High Energy Physics Insights: Black Hole Torsion & Geometrization Conjecture

New Trends in High-Energy Physics (May 2019), Odessa Ukraine Black Hole Torsion Effect and its Relation to Information Ioannis Gkigkitzis, Ioannis Haranas Departments of Mathematics, East Carolina, NC, USA Department of Physics and Astronomy, York University Toronto, Ontario, Canada

Geometrization conjecture Every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. Implies several other conjectures, such as the Poincar conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery. The eight Thurston geometries: ?3,?3,?3,?2 ?,?2 ?, ??(2,?),???,????

Methods/Techniques g ij = 2 R Program for a proof of the conjecture: The central idea is the notion of the Ricci flow Hamilton's fundamental idea was to formulate a "dynamical process" in which a given three-manifold is geometrically distorted such that this distortion process is governed by a differential equation analogous to the heat equation. Under the Ricci flow, concentrations of large curvature will spread out until a uniform curvature is achieved over the entire three-manifold. ij t

g g g 1 jk ij = + h ij hk ik ( ) g The Levi Civita connection is given by the Christoffel symbols i j k 2 x x x i jk j = + and the Riemannian curvature tensor is h ijk h ip p jk h jp p ik R ik i j x g x R = ik R R = jlR g and the contracted Ricci curvature tensor is jl ijkl jl i R lm ijkl R = h h h ijk g k For the interchange of two covariant derivatives (and similar formulas for more complicated tensors) we have i j j = i j k j i k m = i pq i T g T The Laplacian of a tensor is jk p q jk ( ) ( ) + + = + + + pq 2 R B B B R R R R g R R R R ijkl ijkl ijlk ikjl i k jl i l jk j k il j i kl pjkl qi ipkl qj = + ij kl 2 R R g g R R = + pr qs pq 2 2 R R g g R R g R R ik jl t ik ik piqk rs pi qs t ( ) 1 = + / 1 2 / 1 2 R g R g R g R g R g g g g 2 2 2 + d d d 1 1 i i i T C T T ijkl ik jl il jk jk il jl ik ik jl il jk 2

Monotonicity formulas of the entropy of black holes Definition: The entropy of Ricci black holes are defined as Perelman s functional and functional as follows ( 3 M ) ( ( ) )( ) 2 2 / 2 n = + = + + fd f ( , ) F g f R f e ( , , ) 4 W g f R f f n t e d g g 3 M Theorem: (Monotonicity formulas of the entropy of black holes) Under the evolution system of black holes, the entropy increases. g g ij ij = = 2 2 R R ij ij f f t t n 2 2 = + + = + R f f R f f dW dF 2 t t 0 0 dt dt

No Local Collapsing Theorem and the Singularity of the Black Hole Theorem: Given ? 0,+ ,k > 0, we say that a Riemannian manifold ??,? is k no collapsed below the scale ?, if for any metric ball ? ?,? with ? < ?, satisfying ??? < ? 2, we have ?? < ? Let ??,? , t 0,? , be a solution to the Ricci flow on a closed manifold. Then for any scale ? 0,+ , there exists k = k ? 0 ,?,? > 0, such that ? ? is no collapsed below that scale, for all t 0,? . ????? ?,?

Einstein field equations The stress energy momentum tensor or energy momentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in space-time The stress energy tensor is defined as the tensor ??? of order two that gives the flux of the ?? component of the momentum vector across a surface with constant ?? coordinate. The Einstein field equations relate local space-time curvature (expressed by the Einstein tensor) with the local energy and momentum within that space-time (expressed by the stress energy tensor). c 1 8 G 4 + = R Rg g T 2

The Cross Curvature Flow of 3-Manifolds 1 = 2 g h = T R Rg ij ij ij ij ij t 2 1 = ij ik jl ij T g g R Rg = ij kl ij il jk ij ij det P P P P P P g H P kl 2 k l k l t ( ( ) ) kl det T ( ) = = det h T V V ij ij ij kl det g In an orthonormal basis where ???= ??? The eigenvalues of Sectional Curvature are ?2323= ?, ?1313= ?, ?1212= ? The eigenvalues of Ricci Curvature are ?11= (? + ?), ?22= (? + ?), ?33= (? + ?) The eigenvalues of the Cross Curvature are 11= ??, 22= ??, 33= ?? The eigenvalues of the Einstein Tensor are ?11= ?, ?22= ?, ?33= ?

The Cross Curvature Flow of 3-Manifolds In the case of negative sectional curvature, some monotonicity formulas support the conjecture that after normalization, for initial metrics on closed 3-manifolds with negative sectional curvature, the solution exists for all time and converges to a hyperbolic metric. This conjecture is still open at the present time. dJ 1 ( ) ( ) dVol T M 3 / 1 = ij = 0 det J g T T d ( ) det Vol T T d 0 ij g g dt 3 dt By the arithmetic-geometric mean inequality (applied in a basis in which ???is diagonal and ???= ???), the integrand is nonnegative, and identically zero if and only if ???= 1 3????, which implies constant curvature (Bennett Chow, Richard S. Hamilton, 2004)

Spin Spin interaction in Strong Gravity and its Consequences The covariant derivative in the curved space Dirac equation generates extra interaction terms analogous to the Pauli terms in electrodynamics. Singularities in gravitational collapse may be prevented by the direct influence of spin on space-time geometry and for small distances the spin density plays the role of a repulsive potential counteracting the usual attractive gravitational force, arresting collapse to a singularity (Sivaram and Sinha, 1978) Possible balance between mass and spin effects by a simple calculation within the Newtonian approximation where is the spin density of dimension ? ?? 1 ??? 1 and is the mass density. ( ) ( = ) 2 2 4 ) ( + G K f f R 2 f 2 3 G M G S 4 4 R = f = + = = = x 3 3 - t M R S R 4 5 2 3 3 2 R c R R

Spin Effects in Gravitation To understand spin effects in gravitation, we can use torsion. Let us first write a Schwarzschild that includes torsion effects (De Sabbata and Zhang 1992) The modification of the metric becomes: 2 2 2 3 GM G s = 2 1 g00 c 2 4 6 2 c r r c 1 ( ) 2 2 2 2 2 3 2 3 GM G s GM G s 2 2 2 2 2 2 2 2 = + d d 1 d 1 d d sin s c t r r 2 4 6 2 4 6 2 2 c r r c c r r c The surface gravity of a black hole with torsion can be written as c 2 2 G 3 GM G s = mp = = + constant X p G 3 c 2 4 5 2 R c R ( )( ) ( ) / 3 2 / 1 2 = = 2 4 3 2 3 3 s c G c G G c zero surface gravity would correspond to For a spin proportional to for such a black hole, the surface gravity and hence the temperature vanishes. The torsion effects which enter with opposite sign, cancel those of gravity.

2 / 3 6 2 c ( ) / 1 3 4 5 4 8 6 6 10 4 2 2 + + 9 2 81 c G M c G c G M 1 = rH 3 ( ) / 1 3 / 1 2 3 4 5 4 8 6 6 10 4 2 2 + + 9 2 81 c G M c G c G M 2 2 G k = MA 2 4 S H p ( ) ( ) r A ( ) ( ) r A ( ) ( ) r A ( ) ( ) r ( ) ( ) r A 2 2 2 2 r A r A r r A r B r r A r ( ) r = + + + 1 R B r r ( ) r 2 2 4 4 B r B 2 2 R ~ Sch R M c

Spin Spin interaction in Strong Gravity and its Consequences In the case where GR must be extended to include microphysics, matter must be considered and to describe it using mass and spin density. Striking resemblances between the properties-parameters of black holes, being mass, charge and spin, and the same properties of elementary particles, where they occur in quantized units. The spin plays the same role in describing gravitation as charge does for electromagnetism. Spin-spin contact interaction in the gravitational potential between two spin particles is a feature of the curved space Dirac equations. Singularities in gravitational collapse may be prevented by the direct influence of spin on the space-time geometry, and for small distances the spin density in Cartan s equations plays the role of repulsive potential counteracting the usual attractive gravitational force, arresting the collapse to a singularity.