# Could there be a magnetic version of Hawking radiation?

Could there be a magnetic version of Hawking radiation?
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For the process of particle creation to be possible, conservation laws must be respected. These include charge conservation and energy/momentum conservation. Charge conservation means that the total charge of particles created must be zero, so charged particles are created in pairs, particle and its antiparticle, for example a pair of electron + positron). Energy conservation law means that some mechanism must exist for a virtual pair of particles to obtain enough energy (at least $2m{c}^{2}$, the rest energy of two particles of mass m) in order to become real.
Uniform and stationary magnetic field alone does not provide such mechanism, since the energy of a charged particle in such a field is the same (more or less) as without magnetic field.
But strong enough electric field does have a mechanism for virtual charged particle pairs to become real, this is called the Schwinger effect.
Let us consider a pair with masses m each and charges $±e$ appearing side by side in a region with electric field of strength E. For the process to be possible the total energy of the pair must be zero. But this total energy consists of rest energy, kinetic energy and electrostatic potential energy for each member of the pair:
where $\varphi \left({\mathbf{x}}_{1,2}\right)$ is the scalar potential of EM field at the position of each particles. Kinetic energy must be positive, and since particles have opposite charges the sum of potential energies of the pair is determined by their separation along the direction of electric field. Let us denote this separation as 2d. Our energy balance equation becomes:
$0\le {\mathcal{E}}_{\text{kin}}=-m{c}^{2}+edE.$
So, it is possible for a pair to appear as long as the potential difference is large enough to provide both rest and kinetic energy. But virtual particle pairs could appear efficiently only if their separation is about Compton wavelength ($ƛ$) or less. So separation d must be less than $\hslash /mc$. This gives us the Schwinger limit:
${E}_{\text{Schw}}=\frac{m{c}^{2}}{eƛ}=\frac{{m}^{2}{c}^{3}}{\hslash e}.$
Schwinger mechanism allows pair creation by static electric field if its strength is comparable to this Schwinger limit. If the field strength is significantly lower the rate of pair production is exponentially suppressed.