3+1 formulation of light modes in nonlinear electrodynamics Open Access

Light propagation in curved spacetimes has long been a topic of great theoretical interest.¹ Recently, the Event Horizon Telescope (EHT) observed the shadow of supermassive black holes from light propagating around the event horizon and has provided a way to understand black hole geometries.^2–4 Nonlinear electrodynamics has also provided a nontrivial background for light propagation. Born and Infeld introduced a nonlinear Lagrangian for the Maxwell scalar and its pseudoscalar, which reduces to the Maxwell Lagrangian in the weak-field limit but leads to significant modifications of the physics of light–matter and light–light interactions in the strong-field limit.⁵ A detailed study has been made of the causal properties of the Born–Infeld Lagrangian.⁶ Heisenberg and Euler obtained an exact one-loop effective Lagrangian for electrons in a constant electromagnetic field.⁷ Later, Schwinger introduced the proper-time method for quantum field theory and obtained a one-loop effective Lagrangian for spinless charged bosons and spin-1/2 fermions in a constant electromagnetic field.⁸ 

The prominent features of the Heisenberg–Euler–Schwinger (HES) Lagrangian in quantum electrodynamics (QED) are vacuum polarization^9,10 and Sauter–Schwinger pair production of electrons and positrons.^8,11 Interestingly, in strong electric fields, the vacuum becomes unstable due to pair production, which is a consequence of the imaginary part of the HES Lagrangian.^12,13 The polarized vacuum in a strong electromagnetic field provides a nontrivial background for propagation of probe photons (i.e., light). In recent years, nonlinear electrodynamics (NED) has been intensively studied, partly because of the development of ultra-intense lasers using chirped pulse amplification (CPA) technology^14,15 and partly because of astrophysical observations of highly magnetized neutron stars and magnetars with magnetic fields comparable to or stronger than the critical field strength B_c = m²c²/eℏ = 4.4 × 10¹³ G.¹⁶ Such a strong electromagnetic field makes the vacuum into a polarized medium, and this polarized vacuum behaves similarly to a dielectric or ferromagnetic medium. NED in general exhibits, in addition to electric permittivity and magnetic permeability, the magneto-electric effect, in which a magnetic field induces electric polarization while an electric field induces magnetization.^17–19 A strong electric field, on the other hand, creates pairs of charged particles and antiparticles through Sauter–Schwinger pair production.

NED thus exhibits a rich structure of vacuum polarization, involving phenomena such as vacuum birefringence, photon propagation, and the magneto-electric effect. In a strong magnetic field, the polarized vacuum acquires nontrivial refractive indices, which cause birefringence in the photon propagation either along the magnetic field or in the perpendicular plane.^20,21 Light propagation has been intensively studied in the NED of Plebanski-type Lagrangians,²² according to which a low-energy probe photon satisfies a wave equation in a nonlinear background. Most studies in the literature have employed the covariant four-vector formulation of light propagation (see, e.g., Refs. 20 and 23–27), while the 3+1 formulation has been used for post-Maxwellian theory in weak electromagnetic fields.²⁸ 

In a previous study, we investigated light propagation in a supercritical magnetic field and a weak electric field, in particular for the case of a nonzero electric field along the magnetic field (an “electromagnetic wrench”).^29,30 In contrast to the pure magnetic field case, the electromagnetic wrench introduces magneto-electric effects and thus significantly modifies the light modes. In our previous work,^29,30 we focused on a configuration with parallel electric and magnetic fields to study the effect in the simplest setting. However, given the complicated field configurations around neutron stars or in the focal regions of ultra-intense lasers, we need to extend the previous formulation to an arbitrary field configuration. Furthermore, it is worthwhile making the formulation applicable to other Lagrangians considered in the context of general NED.

In Minkowski spacetime with metric (+, −, −, −), the timelike Killing vector ∂_t makes the spacetime manifold foliate into one-parameter spacelike hypersurfaces. On each hypersurface Σ_t, the energy–momentum of a photon is given by k^μ = (ω, k) and the field by F^μν. In curved spacetime, the 3+1 formulation of Maxwell theory^32–34 has often been preferred for modeling in astrophysics and cosmology. In the 3+1 formulation, the propagation and polarization of probe photons (light) are expressed in terms of directly measurable electromagnetic fields (E, B) and the photon direction k on each Σ_t. In practice, the 3+1 formulation has the advantage of providing the light modes in general NED in the familiar language of conventional optics, facilitating the design of experiments and observations.

The remainder of this paper is organized as follows. In Sec. II, we present the general 3+1 framework for analyzing light modes in a medium in the presence of strong background fields. Then, in Sec. III, this is compared with the formulation used in materials science to deal with the magneto-electric effect. In Sec. IV, we apply the framework to the NED of Plebanski-type Lagrangians and compare it with the alternative covariant formulation. In Sec. V, we present specific examples with the representative NED Lagrangians: the post-Maxwellian (PM), Born–Infeld (BI), ModMax (modified Maxwell, MM), and the HES QED Lagrangians. For these Lagrangians, the expressions for the light modes in an arbitrarily strong magnetic field are worked out. Finally, in Sec. VI, summarizing the results, we draw our conclusions, discussing the implications of the 3+1 formulation for research on vacuum birefringence, in the context, for example, of terrestrial experiments using ultra-intense lasers in the weak-field regime and of astrophysical observations of highly magnetized neutron stars in the strong-field regime.

In this section, we set up a 3+1 framework to obtain the light propagation modes in an arbitrary medium in the presence of strong background fields. When applied to NED Plebanski-type Lagrangians, as shown in Sec. IV, this framework yields refractive indices and polarization vectors for a given propagation direction.

In this section, we derive the general expressions for the light propagation modes in NED described by Plebanski-type Lagrangians. The general framework of mode analysis in Sec. II is used to find the refractive indices and polarization vectors for a given propagation direction.

In general, the Lagrangian may have an imaginary part, which leads to absorption of probe light.⁴³ For example, the HES Lagrangian has an imaginary part when either only an electric field exists or the electric field has a component parallel to the magnetic field. This imaginary part becomes significant as the purely electric field or the electric field component along the magnetic field approaches or surpasses the critical field strength $Ec=me2/e=1.32×1016V/cm2$⁠.⁴⁴ As a consequence, electron–positron pairs are produced by strong electromagnetic fields, in Sauter–Schwinger pair production or vacuum breakdown.^8,11 In the present work, we do not take such an imaginary part into account, to assume a stable medium. In practice, the imaginary term can be neglected when the electric field is weaker than E_c/3 and the magnetic field.³⁰ 

The linear response tensors in (19) exhibit some symmetries. First, $ϵE=ϵET$⁠, $μ̄B=μ̄BT$⁠, and $ϵB=−μ̄ET$⁠, i.e., (13). Second, $ϵE↔μ̄B$ and $ϵB↔−μ̄E$ under the duality transformation (E, B) → (−iB, iE), which keeps F, G, and $L$ the same. Finally, ϵ_E and $μ̄B$ have even parity, while ϵ_B and $μ̄E$ have odd parity: $L$ should be an even function of G, i.e., $L(F,−G)=L(F,G)$⁠, to respect inversion symmetry.

The form of Λ in (22) implies that the reciprocal vectors of X, Y, and Z may be used as basis vectors to represent δE, as in crystallography. However, this requires nondegeneracy, i.e., the condition that the volume V ≔ X × Y · Z is not zero. The degeneracy (V = 0) is caused by either acd = 0 or $P×Q⋅n̂=0$⁠, as can be found from (21) and (23). The first condition acd = 0 is independent of $n̂$ and depends only on the functional form of $L(F,G)$ and the background field. In fact, the condition reduces to c = 0, because a² = L_FF and d² = −L_F do not vanish for nontrivial Lagrangians. An example of having c = 0 is the MM Lagrangian, as will be shown in Sec. V C. By contrast, the second condition $P×Q⋅n̂=0$ depends also on $n̂$⁠. The second condition is met by a configuration satisfying $Bt=n̂×Et$⁠,⁴⁸ which we call the free-propagation configuration (FPC). In the FPC, n is unity, and S ≔ E × B heads along $n̂$⁠, and any vector perpendicular to $n̂$ is a polarization vector. In other words, the light in FPC propagates as if it were in a free vacuum. In the nondegenerate case, the FPC is the only configuration with n = 1. In the degenerate case of c = 0, however, another configuration has n = 1, as shown in Sec. IV B 2. A special case of FPC is the collinear configuration: $E‖B‖n̂$⁠. Below, we deal with the nondegenerate case of c ≠ 0 first and then the degenerate case of c = 0, disregarding the FPC, owing to its triviality.

In contrast to the photon in a free vacuum, a photon in the NED of a Plebanski-type Lagrangian acquires a longitudinal component when r ≠ 0: δE · n = r/d from (23), (25), and (26). Such a longitudinal component appears in anisotropic dielectric media.³⁸ 

So far, we have implicitly assumed that n ≠ 1. When n = 1, the dyadic Λ in (22) becomes Λ = XX + YY + ZZ because of e = 0. Then, the condition MN − H² = 0 reduces to P_t = Q_t = 0, which is satisfied only for the FPC. In such a case, the light sees no effect of NED.

In contrast to the nondegenerate case, the degenerate case always has a non-FPC with n = 1. When n = 1, Λ = XX + ZZ. When Z ∦ X (non-FPC), Z × X becomes the polarization vector, because Λ · δE = 0 is satisfied by construction. The case Z‖X corresponds to the FPC. An advantage of our 3+1 formulation is a transparent analysis of the degenerate case compared with that of the covariant formulation.

The procedure to find light modes in nondegenerate nonlinear electrodynamics is outlined in Table II. Here, we succinctly present the essential formulas used for the procedure in order. It is assumed that the formulas for {L_F, L_FF, L_FG, L_GG}, and thus {a², ab, b², c², d²} in (24) are known, regardless of the field configuration.

In summary, light modes are completely determined by the formulas (32), (34), and (40)–(42) once {a², ab, b², c², d²} (24) and the configuration parameters (39) are known.

However, the 3+1 formulation has a more direct connection to the usual concepts in optics, such as the permittivity, permeability, and magneto-electric response tensors (19), and provides the refractive indices and polarization vectors in terms of familiar quantities E, B, and $n̂$⁠. Thus, it can allow a richer comparison of the nonlinear vacua with conventional optical media. The formulation also reveals the unique mathematical structure of the nonlinear vacuum’s response: the Fresnel matrix Λ consists of self-conjugate dyadics (22), which facilitates the construction of polarization vectors. Finally, the formulation is related to the 3+1 formulation in curved spacetime,³² which is popular in numerical simulation of general relativistic phenomena.⁵⁰ In our 3+1 formulation in Minkowski spacetime, we use the Killing vector ∂_t to decompose the spacetime into one-parameter spacelike hypersurfaces Σ_t. Each observer on Σ_t measures the energy–momentum of photons as k^μ and the electromagnetic field E, B.

In this section, we apply the 3+1 formulation to several NED Lagrangians for the case of a purely magnetic background field: post-Maxwellian (PM),²⁸ Born–Infeld (BI),⁵ ModMax (MM),^51,52 and Heisenberg–Euler–Schwinger (HES) QED⁵³ Lagrangians. The PM Lagrangians are frequently used in X-ray polarimetry for light propagation near pulsars and also in black hole physics. Recently, these NED Lagrangians and Plebanski-type Lagrangians, in general, have been actively used for theoretical black holes.⁵⁴ Theoretical magnetic black holes whose magnetic field strength is at electroweak scales have been proposed,⁵⁵ and these require QED corrections in the context of general relativity.

Without loss of generality, we assume that the magnetic field is along the z axis, and the light’s propagation direction vector lies on the xz plane: $B=B0ẑ$ and $n̂=(sin⁡θ,0,cos⁡θ)$⁠, as shown in Fig. 1. The light modes for a given Lagrangian, i.e., n and δE, are found by following the procedure in Table II and Sec. IV C, as summarized in Table III.

Interestingly, the light propagation modes of the four Lagrangians exhibit a common structure, as manifestly parameterized in Table III, owing to the fact that these Lagrangians are of Plebanski type. The polarization vector is either perpendicular to the plane (⊥) formed by the magnetic field and the propagation direction or on the plane (‖). As a consequence of the common structure, the photon propagation at special angles is universal. When θ = 0, n_⊥ = n_‖ = 1 with δE_‖ = (1, 0, 0): a photon propagating along the magnetic field sees no background field effect. Note that δE_⊥ = (0, 1, 0) regardless of θ. When θ = π/2, δE_‖ = (0, 0, 1): a photon propagating perpendicular to the magnetic field has no longitudinal component. In general, nonzero ϵ implies the presence of a longitudinal component: $δE‖⋅n̂=(sin⁡θ⁡cos⁡θ)ϵ/(1+ϵ)$⁠. The details specific to each Lagrangian are discussed below.