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

1   Quantum Hall Effect

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 Iy is flowing through the plate. Perpendicularly the plate is penetrated by a magnetic field with a strength Bz. As a result the Lorentz force

KL=lBzIy/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 EH whose force

KH=-eEH       (1.1)

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

UH=bEH=-blBzIy/(Ne) .

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

N=eblBzi/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

UH=-RHIy

with the Hall resistance

RH=h/(ie2) .

2   Relativistic Quantum Hall Effect

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

Kµ=-eFµνuν

where

(uν)=(u0,u1,u2,u3

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

c2=uνuν=gµνuµuν .

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

(uν)=(u0,0,0,0)

and

c2=g00(u0)2 .

The result for uν is:

(uν)=[c/(g00)1/2,0,0,0] .

Then we obtain

Kµ=-eFµ0u0

or in a to (1.1) analogue component writing:

KH=-eEH/(g00)1/2 .

The value for g00 we get from Schwarzschild metrics

(gµν)= 1-rs/r000
0-1/(1-rs/r)00
00-r20
000-r2(sinθ)2

with the Schwarzschild radius

rs=2GM/c2 .

The result is a modification of the value for RH:

R'H=RH(1-rs/r)1/2≈RH(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.

3   References

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

4   Imprint

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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