Universe, Black Holes, and Particles in Spacetime with Torsion

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

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? PHYSICS TODAY

What is Torsion? Differentiation of tensors in curved spacetime requires geometrical structure: affine connection ½ ¹º Covariant derivative r º V ¹ = º V ¹ + ¹ ½ºV ½ Curvature tensor R ½ ¾¹º = ¹ ½ ¾º - º ½ ¾¹ + ½ ¹ ¾º - ½ º ¾¹ Torsion tensor antisymmetric part of affine connection Contortion tensor

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 More degrees of freedom ECSK Gravity (simplest theory with torsion) Dynamical variables: matter fields + metric + torsion

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

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)

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)

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!

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!

Universe in a black hole

Every black hole forms new universe

Black holes are Einstein-Rosen bridges

Black holes are Einstein-Rosen bridges NJP, Phys. Lett. B 687, 110 (2010)

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 event horizon future past Information not lost Arrow of time in the universe fixed by time-asymmetric collapse of matter through EH (before expansion)

How to test that every black hole contains a hidden universe? To boldly go where no one has gone before

Preferred direction NJP, Phys. Lett. B 694, 181 (2010) 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! 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)

1. Einstein-Cartan-Sciama-Kibble theory of gravity prevents singularities

ECSK gravity Riemann-Cartan spacetime metricity r ½ g ¹º = 0 connection ½ ¹º = { ½ ¹º} + C ½ ¹º 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)

ECSK gravity Curvature tensor = Riemann tensor + tensor quadratic in torsion + total derivative Stationarity of action under ±g ¹º -> Einstein equations R {} ¹º - R {} g ¹º /2 = k(t ¹º + U ¹º ) U ¹º = [C ½ ¹½C ¾ º¾ - C ½ ¹¾C ¾ º½ - (C ½¾ ½C ¾ - C ¾½ C ½¾ )g ¹º /2] /k Stationarity of action under ±C ¹º ½ -> Cartan equations S ½ ¹º - S ¹ ± ½ º + S º ± ½ ¹ = -ks ¹º ½ /2 S ¹ = S º ¹º Same proportionality constant k! Cartan equations are algebraic and linear ECSK torsion does not propagate (unlike metric/curvature)

ECSK gravity Field equations with full Ricci tensor R ¹º - Rg ¹º /2 = ¹º Canonical energy-momentum tensor Belinfante-Rosenfeld relation ¹º = T ¹º + r ½ ½(s ¹º + s ½ º¹ + s ½ ¹º) /2 r ½=r ½ - 2S ½ Conservation law for spin r ½ ½s ¹º = ( ¹º - º¹ ) Cyclic identities R ¾ ¹º½ = -2r ¹ S ¾ º½ + 4S ¾ ¹S º½ (¹, º, ½ cyclically permutated)

ECSK gravity Bianchi identities (¹, º, ½ cyclically permutated) r ¹ R ¾ º½ = 2R ¾ ¼¹S ¼ º½ Conservation law for energy and momentum D º ¹º = C º½¹ º½ + s º½¾ R º½¾¹ /2 D º =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 U ¹º» T ¹º For fermionic matter (quarks and leptons) ½ > 10 45 kg m -3 Nuclear matter in neutron stars ½» 10 17 kg m -3 Gravitational effects of torsion negligible even for neutron stars Torsion significant only in very early Universe and in black holes

Spin fluids Papapetrou (1951) multipole expansion -> equations of motion Matter in a small region in space with coordinates x ¹ (s) Motion of an extended body world tube Motion of the body as a whole wordline X ¹ (s) ±x = x - X ±x 0 = 0 u ¹ = dx ¹ /ds - spatial coordinates M ¹º½ = -u 0 s±x ¹ º½ (-g) 1/2 dv N ¹º½ = u 0 ss ¹º½ (-g) 1/2 dv Four-velocity Dimensions of the body small -> neglect higher-order (in ±x ¹ ) integrals and omit surface integrals

Spin fluids Conservation law for spin -> M ½¹º - M ½º¹ = N ¹º½ - N ¹º0 u ½ /u 0 Average fermionic matter as a continuum (fluid) Neglect M ½¹º -> s ¹º½ = s ¹º u ½ s ¹º u º = 0 Macroscopic spin tensor of a spin fluid Conservation law for energy and momentum -> ¹º = c ¹ u º - p(g ¹º - u ¹ u º ) ² = c ¹ u ¹ s 2 = s ¹º s ¹º /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)

Spin fluids -> Dynamical energy-momentum tensor for a spin fluid Energy density Pressure for random spin orientation F. W. Hehl, P. von der Heyde & G. D. Kerlick, Phys. Rev. D 10, 1066 (1974) Barotropic fluid s 2 / ² 2/(1+w) dn/n = d²/(²+p) p = w² n / ² 1/(1+w) Spin fluid of fermions with no spin polarization -> I. S. Nurgaliev & W. N. Ponomariev, Phys. Lett. B 130, 378 (1983)

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

Cosmology with torsion Friedman conservation -> ² / a -3(1+w) (like without spin) Spin-torsion contribution to energy density negative & dominant at small a gravitational repulsion o Independent of w o Consistent with the particle conservation n / a -3 o ² S decouples from ² spin fluid = perfect fluid + exotic fluid (with p = ² = -ks 2 /4 < 0) Very early universe: w = 1/3 (radiation) ² ¼ ² R» a -4 Total energy density

Big bounce from torsion Friedman equation NJP, Phys. Lett. B 694, 181 (2010) 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)

Cosmology with torsion NJP, Phys. Lett. B 694, 181 (2010) Velocity of antipodal point WMAP parameters of the Universe = 1.002 H 0-1 = 4.4 10 17 s R = 8.8 10-5 a 0 = 2.9 10 27 m Background neutrinos most abundant fermions in the Universe n = 5.6 10 7 m -3 for each type S = 8.6 10-70 (negative, extremely small in magnitude)

Cosmology with torsion GR S = 0 and a m = 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?

Cosmology with torsion ECSK S < 0 and a m > 0 Minimum 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) NJP, Phys. Lett. B 694, 181 (2010)

Cosmology with torsion ECSK S < 0 and a m > 0 1/v 2 a Maximum of v a Closed Universe causally connected at t < 0 remains causally connected through t = 0 until v a = c Universe contains N» (v a /c) 3 causally disconnected volumes S» -10-69 produces N ¼ 10 96 from a single causally connected region torsion solves horizon problem NJP, Phys. Lett. B 694, 181 (2010) Bounce cosmologies free of horizon problem

Cosmology with torsion

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)

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)

Black holes with torsion Friedman eqs. negligible, initially s 2 negligible As a decreases, k becomes less significant Stiff matter -> -> -> -> -> Number of fermions

Black holes with torsion Mass of BH Mass of neutron -> 2 solutions for a: At bounce Bounce due to torsion NJP, arxiv:1103.4192

Black holes with torsion After bounce, universe in BH expands Torsion becomes negligible, k = 1 becomes significant Expansion = time reversal of contraction? Universe reaches a 0 and contracts -> cyclic universe (between a min and a 0 ) 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)

Black holes with torsion Expansion to infinity if Cosmological models with k = 1, 0 H. Bondi, Cosmology (Cambridge Univ. Press, 1960) E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, 1976)

Black holes with torsion Universe in a black hole may oscillate until its mass exceeds M c Then it expands to infinity (Binary IC 10 X-1 24-33) Mass of our Universe

Black holes with torsion 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

Cosmological perturbations Observed scale-invariant spectrum of cosmological perturbations produced by thermal fluctuations in a collapsing black hole if Background Stiff matter w = 1 -> w r = 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)

2. Einstein-Cartan-Sciama-Kibble theory of gravity nonlinearizes Dirac equation

Spinor fields with torsion Dirac matrices Spinor representation of Lorentz group Tetrad Spinors Spinor connection Spin connection Covariant derivative of spinor Metricity -> -> Fock-Ivanenko coefficients (1929)

Hehl-Datta equation Dirac Lagrangian density ; covariant derivative with affine connection : with Christoffel symbols Spin density Totally antisymmetric Variation of C variation of ω Dirac spin pseudovector Cartan eqs. ->

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)

Dark energy from torsion 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

Dark energy from torsion Cosmological constant from QCD vacuum and ECKS torsion This value would agree with observations if (Zel dovich) 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 torsioninduced cosmological constant would agree with observations Simplest model predicting positive cosmological constant and ~ its energy scale does not use new fields

Dark energy from torsion

Matter-antimatter asymmetry Hehl-Datta equation NJP, Phys. Rev. D 83, 084033 (2011) Charge conjugate Adjoint spinor Satisfies Hehl-Datta equation with opposite charge and different sign for the cubic term! Energy levels (effective masses) Fermions Antifermions (-> NR) HD asymmetry significant when torsion is -> baryogenesis -> dark matter? Inverse normalization for spinor wave function

Matter-antimatter asymmetry

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

Summary 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. Thank you!

Universe in a black hole