general relativity in pseudo-complex form · contrary to einstein‘s general relativity in quantum...
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There are no Black HolesGeneral Relativity in Pseudo-Complex Form
Walter Greiner Peter Hess
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God did not create the World
in order to exclude himself
from certain parts of it...from certain parts of it...
Walter Greiner
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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
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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 !
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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 !
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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
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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
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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+
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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
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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
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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
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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]
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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
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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
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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
∂∂
=
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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!
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Geodesics
Taking a length element as a Lagrangian
ω=L
we obtain the equation for geodesics
DivisorZeroXXX v ∈
+ λµ
λνµ
&&&&
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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
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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
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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
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Astronaut
A fatal fall into the black hole (tidal forces)
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Binary Star
Binary Star with one Visible and one Black Hole Component
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Binary Star
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Typical Black Hole
Schwarzschild Solution embedded into the Euclidean Space
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There are no Black Holes
Antigravitation in the center