Analytical solutions of CPT-odd Maxwell equations in Schwarzschild spacetime

Lorentz symmetry (LS) is a fundamental symmetry in both general relativity (GR) [1] and the Standard Model [2] of particle physics. Some candidate theories of quantum gravity, such as certain formulations of loop quantum gravity [3,4] and modifications of string theory [5,6], allow for small deviations from exact Lorentz invariance at very high energies. In some scenarios, this could lead to extremely small Lorentz-violating effects at lower energies. However, Lorentz-violating signals at energy scales accessible in high-energy astrophysical observations ( $ \sim 10^{11} \text{GeV} $ [7]) are expected to be extremely small and are generally suppressed by a small ratio involving the Planck scale $ 1.22 \times 10^{19} \text{GeV} $ , as suggested by dimensional analysis and observational constraints.

Extremely small Lorentz violation (LV) effects may accumulate over long distances and at high energies in certain models, making them potentially detectable through terrestrial observations. Additionally, some experiments and astrophysical observations can test processes that are strictly forbidden in standard Lorentz-invariant (LI) physics but may occur in LV scenarios, such as vacuum birefringence [8,9], photon decay [10], and photon splitting [11]. These observations have established stringent constraints on LV with high precision, particularly through high-energy observatories such as Pierre Auger and LHAASO [12,13]. Moreover, most of these observatories primarily rely on multi-wavelength observations and long-distance photon propagation to study astrophysical events.

As a comprehensive framework of effective field theory, the Standard Model Extension is capable of describing both small deviations from LS in flat spacetime [6,14] and violations of local LS in gravitational contexts [15], thus encompassing both high-energy phenomena in flat spacetime [6] and gravitational effects or particle motions in curved spacetime [16,17]. In recent years, there has been increasing interest in probing LV in astrophysics through observations of CMB photons [18,19], neutrinos [20], and gravitational waves [21]. A natural question is whether interesting LV effects manifest in intrinsically curved spacetime. Regarding the constraints on LV from studying the cosmological propagation of GRB or CMB photons, a key assumption is that spacetime is described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is curved but conformally flat. Here, we investigate LV electrodynamics in the simplest non-conformally flat curved spacetime: the Schwarzschild metric.

There has been a growing number of studies on photon behavior in curved spacetime, particularly following the successful capture of black hole (BH) images by the Event Horizon Telescope (EHT) Collaboration [22]. The photon sphere and subrings at the edge of the black hole shadow may encode crucial information about potential new physics beyond GR [23]. This highlights the necessity of studying photon behavior in curved spacetime near BHs.

As a first attempt, we aim to investigate the asymptotic behavior of CPT-violating (CPTV) photons in the Schwarzschild geometry using the null formalism. Intuitively, employing the null formalism to study massless particles is, in some sense, analogous to describing the motion of massive particles — such as a gyroscope — using an orthonormal tetrad in its instantaneous rest frame, even though massless particles do not possess a rest frame. However, the underlying principle remains the same: null tetrads naturally accommodate massless particles that follow null trajectories. Moreover, the null formalism offers unique advantages in analyzing the asymptotic behavior of massless particles, particularly photons in this context. It significantly simplifies the description of both the tangent of the null geodesic for photons and gravitons, as well as their polarization states [24]. Consequently, it provides a coordinate-independent framework for studying photon dynamics with a clear geometric interpretation, especially in highly curved spacetimes. Notably, it also offers a physically transparent decomposition of the Faraday tensor into ingoing, outgoing, and Coulomb modes.

In the presence of CPT or Lorentz violation, contrary to conventional expectations, the asymptotic behavior of photons may be qualitatively altered [25]. For instance, vacuum birefringence can be understood in terms of the modified topology of the light cone structure, where different helicities of CPT-odd photons experience distinct causal cones [26].

Pioneering studies on exact solutions for LI photon fields as perturbations in given background geometries include vacuum solutions for photon fields in the Kerr spacetime [27] as well as solutions for a point charge near a Schwarzschild BH [28] and a Kerr BH [29]. Bičák et al. studied photon fields in curved spacetime within the Newman-Penrose (NP) framework [30], including the Schwarzschild [31], ReissnerNordström (R-N) [32], and Kerr [33] BH backgrounds. As a preliminary attempt, here we study the behavior of CPT-odd photons in the Schwarzschild geometry following a similar approach.

In Sec. II, we review the Newman-Penrose (NP) formalism and discuss some earlier studies on the LI Maxwell equations in curved spacetime using the NP framework. Then, we examine the CPT-violating Maxwell equations within the NP formalism in curved spacetime. Next, we present a method to solve the coupled Maxwell equations and provide special solutions in Sec. III. In the last section, we summarize our results and provide a short conclusion. In this paper, the signature of the metric tensor $ g_{\mu \nu} $ is chosen to be $ (+,-,-,-) $ , and we use geometric units with $ \epsilon_0=\mu_0=c=G=\hbar=1 $ . The notation conventions are as follows: spacetime indices are represented by Greek letters such as $ \mu, \nu, \rho $ , while null tetrad indices are represented by Latin letters such as $ a, b, c $ .

We study LV (more specifically CPTV) photon behavior within the photon sector of the minimal SME [6]. The action is given by