The force-free dipole magnetosphere in non-linear electrodynamics

TL;DR

This paper investigates how quantum electrodynamics-induced non-linear electrodynamics modifies the force-free magnetosphere structure of pulsars, deriving corrected solutions that show stronger polar magnetic fields.

Contribution

It derives the first-order non-linear corrections to the pulsar magnetosphere in a general framework, extending classical models to include quantum effects.

Findings

Corrected dipole fields converge towards the rotational axis.

Polar magnetic fields are stronger in the non-linear models.

Solutions provide a more accurate description of magnetar and pulsar magnetospheres.

Abstract

Quantum electrodynamics (QED) effects may be included in physical processes of magnetar and pulsar magnetospheres with strong magnetic fields. Involving the quantum corrections, the Maxwell electrodynamics is modified to non-linear electrodynamics. In this work, we study the force-free magnetosphere in non-linear electrodynamics in a general framework. The pulsar equation describing a steady and axisymmetric magnetosphere is derived, which now admits solutions with corrections. We derive the first-order non-linear corrections to the near-zone dipole magnetosphere in some popular non-linear effective theories. The field lines of the corrected dipole tend to converge on the rotational axis so that the fields in the polar region are stronger compared to the pure dipole case.

Equations

$$S=\int \sqrt{-g}\left[\frac{1}{4\pi }{\mathcal{L}}_{\text{EM}}(s,p)+A_{\mu }{J}^{\mu }\right]d^{4}x,$$

$$s\equiv \frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }=\frac{1}{2}({\mathbf{B}}^2-{\mathbf{E}}^2),$$

$$p\equiv \frac{1}{4}{\tilde{F}}^{\mu \nu }{F}_{\mu \nu }=\mathbf{E}\cdot \mathbf{B}.$$

$${\nabla }_{\mu }{\tilde{F}}^{\mu \nu }=0.$$

$$\nabla \times \mathbf{E}=-\dot{\mathbf{B}},$$

$$\nabla \cdot \mathbf{B}=0.$$

$${\nabla }_{\mu }{G}^{\mu \nu }=4\pi J^{\nu },$$

$$G^{\mu \nu }=S{F}^{\mu \nu }+P{\tilde{F}}^{\mu \nu },$$

$$S\equiv {\partial }_s{\mathcal{L}}_{\text{EM}},P\equiv {\partial }_p{\mathcal{L}}_{\text{EM}}.$$

$$\nabla \cdot \mathbf{D}=4\pi \rho ,$$

$$\nabla \times \mathbf{H}=4\pi \mathbf{j}+\dot{\mathbf{D}},$$

$$\mathbf{D}=-S\mathbf{E}+P\mathbf{B},$$

$$\mathbf{H}=-S\mathbf{B}-P\mathbf{E}.$$

$$T_{\text{EM}}^{\mu \nu }=-\frac{1}{4\pi }[S{F}_{\alpha }^{\mu }{F}^{\nu \alpha }+P{\tilde{F}}_{\alpha }^{\mu }{F}^{\nu \alpha }-g^{\mu \nu }{\mathcal{L}}_{\text{EM}}].$$

$${\nabla }_{\mu }{T}_{\text{EM}}^{\mu \nu }=J_{\mu }{F}^{\mu \nu }.$$

$$J_{\mu }{F}^{\mu \nu }=0.$$

$$\mathbf{j}\cdot \mathbf{E}=0,\rho \mathbf{E}+\mathbf{j}\times \mathbf{B}=0,$$

$$p=\mathbf{E}\cdot \mathbf{B}=0.$$

$${\partial }_r{A}_0{J}^r+{\partial }_{\theta }{A}_0{J}^{\theta }=0,$$

$${\partial }_r{A}_0{J}^0+F_{r\theta }{J}^{\theta }+{\partial }_r{A}_{\varphi }{J}^{\varphi }=0,$$

$${\partial }_{\theta }{A}_0{J}^0-F_{r\theta }{J}^r+{\partial }_{\theta }{A}_{\varphi }{J}^{\varphi }=0,$$

$${\partial }_r{A}_{\varphi }{J}^r+{\partial }_{\theta }{A}_{\varphi }{J}^{\theta }=0.$$

$$[A,B]\equiv {\mathcal{L}}_{T}B={\partial }_{r}A{\partial }_{\theta }B-{\partial }_{\theta }A{\partial }_{r}B,$$

$$[A_0,A_{\varphi }]=0.$$

$$d{A}_0=-\Omega (A_{\varphi })d{A}_{\varphi }.$$

$$J^0=-\frac{1}{4\pi }\nabla \cdot \left(S\nabla A_0\right),$$

$$J^r=-\frac{1}{4\pi r^2\sin \theta }{\partial }_{\theta }(\sin \theta S{F}_{r\theta }),$$

$$J^{\theta }=\frac{1}{4\pi r^2\sin \theta }{\partial }_r(\sin \theta S{F}_{r\theta }),$$

$$J^{\varphi }=\frac{1}{4\pi }\nabla \cdot \left(\frac{S\nabla A_{\varphi }}{r^2{\sin }^2\theta }\right).$$

$$[A_0,\sin \theta S{F}_{r\theta }]=[A_{\varphi },\sin \theta S{F}_{r\theta }]=0.$$

Full text

1 Introduction

It is inferred that pulsars and magnetars possess very strong magnetic fields. The field strength can even exceed the critical value $B_{Q}=m_{e}^{2}c^{3}/\hbar e\simeq 4.4\times 10^{13}$ G, above which the quantum electrodynamics (QED) effects should not be ignored.

QED effects will affect the polarization and spectra of the thermal radiation from the surface of magnetars and pulsars with strong magnetic fields. In the magnetospheres, the photon polarizations can be decomposed into two modes according to the propagating direction and the local magnetic field direction. Due to the resonance between the vacuum and plasma birefringence, one of the polarization modes of the X-ray photon can be converted into the other. This gives rise to an energy-dependent polarization signature for the observed quiescent non-thermal X-ray emission [1, 2, 3, 4, 5, 6, 7]. The magnetar magnetospheres are opaque to high energy photons due to the attenuation by the magnetic photon splitting (below the energy threshold $2m_{e}c^{2}$) and pair production (above the threshold). This will distort the blackbody spectra of the surface thermal radiation and can be tested with future precise observations of the spectra and the polarizations [8, 9], which meanwhile provides information on the surface magnetic fields in the magnetar magnetospheres.

Thus, the strength and geometry of the magnetic fields are crucial in these QED processes. It is expected that these processes more effectively work at low altitudes where the magnetic fields are higher. In the near-zone regions, the geometry of the pulsar magnetospheres is usually taken to be of a force-free dipole structure. When including the QED corrections, the Maxwell electrodynamics should be modified with additional non-linear terms and the dipole magnetosphere must be corrected with non-linear contributions, as analyzed in previous works [10, 11].

In this work, we consider the force-free magnetosphere in general non-linear electrodynamics. In contrast to previous treatments [12, 13], we shall follow the traditional approach to do so. We first derive the pulsar equation, describing the steady and axisymmetric magnetospheres, in non-linear electrodynamics. We then obtain the corrected dipole magnetosphere to leading order from the pulsar equation in some popular non-linear effective theories, like the Euler-Heisenberg (EH) theory, Born-Infeld (BI) theory and the logarithmic theory.

2 Non-linear electrodynamics

The action of general electrodynamics takes the form

[TABLE]

where $\mathcal{L}_{\textrm{EM}}(s,p)$ is the general Lagrangian of the electromagnetic fields with

[TABLE]

[TABLE]

The dual field strength $\widetilde{F}^{\mu\nu}=(1/2)\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$.

The equations of motion can be derived from the action. It is obvious that the Bianchi identity is automatically satisfied:

[TABLE]

In Minkowski spacetime, the equation can be decomposed into

[TABLE]

[TABLE]

The relation between the current and fields is given by

[TABLE]

where $J^{\nu}$ is the conserved current and

[TABLE]

with

[TABLE]

When $S=-1$ and $P=0$, the equation reduces to the Maxwell theory case. In Minkowski spacetime, the equation can be re-expressed as:

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

3 The force-free condition

The derivative of the Lagrangian with respect to the metric gives the energy-momentum tensor of the electromagnetic fields

[TABLE]

The tensor satisfies

[TABLE]

The equation (15) relating the divergence of the EM energy-momentum to the Lorentz force takes the same form as in the Maxwell theory. It determines the change of the momenta of the charged particles in the system.

It is usually assumed that, in a steady magnetosphere filled with plasma, the charged particles in the magnetospheres with the strong EM fields should feel no net force (at least in most regions). This means that the Lorentz force in Eq. (15) should vanish:

[TABLE]

This is the force-free condition in general non-linear electrodynamics, also of the same form as in the Maxwell theory. This condition also says that the dynamics of the energy density in the system is dominated by the electromagnetic fields and the inertial of the plasma in the system can be ignored. That is, $T_{\textrm{EM}}^{\mu\nu}$ can be approximately taken as the energy-momentum density of the whole system so that it is conserved.

In components, the force-free equation is decomposed into

[TABLE]

which implies

[TABLE]

So we simply have $\mathcal{L}_{\textrm{EM}}(s,p)=\mathcal{L}_{\textrm{EM}}(s)$ and $G^{\mu\nu}=SF^{\mu\nu}$ under the force-free condition.

4 The pulsar equation

Under the force-free condition, the equations describing a force-free magnetosphere can be derived. As usual, we consider the simplest case: the axisymmetric and steady magnetospheres in Minkowski spacetime. On the spherical coordinates, the force-free condition (16) reads:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For convenience, let us define the Poison bracket as in [14] (and also [15]):

[TABLE]

where the tangent vector $T=\partial_{r}A\partial_{\theta}-\partial_{\theta}A\partial_{r}$. The necessary condition that $A$ is a function of $B$ (vice versa) is that $[A,B]=0$.

From Eqs. (19) and (22), we can find that

[TABLE]

So $A_{0}$ should be a function of $A_{\phi}$. We can define:

[TABLE]

As known, $\Omega$ is the angular velocity of a magnetic field line. It is constant along the magnetic field line.

The Maxwell equations (7) with non-linear corrections are expressed as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From Eqs. (19), (22), (27) and (28), we find that

[TABLE]

So $\sin\theta SF_{r\theta}$ is also a function of $A_{\phi}$. Let us denote

[TABLE]

Then the charge density is expressed as

[TABLE]

From Eq. (20) or (21), we have

[TABLE]

where the prime denotes the derivative with respect to $\psi$.

By comparing Eqs. (29) and (33), we can derive the general pulsar equation:

[TABLE]

Specifically, the equation on spherical coordinates can be written as

[TABLE]

When $S=-1$, this reduces to the pulsar equation in Maxwell’s theory.

The electromagnetic fields in the unit basis of spherical coordinates are expressed as:

[TABLE]

[TABLE]

With them, we have

[TABLE]

which must be non-negative. The expression on the cylinder coordinates ($x=r\sin\theta$ and $z=r\cos\theta$) given in the second line indicates that the translational symmetry along the rotation axis remains in non-linear electrodynamics, i.e., the action and the equations are invariant with the transformation: $z\rightarrow z^{\prime}=z+\epsilon$.

The spin-down rate is obtained:

[TABLE]

where the Poynting flux is

[TABLE]

So the torque takes the same form as in the Maxwell theory.

5 The near-zone dipole magnetospheres

The pulsar equation is hard to solve even in the Maxwell theory. So it is expected that numerical methods are needed to solve the equation (34) in non-linear electrodynamics. But here we do not need to seek for the global solutions. We only need to focus on the magnetosphere at low altitudes where the electromagnetic fields are strong and the non-linear corrections may be important.

As done in the Maxwell theory, the near-zone magnetospheres on pulsars are usually taken as a dipole structure, which serves as the inner boundary condition in numerical simulations of pulsar magnetospheres [16, 17, 18]. This structure can be obtained from the pulsar equation at $r\rightarrow 0$, where the rotational velocities of the magnetic field lines are much less than the speed of light and the electric current is negligible by setting $I=0$. In this limit, the equation (4) with $S=-1$ reduces to:

[TABLE]

The equation is solved by the general form

[TABLE]

where $\psi_{-n}(\theta)$ is related to the associated Legendre polynomials for different $n$. For $n=0$, it is a monopole, and, for $n=1$, it is a pure dipole $\psi=\sin^{2}\theta/r$. The rotational effects in outer regions just deform this basic dipole geometry.

In what follows, we determine the force-free magnetospheres in the near regions in different non-linear theories.

5.1 The BI theory

The BI effective theory is a well regularized non-linear theory, leading to finite self-energy of point-like charge and absence of birefringence. It also arises from the worldvolume action of D-branes in string theory. Some aspects of pulsar magnetospheres in BI effective theory were discussed previously in [2, 19]. Here, we consider the corrected dipole geometry in the theory.

The Lagrangian of electromagnetic fields in the BI effective theory takes the form

[TABLE]

where the only parameter is of dimension of mass squared: $b=M^{2}$. The lower bound of $M$ is constrained to be $4\times 10^{-4}$ GeV by PVLAS [20] and 100 GeV by ATLAS in LHC [21] (see also [19]).

From the Lagrangian, we can obtain the expression of $S$:

[TABLE]

At large distance $r$, $S^{2}\rightarrow 1$ and so the pulsar equation recovers the one in Maxwell’s theory. Inserting it into the non-linear pulsar equation (4), we can basically derive solutions. But it is difficult to do so. There even does not exist solutions that are only dependent on $\theta$ (like Michel’s monopole solution in the Maxwell theory).

Let us take the near-zone limit with approximately vanishing $\Omega$ and $I$, for which the equation (4) with (44) is simplified to:

[TABLE]

There exist exact solutions that are independent of the parameter $b$: $\psi=\cos\theta$ ($n=0$), $\psi=r\cos\theta$ ($n=1$) and $\psi=r^{2}\sin^{2}\theta$ ($n=2$), which are also solutions to the pulsar equation in the Maxwell theory. So the non-linear terms do not alter the non-rotating monopole solution.

For the dipole solution, the situation is different. From the above equation, we can determine that the solution can be expanded in powers of $b^{-2}$:

[TABLE]

The zero-th order part $\psi^{(0)}=m\sin^{2}\theta/r$ is the pure dipole solution. $\psi^{(1)}$ is the first order correction at the order $\mathcal{O}(b^{-2})$. Inserting the expression into Eq. (5.1), we get the leading order equation:

[TABLE]

Thus, the dependence of $\psi^{(1)}$ on $r$ should be of the form $\sim r^{-7}$. Up to the first order, the final solution is:

[TABLE]

where the first-order characteristic distance

[TABLE]

So the correction becomes unimportant sharply at $r\gg r_{1}$. The first-order corrected part $\psi^{(1)}$ becomes important for $r_{1}\gtrsim r\gg r_{2}$, where $r_{2}$ is the characteristic distance of $\psi^{(2)}$ (not derived here). The distribution of the magnetic field lines from the solution is displayed in Fig. (1). Compared with the dipole magnetosphere, the field lines around $r\sim r_{1}$ tends to converge on the rotation axis. The curvature of the field lines becomes larger in the distance less than but near the characteristic distance $r_{1}$.

5.2 The Logarithmic theory

In the Logarithmic theory (e.g., see [22, 23]), the self-energy of point-like charge is finite and the birefringent phenomenon appears. Its action takes a logarithmic form

[TABLE]

The equation (4) with $I=\Omega=0$ reduces to

[TABLE]

It is interesting that the equation has the same three exact solutions as Eq. (5.1). The first order solution $\psi^{(1)}$ also takes the same form as Eq. (47). So the geometries of the field lines are the same and the solutions can not be discriminated in the two theories up to the first order.

5.3 The EH theory

The various QED effects on the physical processes in magnetar magnetospheres, including the vacuum birefringence, photon splitting and pair production, were mostly discussed based in the EH effective theory [24, 25]. In the weak field limit, the Lagrangian expanded to leading orders is

[TABLE]

where $\beta=e^{2}/(45hcB_{K}^{2})$ with $B_{k}=m_{e}^{2}c^{3}/(\hbar e)$. So the leading order terms are the same as in the BI (43) and the Logarithmic (50) lagrangians under the force-free condition, just with different parameters $b$ and $\beta$. The first-order corrected dipole geometry should be also the same as given in Fig. (1). So, within the characteristic distance, the dipole structure takes a multipole-like structure, consistent with the corrected dipole structure to leading orders in the EH theory [10, 11].

6 Conclusions

The pulsar equation in general non-linear electrodynamics is derived. The corrected dipole solutions in some popular non-linear effective theories are obtained and discussed. These solutions take the same form up to the first order, which indicate that the field lines tend to converge on the rotation axis. So the fields are stronger in polar region and have larger curvature within the characteristic distance than in the pure dipole magnetosphere. This discrepancy should be taken into account in considering the quantum effects in the radiative transfer process of the surface emission.