# 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/r 0 0 0 0 -1/(1-rs/r) 0 0 0 0 -r2 0 0 0 0 -r2(sinθ)2

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