserved here, not the current of particles. Thus, integrating over a surface through which the field lines pierce is sufficient to find the number of these lines that are passing through; so the charge is integrated on surfaces, rather than on the volume. After fixing a particular gauge for the fields $a_μ$ and $b_{μν}$, we obtain a generalized black-hole metric. We find that on hypersurfaces satisfying $(r-t)=$ constant, this metric reduces locally to a Reissner Nordstr$ö$m geometry with an effective charge parameter depending on the constant $(r-t)$.

• The paper introduces a novel extension of Einstein-Maxwell theory by replacing the Maxwell term with a higher-form invariant combination, leading to dynamic charge profiles.
• It employs a static, spherically symmetric ansatz to derive black hole solutions that differ from Reissner-Nordström metrics by featuring arbitrary retarded charge functions.
• The work explores implications for holographic duals and astrophysical models, suggesting new dynamical regimes in force-free electrodynamics coupled to gravity.

Coupling Higher Form Structures of the EFT of Force Free Electrodynamics to Gravity

Overview

This work introduces and analyzes a generalization of the Einstein-Maxwell theory in which the standard Maxwell field strength is replaced by a gauge-invariant combination of a two-form field and a one-form gauge field, reflecting the higher-form symmetry structures found in the effective field theory (EFT) of Force-Free Electrodynamics (FFE). The central result is the derivation of static, spherically symmetric black hole solutions in this higher-form-coupled gravitational theory, with particular attention to how these solutions differ structurally and dynamically from standard Reissner-Nordström solutions.

Motivation and Formalism

The EFT approach to FFE, as established by Gralla and Iqbal (Gralla et al., 2018), encodes the theory in terms of a conserved symmetric stress tensor $T^{\mu\nu}$ and a two-form current $J^{\mu\nu}$. This formalism utilizes background two-form fields $b_{\mu\nu}$ to couple to the magnetic flux conservation, with field strengths $$F_{\mu\nu} = b_{\mu\nu} - \partial_\mu a_\nu + \partial_\nu a_\mu$$, where $a_\mu$ is a one-form gauge field. The higher-form symmetries in these systems, $b \rightarrow b + d\Lambda$ and $a \rightarrow a + \Lambda$ for arbitrary one-form $\Lambda$, underpin a generalized global symmetry structure, distinguishing this setup from the simple $U(1)$ gauge symmetry of Maxwell theory.

Coupling this framework to gravity, one seeks an extension of the Einstein-Hilbert action where the electromagnetic sector is governed by the higher-form-invariant kinetic term: $$F^2 = (b_{\mu \nu} - \partial_\mu a_\nu + \partial_\nu a_\mu)^2,$$ with the action

$J^{\mu\nu}$0

The cosmological constant is given by $J^{\mu\nu}$1. This action preserves diffeomorphism invariance and higher-form gauge symmetry by construction.

Solution Structure

For $J^{\mu\nu}$2 spacetime dimensions, the ansatz used is a static, spherically symmetric metric: $J^{\mu\nu}$3 Field components are chosen such that only the $J^{\mu\nu}$4-component of $J^{\mu\nu}$5 and $J^{\mu\nu}$6 are non-zero, simplifying the problem to radial dependence only.

The stress tensor derived from this action takes the form analogous to the Maxwell case but with $J^{\mu\nu}$7 replaced by the new gauge-invariant field strength. The Einstein equations,

$J^{\mu\nu}$8

lead to a structure where the only nontrivial field strength components depend on $J^{\mu\nu}$9, with $b_{\mu\nu}$0 the radial profile of $b_{\mu\nu}$1.

Imposing field equations for $b_{\mu\nu}$2 leads to

$b_{\mu\nu}$3

where $b_{\mu\nu}$4 is an arbitrary function of $b_{\mu\nu}$5, reflecting the enhanced gauge freedom of the higher-form symmetry compared to the standard Maxwell case.

Solving the $b_{\mu\nu}$6-component gives a master equation for the metric function,

$b_{\mu\nu}$7

This is to be contrasted with the Reissner-Nordström case, where $b_{\mu\nu}$8 is simply a constant electric/magnetic charge.

The explicit solution is

$b_{\mu\nu}$9

revealing that the effective 'charge' sourcing the gravitational potential is now permitted to be an arbitrary function of $$F_{\mu\nu} = b_{\mu\nu} - \partial_\mu a_\nu + \partial_\nu a_\mu$$0. This is in strong contrast to the Reissner-Nordström-AdS solution, where the charge parameter is constant.

Physical and Theoretical Implications

This construction has significant implications:

• Deformation of the Einstein-Maxwell Sector: The substitution of the Maxwell kinetic term by a higher-form-invariant combination yields spacetime solutions that can exhibit nontrivial, possibly time-dependent or retarded charge profiles. The freedom in selecting $$F_{\mu\nu} = b_{\mu\nu} - \partial_\mu a_\nu + \partial_\nu a_\mu$$1 is a direct consequence of the underlying higher-form symmetry.
• Holographic and Astrophysical Contexts: The model provides a candidate gravity dual for force-free electrodynamics in curved spacetime, potentially relevant for systems like magnetar magnetospheres, pulsar winds, or holographic duals for strongly correlated cold string/flux line states.
• On-shell Solution Space: On constant $$F_{\mu\nu} = b_{\mu\nu} - \partial_\mu a_\nu + \partial_\nu a_\mu$$2 slices, the metric reduces locally to standard Reissner-Nordström-like solutions with 'charge' fixed by $$F_{\mu\nu} = b_{\mu\nu} - \partial_\mu a_\nu + \partial_\nu a_\mu$$3. The physical interpretation for this behavior remains open and warrants further exploration.

One theoretical conjecture is that the presence of arbitrary functions of retarded time in the charge parameter might point to new, unexplored sectors in the solution space of gravity coupled to higher-form field theories, relevant for dynamical magnetic flux tubes, topological string fluids, or time-dependent backgrounds in holography.

Conclusion

The paper systematically constructs black hole solutions in a gravitational theory where the electromagnetic sector is governed by the higher-form structure inherent to the effective field theory of force-free electrodynamics. The main outcome is the identification of static, spherically symmetric solutions whose charge parameter can be an arbitrary function of $$F_{\mu\nu} = b_{\mu\nu} - \partial_\mu a_\nu + \partial_\nu a_\mu$$4, in stark contrast with traditional Einstein-Maxwell theory, where the charge is a constant of integration. This work rigorously extends the connection between higher-form symmetries, effective field theories for cold string fluids, and their gravitational realizations, providing a concrete framework for subsequent study of dynamical higher-form gauge structures in general relativity and applications in both high energy and astrophysical contexts.

Speculatively, future directions include a deeper analysis of the physical interpretation of variable-charge solutions, possible instabilities, implications for black hole uniqueness theorems, and the construction of dynamical, non-stationary solutions relevant for astrophysical jets and holographic models.