The relativistic Quantum Hall Effect

This effect is a version of the Quantum Hall Effect, which describes an influence of gravitation in Hall resistance. This effect could be used to measure the value of the gravity constant.

By Tilmann Schneider

Contents

1 Quantum Hall Effect
2 Relativistic Quantum Hall Effect
3 References
4 Imprint

The Quantum Hall Effect makes possible a very precisely measurement of resistance, because the Hall resistance only depends on nature constants. For derivation we consider an electrical conducting plate whith a thickness d, a width b and a length l. Lengthwise a direct current I_y is flowing through the plate. Perpendicularly the plate is penetrated by a magnetic field with a strength B_z. As a result the Lorentz force

K_L=lB_zI_y/N

is exerted on a single electron (N = number of all electrons in the plate). The electrons which are retained and exhausted respectively at the boundary cause an electric field E_H whose force

K_H=-eE_H       (1.1)

to the electrons compensates the Lorentz force. The result for Hall voltage is

U_H=bE_H=-blB_zI_y/(Ne) .

Applies d<<l, d<<b to the plate dimensions there is a 2 dimensional electron gas. The number of electrons is

N=eblB_zi/h,  i=1,2,3,...

(h = Planck's constant). The quantisation is a result of a circle movement of the electrons in the magnetic field (cyclotron resonance). Quantum mechanically allowed along these circle curves are only standing waves. Then the result for Hall voltage is

U_H=-R_HI_y

with the Hall resistance

R_H=h/(ie²) .

To describe the relativistic Quantum Hall Effect equation (1.1) must be generalised. With the electromagnetic field tensor F_µν the covariant form of force is

K_µ=-eF_µνu^ν

where

(u^ν)=(u⁰,u¹,u²,u³) 

is the 4-velocity. To speed of light applies the following relation:

c²=u_νu^ν=g_µνu^µu^ν .

To measure the Quantum Hall Effect in a gravity field, we consider a resting experiment facility e.g. on the surface of earth. Then we have

(u^ν)=(u⁰,0,0,0)

and

c²=g₀₀(u⁰)² .

The result for u^ν is:

(u^ν)=[c/(g₀₀)^1/2,0,0,0] .

Then we obtain

K_µ=-eF_µ0u⁰

or in a to (1.1) analogue component writing:

K_H=-eE_H/(g₀₀)^1/2 .

The value for g₀₀ we get from Schwarzschild metrics

(gµν)=		1-rs/r	0	0	0
		0	-1/(1-rs/r)	0	0
		0	0	-r2	0
		0	0	0	-r2(sinθ)2

with the Schwarzschild radius

r_s=2GM/c² .

The result is a modification of the value for R_H:

R'_H=R_H(1-r_s/r)^1/2≈R_H(1-GM/rc²) .

For M and r the mass of earth and the earth radius have to be inserted. For GM/rc² we obtain a value of 7·10^-10. If we could increase the precision of measurements of the Quantum Hall Effect up to 10^-12 the gravity constant G could be determined.

Hermann Weyl, “Space, Time, Matter”, Dover Publications
Torsten Fließbach, “Allgemeine Relativitätstheorie”, 2nd edition, Spektrum Akademischer Verlag, 1995
Hajdu, Kramer, “Der Quanten-Hall-Effekt”, Phys. Bl., 41, (1985), 401-406

Title: The relativistic Quantum Hall Effect
Author: Tilmann Schneider
URL: http://www.relativistische-asynchronmaschine.de
E-Mail: admin@relativistische-asynchronmaschine.de
Rev. 3.0, 14.11.2009
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