The particle production at the event horizon of a black hole as gravitational Fowler-Nordheim emission in uniformly accelerated frame, in the non-relativistic scenario

Abstract

In the conventional scenario, the Hawking radiation is believed to be a tunneling process at the event horizon of the black hole. In the quantum field theoretic approach the Schwinger’s mechanism is generally used to give an explanation of this tunneling process. It is the decay of quantum vacuum into particle anti-particle pairs near the black hole surface. However, in a reference frame undergoing a uniform accelerated motion in an otherwise flat Minkowski space-time geometry, in the non-relativistic approximation, the particle production near the event horizon of a black hole may be treated as a kind of Fowler-Nordheim field emission, which is the typical electron emission process from a metal surface under the action of an external electrostatic field. This type of emission from metal surface is allowed even at extremely low temperature. It has been noticed that in one-dimensional scenario, the Schrödinger equation satisfied by the created particle (anti-particle) near the event horizon, can be reduced to a differential form which is exactly identical with that obeyed by an electron immediately after the emission from the metal surface under the action of a strong electrostatic field. The mechanism of particle production near the event horizon of a black hole is therefore identified with Schwinger process in relativistic quantum field theory, whereas in the non-relativistic scenario it may be interpreted as Fowler-Nordheim emission process, when observed from a uniformly accelerated frame.

Appendices

Appendix A

Consider the differential equation

$$ \frac{d^{2}X}{d\xi^{2}}+\xi X=0 $$

(12)

To get a solution, let us substitute $X(\xi)=\xi^{n}\psi(\xi)$, where $n$ is an unknown quantity. Then the above differential equation reduces to

$$ \xi^{2} \frac{d^{2} \psi}{d \xi^{2}} +2n \xi\frac{d\psi}{d\xi}+\bigl[n(n-1)+ \xi^{3}\bigr]\psi=0 $$

(13)

Let $\xi=\beta z^{2/3}$, where $\beta$ is another unknown quantity. Then we have the reduced form of the above equation as

$$ z^{2}\frac{d^{2} \psi}{dz^{2}}+ \biggl( n+\frac{1}{4} \biggr) \frac{4}{3}z \frac{d \psi}{dz} +\frac{4}{9} \bigl[n(n-1)+ \beta^{3}z^{2}\bigr]\psi(z)=0 $$

(14)

Let us choose $n=1/2$, then we have

$$ z^{2}\frac{d^{2}\psi}{dz^{2}}+z\frac{d\psi}{dz}+ \biggl[ \frac{4}{9} \beta^{3} z^{2}- \frac{1}{9} \biggr] \psi(z)=0 $$

(15)

Finally choosing $\beta=(9/4)^{1/3}$, we get

$$ z^{2}\frac{d^{2}\psi}{dz^{2}}+z\frac{d\psi}{dz} + \biggl( z^{2} -\frac{1}{9} \biggr)\psi(z)=0 $$

(16)

Comparing this differential equation with the standard form of Bessel equation

$$ z^{2}\frac{d^{2}\psi}{dz^{2}}+z\frac{d\psi}{dz} + \bigl( z^{2} - \nu^{2} \bigr)\psi(z)=0 $$

(17)

whose solution is $J_{\nu}(z)$, Bessel function of order $\nu$ (Bessel function with negative order has no relevance) or $H_{\nu}^{(2)}(z)$, the second kind Hankel function of order $\nu$. Then depending on the physical situation, we have the appropriate solution of Eq. (17) as

$$ \psi(z)=J_{1/3}(z)\quad \mbox{or}\quad \psi(z)=H_{1/3}^{(2)}(z) $$

(18)

Appendix B

In this Appendix using some of the established useful formulas of special relativity with uniform accelerated motion (see Socolovsky 2013; Torres del Castillo and Perez Sanchez 2006; Huang and Sun 2007) we shall obtain the single particle Lagrangian and Hamiltonian in Rindler space. Using the results from Socolovsky (2013); Torres del Castillo and Perez Sanchez (2006); Huang and Sun (2007) the Rindler coordinates are given by

$$\begin{aligned} & ct = \biggl(\frac{c^{2}}{\alpha}+x^{\prime}\biggr)\sinh\biggl(\frac{\alpha t^{\prime}}{c} \biggr)\quad \mbox{and}\quad \\ & x = \biggl(\frac{c^{2}}{\alpha}+x^{\prime}\biggr)\cosh\biggl(\frac{\alpha t^{\prime}}{c} \biggr) \end{aligned}$$

(19)

Hence one can also express the inverse relations

$$ ct^{\prime}=\frac{c^{2}}{2\alpha}\ln\biggl(\frac{x+ct}{x-ct} \biggr) \quad\mbox{and}\quad x^{\prime}=\bigl(x^{2}-(ct)^{2}\bigr)^{1/2}-\frac{c^{2}}{\alpha} $$

(20)

The Rindler space-time coordinates, given by Eqs. (19) and (20) are then just an accelerated frame transformation of the Minkowski metric of special relativity. The Rindler coordinate transform the Minkowski line element

$$\begin{aligned} & ds^{2} = d(ct)^{2}-dx^{2}-dy^{2}-dz^{2} \quad \mbox{to}\quad \\ & ds^{2} = \biggl(1+\frac{\alpha x^{\prime}}{c^{2}} \biggr)^{2}d \bigl(ct^{\prime}\bigr)^{2}-{dx^{\prime}}^{2} -{dy^{\prime}}^{2}-{dz^{\prime}}^{2} \end{aligned}$$

(21)

The general form of metric tensor may then be written as

$$ g^{\mu\nu}=\mathrm{diag} \biggl( \biggl(1+\frac{\alpha x}{c^{2}} \biggr)^{2},-1,-1,-1 \biggr) $$

(22)

Now following the concept of relativistic dynamics of special theory of relativity (Landau and Lifshitz 1975), the action integral may be written as (see also Huang and Sun 2007)

$$ S=-\alpha_{0} \int_{a}^{b} ds\equiv \int_{a}^{b} Ldt $$

(23)

Then using Eqs. (19)–(22) and putting $\alpha_{0}=-m_{0} c$, where $m_{0}$ is the rest mass of the particle, the Lagrangian of the particle is given by

$$ L=-m_{0}c^{2} \biggl[ \biggl( 1+\frac{\alpha x}{c^{2}}\biggr)^{2} -\frac{v^{2}}{c^{2}} \biggr] $$

(24)

where $\vec{v}$ is the three velocity of the particle. The three momentum of the particle is then given by

$$\begin{aligned} & \vec{p}=\frac{\partial L}{\partial\vec{v}}, \quad \mbox{or} \\ & \vec{p}=\frac{m_{0}\vec{v}}{ [ (1+\frac{\alpha x}{c^{2}} )^{2} - \frac{v^{2}}{c^{2}} ]^{1/2}} \end{aligned}$$

(25)

Hence the Hamiltonian of the particle is given by

$$\begin{aligned} & H=\vec{p}.\vec{v}-L\quad \mbox{or} \\ & H=m_{0}c^{2} \biggl(1+\frac{\alpha x}{c^{2}} \biggr) \biggl(1+\frac{p^{2}}{m_{0}^{2}c^{2}} \biggr)^{1/2} \end{aligned}$$

(26)

which is Eq. (3) in the main text.