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The Limiting Curvaturehypothesis
A new principle of physics
Dr. Yacoub I. Anini
Shortcut (2) to Presentation.dvi.lnk
Is there a limit to the strength of gravitational force?
• The surface gravity of the earth : 9.8 m/s²• The surface gravity of the sun : 270 m/s²• The surface gravity of a white dwarf : 5000,000 m/s²• The surface gravity of a neutron star:• 5,000,000,000,00 m/s²• The surface gravity of a stellar black hole:• 2,000,000,000,000 m/s²
• The surface gravity of a premordial black hole• 7,000,000,000,000,000,000,000,000,000,000,0 m/s²
The limiting curvature hypothesis
• The curvature of spacetime (all curvature
invariants) at any point have a maximum
limiting value. Moreover, when the
the curvature approaches its limiting value,
The spacetime geometry approaches
The perfectly regular de sitter geometry.
The Lagrangian
• It is possible to implement the limitingCurvature hypothesis by introducing the following
lagrangian:
L = (R + Λ/2 ) – (Λ/2)(√1 - R²/Λ²),
where R is the Ricci scalar curvature andΛ is the limiting value of the curvature.
The contracted field equations
• -R -Λ (1 – U ) = -8π G T,
• U = √(1- R²/Λ²)
• Introducing the following notation :• Β =R/Λ• γ = 8πG (T/Λ)
• The contracted field equations take the form
• 2β² + 2(1-γ)β + γ² - 2 γ = 0
Expressing β in terms of γ
Β = - ½ (1 – γ ) ±½√1 - γ² + 2γ
It is clear that if β is to be real then there will
Be a limit on the allowed values of γ
(1-√2 ) < γ < (1+ √2 )
By varying the gravitational action with respect to the metric we obtain
the new field equations
• Gμν - ¼ [1 - √(1- R²/Λ²)] gμν = - 8π G Tμν
Spaces of constant curvature
• Writing the field equation for
A homogeneous and isotropic space
• The cosmological Case
The limiting geometry
• The limiting gravitational state
• The limiting value of curvature
• The limiting state of matter
• The limiting value of density
De sitter spacetime
Some Cosmological solutions
• The limiting de sitter geometry
• The radiation filled universe
• The matter filled universe
• The general case ( radiation + matter )
Spherically symmetric solutions
• Writing the field equations for spherically Symmetric solutions
• Non- singular black holes
Singular geometry
The numerical value of the limiting curvature
• Low curvature limiting value (effective gravity theory)
• Planck –scale limiting curvature (quantum
Gravity scale)
Spherically Symmatric solutions
• Non- Singular Black Holes
The gravitational field lines inside a collapsing star
Accretion of matter into a black hole
References
references
• Collapse to a Black Hole (Movie)_files
• Falling to the Singularity of the Black Hole (Movie)_files
• Sphere collapsing to a black hole_files
White Holes and Wormholes.htm
• paper.pdf
• Falling to the Singularity of the Black Hole (Movie).htm
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