The Limiting Curvaturehypothesis

A new principle of physics

Dr. Yacoub I. Anini

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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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