universe, black holes, and particles in spacetime...

Universe, Black Holes, and Particles in Spacetime with

Torsion

Nikodem J. Popławski

Department of Physics, Indiana University, Bloomington, IN

CTA Theoretical Astrophysics and General Relativity Seminar

University of Illinois at Urbana-Champaign, Urbana, IL 27 April 2011

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Outline A. 1. Torsion 2. Universes in nonsingular black holes B. 3. Einstein-Cartan-Sciama-Kibble gravity 4. Spin fluids 5. Big-bounce cosmology without inflation C. 6. Nonlinear Dirac equation 7. Dark energy from torsion 8. Matter-antimatter asymmetry from torsion

What is Torsion?

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PHYSICS TODAY

What is Torsion? • Differentiation of tensors in curved spacetime requires geometrical structure: affine connection ¡½

¹º

• Covariant derivative rºV

¹ = ºV¹ + ¡¹

½ºV½

• Curvature tensor R½

¾¹º = ¹¡½¾º - º¡

½¾¹ + ¡½

¿¹¡¿¾º - ¡

½¿º¡

¿¾¹

• Torsion tensor – antisymmetric part of affine connection

• Contortion tensor

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Einstein-Cartan-Sciama-Kibble gravity Special Relativity - no curvature & no torsion Dynamical variables: matter fields

General Relativity - no torsion Dynamical variables: matter fields + metric tensor

ECSK Gravity (simplest theory with torsion) Dynamical variables: matter fields + metric + torsion

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More degrees of freedom

Why ECSK gravity? GR ECSK MTG

Avoids curvature singularities for ordinary matter NO YES! -

Takes into account intrinsic angular momentum (spin) of matter NO YES -

Dirac equation is linear YES NO -

Free parameters NO NO! YES

• ECSK ≠ GR at densities >> nuclear -> ECSK passes all GR tests

• ECSK -> dark energy & matter-antimatter asymmetry

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History of Torsion

• E. Cartan (1921) – asymmetric affine connection -> torsion

• Sciama and Kibble (1960s) – spin generates torsion (energy & momentum generate curvature)

T. W. B. Kibble, J. Math. Phys. 2, 212 (1961) D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964) F. W. Hehl, Phys. Lett. A 36, 225 (1971); Gen. Relativ. Gravit. 4, 333 (1973); 5, 491 (1974) F. W. Hehl, P. von der Heyde, G. D. Kerlick & J. M. Nester, Rev. Mod. Phys. 48, 393 (1976) E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, 1976)

• (1970s) – torsion may avert cosmological singularities (polarized spins)

W. Kopczyński, Phys. Lett. A 39, 219 (1972); 43, 63 (1973) A. Trautman, Nature (Phys. Sci.) 242, 7 (1973) J. Tafel, Phys. Lett. A 45, 341 (1973)

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History of Torsion

• Hehl and Datta (1971) – Dirac equation with torsion is nonlinear (cubic in spinors) -> Fermi-like four-fermion interaction

F. W. Hehl & B. K. Datta, J. Math. Phys. 12, 1334 (1971)

• (1970s) – torsion averts cosmological singularities (spin fluids with unpolarized spins) -> bounce cosmology

F. W. Hehl, P. von der Heyde & G. D. Kerlick, Phys. Rev. D 10, 1066 (1974) B. Kuchowicz, Gen. Relativ. Gravit. 9, 511 (1978) M. Gasperini, Phys. Rev. Lett. 56, 2873 (1986)

• (1991) – macroscopic matter with torsion has spin-fluid form

K. Nomura, T. Shirafuji & K. Hayashi, Prog. Theor. Phys. 86, 1239 (1991)

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Big-bounce cosmology • Origin of Universe? Torsion combines both problems

• Nature of black-hole interiors?

Torsion in ECSK gravity averts big-bang singularity and singularities in black holes via gravitational repulsion at high densities

-> Energy conditions in Penrose-Hawking theorems not satisfied

Big bounce instead of big bang M. Bojowald, Nature Phys. 3, 523 (2007) M. Novello & S. E. Perez Bergliaffa, Phys. Rept. 463, 127 (2008) A. Ashtekar & D. Sloan, Phys. Lett. B. 694, 108 (2010) R. H. Brandenberger, arXiv:1103.2271; W. Nelson & E. Wilson-Ewing, arXiv:1104.3688

Loop Quantum Gravity -> big bounce!

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Universe in a black hole Our Universe was contracting before bounce – from what?

Idea: every black hole produces a nonsingular, closed universe Our Universe born in a black hole existing in another universe • Idea not new (1970s)

R. K. Pathria, Nature 240, 298 (1972) V. P. Frolov, M. A. Markov & V. F. Mukhanov, Phys. Rev. D 41, 383 (1990) L. Smolin, Class. Quantum Grav. 9, 173 (1992) W. M. Stuckey, Am. J. Phys. 62, 788 (1994) D. A. Easson & R. H. Brandenberger, J. High Energy Phys. 0106, 024 (2001) J. Smoller & B. Temple, Proc. Natl. Acad. Sci. USA 100, 11216 (2003)

Simplest mechanism – torsion!

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Universe in a black hole

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Every black hole forms new universe

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Black holes are Einstein-Rosen bridges

Black holes are Einstein-Rosen bridges

NJP, Phys. Lett. B 687, 110 (2010)

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Arrow of time

• Why does time flow only in one direction?

• Laws of ECKS gravity (and GR) are time-symmetric

• Boundary conditions of a universe in a BH are not: motion of matter through event horizon is unidirectional can define arrow of time

• Arrow of time in the universe fixed by time-asymmetric collapse of matter through EH (before expansion)

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event horizon

future past

Information not lost

How to test that every black hole contains a hidden universe?

To boldly go where no one has gone before

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Preferred direction

• Stars rotate -> Kerr black holes

Universe in a Kerr BH inherits its preferred direction

Small corrections to FLRW metric – Kerr length a=L/(Mc)

Heaviest and fastest spinning stellar BH: GRS 1915+105 a ≈ 26 km • Source of Lorentz-violating parameters of Standard Model

Extension -> matter-antimatter asymmetry in neutrino and neutral-meson oscillations?

Preferred-frame parameter: -2.4 × 10−19 GeV ~ 820 m close!

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J. E. McClintock et al., Astrophys. J. 652, 518 (2006) T. Katori, V. A. Kostelecký & R. Tayloe, Phys. Rev. D 74, 105009 (2006) V. A. Kostelecký & R. J. Van Kooten, Phys. Rev. D 82, 101702(R) (2010)

NJP, Phys. Lett. B 694, 181 (2010)

1. Einstein-Cartan-Sciama-Kibble

theory of gravity prevents

singularities

ECSK gravity • Riemann-Cartan spacetime – metricity r½g¹º = 0

→ connection ¡½¹º = {½

¹º} + C½¹º

Christoffel symbols contortion tensor • Lagrangian density for matter

Dynamical energy-momentum tensor Spin tensor

Spin tensor different from 0 for Dirac spinor fields

Total Lagrangian density (like in GR)

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ECSK gravity • Curvature tensor = Riemann tensor + tensor quadratic in torsion + total derivative • Stationarity of action under ±g¹º -> Einstein equations

R{}¹º - R

{}g¹º /2 = k(T¹º + U¹º)

U¹º = [C½¹½C

¾º¾ - C

½¹¾C

¾º½ - (C

½¾½C

¿¾¿ - C

¾½¿C¿½¾)g¹º /2] /k

• Stationarity of action under ±C¹º

½ -> Cartan equations

S½¹º - S¹±

½º + Sº±

½¹ = -ks¹º

½ /2

S¹ = Sº¹º

• Cartan equations are algebraic and linear

ECSK torsion does not propagate (unlike metric/curvature)

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Same proportionality constant k!

ECSK gravity • Field equations with full Ricci tensor

R¹º - Rg¹º /2 = £¹º

Canonical energy-momentum tensor • Belinfante-Rosenfeld relation

£¹º = T¹º + r¤½(s¹º

½ + s½º¹ + s½

¹º) /2 r¤½=r½- 2S½

• Conservation law for spin

r¤½s¹º

½ = (£¹º - £º¹)

• Cyclic identities

R¾¹º½ = -2r¹S

¾º½ + 4S¾

¿¹S¿º½ (¹, º, ½ cyclically permutated)

ECSK gravity • Bianchi identities (¹, º, ½ cyclically permutated)

r¹R¾¿º½ = 2R¾

¿¼¹S¼º½

• Conservation law for energy and momentum

Dº£¹º = Cº½

¹£º½ + sº½¾Rº½¾¹/2 Dº=r{}

º

Equations of motion of particles F. W. Hehl, P. von der Heyde, G. D. Kerlick & J. M. Nester, Rev. Mod. Phys. 48, 393 (1976) E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, 1976) NJP, arXiv:0911.0334

ECSK gravity • No spinors -> torsion vanishes -> ECSK reduces to GR • Torsion significant when U¹º » T¹º

For fermionic matter (quarks and leptons)

½ > 1045 kg m-3

Nuclear matter in neutron stars

½ » 1017 kg m-3

Gravitational effects of torsion negligible even for neutron stars Torsion significant only in very early Universe and in black holes

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Spin fluids • Papapetrou (1951) – multipole expansion -> equations of motion

Matter in a small region in space with coordinates x¹(s)

Motion of an extended body – world tube

Motion of the body as a whole – wordline X¹(s)

• ±x® = x® - X® ±x0 = 0

u¹ = dX¹/ds ® - spatial coordinates

M¹º½ = -u0 s±x¹ £º½(-g)1/2 dV

N¹º½ = u0 ss¹º½(-g)1/2 dV

• Dimensions of the body small -> neglect higher-order (in ±x¹) integrals and omit surface integrals

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Four-velocity

Spin fluids • Conservation law for spin ->

M½¹º - M½º¹ = N¹º½ - N¹º0 u½/u0

• Average fermionic matter as a continuum (fluid)

Neglect M½¹º -> s¹º½ = s¹ºu½ s¹ºuº = 0

Macroscopic spin tensor of a spin fluid

• Conservation law for energy and momentum ->

£¹º = c¦¹uº - p(g¹º - u¹uº) ² = c¦¹u¹ s2 = s¹ºs¹º /2

Four-momentum Pressure Energy density density

J. Weyssenhoff & A. Raabe, Acta Phys. Pol. 9, 7 (1947) K. Nomura, T. Shirafuji & K. Hayashi, Prog. Theor. Phys. 86, 1239 (1991)

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Spin fluids

• -> Dynamical energy-momentum tensor for a spin fluid

Energy density Pressure

F. W. Hehl, P. von der Heyde & G. D. Kerlick, Phys. Rev. D 10, 1066 (1974)

• Barotropic fluid s2 / ²2/(1+w)

dn/n = d²/(²+p) p = w² n / ²1/(1+w)

• Spin fluid of fermions with no spin polarization ->

I. S. Nurgaliev & W. N. Ponomariev, Phys. Lett. B 130, 378 (1983)

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for random spin orientation

Cosmology with torsion • A closed, homogeneous and isotropic Universe

Friedman-Lemaitre-Robertson-Walker metric (k = 1)

Distance from O to its antipodal point A: a¼

• Friedman equations for scale factor a

Conservation law M. Gasperini, Phys. Rev. Lett. 56, 2873 (1986)

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a

O

A

Cosmology with torsion

• Friedman conservation -> ² / a-3(1+w) (like without spin)

Spin-torsion contribution to energy density

o Independent of w

o Consistent with the particle conservation n / a-3

o ²S decouples from ²

spin fluid = perfect fluid + exotic fluid (with p = ² = -ks2/4 < 0) • Very early universe: w = 1/3 (radiation) ² ¼ ²R » a-4

Total energy density

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negative & dominant at small a gravitational

repulsion

Big bounce from torsion • Friedman equation

• Gravitational repulsion from spin & torsion (S<0)

No singularity & no big bang -> big bounce! (t=0)

Universe starts expanding from minimum radius (when H = 0)

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NJP, Phys. Lett. B 694, 181 (2010)


Velocity of antipodal point

• WMAP parameters of the Universe = 1.002 H0

-1 = 4.4£1017 s R = 8.8£10-5

a0 = 2.9£1027 m • Background neutrinos – most abundant fermions in the Universe

n = 5.6£107 m-3 for each type • S = – 8.6£ 10-70 (negative, extremely small in magnitude)

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NJP, Phys. Lett. B 694, 181 (2010)


GR

S = 0 and am = 0

» 1 today -> (a) at GUT epoch must be tuned to 1 to a precision of > 52 decimal places

Flatness & horizon problems in big-bang cosmology Solved by cosmic inflation – consistent with cosmic perturbations

Problems: - Initial (big-bang) singularity exists - Needs new physics (scalar fields), specific forms of potential, free

parameters - Eternal inflation Why » 1 before inflation?

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ECSK

S < 0 and am > 0

Appears tuned to 1 to a precision of » 63 decimal places!

No flatness problem – advantages: - Nonsingular bounce instead of initial singularity - No new physics, no additional assumptions, no free parameters - Smooth transition: torsion epoch -> radiation epoch (torsion becomes negligible)

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NJP, Phys. Lett. B 694, 181 (2010)

Minimum


ECSK

S < 0 and am > 0

• Closed Universe causally connected at t < 0 remains causally connected through t = 0 until va = c

• Universe contains N » (va/c)3 causally disconnected volumes

• S » -10-69 produces N ¼ 1096 from a single causally connected region – torsion solves horizon problem

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NJP, Phys. Lett. B 694, 181 (2010)

Maximum of va

1/v2a

Bounce cosmologies free of horizon problem


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Black holes with torsion Q: Where does the mass of the Universe come from? • Possible solution: stiff equation of state p = ɛ • Strong interaction of nucleon gas

-> ultradense matter has stiff EoS

Y. B. Zel’dovich, Sov. Phys. J. Exp. Theor. Phys. 14, 1143 (1962) J. D. Walecka, Ann. Phys. 83, 491 (1974); Phys. Lett. B 59, 109 (1975) S. A. Chin & J. D. Walecka, Phys. Lett. B 52, 24 (1974) V. Canuto, Ann. Rev. Astr. Astrophys. 12, 167 (1974); 13, 335 (1975)

• Stiff matter – possible content of early Universe

Y. B. Zel’dovich, Mon. Not. R. Astr. Soc. 160, 1 (1972) D. N. C. Lin, B. J. Carr & S. M. Fall, Mon. Not. R. Astr. Soc. 177, 51 (1976) J. D. Barrow, Nature 272, 211 (1977) R. Maartens & S. D. Nel, Commun. Math. Phys. 59, 273 (1978) J. Wainwright, W. C. W. Ince & B. J. Marshman, Gen. Relativ. Gravit. 10, 259 (1979)

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Black holes with torsion

• Collapse of BH – truncated, closed FLRW metric

• X-ray emission from neutron stars -> NS composed of matter with stiff EoS -> BHs expected to obey stiff EoS

V. Suleimanov, J. Poutanen, M. Revnivtsev & K. Werner, arXiv:1004.4871

• Friedman conservation law -> mass of collapsing BH increases (external observers do not see it)

• May be realized by intense particle production in strong fields

L. Parker, Phys. Rev. 183, 1057 (1969) Y. B. Zel’dovich, J. Exp. Theor. Phys. Lett. 12, 307 (1970) Y. B. Zel’dovich & A. A. Starobinsky, J. Exp. Theor. Phys. Lett. 26, 252 (1978)

• Total energy (matter + gravity) remains constant

F. I. Cooperstock & M. Israelit, Found. Phys. 25, 631 (1995)


• Friedman eqs.

• ¤ negligible, initially s2 negligible

As a decreases, k becomes less significant

• Stiff matter

-> -> -> ->

->

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Number of fermions


-> 2 solutions for a: At bounce

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Mass of BH

Mass of neutron

Bounce due to torsion

NJP, arXiv:1103.4192


• After bounce, universe in BH expands

Torsion becomes negligible, k = 1 becomes significant

Expansion = time reversal of contraction?

Universe reaches a0 and contracts -> cyclic universe (between amin and a0)

• Particle production increases entropy

Effective masses of fermions increase due to Hehl-Datta equation

Fermionic matter may be dominated by heavy, NR particles

Particle physics at extremely high energies? Matter creation?

-> Expansion of Universe ≠ time reversal of contraction -> Mass of Universe not diluted during expansion

F. Hoyle, Mon. Not. R. Astron. Soc. 108, 372 (1948); 109, 365 (1949)

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Cosmological models with k = 1, ¤ ≠ 0

H. Bondi, Cosmology (Cambridge Univ. Press , 1960) E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, 1976)

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Expansion to infinity if


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Universe in a black hole may oscillate until its mass exceeds Mc Then it expands to infinity Mass of our Universe

(Binary IC 10 X-1 – 24-33)


• A new universe in a BH invisible for observers outside the BH (EH formation and all subsequent processes occur after ∞ time)

• As the universe in a BH expands to infinity, the BH boundary becomes an Einstein-Rosen bridge (Flamm 1916, Weyl 1917, Einstein & Rosen 1935) connecting this (child) universe with the outer (parent) universe

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Cosmological perturbations • Observed scale-invariant spectrum of cosmological perturbations

produced by thermal fluctuations in a collapsing black hole if

• Stiff matter w = 1 -> wr = 0 (NR gas of fluctuating particles)

Work in progress

Y.-F. Cai, W. Xue, R. Brandenberger & X. Zhang, J. Cosm. Astropart. Phys. 06, 037 (2009)

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Background

2. Einstein-Cartan-Sciama-Kibble

theory of gravity nonlinearizes

Dirac equation

Spinor fields with torsion

• Dirac matrices

• Spinor representation of Lorentz group

• Spinors

• Covariant derivative of spinor

Metricity -> ->

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Spinor connection

Fock-Ivanenko coefficients (1929)

Tetrad

Spin connection

Hehl-Datta equation

• Dirac Lagrangian density • Spin density

• Cartan eqs. ->

•

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Variation of C variation of ω

Totally antisymmetric

Dirac spin pseudovector

; – covariant derivative with affine connection

: – with Christoffel symbols

Dark energy from torsion • Observed cosmological constant

• Zel’dovich formula

Y. B. Zel’dovich, J. Exp. Theor. Phys. Lett. 6, 316 (1967)

• Spin-torsion coupling reproduces Zel’dovich formula!

Effective Lagrangian density for Dirac field contains axial-axial four-fermion interaction (Kibble-Hehl-Datta)

HD energy-momentum tensor

GR part cosmological term Effective cosmological constant NJP, Annalen Phys. 523, 291 (2011)

Vacuum energy density

Not constant in time, but constant in space at cosmological distances for homogeneous and isotropic Universe

Dark energy from torsion

Cosmological constant if spinor field forms condensate with nonzero vacuum expectation value like in QCD

Vacuum-state-dominance approximation

M. A. Shifman, A. I. Vainshtein & V. I. Zakharov, Nucl. Phys. B 147, 385 (1979)

For quark fields

Axial vector-axial vector form of HD four-fermion interaction gives positive cosmological constant


Cosmological constant from QCD vacuum and ECKS torsion This value would agree with observations if

• Energy scale of torsion-induced cosmological constant from QCD vacuum only ~ 8 times larger than observed

• Contribution from spinor fields with lower VEV like neutrino condensates could decrease average such that torsion- induced cosmological constant would agree with observations • Simplest model predicting positive cosmological constant and ~ its energy scale – does not use new fields


(Zel’dovich)


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Matter-antimatter asymmetry • Hehl-Datta equation

• Charge conjugate

Satisfies Hehl-Datta equation with opposite charge and different sign for the cubic term!

HD asymmetry significant when torsion is -> baryogenesis -> dark matter?

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NJP, Phys. Rev. D 83, 084033 (2011)

Adjoint spinor

Energy levels (effective masses)

Fermions (-> NR) Antifermions

Inverse normalization for spinor wave function

Matter-antimatter asymmetry

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Acknowledgments James Bjorken Yi-Fu Cai Chris Cox Shantanu Desai Luca Fabbri

The Einstein-Cartan-Kibble-Sciama gravity accounts for spin of elementary particles, which equips spacetime with torsion.

For fermionic matter at very high densities, torsion manifests itself as gravitational repulsion that prevents the formation of singularities in black holes and at big bang.

Torsion allows for a scenario in which every black hole produces a new universe inside, explaining arrow of time and predicts preferred direction (ν oscillations).

Our own Universe may be the interior of a black hole existing in another universe. Big bounce instead of big bang.

Bounce cosmology solves flatness and horizon problems without inflation.

Dirac equation with torsion becomes nonlinear Hehl-Datta equation, which may be the origin of dark energy.

Hehl-Datta equation causes matter-antimatter asymmetry, which may be the origin of baryogenesis and dark matter.

Future work: cosmological perturbations and QFT from nonlinear spinors.

Torsion may be a unifying concept in physics that may resolve most major current problems of theoretical physics and cosmology.

Summary Thank you!

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Universe in a black hole

universe, black holes, and particles in spacetime...

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