BERLIN ICRANet Pescara

Dark Matter in Einstein´s Field Equations

Hagen Kleinert, FU BERLIN

BERLIN ICRANet Pescara ICRANet Pescara Nizza

Universe

All Charged Particles in

Charge:

Homogeneous Poisson Equation:

Aha:

Solutions:

Radial Version

Charge:

Homogeneous Poisson Equation: Solution:

Radial form:

Gravitational Field

Schwarzschild

or

Kerr

Einstein Equation

Schwarzschild

closed bosonic string

Now: Singular Surfaces

Note: Gravitational Singularities have Finite Energies Electrostatic

Singularities have Infinite Energies

Explains Dominance of Dark Matter

Why old String Theory failed:

Short Strings are MIT-Bags

Quantisation of Relativistic Particle

Replace by harmonic Action

Klein-Gordon Propagator

SurfacesReplace Nambu-Goto Action by

harmonicPolyakov Action

= Nambu-Goto Action

Removes Tachyons

Polyakov adds Liouville Action

IN FACT: Any interacting field

theory is freeof tachyons if energy has

bottom

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Single-Valued Fields

Multi-Valued Fields

Example: Complex Scalar Field

FALSE!Chain Rule:

set

Wrong Universality Class

Jumps!

In 1D, can be removed by going to covering group U(1)

In >1D impossible

Correct Chain Rule:

Vortex Gauge Field

Axial Gauge

Invariant Field Strength:

Simplest MULTIVALUED FIELD in 2D

Solve:

NOTE:

Mother of Two Important Green Functions

NOTE:

Mother of Two Important Green Functions

Now : Nontrivial Geometry from Nonholonomic Coordinate Transformations

Burgers vector b

Translational DefectsDISLOCATIONS

Rotational Defects (Disclinations)

ANALOG MODEL: Induced Gravity in `World Crystal´

Canonical Form

Momentum Conservation

Enforced as Bianchi Idty:

Double Gauge Theory

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?

BUT NEED

yields Quantum Gravity of

Dark Matter

SUMMARY:

String Theory obtained from Einstein Action of Surface

Singularites

World is Cracking Open(as Gell-Mann used to say in 1972)

Path Integrals fix Products of Distributions

Distribution

Distribution

PERTUBATION SERIES:

Multiplication Rules for Distributions

CURIOSITY: Induced Gravity in `World Crystal´

Canonical Form

Momentum Conservation

Enforced as Bianchi Idty:

Double Gauge Theory

Dual Representation

BUT NEED

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?

Interdependence

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All Charged Particles from

Charge:

Homogeneous Poisson Equation: Solution:

Radial form:

Recall OLD FEYNMAN ARUMENT

Converges for

R.P. Feynman (1918-1988)

General Coordinate Transformation

Basis Tetrads

Affine Connection

Multivalued infinitesimal coordinate transformation

INTEGRABILITY CONDITIONS

Bianchi Identities

Conservation Laws

Define Torsion

Linearized Fundamental Identity

Define Einstein Tensor

Linearized Bianchi Identity

Volterra Construction

INTEGRABILITY CONDITIONS

Define Curvature Tensor:

Then above integrability implies:

(linearized Biachi identitiy)

Multivalued Basis Tetrads

INTEGRABILITY CONDITIONS

Bianchi Identities

Rewrite as

General , then Bianchi Identities

Palatini tensor

Gravitational field version of conservation laws

Minimal Coupling from Nonholonomic Coord. Tranfs.

Holonomic vierbeintransforming tononholonomic Coordinates

Multivalued infinitesimal coordinate transformation

INTEGRABILITY CONDITIONS

Bianchi Identities

Derivation from Nonholonomic Mapping Principle for Dirac ElectronFlat Space

Local Lorentz Transformations

“ EXPERIMENTAL “ SITUATION

Hydrogen Atom in Momentum Space

Eliminates candidates

Harald`s 65th birthday Munich 2008

Murray`s Berlin visit 2005

Anachronism