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Es gibt keine schwarzen Löcher
Von Einstein zu Zweistein
Walter Greiner
Frankfurt Institute for Advanced Studies
Prof. Senator E. h. Carlo Giersch Senatorin E.h. Karin Giersch
1
Special Message to Nicola up in Heaven:Special Message to Nicola up in Heaven:
There are no Black Holes!There are no Black Holes!
Albert EinsteinAlbert Einstein Karl SchwarzschildKarl Schwarzschild
( )22 drm
Metric:Metric:
( )22222
22 sin2
1
21 φθθω ddr
r
m
drdt
r
md +−
−−
−≈
Singularity:Singularity: light cannot escape from within light cannot escape from within
the Schwarzschild radiusthe Schwarzschild radius
A fatal fall into the gravitational center
(tidal forces)
Astronaut
GravitationskraftGravitationskraftGravitationskraftGravitationskraft
Isaac Newton
1M 2Mr
Isaac Newton
* 04.01.1643-
† 31.03.17272
21
r
MMK γ=
Relative Relative Relative Relative GrößenGrößenGrößenGrößen von von von von SonneSonneSonneSonne und und und und IhrenIhrenIhrenIhren PlanetenPlanetenPlanetenPlaneten
Sonnenradius ≈ 695000 km Erdradius ≈ 6400 km
PlanetenPlanetenPlanetenPlaneten und und und und derenderenderenderen UmlaufbahnenUmlaufbahnenUmlaufbahnenUmlaufbahnen
Solar SystemSolar SystemSolar SystemSolar System
E
Veff
BahntypenBahntypenBahntypenBahntypen imimimim effektiveneffektiveneffektiveneffektiven PotentialPotentialPotentialPotential
EEllipse
EHyperbel
r
rmin rmax
rKreis
2LMV +−=
γ
EKreis
Drehimpuls Null
22mr
L
r
MVeff +−=
γ
2
2r
mTkin &=
FluchtgeschwindigkeitFluchtgeschwindigkeitFluchtgeschwindigkeitFluchtgeschwindigkeit
Die Gesamtenergie ist gleich auf derErdoberfläche und weit weg von der Erde
0)(2
1 2 ≅∞→=−= ⊕ rER
mMmVE F
γ2 ⊕R
F
⊕
⊕=⇒R
MVF
γ2ist die kleinste nötige
Geschwindigkeitum Gravitationsfeld
verlassen zukönnen.
Himmelskörper Fluchtgeschwindigkeit km/s
Erde 11.2
Venus 10.2
Jupiter 59.6
Sonne 617.3
können.
Schwarzschild RadiusSchwarzschild RadiusSchwarzschild RadiusSchwarzschild Radius
Naive Rechnung: die Fluchtgeschwindigkeit
darf die
V(r)
rR RSchwarzschild
darf die Lichtgeschwindigkeit nicht
überschreiten
⇒== cR
MVF
γ22
2
c
MR ildSchwarzsch
γ=
Himmelskörper-Masse Schwarzschild Radius
Erde 1 cm
Jupiter 3.2 m
Sonne 3 km
Syrius A 6.3 km
R c
GeschlosseneGeschlosseneGeschlosseneGeschlossene UmlaufbahnenUmlaufbahnenUmlaufbahnenUmlaufbahnen und und und und RosettenlaufbahnRosettenlaufbahnRosettenlaufbahnRosettenlaufbahn
Jupiter MoonsJupiter MoonsJupiter MoonsJupiter Moons
Die Die Die Die MilchstraßeMilchstraßeMilchstraßeMilchstraße GalaxieGalaxieGalaxieGalaxie
SONNESONNESONNESONNE
IM ORIONNEBELIM ORIONNEBELIM ORIONNEBELIM ORIONNEBEL
100000 100000 100000 100000 LichtjahreLichtjahreLichtjahreLichtjahre
Beinhaltet ungefähr Beinhaltet ungefähr Beinhaltet ungefähr Beinhaltet ungefähr 2222ˣ1010101011111111 SonnenmassenSonnenmassenSonnenmassenSonnenmassen
Die Die Die Die nächstenächstenächstenächste UmgebungUmgebungUmgebungUmgebung derderderder SonneSonneSonneSonne
FRANKFURT560 km
Barnards
Pfeilstern
α Centauri
61 Cygni
Prokyon
FRANKFURT
Sirius
61 Cygni
Die Die Die Die MilchstraßeMilchstraßeMilchstraßeMilchstraße
MagellanscheMagellanscheMagellanscheMagellansche WolkenWolkenWolkenWolken kreisenkreisenkreisenkreisen um um um um MilchstraßeMilchstraßeMilchstraßeMilchstraße
This galaxy known as Messier101 (Feuerrad-Galaxie) helped measuring the expansion rate
of the universe within the Hubble project
Interstellar gases are collecting into Interstellar gases are collecting into Interstellar gases are collecting into Interstellar gases are collecting into newly forming stars newly forming stars newly forming stars newly forming stars
Sometimes they reach 100 LY in diameter and have the mass of 6ˣ106 solar masses (T=10 K).
"Eye of God“ - the Helix Nebula is actually the glowing gas remnants of a dying star
GravitationslinseGravitationslinseGravitationslinseGravitationslinse
Einsteinring
BeschleunigteBeschleunigteBeschleunigteBeschleunigte Expansion des Expansion des Expansion des Expansion des UniversumsUniversumsUniversumsUniversumsin Supernovae in Supernovae in Supernovae in Supernovae EntfernungsmessungenEntfernungsmessungenEntfernungsmessungenEntfernungsmessungen....
DunkleDunkleDunkleDunkle MaterieMaterieMaterieMaterie????
Blaue Linie zeigt die Messungen. Grüne Linie entsprichtdem Weltraum, der sich weder beschleunigt noch verzögert.
Andere Linien zeigen aktuelle theoretische Modelle.
Clusters of Matter in the Cosmos
DARK ENERGY:
ANTI-
Gravitation!Gravitation!
-Drives the
Universe apart:
Expansion
Large - scale Large - scale
cosmic matter
distribution
(simulation)
““““SchwarzesSchwarzesSchwarzesSchwarzes Loch”Loch”Loch”Loch”
““““SchwarzesSchwarzesSchwarzesSchwarzes Loch”Loch”Loch”Loch”
““““SchwarzesSchwarzesSchwarzesSchwarzes Loch”Loch”Loch”Loch”
Gibt es Schwarze Löcher?
Erde Schwarzes Loch? Nein!
Erd-Masse mit Durchmesser kleiner als 2 cm!
Jedoch: Im Zentrum unserer Milchstraße sitzt
Sonne Schwarzes Loch? Nein!
Sonnen-Masse mit Durchmesser kleiner als 6 km…
Jedoch: Im Zentrum unserer Milchstraße sitzt
ein gewaltiges Schwarzes Loch von
M = 3,6 Millionen Sonnenmassen !
Ein Monster im Sternbild Schütze: wurde entdeckt von Reinhard Genzel aus München
(MPI für Astrophysik)
Ein Schwarzes Loch von Ein Schwarzes Loch von
M =3,6 Millionen Sonnenmassen im
Zentrum unserer Milchstraße
Bewegung von Sternen um das Schwarze Loch
beobachtet !
Schwarzes Loch im Zentrum unserer Milchstraße M=3,6
Millionen Sonnenmassen
Bahnen von Sternen 1992 – 2006 innerhalb von 10
Lichttagen um das Zentrum der Milchstraße Sagitarius A*,
Sternbild Schütze: Genzel et al.
Das Zentrum der
Milchstraße ist Milchstraße ist
25 000 Lichtjahre von
der Sonne entfernt.
Sichtbares Licht wird
an Staub absorbiert.
Wir können bis zum
Zentrum mit
infrarotem Licht
sehen.
Aufbau der aktiven Galaxiekerne
ähnlich wie galaktische scharze Löcher,
aber insgesamt etwas beeindruckender...
•supermassives schwarzes Loch (107 M )•supermassives schwarzes Loch (107 Mʘ)
•Akkretionsrate der Scheibe (1-2 Mʘ/Jahr)
•Leuchtkraft hoch (L ≈ 1010 Lʘ)
•Schwarzschildradius jetzt ≈ 1 AU
Amand Fassler, Tübingen
Computer-Simulation des
Proton-Proton-Stoßes am LHC.
Spekulationen über Mikro Schwarze Löcher
The Lord did not create the World
in order to exclude himself
from certain parts of it...
Walter Greiner
PSEUDO-COMPLEX GENERAL
RELATIVITY
Peter O. Hess (ICN-UNAM and FIAS)
and
Walter Greiner (FIAS)
34
FROM KLEI-GORDO TO DIRAC
( )
22
γ=β,γ=βα
βmc+pα=c
Ecm+cp±=E
rr⋅→422
( )0ii γ=β,γ=βα
35
The Dirac equation yields an explanation for the vacuum.
Remember the historical fact2 2 = 2 + 02 2
Historical Relativistic Equations
2 2 = 2 + 02 2
1 2 2Ψ2 = 22 + 22 + 22 − 02 2ℏ2 Ψ
Klein-Gordon Equation
1 2 Ψ2 = 2 + 2 + 2 − ℏ2 Ψ
ℏ Ψ = ℏ !"#1 1 + "#2 2 + "#3 3% + 0 2&' ( Ψ
Dirac Equation
Recall the Dirac Matrices
"# = )0 0 0 10 0 1 00 1 0 0* "# = )0 0 0 −0 0 00 − 0 0 *
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 *
"#3 = )0 0 1 00 0 0 −11 0 0 00 −1 0 0 * &' = )1 0 0 00 1 0 00 0 −1 00 0 0 −1* "#3 = )0 0 1 00 0 0 −11 0 0 00 −1 0 0 * &' = )1 0 0 00 1 0 00 0 −1 00 0 0 −1*
And the Dirac Gamma-Matrices
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*
+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 * +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 *
The Dirac Equation in a Covariant Form
Dirac Equation
The Dirac Equation in a Covariant Form
Ψ=Ψ
∂∂
+∂∂
+∂∂
+∂∂
mcxxxx
i3
3
2
2
1
1
0
0 γγγγh
Klein-Gordon Equation contains no spin
From Klein-Gordon to Dirac Equation
Dirac Equation describes particles spin 1/2
E
mc2
The Dirac equation predicts
-mc2
0
Dirac Sea
The Dirac equation predicts
the existence of antiparticles
and yields the model
for the vacuum
CONTENT:
Pseudo-complex variables
Pseudo-complex General Relativity
Schwarzschild metric as an example (“black holes”, perihelion
shift of Mercury)
Pseudo-complex Robertson-Walker Metric
Conclusions and outlook
41
FIRST ATTEMPTS.
A. Einstein, Ann.Math. 46 (1945), 518.
A. Einstein, Rev. Mod. Phys. 20 (1948), 35.
(Unification of gravitation and electrodynamics)
C. Mantz, T. Prokopec, (2008); arXiv:0804.0213
(hermitian gravity and cosmology)
12 −+ =ipm
lix=X µµµ
(Introduction of the Planck length, l)
42
m
[ ] νµµνδ dxdxgdsipx ki
jk == 2, hBorn’s equivalence
principle:but
(M. Born, Proc. Roy. Soc. A 165 (1938), 291 and
M. Born, Rev. Mod. Phys. 21 (1949), 463.)
PROPOSAL(M. BORN)
2 1dτ
du
dτ
dul+dxdxdudul+dxdxdΩ
µµ2µ
µ
µ
µ
2µ
µ
==
( )221 aldxdx
dτdτ
µ
µ
µµµ
−→
ττ
µµµ
µd
du
d
duaaa =−=2
2
2 1
al ≤
44
Maximal acceleration!!!
E.R. Caianiello (1981), H.E. Brandt, R.G. Beil (1980’s)
S.G. Low (1990’s and more recently: representation theory)
ττ dd
l is a minimal length
PSEUDO-COMPLEX VARIABLES
1, 2
21 =+= IIXXX
Alternative: −−++± +=⇒±= σσσ XXXI )1(2
1
2
Permits the independent treatment of the two components!
Pseudo-Complex conjugate: )(*
mσσσX+σX=IXXX ±++21 =−= −−
Null-Norm : !=XX=XσX=X 0*2⇒
±± =σσ 20=−+σσ
45
Null-Norm : !=XX=XσX=X ±± 0*2⇒
(Other names in the literature: Para-complex, hyperbolic,
hypercomplex, semi-complex Variables).
Plane of the pseudo-complex variable X. Shown are the
pseudo-real, pseudo-imaginary components and the
zero divisor basis.46
)σF(X+)σF(X=F(X) ++ −−
Rules for manipulation are similar as with
real and complex numbers/variables:
F(X)∆X)+F(XDF(X)
σ)G(X
)F(X+σ
)G(X
)F(X=
(X)G(X)G
(X)F(X)G=
G(X)
F(X)
)σ)G(XF(X)σ)G(XF(X=F(X)G(X)
+
+
+
+++
++
−
+
−−
−
−−−
−−
*
*
∆X
F(X)∆X)+F(X=
DX
DF(X)∆X
−→0lim
47
CONSEQUENCE OF THE PSEUDO-COMPLEX
EXTENSION FOR THE LORENTZ SYMMETRY
Finite transformation:µν
µνω Λie
21 σωσωωωω +=+= −+I
=Λµν Normal Lorentz-
transformation
21
21
µνµνµν
µνµνµνµνµν
ωωω
σωσωωωω
±=
+=+=±
−−
++I
transformation
)!=σ(σ)(SO)(SOσΛ
e+σΛ
e=Λ
e ++
µνµν
iω
+
µν+µν
iωµν
µνiω
03,13,1 −−−
−
→
Direct product of two Lorentz groups because both commute with each
other. The normal Lorentz group is obtained for a vanishing
pseudo-complex part in omega.
48
pseudo-complex part in omega.
consequence: Appearance of a minimal Length“l”.
It is a scalar parameter!
Minimal Length = maximal acceleration
µµµ IluxX +=Pseudo-complex world line:
Pseudo-complex world line:µµµ IluxX +=
Remark:
IluxX +=
µx
µu
= position in 4-space
= basis vector of the tangent space at µx
with units of the 4-velocity
49
NEW VARIATIONAL PRINCIPLE:
τdLS ∫=Define the action through the Lagrangian
for the variation we requirefor the variation we require
DivisorZerodLS ∈= ∫ τδδ
this results in the equations of motion
)(conventionS −= ξσδ
50(F. Schuller, PhD thesis, University of Cambridge (2003);
F. Schuller, Ann. Phys. (N.Y.) 299 (2002), 174,
F. S. has proposed this general variation principle in his thesis at Cambridge.)
this results in the equations of motion
DivisorZeroDX
DL
DX
DL
Ds
D∈
−
µµ
Fields, variables and masses are pseudo complex
Scalar Field
Pseudo Complex Field Theory
( )22
2
1Φ−ΦΦ= MDD µ
µL
Scalar Field
Dirac Fieldµµ
XD
∂∂
=
( )Ψ−Ψ= MDi µµγL
Dirac Field
Propagator for the scalar field
11−
Field Propagators
2222
−+ −−
− MpMp
Propagator for the Dirac field
−−
− MpMp µµ γγ11
−+ −− MpMp µµ
µµ γγ
Regularization via Pauli-Villars is automatically
included within the theory!
Can one extend the General Can one extend the General
Relativity to pseudo-complex
coordinates?
53
EXTENSION OF THE THEORY OF
GENERAL RELATIVITY:
•The metric is pseudo-complex, without torsion:•The metric is pseudo-complex, without torsion:
* pseudo-complex length element
νµµνµνµνµν σσ ggggg =+= −−
++ ,
=ω νµµν DXDXgd 2
54
−−−−
++++ +=
=
σσ
ωνµ
µννµ
µν
µν
DXDXgDXDXg
DXDXgd
Parallel Transport and Christophel symbols:
55
Covariant derivative of a contravariant 4-vector:Covariant derivative of a contravariant 4-vector:
56
The covariant derivative of the metric is zero.
This assures a universal metric
or57
COORDINATES:
58
THE LENGTH ELEMENT:
−−−−
++++ += σσω νµ
µννµ
µν DXDXgDXDXgd 2
Condition of reality:
µµ
µµ
µµ
µµ
ω duduldxdxd
dxdududxl
22
0)(
+=
=+
Condition of reality:
Dispersion relation!
(will be used as a
constriction) (see next slide)
59
µµ
µµω duduldxdxd 22 +=
Limit of standard GR (pseudo-complex part=0): old results,
where the dispersion relation is AUTOMATICALLY fulfilled.
ds
dx=u
ds
dx=u
µµ
µ ;µ
4-velocity:
Dispersion relation: constuu =µDispersion relation: constuu =µ
µ
Differentiation:
( ) dxdx µ( )0
00
=dudx+dxdu
=duds
dx
ds
dxdu=uud
µ
µ
µ
µ
µµµ
µ
µ
µ
⇒
+⇒
60
THE LENGHT ELEMENT (CONT.)
0 =+ λνλν δhhgg
Problems:
00
0
0
0
=+
=+λν
µλλν
µλ
µνλν
µλλν
µλ δ
hggh
hhgg
61
Problems:
The g- und h-”matrix”, alone, are no tensors!
Setting we getRgL −=
Equations for the matter free space
DivisorZeroRgRG ∈−= µνµν 1
Equations for the matter free space
DivisorZeroRgRG ∈−= µνµνµν
2
1
This suggests that the right hand side is
related to the energy momentum tensor
EQUATIONS OF MOTION IN THE
SCHWARZSCHILD PROBLEM
−=
+ σξλν
µ µλνµ XXX &&&& (1)
λν
(2)
( ) −−−−
−
=−−′+′+
=′+
σξϕϑϑνλ
σξν
λλλν ReRReXeRR
XRX
222202
000
sin2
1
2
1&&&&&&
&&&&
−=−+ σξϕϑϑϑϑ ϑ&&&&& R
Rcossin
2 2
63
., etc,Ds
DR=R
DR
Dν=ν' &
Comparison of (1) with (2) Christoffel symbols of the 2. kind
−=++ σξϕϑϕϑϕ ϕ&&&&&& R
R
R
2cot2
CHRISTOFFEL SYMBOLS OF THE 2.KIND
ν ′=
=
2
1
10
0
01
0 λνν −′=
e2
1
00
1λ ′=
2
1
11
1
21001 200 211
λ−−=
Re22
1
R
1
21
2
12
2=
=
λϑ −−=
eR 2sin33
1
ϑϑcossin2
−=
ϑcot33
=
=
64
ϑϑcossin33
−=
ϑcot
2332=
=
R
1
13
3
31
3=
=
THE RICCI-TENSOR:
65
SOLUTION:
[ ]22222202 )(sin)()()( ϕϑϑω λν DDRDReDXed +−−=
01 ξξ =
[ ]−
−−
+− ∫+= −−−+ σσ
ξξννλ DRR
eee2/)( 10
The integral in the metric should vanish at large distances. In order
to achieve this, the simplest way is to make the following assumption:
66
Assumption: νλ ee =⇒ −
νννλ eee=e =+ −+− −+ σσ
SOLUTION:
++λ
R=e
2M1−
−:component−+σ
+R
:component−−σ
definition:
mM =±
67
In order to approach the Schwarzschild solution for large distances,
applying the before mentioned reality condition on the length
element, leads to the identification of:
FURTHER PROBLEMS:
Ω=?Ω should fall off with distance sufficiently, such that it does not
contradict current measurements (solar system experiments).
2
3
2
−−
==R
mb
R
BΩ
( )≈
k
+k
k R
mb
R
BΩ
−−
==1
contradict current measurements (solar system experiments).
This leads to the minimal ansatz:
But it also can be:
Using: ( )rrl±r=Rl ± ≈&1,<<
−
−= 12m
exp2
3 kr
r
mbΩ α
68Also possible:
here the
contribution is
arbitrary small
for r>2m
Using:
The spherically symmetric
Schwarzschild Solution
Equations for the matter free space
0RR == &0µν
2 Bm
+−
We obtain the isotropic Schwarzschild solution
2
2
22
2
2
21
21
2
21
2
21 dr
r
B
r
m
r
mr
B
r
m
dtr
B
r
md
+−
−
+−−
+−≈ω
The Red Shift g factor:
1
gdtdtr
B
r
mdtgd ≡
+−=≈3
0
002
21τ
The Red Shift
The metric component
of the time must be positive
00
00 >g3
27
64mB >
Antigravitation below
g0.4
0.6
0.8
1
repulsion
attraction
Antigravitation below
2/3 of the Schwarzschild
radius!r/(2m)1 10 1000.1
0
0.2
A fatal fall into the black hole (tidal forces)
Astronaut
Binary Star with one Visible and one Black Hole Component
Binary Star
Schwarzschild Solution embedded
into the Euclidean Space
Typical Black Hole
There are no Black Holes!
Antigravitation in the center
EFFECTIVE POTENTIAL(MISNER ET AL., GRAVITATION)
75
(B=5m3)
PARAMETRIZED-POST-NEWTON (PPN)-FORMALISM:
ISOTROPIC COORDINATES
[ ]22222222 dΩrrdf)(dXg=dΩr(dr)g+)(dXgdω 200
00
0
rr
00
00
2 −+−=
= ∫ drg
rrr
exp
2
222022
r
r=frdf=drgdΩrf=dΩr 22
rr
2 ⇒
32
31
2
3221 dt
r
mb
r
m+
r
mdω 22
−−
−≈
= ∫ dr
rr exp
76
[ ]222
2
2
321 dΩr+rd
r
m+
r
m+
−
The correction appears only in the time element-
Perihelion shift, everything else is identical to standard GR.
(approx. 10-7) in the perihelion shift
PSEUDO-COMPLEX
ROBERTSON-WALKER
UNIVERSE
with
77
with
EQUATIONS OF MOTION
78
Christophel symbols:Christophel symbols:
79
EQUATIONS OF MOTION-2:
80
Metric:
Homogeneity: 321 , ξξξ ==== TTTHomogeneity:321
3
3
2
2
1
1 , ξξξ ==== TTT
−= σξπκ
cρ Λ 0
2
8Dark energy density:
(Pseudo-complex energy density)
81
PSEUDO-COMPLEX PRESSURE
Shifting ξ to the left hand side:
−
−
−=
−−−→
−
σξπκ
cp
c
p
c
p
c
πκσξ
πκ
c+T
c
πκ
kξ
ξ
k
k
k
8
8
8
8
4
222
2
2
with
negative pressure!
82
−−= σξπκ
p kξ8
negative pressure!
83
k=0 flat universe
( )20122
33ξ
2
13p4
Rc
Rσξ
c+ρ
c
πκ ′′−−=
−
( )22
2
0122
20122
2
14
2
Rc
RRRσξξ
c
p+ρ
c
πκ
Rccc
′′−′−−=
−
−
84
ESTIMATION OF ξk -FUNCTIONS:
R
Rc
Rξ
2
20
R2
3
′′′
′′=
R=rrl±r=R ≈&
RcRc
Rξ
222
2
1
R2 ′′+
′=
[ ] [ ] HH+H=HH+H
H=
R
RR+H=
R
R)R(=
R
R
R
R=
R
R
r
rr
r
rr
r
r
r
r
r
r ≈′
′′′′′′′
′′′′′
lnlnln
Using the approximations:
=radius of the universe
( )...3
0
2
001 +++= δξγξβξξ 85 (Ansatz)
r± R=rrl±r=R ≈&
(k=0)
=radius of the universe
we obtain:
LOCAL ENERGY CONSERVATION
⇒
01 βξξ = )1(3
0
−−Λ= βξ RSolution for ξ: assumption
Condition: (Energy + energy due to pressure) = 0dt
d
0 −
86
Solution for ρ: p=αρ
+−2
13
c
α
Rρ=ρ r0
(α=0 dust, =1/3 radiation)
ACCELERATION:
Defining a rescaling, for notational reasons:
2~ AcA=
we obtain for the accelereation (rescaled):
( ) 1/13
2
1)1(3
0
2
1)13(~
4
3~ ++−+−
+−−=′′
=′′ c
rrr
r Rc
RAk
RR αβ α
βρπ
016 ρπkA=
87
Contribution to dark energy classical part
yields acceleration yields de-acceleration
if β>(1/3)
204
rrrckπ
DIFFERENT CASES CASES:
• β <1/3 de-acceleration• β <1/3 de-acceleration
2) 2/3> β >1/3 acceleration, with diminishing strength in R
88
3)β >2/3 acceleration, with increasing strength in R
β =1/2 , A=3
maximum
1. phase: de-acceleration
2. phase: increasing acceleration, until a
maximum is reached
3. phase: decreasing acceleration89
β =2/3 , A=4
No maximum!
1. phase: de-acceleration1. phase: de-acceleration
2. phase: increasing acceleration, which for
Rinfinity approaches a constant value
In the standard GR, with a constant Λ, the asymptotic
behavior is linear in R.90
ESTIMATION FOR β : RELATION OF TO THE HUBBLE
CONSTANT H AND ITS DERIVATIVE
RRξ
Rc
Rξ
2
20
2
3
′′+
′=
′′=
Rc
R
Rc
Rξ
2221
2 ′′+
′=
[ ] [ ]
H+H=HH+H
H=
R
RR+H=
R
R)R(=
R
R
R
R=
R
R
r
rr
r
rr
r
r
r
r
r
r
′
′
′′′′′′
′′′′′
lnlnln
2
2
221
3
21
32
H
H+β
Hc
ββξHc
+ξξ 00
′=⇒
==′=⇒
91
H+H=HH+H
= ′
UNIVERSE WITH DUST AND
RADIATION:
[ ]'R
(P. Peebles, Rev. Mod.75 (2003), 559.
[ ]
2)1()1()(
)()1()1(
)1(3)1(3
432
0
'
−=⇒+=+=
Ω++Ω++Ω=
+−
Λ
ββ wzzzf
zfzzHR
R
w
rd
r
r
dust radiation dark energy
within our theory
92
z)+(R
R
r
r0 1=with: and z as the redshift variable
CONCLUSIONS AND OUTLOOK
Variables = pseudo-complex
Modified variational principle dark energy
Schwarzschild solution: Dark energy accumulates towards smaller
radial distances Collaps is avoided
Pseudo-complex Robertson-Walker- Universe: different, old and
new solutions, Estimation of ξ_k as a function of the Hubble
constant H and its derivative.
If , then w must be different from -1!
0' ≠H
93
OUTLOOK
Application of the PPN-formalism
Results only different to standard GR in the perihelion shift
Determine perihelion shift of … the moon???... (difficult due to the
influence of the sun)
50
