Electromagnetic Wave Propagation in Pulsar Magnetospheres

A new nonlinear electromagnetic wave mode in a magnetized plasma is predicted. Its existence depends on the interaction of an intense circularly polarized electromagnetic wave with a plasma, where quantum electrodynamical photon–photon scattering is taken into account. This scattering gives rise to a new coupling between the matter and the radiation. Specifically, we consider an electron–positron plasma, and show that the propagation of the new mode is admitted. It could be of significance in pulsar magnetospheres, and result in energy transport between the pulsar poles. Subject headings: Plasmas — pulsars: general — stars: neutron — waves Astrophysical environments can be most violent and energetic. Physics considered ‘exotic’ in Earth based laboratory applications can be common throughout our Universe, and sometimes even vital for the existence of certain observed phenomena. Pulsars, surrounded by strong magnetic fields, are most prolific sources of exotic physics. Quantum electrodynamics (QED) is an indispensable explanatory model for much of the observed pulsar phenomena. Scattering of photons off photons is predicted by QED, and it can be a prominent component of pulsar physics, since pulsars offer the necessary energy scales for such scattering to occur. Related to the scattering of photons is the concept of photon splitting in strong magnetic fields (Adler 1971). It has been suggested that such effects could be important in explaining the radio silence of magnetars (Kouveliotou 1998; Baring & Harding 2001). In the present Letter we will point out the existence of a new electromagnetic wave that may exist in pulsar magnetospheres, due to the interaction of photons with the quantum vacuum. A discussion of the properties of this electromagnetic wave using parameters relevant to strongly magnetized pulsars will be given. The weak field theory of photon–photon scattering can be formulated in terms of the effective Lagrangian density L = L0 + LHE, (1) where L0 = − 1 4ǫ0FabF ab = 12ǫ0(E 2 − cB) is the classical free field Lagrangian, and LHE = κǫ 2 0 [ 4 ( 1 4FabF ab )2 + 7 ( 1 4FabF̂ ab )2] , (2) is the Heisenberg–Euler correction (Heisenberg & Euler 1936; Schwinger 1951), where F̂ab = 1 2ǫabcdF , and 1 4 F̂abF ab = −cE · B. Here κ ≡ 2α~/45mec 5 ≈ 1.63 × 10 ms/kg, α is the fine-structure constant, ~ is the Planck constant, me is the electron mass, and c is the speed of light in vacuum. With Fab = ∂aAb − ∂bAa, A being the four-potential, we obtain, from the Euler– Lagrange equations, the field equations ∂b[∂L /∂Fab] = 0, i.e. (see, e.g. Shukla et al. 2004) ∂bF ab = 2ǫ0κ∂b [ (FcdF )F ab + 74 (FcdF̂ )F̂ ab ] + μ0j , (3) where j is the four current. For a circularly polarized waveE0 = E0(x̂±iŷ) exp(ik· x − iωt) propagating along a constant magnetic field B0 = B0ẑ, the invariants satisfy FcdF cd = −2E 0 ( 1− kc ω2 ) + 2cB 0 and FcdF̂ cd = 0, (4) where k is the wave number and ω the frequency of the circularly polarized electromagnetic wave. Thus, Eq. (3) can be written as 2A = −4ǫ0κ [ E 0 ( 1− kc ω2 ) − cB 0 ] 2A + μ0j , (5) in the Lorentz gauge, and 2 = ∂a∂ . For circularly polarized electromagnetic waves propagating in a magnetized cold multicomponent plasma, the four current can be ‘absorbed’ in the wave operator on the left-hand side by the replacement