Thomas-Fermi Model in Rindler Space

TL;DR

This paper explores the behavior of an electron gas modeled by Thomas-Fermi theory within Rindler space, revealing unique periodic and discontinuous electron distributions influenced by uniform acceleration.

Contribution

It introduces a two-dimensional Thomas-Fermi model in Rindler space and describes novel electron distribution patterns with physical interpretations.

Findings

Electrons form periodic rectangular strip distributions.

Some regions are void of electrons, others are filled.

Discontinuous electron distributions are observed in Rindler space.

Abstract

In this article we have investigated the Thomas-Fermi model for the electron gas in Rindler space. We have found that if the uniform acceleration is along $+x$-direction, then there is $y-z$-symmetry in space. For the sake of mathematical simplicity, we have assumed two dimensional spatial structure ($x-y$) in Rindler space. It has been observed that in two dimensional spatial coordinates the electrons are distributed discontinuously but in a periodic manner in a number of rectangular strips like domain along $\pm y$-direction. Some of them are having void structure, with no electrons inside such rectangular strips, while some are filled with electrons. We call the later type domain as the normal zone. We have also given physical interpretation for such exotic type electron distribution in Rindler space.

Equations

$$t$$

$$x$$

$$y$$

$$z$$

$$g^{\mu \nu }=\mathrm{diag}\left((1+\alpha x{)}^2,-1,-1,-1\right)$$

$$g^{\mu \nu }=\mathrm{diag}\left((1+\alpha x{)}^2,-1\right)$$

$$d{s}^2={\left(1+\alpha x\right)}^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2$$

$$L=-m_0{c}^2{\left[{\left(1+\alpha x\right)}^2-v^2\right]}^{1/2}$$

$$H=\left(1+\alpha x\right){\left(p^2+m_0^2\right)}^{1/2}$$

$$H=(1+\alpha x)\left(\frac{p^2}{2m_0}+m_0\right)$$

$${ϵ}_e={\mu }_m=g_{00}^{-1/2}=\frac{1}{\left(1+\alpha x\right)}$$

$${ϵ}_e=\mathrm{diag}\left({ϵ}_{xx},{ϵ}_{yy},{ϵ}_{zz}\right)=\mathrm{diag}\left(\frac{1}{\left(1+\alpha x\right)},\gamma ,\gamma \right)$$

$$\nabla .\vec{D}=4\pi \rho$$

$$\gamma \left(\frac{{\partial }^2\varphi }{\partial y^2}+\frac{\text{textbullet}{\partial }^2\varphi }{\partial z^2}\right)+\frac{\partial }{\partial x}\left({ϵ}_{xx}\frac{\partial \varphi }{\partial x}\right)=4\pi e{n}_e$$

$$\left(\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial t^2}\right)\varphi (x,y,z)+\frac{\partial }{\partial x}\left(\frac{1}{1+\alpha x}\frac{\partial \varphi }{\partial x}\right)=4\pi e{n}_e$$

$$\left(\frac{{\partial }^2\varphi }{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial \varphi }{\partial \rho }\right)+\frac{\partial }{\partial x}\left(\frac{1}{1+\alpha x}\frac{\partial \varphi }{\partial x}\right)=4\pi e{n}_e$$

$$n_e=\frac{1}{2\pi }{p}_F^2$$

$$\frac{{\partial }^2\varphi (x,y)}{\partial y^2}+\frac{\partial }{\partial x}\left(\frac{1}{1+\alpha x}\frac{\partial \varphi (x,y)}{\partial x}\right)=4\pi e{n}_e$$

$$\frac{p_F^2}{2m}-e\varphi (x,y)=\mu =\mathrm{constant}$$

$$n_e=\frac{m}{\pi }\left(\mu +e\varphi \right)$$

$$\frac{{\partial }^2\varphi }{\partial y^2}+\frac{\partial }{\partial x}\left(\frac{1}{1+\alpha x}\frac{\partial \varphi }{\partial x}\right)=4me\left(\mu +e\varphi \right)$$

$$\frac{{\partial }^2\varphi }{\partial y^2}\to \frac{1}{e}\frac{{\partial }^2\varphi }{\partial y^2}$$

$$\frac{\partial }{\partial x}\left(\frac{1}{1+\alpha x}\frac{\partial \varphi }{\partial x}\right)\to \frac{1}{e}\frac{\partial }{\partial x}\left(\frac{1}{1+\alpha x}\frac{\partial \varphi }{\partial x}\right)$$

$$\frac{{\partial }^2\varphi }{\partial y^2}+\frac{\partial }{\partial u}\left(\frac{1}{u}\frac{\partial \varphi }{\partial u}\right)=\frac{4m{e}^2}{{\alpha }^2}\varphi (x,y)$$

$$\frac{1}{Y}\frac{d^{2}Y}{d{y}^2}+\delta -\beta a=0$$

$$\frac{1}{U}\frac{d}{du}\left(\frac{1}{u}\frac{dU}{du}\right)-\gamma a-\delta =0$$

$$\frac{d^{2}Y}{d{y}^2}+{\nu }^{2}Y=0$$

$$Y=A\cos \nu y+B\sin \nu y$$

$$\frac{d^{2}U}{d{u}^2}-\frac{1}{u^2}\frac{dU}{du}-uU=0$$

$$A_i^{{}^{\prime }}(u)=-\frac{1}{\pi }\frac{4}{3^{1/2}}{K}_{2/3}(\zeta )$$

$$K_{\nu }(\zeta )=\frac{{\pi }^{1/2}(\frac{1}{2}\zeta {)}^{\nu }}{\Gamma (\nu +\frac{1}{2})}{\int }_1^{\infty }{e}^{-\zeta t}(t^2-1{)}^{\nu -\frac{1}{2}}dt$$

$$Y(y)=A\cos (\nu y)$$

Full text

\pub

Received (Day Month Year)Revised (Day Month Year)

\ccode

PACS Nos.: 03.65.Ge,03.65.Pm,03.30.+p,04.20.-q

The well known Lorentz transformation in special theory of relativity is between two inertial frames, i.e., having a uniform relative velocity between the frames [1, 2]. A similar type transformation is also possible between an inertial frame and a frame undergoing an uniform accelerated motion [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In the later case, it is called the Rindler coordinate transformations. Further in the case of Rindler transformation, the space is called the Rindler space, whereas the former one is the well known Minkowski space. The Rindler transformations are therefore exactly like Lorentz transformation. The only difference is that the frame is undergoing an uniform accelerated motion. The Rindler space is therefore also flat like the Minkowski space.

Now from the principle of equivalence, an accelerated frame in absence of gravity may be replaced by a frame at rest but in presence of a gravitational field. The strength of the gravitational field is exactly equal to the magnitude of the acceleration of the moving frame. Therefore in the case of a frame undergoing an accelerated motion along positive $x$-direction with an uniform acceleration $\alpha$ is equivalent to a frame at rest in presence of a constant gravitational field $\alpha$ [5]. Then following the references [5, 6, 7, 8, 9, 11, 12], the Rindler coordinate transformations are given by

[TABLE]

where the primed coordinates are in the non-inertial frame. Hence it is trivial to show that the metric tensor In $3+1$ dimension is given by (here we have discarded the prime symbols and considered the natural units with $c=\hbar=1)$

[TABLE]

whereas in $1+1$ dimension it can be expressed in the following form

[TABLE]

where $\alpha$ is the uniform acceleration of the moving frame along positive $x$-direction, which is also the constant gravitational field along negative $x$-direction for the reference frame at rest. From the Rindler transformations, one can show very easily that the square of the length element in four dimension is given by

[TABLE]

This four length element can be shown to be invariant under Rindler transformations. Further, assuming the direction of motion with uniform acceleration along positive $x$-direction, then we have $dy=dz=0$. Using $ds^{2}$ from eqn.(7) for $1+1$ dimension and following Landau and Lifshitz [1], it can be shown that the single particle Lagrangian is given by:

[TABLE]

where $m_{0}$ is the rest mass of the particle and $v$ is the single particle velocity along $x$-direction. Hence using the standard technology of classical mechanics, the single particle Hamiltonian in the relativistic form is given by

[TABLE]

Whereas in the non-relativistic scenario we have

[TABLE]

Now it has been shown in the literature that the free Riemann space transforms to a refracting medium in presence of strong gravitational field [1, 14, 15]. Further, if the gravitational field changes from point to point then the free space will behave like overlapping refracting media with continuously varying refractive index. Stated otherwise, the free space will behave like a number of overlapping refracting media with continuously varying refractive index. With such a concept of varying refractive index, one can very easily explain the phenomena of bending of electromagnetic as well as de Broglie waves (matter waves) near strongly gravitating objects. This may be treated as an alternative explanation for bending. This concept of refracting media in presence of varying gravitational field is also applicable in Rindler space. In the later case we have to consider a number of reference frames with various values of uniform gravitational field along the same direction. In other wards these reference frames are assumed to be undergoing accelerated motion with different values of uniform accelerations.

Then following Landau and Lifshitz [1] (see also [16, 17, 18]) it is trivial to show that in the Rindler space both the electric permittivity $\epsilon_{e}$ (or dielectric constant) and the magnetic permeability $\mu_{m}$ are given by

[TABLE]

In the present work we are interested only in the gravity induced electrical permittivity or dielectric constant. We further assume that the tensor quantity $\epsilon_{e}$ is diagonal in nature and since the direction of gravitational field is along $x$-axis, we can write

[TABLE]

where $\gamma$ is a constant $\sim 1$. Considering this diagonal form of tensorial structure for $\epsilon_{e}$, the Poisson’s equation in CGS Gaussian unit may be written as

[TABLE]

where $\rho=en_{e}$, the charge density of electrons in the Rindler space and $n_{e}$ is the electron number density.

Expressing the displacement vector $\vec{D}$ in terms of electric field vector $\vec{E}$ in the component form, given by $D_{i}=\epsilon_{ij}E_{j}$, with $\epsilon_{ij}$ is diagonal in nature and finally writing each component of electric field in terms of electric potential $\phi(x,y,z)$ in the form $E_{i}=\partial\phi(\{q_{i}\})/\partial q_{i}$, where $q_{i}=x$, $y$ or $z$, we have

[TABLE]

Absorbing the constant $\gamma$ in $y$ and $z$ derivatives we can write

[TABLE]

In the cylindrical coordinates with axial symmetry, this equation can be written in the form

[TABLE]

Since the uniform acceleration is along $x$ direction, the electron distribution in Rindler space has $y-z$ symmetry. Therefore for the sake of mathematical simplicity, or to say to get an analytical solution, we assume the distribution of electrons in two dimension Rindler space. The two dimensional space is indicated by $x-y$ coordinates. Our intention in this work is to investigate the distribution of degenerate electron gas in two-dimensional Rindler space with Thomas-Fermi approximation. To the best of our knowledge this study has not been done before.

Further, in two dimensional momentum space the electron density $n_{e}$ is given by

[TABLE]

where $p_{F}$ is the electron Fermi momentum. Then the Poisson differential equation (eqn.(15)) in $2$-dimensional Cartesian form may be written as

[TABLE]

Now the well known Thomas-Fermi condition is given by [19, 20]

[TABLE]

Before we proceed further let us now give a physical interpretation of Thomas-Fermi equation (eqn.(18)). When the system consists of a large number of electrons (here in Rindler space) and is in a stationary state, the value of $\mu$, which is the total energy of an electron should be same throughout the system. Then the electrons anywhere in the system do not have an overall tendency to move towards other parts of the system where the single particle energy value is less. Since for any natural system the tendency is to have minimum energy configuration, therefore if the total energy is not same throughout the system, then the electrons will try to occupy the spatial region where the energy is minimum. An instability will therefore grow in this situation.

We would also like to compare eqn.(18) with the eqns.(66)-(68) of the reference [21]. In this extremely interesting and important piece of work the authors have generalized the isothermal Tolman condition and the constancy of the Klein potential for charge neutral neutron-proton-electron system in $\beta$-equilibrium in compact neutron stars in presence of gravity. The authors have considered $\sigma-\omega-\rho$ meson exchange type interactions for the baryons, i.e., for the neutrons and protons. For the electrons, the electromagnetic interaction has been considered. They have given a formalism to solve these equations self-consistently in the general relativistic case. They have considered Einstein-Maxwell-Thomas-Fermi equations and obtained the Thomas-Fermi conditions for the constituents of the neutron stars from the constancy of the Klein potentials for each particle species. However, in our simplified model we have considered only electrons in presence of an electrostatic potentail in Rindler space, which is basically flat in nature and the gravitational field is assumed to be uniform in some limited spatial region. The formulation given in [21] can be used to investigate the effect of strong quantizing magnetic field of neutron stars on the charge particles (protons and electrons) when the Landau levels of these particles will be populated [22]. The formalism can be developed for both Rindler space, which is flat in nature and with uniform gravitational field in a limited region and also for the general relativistic scenario with Schwarzschild metric.

Now using eqns.(16) and (18), the expression for the electron density can be written as

[TABLE]

Then we have from eqn.(17)

[TABLE]

Redefining $\mu+e\phi$ as $\phi$, we have

[TABLE]

and

[TABLE]

Now making the coordinate transformation $1+\alpha x=u$ and finally redefining $\alpha y\rightarrow y$, we have

[TABLE]

Where $4me^{2}/\alpha^{2}=a$ is a constant. Writing the two dimensional potential $\phi(x,y)$ in the separable form, given by $\phi(x,y)=U(x)Y(y)$ We have the differential equations satisfied by $Y(y)$ and $U(u)$ as:

[TABLE]

and

[TABLE]

where $\delta$ is a positive constant assumed to be $>\beta a$ and $\beta+\gamma=1$. Defining $\nu^{2}=\delta-\beta a$, which is greater than zero, otherwise we get unphysical results, the first differential equation reduces to:

[TABLE]

with the solution

[TABLE]

where $A$ and $B$ are two real constants. The other differential equation is given by

[TABLE]

where in the above differential equation we have redefined $bu\rightarrow u$, with $b^{3}=\gamma a+\delta>0$ This differential equation is satisfied by the derivative of Airy function $A_{i}^{{}^{\prime}}(u)$ and is given by [23]

[TABLE]

Where $\zeta=\frac{2}{3}u^{3/2}$ with $u$ is a real positive variable and $K_{\nu}(\zeta)$ is the modified Bessel function of second kind, given by

[TABLE]

Eqns.(29) and (30) are used to evaluate numerically the derivative of Airy function $A_{i}^{\prime}(u)$ and thereby to obtain the potential $U(u)$ as a function of $u$.

Now before we go to the numerical evaluation of the derivative of Airy function, let us analyze the nature of the solutions given by eqns.(27) and (29). Since n $2$-dimension $n_{e}\propto\phi(x,y)$ and also $n_{e}\propto p_{F}^{2}$, the potential $\phi(x,y)=U(u)Y(y)$ should be positive definite. Since the argument of $U(u)$ is positive the derivative of Airy function is negative in nature. Therefore the other part which is $Y(y)$ should also be negative, otherwise the space will be forbidden for the electron distribution, or in other wards that particular spacial region will behave like a void. In fig.(1) we have shown a few such zones in two dimension, including the central zone, spread along both positive and negative directions of $y$-axis. In this figure we have shown the surface plot of the potential $\phi(u,y)$. The deep lines are for $\phi(u,y)=-4.5\times\exp(-3.5x)\times\cos(y)$, whereas the curves with relatively light impression are for $\phi(u,y)=-4.5\times\exp(-3.5x)\times\sin(y)$. For the sake of illustration we have taken $A=B=\nu=1$. Further in the above expressions, the functional form of $U(u)$ is obtained by fitting the numerical data using a search routine. Let us now consider the solution

[TABLE]

Since $Y(y)$ is positive in central domain having boundary from $-\pi/2$ to $+\pi/2$ along y-axis, the central rectangular strip is a void. Along $x$ or $u$ direction, the lower limit is $u_{0}$ for $x=0$ and the maximum value of $u$ is fixed by the vanishingly small value of $U(x)$ (is fixed at $10^{-7}$). Next from $+\pi/2$ to ${+3\pi}/2$, $Y(y)$ is negative. As a consequence the overall nature of the potential is positive in nature, which allows the presence of electrons in this domain and the distribution of electrons is symmetric about $y=0$. In this way one can get periodically symmetric distribution of the voids and normal structures of the domains along $\pm y$ direction. The non-uniform nature of electron distribution in the normal zones along $y$-direction will be determined by the variation of $Y(y)$ with $y$. For the other solution along $y$-axis, given by

[TABLE]

the region from $\nu y=0$ to $\nu y=\pi$ is not allowed, whereas $\nu y=0$ to $\nu y=-\pi$ is the normal zone. Unlike the previous case the spatial structure along $y$-direction is asymmetric about $y=0$. Therefore half of the central strip is void, and the other half is the allowed zone. Now for the symmetric solution along $y$-axis (solution given by eqn.(31)), the potential $Y(y)$ will vanish for

[TABLE]

where $n$ is an odd integer. Since the parameter $a\sim\alpha^{-2}$, the breadth of both the normal zones and the voids will decrease with the increase in the strength of gravitational field $\alpha$. In the extreme case, for $\alpha\rightarrow\infty$,

[TABLE]

For the case of asymmetric solution of $Y(y)$, given by eqn.(32), we have $Y=0$ at

[TABLE]

where $n=0$ or any integer number. In this case also the breadth of both physical and un-physical region will decrease with the increase in the strength of the gravitational field $\alpha$. Unlike the symmetric solution case, here in the extreme case for $\alpha\rightarrow\infty$, we have

[TABLE]

Along the longitudinal direction also the length of the strips decreases with increase in $\alpha$.

The physical nature of such periodic nature of voids and normal zones is simply because of the minimization of electrostatic potential energy for the electrons. In the voids the potential $\phi(u,y)$ are negative. The force acting on the electrons will then be of repulsive in nature. As a consequence the potential energy of the electrons will be positive, but reverse is the case for the normal zones with $\phi(u,y)>0$, with negative values for electron potential energy. This phenomena may be compared with the existence of charges on the outer surface of a hollow metallic charged sphere.

Acknowledgment: We would like to thank Prof. B.K. Talukdar for some valuable discussion.