Download - Dark Matter in Einstein´s Field Equations
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BERLIN ICRANet Pescara
Dark Matter in Einstein´s Field Equations
Hagen Kleinert, FU BERLIN
BERLIN ICRANet Pescara ICRANet Pescara Nizza
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Universe
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All Charged Particles in
Charge:
Homogeneous Poisson Equation:
Aha:
Solutions:
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Radial Version
Charge:
Homogeneous Poisson Equation: Solution:
Radial form:
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Gravitational Field
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Schwarzschild
or
Kerr
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Einstein Equation
Schwarzschild
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closed bosonic string
Now: Singular Surfaces
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Note: Gravitational Singularities have Finite Energies Electrostatic
Singularities have Infinite Energies
Explains Dominance of Dark Matter
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Why old String Theory failed:
Short Strings are MIT-Bags
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Quantisation of Relativistic Particle
Replace by harmonic Action
Klein-Gordon Propagator
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SurfacesReplace Nambu-Goto Action by
harmonicPolyakov Action
= Nambu-Goto Action
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Removes Tachyons
Polyakov adds Liouville Action
IN FACT: Any interacting field
theory is freeof tachyons if energy has
bottom
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Or read my elder book
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Single-Valued Fields
Multi-Valued Fields
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Example: Complex Scalar Field
FALSE!Chain Rule:
set
Wrong Universality Class
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Jumps!
In 1D, can be removed by going to covering group U(1)
In >1D impossible
Correct Chain Rule:
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Vortex Gauge Field
Axial Gauge
Invariant Field Strength:
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Simplest MULTIVALUED FIELD in 2D
Solve:
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NOTE:
Mother of Two Important Green Functions
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NOTE:
Mother of Two Important Green Functions
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Now : Nontrivial Geometry from Nonholonomic Coordinate Transformations
Burgers vector b
Translational DefectsDISLOCATIONS
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Rotational Defects (Disclinations)
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ANALOG MODEL: Induced Gravity in `World Crystal´
Canonical Form
Momentum Conservation
Enforced as Bianchi Idty:
Double Gauge Theory
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Newton OK if we modify action to
FLOPPY CRYSTAL. Directional memory lost by Kosterlitz-Thouless type Fluctuations in D=4:
2.) Are Strings the World Lines of Defects in a Transplanckian World Crystal
New: 1.) High-curvature regime molten phase?
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BUT NEED
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yields Quantum Gravity of
Dark Matter
SUMMARY:
String Theory obtained from Einstein Action of Surface
Singularites
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World is Cracking Open(as Gell-Mann used to say in 1972)
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Path Integrals fix Products of Distributions
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Distribution
Distribution
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PERTUBATION SERIES:
Multiplication Rules for Distributions
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CURIOSITY: Induced Gravity in `World Crystal´
Canonical Form
Momentum Conservation
Enforced as Bianchi Idty:
Double Gauge Theory
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Dual Representation
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BUT NEED
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Newton OK if we modify action to
FLOPPY CRYSTAL. Directional memory lost by Kosterlitz-Thouless type Fluctuations in D=4:
2.) Are Strings the World Lines of Defects in a Transplanckian World Crystal
New: 1.) High-curvature regime molten phase?
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Interdependence
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If you want to know more, read my newest book
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All Charged Particles from
Charge:
Homogeneous Poisson Equation: Solution:
Radial form:
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Recall OLD FEYNMAN ARUMENT
Converges for
R.P. Feynman (1918-1988)
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General Coordinate Transformation
Basis Tetrads
Affine Connection
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Multivalued infinitesimal coordinate transformation
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INTEGRABILITY CONDITIONS
Bianchi Identities
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Conservation Laws
Define Torsion
Linearized Fundamental Identity
Define Einstein Tensor
Linearized Bianchi Identity
Volterra Construction
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INTEGRABILITY CONDITIONS
Define Curvature Tensor:
Then above integrability implies:
(linearized Biachi identitiy)
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Multivalued Basis Tetrads
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INTEGRABILITY CONDITIONS
Bianchi Identities
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Rewrite as
General , then Bianchi Identities
Palatini tensor
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Gravitational field version of conservation laws
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Minimal Coupling from Nonholonomic Coord. Tranfs.
Holonomic vierbeintransforming tononholonomic Coordinates
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Multivalued infinitesimal coordinate transformation
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INTEGRABILITY CONDITIONS
Bianchi Identities
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Derivation from Nonholonomic Mapping Principle for Dirac ElectronFlat Space
Local Lorentz Transformations
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“ EXPERIMENTAL “ SITUATION
Hydrogen Atom in Momentum Space
Eliminates candidates
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Harald`s 65th birthday Munich 2008
Murray`s Berlin visit 2005
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Anachronism