There are no Black HolesGeneral Relativity in Pseudo-Complex Form

Walter Greiner Peter Hess

God did not create the World

in order to exclude himself

from certain parts of it...from certain parts of it...

Walter Greiner

Historical relativistic equations

Remember the historical fact�2�2 = ��2 + �02�2

Klein-Gordon Equation

1�2 �2Ψ� 2 = � �2��2 + �2��2 + �2��2 − �02�2ℏ2 � Ψ

�ℏ �Ψ� = � ℏ�� ���1 ���1 + ��2 ���2 + ��3 ���3� + �0�2�� � Ψ

Klein-Gordon Equation

Dirac Equation

Dirac Matrices

Recall the Dirac Matrices

��1 = 0 0 0 10 0 1 00 1 0 01 0 0 0! ��2 = 0 0 0 −�0 0 � 00 −� 0 0� 0 0 0 ! ��1 = 0 0 1 00 1 0 01 0 0 0! ��2 = 0 0 � 00 −� 0 0� 0 0 0 !

��3 = 0 0 1 00 0 0 −11 0 0 00 −1 0 0 ! �� = 0 0 0 −�0 0 � 00 −� 0 0� 0 0 0 !

Gamma Matrices

And the Dirac Gamma-Matrices

"0 = 1 0 0 00 1 0 00 0 −1 00 0 0 −1! "1 = 0 0 0 10 0 1 00 −1 0 0−1 0 0 0! " = 0 1 0 00 0 −1 00 0 0 −1! " = 0 0 1 00 −1 0 0−1 0 0 0!

"2 = 0 0 0 −�0 0 � 00 � 0 0−� 0 0 0 ! "3 = 0 0 1 00 0 0 −1−1 0 0 00 1 0 0 !

Dirac Equation

The Dirac Equation in a Covariant Form

Ψ=Ψ

∂

+∂

+∂

+∂

mci 3210 γγγγh Ψ=Ψ

∂∂

+∂∂

+∂∂

+∂∂

mcxxxx

i3

3

2

2

1

1

0

0 γγγγh

From Klein-Gordon to Dirac Equation

Klein-Gordon Equation contains no spin

Dirac Equation describes particles spin 1/2

E

mc2

-mc2

0

Dirac Sea

The Dirac equation predicts

the existence of antiparticles

and yields the model

for the vacuum

Complex Numbers vs Pseudo-Complex Numbers

Complex numbersPseudo-complex numbers

(hypercomplex, hyperbolic numbers)� + �� �2 = −1 #� + ��$∗ = � − ��

& + '( '2 = +1 #& + '($∗ = & − '(

I can be a matrix (a Pauli or a Gell-Mann matrix or ...)

#� + ��$∗ = � − ��

'2 = +1#& + '($∗ = & − '(

)� = *0 11 0+ )� = *0 −�� 0 + )� = *1 00 −1+

The Zero Divisor Basis

Pseudo-complex number in the zero divisor basis

)± = 12 #1 ± '$ )2± = 1

& = �1 + '�2 & = &+)+ + &−)− &± = �1 ± �2

)2± = 1)+)− = )−)+ = 0 &± = �1 ± �2

The property of the zero divisor

& = &±)± |&|2 = &∗& = 0

Einstein vs new theory

Einstein uses the Riemann metric

νµµνµνµνµν σσ ggggg =+= −−

++ ,

)(xgµννµ

µν dxdxxgds )(2 =

We use the pseudo complex metric

νµµνµνµνµν σσ ggggg =+= −+ ,

ννν lIuxX +=

τ

νν

d

dxu = is the four-velocity.

We use the pseudo complex numbers to unify

the space-time with the four-momentum (four velocity)

where

Minimal length in the new theory

The differential length element is now given by

0)1( 222 ≥−=== νν

νµµν

ννω dxdxaldXdXgdXdXd

where is the four-accelerationµa

ττ

µµµ

µd

du

d

duaaa =−=2

2

2 1

al ≤ is a minimal length

is a maximal accelerationa

l

Born’s Reciprocity Theorem

Contrary to Einstein‘s General Relativity in Quantum Mechanics

there is complete symmetry between coordinates and momenta

Thus suggests introducing the length element [Born, 1938]

[ ] ijj

i ipx δh=, [ ] 0, =ji xx [ ] 0, =ji pp

Lead by pure symmetry and dimensional arguments

Born has introduced a scalar length parameter,

which is unaffected by Lorentz transformations.

)(2

22 νµνµ

µν dpdpm

ldxdxgdS +=

Thus suggests introducing the length element [Born, 1938]

Christoffel Symbols

++= κ

µνµ

νκνµκκµν

DX

Dg

DX

Dg

DX

Dg

2

1],[

Christoffel Symbols of the first kind

(using pseudo complex derivatives)

Christoffel Symbols of the second kind

±

±

± −=

−=Γ ],[ κνµµν

λ λκλµν g

The connection between two types is given by

−

−

+

+

−

−=

−=Γ σµν

λσ

µνλ

µνλλ

µν

The modified variational procedure

τdLS ∫=Define the action through the Lagrangian

for the variation we require

∫ DivisorZerodLS ∈= ∫ τδδ

this results in the equations of motion

DivisorZeroDX

DL

DX

DL

Ds

D∈

−

µµ

(Proportional or )+σ −σ

Pseudo Complex Field Theory

Fields, variables and masses are pseudo complex

( )22

2

1Φ−ΦΦ= MDD µ

µL

Scalar Field

2

( )Ψ−Ψ= MDi µµγL

Dirac Fieldµµ

XD

∂∂

=

Field Propagators

Propagator for the scalar field

2222

11

−+ −−

− MpMp

Propagator for the Dirac fieldPropagator for the Dirac field

−+ −−

− MpMp µµ

µµ γγ

11

Regularization via Pauli-Villars is automatically

included within the theory!

Geodesics

Taking a length element as a Lagrangian

ω=L

we obtain the equation for geodesics

DivisorZeroXXX v ∈

+ λµ

λνµ

&&&&

Equations for the matter free space

Equations for the matter free space

Setting we getRgL −=

DivisorZeroRgRG ∈−= µνµνµν

2

1

Equations for the matter free space

The spherically symmetric Schwarzschild Solution

Equations for the matter free space

0RR == &0µν

( )22222

2

22

2

2 sin2

12

1

2

21

2

21 ϕϑϑω ddrdr

r

B

r

m

r

mr

B

r

m

dtr

B

r

md +−

+−

−

+−−

+−≈

We obtain the isotropic Schwarzschild solution

The Red Shift

The Red Shift g factor:

The metric component

of the time must be positive0.8

1

repulsion

gdtdtr

B

r

mdtgd ≡

+−=≈2

0

002

21τ

00

00 >g 22mB >

Antigravitation below

half of the Schwarzschild

radius!

g

r/(2m)1 10 1000.1

0

0.2

0.4

0.6

attraction

Astronaut

A fatal fall into the black hole (tidal forces)

Binary Star

Binary Star with one Visible and one Black Hole Component

Binary Star

Typical Black Hole

Schwarzschild Solution embedded into the Euclidean Space

There are no Black Holes

Antigravitation in the center