Hawking-like radiation of charged particles via tunneling across the lightcylinder of a rotating magnetosphere

In rotating magnetospheres planted on compact objects, there usually exist lightcylinders (LC), beyond which the rotation speed of the magnetic field lines exceeds the speed of light. The LC is a close analog to the horizon in gravity, and is a casual boundary for charged particles that are restricted to move along the magnetic field lines. In this work, it is proposed that there should be Hawking-like radiation of charged particles from the LC of a rotating magnetosphere from the point of view of tunneling by using the field sheet metric.


Introduction
Particles can be created and radiated from the horizons of black holes and the universe [1,2]. Outside the horizon of a black hole, virtual particle pairs are excited from the vacuum. The positive energy particle may escape from the near horizon region, forming Hawking radiation, while the negative energy particle falls into the black hole. The Hawking radiation from a black hole can also be interpreted as particles tunneling across the horizon [3]. When particle pairs are created inside the horizon, the positive energy particles can tunnel through the horizon and come out, which is forbidden by the classical theory. When pairs are created outside the horizon, the negative energy particles tunnel from outside to the interior.
Analogous horizon also exists in electrodynamics. In constructing magnetospheric theory on rotating and compact objects, we usually encounters a critical surface, called the lightcylinder (LC). Beyond the LC, the rotation speed of the magnetic field lines exceeds the speed of light. In pulsar magentospheres, the LC is a turning surface of geometry and topology of the field lines. It is a horizon for charged particles and Alfvén waves propagating along the magnetic field lines. Once a charged particle moves out of the LC, it will never cross the LC and get inside again. This is quite similar to the case of horizons in gravity.
When a pair of charged particles are excited just inside the LC, one particle may cross the LC and never comes back, left the other particle inside the LC. Similarly, a charged particle outside the LC may also tunnel across the LC and gets inside, left the other particle remain outside the LC. Hence, there should exist radiation of particles from the LC towards both sides. In this work, we shall consider this by using the field sheet metric obtained in [4]. The field sheet is a two dimensional "spacetime" that governs the motion of the charged particles that move along the magnetic field lines. So here only charged particles are relevant. This is different from the case of a horizon in gravity, for which any particles that exist in nature can be radiated.

The field sheet of a rotating magnetosphere
The magnetospheres on compact objects are usually filled with large enough amount of charged particles so that the electric field parallel to the magnetic field line is screened, i.e., the electromagnetic fields satisfy the degenerate condition. Moreover, the electromagnetic fields are very strong on these objects and the inertia of charges can be neglected. So the force-free condition is usually assumed.
As shown in [5,6,7,4], the degenerate electromagnetic field satisfying the condition F ⋆ F = 0 can be expressed as in terms of the two Euler scalars. In components, this means that the electric field and magnetic field lines are perpendicular: E · B = 0, which can be derived from the force-free condition: F µν J ν = 0. The gauge field potential is given by: For stationary and axisymmetric magnetospheres, the Euler potentials take the following forms The functions ψ and f determines the poloidal current I that flows along the magnetic field lines: In the force-free approximation, the angular velocity of the magnetic field lines Ω and the poloidal current I are both functions of ψ generally. They satisfy the stream equation in Minkowski spacetime for constant Ω: The field sheet is defined as the intersection of the constant Euler potentials. It is like the spacetime that governs the propagation of the charged particles and Alfvén waves on the field lines. Substituting the conditions dφ 1 = dφ 2 = 0 into the Minkowski spacetime in spherical coordinates, we get the two-dimensional field sheet metric: For vacuum solutions with I = 0, the metric is static. This two-dimensional metric generally has vanishing Einstein tensor: G ij = T matter ij = 0, where i, j denote the coordinates (t, r). This is analogous to the vacuum spacetime around a black hole. It is consistent with the force-free approximation, for which the inertial of the charged particles is neglected.
This metric in the Eddington-Finkelstein coordinates looks like the de Sitter metric, but is not exactly the same one. The horizon is located at r = 1/(Ω sin θ) with constant θ.
The surface gravity of the horizon is κ = Ω sin θ.
Following the standard procedure [3], we can calculate the emission rate of particles due to the tunneling across the LC by using the above metric. Here the tunneling calculation is similar to the case for de Sitter space [9,10]. The transmission coefficient for crossing the classically forbidden region is given by The equation of motion along the null and radial geodesic iṡ The upper (lower) solution represents the ingoing (outgoing) mode. This is consistent with the fact that the current is null: J 2 = 0, for the monopole solution, which means that the charges travel outwards with the speed of light, probably as a result from the force-free condition. Consider the charged particle pairs are created outside the LC at r = r out . The swave positive energy ω particles cross the LC inward to r = r in inside the LC. Then the imaginary part of the action is Thus, the radiation is thermal with the temperature where ν = Ω/2π is the rotation frequency. So the temperature is proportional to the rotation frequency of the magnetosphere on the compact object, which implies that it should be the rotation of the magnetosphere that provides the energy of radiation. For millisecond pulsars, the temperature is ∼ 7.6 × 10 −9 × ν/(10 3 s −1 )K. This is the radiation observed for an observer at the center r = 0. On the other side (outside the LC), radiation of charged particles should also be observed. The radiation comes from the particle pairs created from vacuum. When there is a net flux of radiation towards r = 0, there should be a net flux towards r → ∞. Unlike the case for a spacetime horizon, this negative-energy particle flux beyond the LC can be "seen" since the LC is just a horizon for the charged particles, but not for the observers. In black holes, the process that the negative-energy particles fall into the horizon and reduce their masses remains a mysterious part of the Hawking radiation theory. The analogous process here is also puzzling since we do not expect to observe particles with negative energies at large r.
In [11], we observe that negative energy indeed can exist in a rotational magnetosphere system in the Minkowski spacetime. The negative energy being dissipated means that the thermal energy is extracted. The emergency of the negative energy is probably due to the observational effect in different frames. It disappears in the co-rotating frame. We guess that here the negative energy of the particles outside the LC could also be taken as an observational phenomenon. These particles could reduce the rotational energy of the magnetosphere via interaction with the latter. They might become normal particles with positive energies at large r by attaining energy from the magnetosphere. But the exact process needs further detailed study.
As the radiation proceeds, the rotation speed of the magnetosphere slows down and so the LC expands. The temperature decreases meanwhile.

Discussion
We have shown that Hawking-like radiation of charged particles could arise from the LC of a rotating magnetosphere. The temperature is proportional to the rotation frequency. It is quite low even for the most rapidly rotating neutron stars. But, this may be theoretically interesting because this provides the possibility to study the analogous Hawking radiation process in an electrodynamic system without any ambiguity.
A possible problem for the temperature is that it is not constant at different angle θ along the LC. As learnt in gravitational theory, a horizon must have constant surface gravity or temperature, satisfying the zero-th law. We think that the difference here may arise from the fact that the charged particles are restricted to propagate only along the magnetic field lines. So the temperature discrepancy on different lines can not be easily erased since the trans-line communication of charged particles is inhibited. This is like the situation of sunspots, which can have lower temperatures than their surrounding photosphere in the presence of magnetic structures.
The temperature is irrelevant to the strength of the electromagnetic fields in the magnetosphere. But the field strength should still play important role. First, a stronger magnetic field is necessary for restricting higher energy charged particles to move along the field lines. Second, the field strength exceeding the Schwinger limit may enhance the radiation because more charged particles can be excited and created within a duration. Then more particles possibly cross or tunnel through the LC. But here we do not know how to reconcile the Schwinger mechanism with the tunneling scenario.
Our discussion above is for the force-free electromagnetic fields in a charge-filled magnetosphere. For vacuum fields, the charge and current I are both zero. For example, for the vacuum solution ψ = Bz with constant and parallel magnetic fields extending in the x direction in the cylinder coordinates (x = r sin θ, z = r cos θ), the metric is ds 2 = −(1 − x 2 Ω 2 )dt 2 + dx 2 . In this case, the above calculation method is invalid. But this does not mean that there is no radiation of charged particles since the LC still exists and the tunneling can still happen. The radiation in this case is suitable to be tested in laboratory. We may test the radiation by simply rotating a magnet rapidly.
Finally, it is worth mentioning that the radiation should be not due to the Unruh effect. It applies only for charged particles. There is no clue indicating that the rotational acceleration can lead to thermal creation of particles [12,13]. The radiation should be due to the existence of the LC or horizon for the charged particles.