Principles of Einstein Finsler Gravity

Transcription

1 Principles of Department of Science University Al. I. Cuza (UAIC), Iaşi, Romania Review Lecture University of Granada Department of Geometry and Topology September 9, 2010 Principles of

2 Differential & Finsler Geometry, Iaşi, Romania Research group "Geometry & Applications in Physics" 100 years traditions on math & applications; supervision/ collaborations by/with D. Hilbert, T. Levi Civita and E. Cartan of PhD of prominent members of Romanian Academy. E. Cartan visit at Iaşi in 1931 induced 80 years of research on Finsler/integral geometry etc, "isolation" after 1944; "Japanese Finsler geometry orientation" after 1968 Alexandru Myller ( ), PhD 1906: D. Hilbert (chair/adviser) and F. Klein, H. Minkowski (commission). Gheorghe Vrǎnceanu ( ), PhD-1924, from Levi Civita, commission head: Volterra; , Rockefeller scholarship for France, E. Cartan, and USA at Harvard & Princeton (Morse, Birkhoff, Veblen) Principles of

3 Differential & Finsler Geometry, Iaşi, Romania (prolongation) Mendel Haimovici ( ); PhD Levi Civita. Radu Miron ( ); 28 monogr., 240 rev. MathSciNet Lagrange Finsler, Hamilton Cartan & higher order, applications to mechanics and relativity etc. Iaşi team and "Romanian Finsler diaspora": M. Anastasiei, D. Bucǎtaru and M. Crâsmâreanu (Iaşi); A.Bejancu(Kuwait);D.Hrimiuc(Canada);V.Sabau(Japan); S. Vacaru (Cernâuţi/Chernivtsy, Chişinâu/ Kishinev, Tomsk, Dubna, Moscow, Kyiv, Bucharest Mǎgurele, Lisbon, Madrid, Toronto, Iaşi) Principles of

4 Outline 1 Nonlinear dispersions from QG and LV Nonholonomic Ricci / Finsler flows Exact off diagonal solutions and cosmology 2 Gravitational field eqs in EFG 3 Nonholonomic Perelman s functionals Finsler branes & cosmological solutions 4 Principles of

5 Nonlinear dispersions Goals Finsler modifications of GR derived for QG theories; Geometric models for quantum contributions and LV Nonholonomic evolutions of (pseudo) Riemannian geometries into Lagrange Finsler ones Canonical models for Einstein Finsler gravity (EFG); principles and axioms Physical implications in EFG: Finsler branes, locally anisotropic cosmology & astrophysics Reviews and new results: S. Vacaru (in CQG, PLB, IJGMMP, JMP, JGP, IJTP) arxiv: ; ; ; ; Principles of

6 Nonlinear dispersions Motivation: nonlinear disps; QG & LV, cosmology 1. Deforms in Minkovski s-t: E 2 = p 2 c 2 + m0 2c4 + ϕ(e, p; µ; M P ) E t, p î, ω = φ xî t kî = φ, ω 2 = cs 2 k 2 + c 2 h s ( xî 2m 0 c s ) 2 k effective c s, (x 1 = ct, x 2, x 3, x 4 ); î, ĵ... = 2, 3, 4; ω 2 = c 2 [gîĵ k îk ĵ] 2 (1 qî1 î 2...î2r y î1...y î2r /r[gîĵ k îk ĵ] 2r ) light velocity in "media/ether" c 2 = gîĵ (x i )y îy ĵ/τ 2 ˇF 2 (y ĵ)/τ 2 fundamental Finsler function F(x i, βy j ) = βf(x i, y j ), β > 0, ds 2 = F 2 (cdt) 2 + gîĵ (x k )y îy ĵ[1 + 1 r qî1 î 2...î2r (xk )y î1...y î2r (gîĵ (xk )y î y ĵ ) r ] + O(q 2 ) F 2 y i y j Finsler "metrics", velocities on TV, F g ij (x i, y j ) = Nonholonomic Ricci flows and mutual transforms of Riemann Finsler geometries. 3. Exact solutions & modified cosmology with generic off diagonal metrics and local anisotropy. Principles of

7 (EFG) Statement I: A (pseudo) Finsler metric, F g ij (x k, y a ), DOES NOT define completely a geometric model (not Riemannian!) Statement II: A model of Finsler geometry is defined on TV by THREE fundamental geometric objects induced by F(x, y) : 1 N connection, Ni a (x, y), splitting F N : TTV = htv vtv canonically, Euler Lagrange for L = F 2 are semi sprays, 2 d connection, N adapted linear connect. F D = (hd, vd), preferred/ canonically induced by F g ij and N a i 3 d metric, F g = hg vg 2 classes: a) nonmetricity, F Q := F D F g, Chern d conn., Ch D b) metricity, F Q = 0, Cartan d conn., Cart D Levi Civita F is NOT adapted to nonholonomic F N. induced by F g : torsion F T, and/or F Q(not Riemann-Cartan) Principles of

8 Spacetime as a nonholonomic manifold/ bundle V := (V, D) (Vrǎnceanu, 1926), or TM, with a non integrable distribution D. Geometric data: Finsler (F : N, D, g) and Riemannian (, g) N anholonomic frames: e ν = (e i = i N a i a, e a = a ) Sasaki d metric: F g = F g ij (u)dx i dx j + F g ab (u) c e a c e b, for c e a = dy a + c N a i (u) dx i. For D, standard Riemannian, Ricci, Einstein d tensors; h-/v splitting. N adapted coef.: Cart D = D = (h D, v D) = { Γ α γτ = ( L i jk, Ca bc )}, L i jk = 1 2 F g ir (e k F g jr + e j F g kr e r F g jk ), C a bc = 1 2 F g ad (e c F g bd + e c F g cd e d F g bc ). Theorem: Equivalent (pseudo) Finsler & Riemannian theories if g D = g + g Z, distortion determined by g = F g. Principles of

9 Analogous Gravity and Lagrange Finsler Geometry Unified formalism for Riemann Cartan, Finsler spaces and geometric mechanics. Alternative works on analogous gravity. "Pseudo" (relativistic) geometric mechanics. ( + ++), local pseudo Euclidian with x 1 = i x 1, i 2 = 1. Lagrange spaces: "Mechanical" modelling of gravitational interactions on semi Riemannian manifolds V, or E = TM, fundamental/generating Lagrange function L(x, y) : Canonical N connection L N i j (x, y) = L G i y j, L g ab = 1 2 L 2 y a y b, det g ab 0. L G i = 1 L g ij 2 L ( 4 y i x k y k L x i ) nonlinear geodesic equations for x i (τ), y i = dxi dτ Principles of

10 Analogous Gravity and Lagrange Finsler Geometry Finsler/Lagrange modelling Theorem: Any Lagrange (Finsler) geometry can be modelled equivalently as a N anholonomic Riemann manifold V, and inversely, with canonically induced by L (F) d metric structure L g = L g ij (u) e i e j + L g ab (u) L e a L e b e i = dx i, L e b = dy b + L N b j (u)dx j ; (not) N adapted connections, L D; equivalently, L. Principles of

11 Analogous Gravity and Lagrange Finsler Geometry Almost Kähler variables/models in, classical and quantum gravity, nonholonomic Ricci flows Almost complex structure determined by the canonical N connection: J(e i ) = e i and J(e i ) = e i L(x, y) induces a canonical 1 form L ω = 1 2 e i y i L g canonical 2 f. L θ(x, Y) L g(jx, Y) = L g ij (x, y)e i e i Almost Kähler models of Lagrange Finsler/Einstein spaces with θ D = D θ DX L g = 0 and θ DX J = 0. Important for deformation quantization (Fedosov) of Einstein and Lagrange Finsler/Hamilton Cartan gravity. L Principles of

12 Analogous Gravity and Lagrange Finsler Geometry Remarks: 1 a unique geometric formalism of nonholonomic deformations and analogous modeling of gravitational, Einstein and Finsler and "pseudo" mechanical models. 2 Key questions: for what types of connections we postulate the field equations and what class of nonholonomic constraints is involved? 3 Different Finsler d connections (for instance) Chern s one Ch Γ γ αβ = ( Li jk, Ĉa bc = 0 ), Ch D F g 0, but Ch T = 0. 4 Nonmetricity is not compatible with standard physics: a. Definition of spinors; b. Conservation laws; c. Supersymmetric / noncommutative generalizations of Finsler like spaces; d. Exact solutions? Principles of

13 Principles: Similarly to GR with g on V construct EFG: with g F g, N F N and Cart D on TV, or V. 1 Generalized equivalence principle: Ideas on Free Fall and Universality of Gravitational Redshift for Cart D. 2 Generalized Mach principle: quantum energy/motion encoded via (N, g, D) for spacetime ether with y a. 3 Principle of general covariance extended on V, or TV, with "mixing of Finsler parametrizations". 4 Motion eqs and conservation laws: Nonholonomc Bianchi identities for F D; i T ij = 0 D α Υ αβ 0. 5 Einstein Finsler gravitational field eqs for F D. 6 Axiomatics: Constructive axiomatic appr. (Ehlers-Pirani Schild, EPS axioms), paradigm "Lorentzian 4 manifold" in GR; nonholon. tangent bundle on "L..." for EFG. Principles of

14 Gravitational field eqs in EFG D, Einstein eqs: E αβ = Υ αβ, h /v components, for R ai = R b aib and R ia = R k ikb : R ij 1 2 (R + S)g ij = Υ ij, R ab 1 2 (R + S)h ab = Υ ab, R ai = Υ ai, R ia = Υ ia, Remark: For Cart D, general off diagonal solutions for EFG, restrictions to GR, g = g αβ (u) du α du β, [ ] gij + N g αβ = i a Nj b h ab Nj e h ae Ni e, where Ni a A a bi h be h (x)y b ab Claim: Compactification/trapping/warping mechanism on velocity/momenta for a "new" QG and LV phenomenology. Principles of

15 Gravitational field eqs in EFG Levi Civita and canonical d connection Levi Civita connection = { g Γ γ αβ }, T α βγ = 0 and g = 0 Canonical d connection D = { g Γγαβ } Dg = 0 and h T(hX, hy ) = 0, v T(vX, vy ) = 0, g Γ γ αβ = g Γγ αβ + g Z γ αβ Distortion g Z γ αβ defined by g, Γ γ αβ ( Li = jk, L ) a bk, Ĉi jc, Ĉa bc, L i jk = 1 2 gir (e k g jr + e j g kr e rg jk ), La bk = e b (N a k ) hac ( e k h bc h dc e b N d k h db e cn d k Ĉ i jc = 1 2 gik e cg jk, Ĉ a bc = 1 2 had (e ch bd + e ch cd e d h bc ). Nontrivial d torsion T γ αβ : T i ja = Ĉi jb, T a ji = Ω a ji, T c aj = L c aj e a(n c If T γ αβ = 0, g Γ γ αβ = g Γγαβ even D ), j ) Principles of

16 General Solutions in Gravity Einstein eqs for the canonical d connection The Einstein equations for a d metric g βδ, also in GR, can be rewritten equivalently using D, R βδ 1 2 g βδ s R = Υ βδ, L c aj = e a (N c j ), Ĉ i jb = 0, Ωa ji = 0, R βδ for Γ γ αβ, s R = g βδ R βδ and Υ βδ κt βδ for D. (2+2) splitting, (u α = (x k, t, y 4 ), ansatz with Killing / y 4, K g = g 1 (x k )dx 1 dx 1 + g 2 (x k )dx 2 dx 2 +h 3 (x k, t)e 3 e 3 + h 4 (x k, t)e 4 e 4 for N 3 i = w i (x k, t), N 4 i = n i (x k, t), e 3 = dt + w i (x k, t)dx i, e 4 = dy 4 + n i (x k, t)dx i Principles of

17 General Solutions in Gravity Theorem 1 (Separation of Eqs) The Einstein eqs for ansatz K g and D are: R 1 1 = R 2 2 = 1 [g 2 g 1 g 2 (g 2 ) 2 + g 1 g 1g 2 (g 1) 2 ] = Υ 4 (x k ) 2g 1 g 2 2g 1 2g 2 2g 2 2g 1 [ ] R 3 3 = R 4 4 = 1 h4 (h 4) 2 h 3h4 = Υ 2 (x k, t), 2h 3 h 4 2h 4 2h 3 R 3k = w k 2h 4 [ R 4k = h 4 2h 3 n k + h4 (h 4) 2 h 3h4 2h 4 2h 3 ( h4 h 3 h h 4 ] ( + h 4 k h 3 + ) kh 4 kh4 = 0, 4h 4 h 3 h 4 2h 4 ) n k 2h 3 = 0, where a = a/ x 1, a = a/ x 2, a = a/ t. Principles of

18 Integration of (non)holonomic Einstein eq Theorem 2 (Integral Varieties) ψ + ψ = 2Υ 4 (x k ) h 4 = 2h 3 h 4 Υ 2 (x i, t)/φ βw i + α i = 0 ni + γni = 0 ( α i = h4 i φ, β = h4 φ, φ = ln h 4, γ = h 3 h 4 ln h 4 3/2 h 3 General solution: g 1 = g 2 = e ψ(xk), h 4 = 0 h 4 (x k ) ± 2 h 3 = ± 1 4 [ h 4 (x i, t) ] 2 exp[ 2 φ(x k, t)] ), h 3,4 0, Υ 2,4 0, w i = i φ/φ, n k = 1 n k ( x i) + 2 n k (x i) [h 3 /( h 4 ) 3 ]dt LC conditions: w i = e i ln h 4, e k w i = e i w k, n i = 0, i n k = k n i Principles of (exp[2 φ(x k, t)]) dt, Υ 2

19 Integration of (non)holonomic Einstein eq General Solutions Dependence on 4th coordinate via ω 2 (x j, t, y) g = g i (x k )dx i dx i + ω 2 (x j, t, y)h a (x k, t)e a e a, e 3 = dy 3 + w i (x k, t)dx i, e 4 = dy 4 + n i (x k, t)dx i, e k ω = k ω + w k ω + n k ω/ y = 0, ω 2 = 1 results in solutions with Killing symmetry. N deformations and exact solutions Polarizations η α and η a i, nonholonomic deformations, g = [ g i, h a, N a k ] η g = [ g i, h a, N a k ]. Deformations of fundamental geometric structures: η g = η i (x k, t) g i (x k, t)dx i dx i + η a (x k, t) h a (x k, t)e a e a, e 3 = dt+η 3 i (x k, t) w i (x k, t)dx i, e 4 = dy 4 +η 4 i (x k, t) n i (x k, t)dx i. Principles of

20 Integration of (non)holonomic Einstein eq Remarks "Almost" any solution of Einstein eqs, g α β, via eα = eα α(x i, y a )e α, g αβ = e α αe β β g α β, expressed g αβ = g 1 + ω 2 (w1 2 h 3 + ω 2 (n1 2 h 4 ) ω 2 (w 1 w 2 h 3 + n 1 n 2 h 4 ) ω 2 w 1 h 3 ω 2 n 1 h 4 ω 2 (w 1 w 2 h 3 + n 1 n 2 h 4 ) g 2 + ω 2 (w2 2 h 3 + n2 2 h 4 ) ω 2 w 2 h 3 ω 2 n 2 h 4 ω 2 w 1 h 3 ω 2 w 2 h 3 h 3 0 ω 2 n 1 h 4 ω 2 n 2 h 4 0 h 4 Concept of general solutions for systems of nonlinear partial differential eqs? Topology, symmetries etc. Arbitrariness, uniqueness, sources? Complex/supersymmetric/ nonholonomic / quantum distributions applications to modern gravity and physics Higher dimensions - "shell by shell". Almost Kähler structures etc, generalized (algebroid etc) symmetries. Nontrivial topology etc Exact solutions in astrophysics, cosmology: black ellipsoids/toruses, wormholes, solitons, Dirac waves, pp waves etc Principles of

21 Nonholonomic Ricci Flows Constrained Ricci Evolution Nonholonomic Perelman s functionals Finsler branes & cosmological solutions (Non) commutative/ supersymmetric Lagrange Finsler, almost Kähler and nonholonomic Ricci flows 1 Families regular Lagrangians L(u, χ) = L(x, y, χ) on TM, or V 2 for instance, g αβ as solutions of Einstein eqs R αβ = λ g αβ 3 g αβ (χ) as solutions of the Ricci flow eqs g αβ = 2R χ αβ real parameter χ, Ricci tensor R αβ for or any metric compatible connection D, Dg = 0, but torsion g,d T 0 4 N adapted evolution: ] χ g ii = 2 [ Rii λg ii χ haa = ) ( Raa 2 λh aa, R αβ = 0, for α β h cc χ (Nc i ) 2, Principles of

22 Ricci Lagrange/ Finsler Evolution (Semi)sprays and N connections: Nonholonomic Perelman s functionals Finsler branes & cosmological solutions dy a dς + 2Ga (x, y) = 0, curve x i (ς), 0 ς ς 0, when y i = dx i /dς. Regular Lagrangian: L(x, y) = L(x i, y a ), L g ij = 1 2 N a i ( = Ga y i, 4G j = L g ij 2 L y i x k y k L x i 2 L y i y j ), L g = L g ij (x, y) [e i e j + e i e j] e α = [e i = dx i, e a = dy a + N a i (x, y)dx i ]. Principles of

23 Ricci Lagrange/ Finsler Evolution Hamilton s evolution eqs: Nonholonomic Perelman s functionals Finsler branes & cosmological solutions g αβ (χ) χ = 2 R αβ (χ) for a set of (semi) Riemannian metrics g αβ (χ), real parameter χ, Ricci tensors R αβ (χ) for the Levi Civita connection. Perelman s functionals for flows of Riemannian metrics ( F(L, f ) = R + f 2) e f dv, W(L, f, τ) = V V [ ] τ ( R + f ) 2 + f 2n µ dv, volume form of L g, dv, integration over compact V, function f for gradient flows with different measures, scalar curvature for, R. For τ > 0, V µdv = 1, µ = (4πτ) n e f. Principles of

24 Nonholonomic Perelman s functionals Finsler branes & cosmological solutions Claim: For Lagrange spaces, Perelman s functionals for D, F(L, f ), Ŵ(L, f, τ) are F = Ŵ = V V ( R + S + D f 2) e f dv, [ ( τ R + S + ) ] h D f + v 2 D f + f 2n µ dv, R and S are h- and v components of curvature scalar of D = ( h D, v D), D f 2 = h D f 2 + v D f 2, f satisfies V µdv = 1 for µ = (4πτ) n e f and τ > 0. Principles of

25 Nonholonomic Perelman s functionals Finsler branes & cosmological solutions Proofs for N adapted evolution eqs Theorem: If a Lagrange (Finsler) metric L g(χ) and functions f (χ) and τ(χ) evolve for τ χ = 1 and constant V (4π τ) n e f dv as solutions of g ij χ = 2 R g ij, ab χ = 2 R ab, f χ = f + D f 2 R S + ṋ τ, then L χŵ( g(χ), f (χ), τ(χ)) = 2 + R ab + D a D b f 1 2 τ g ab 2 ](4π τ) n e f dv. V τ[ R ij + D i D j f 1 2 τ g ij 2 Principles of

26 Nonholonomic Perelman s functionals Finsler branes & cosmological solutions Corollary: The evolution, for all τ [0, τ 0 ), of N adapted frames e α (τ) = eα α (τ, u) α is defined by [ ] eα α (τ, u) = e i i (τ, u) Ni b (τ, u) e a b (τ, u) 0 ea a, (τ, u) with L g ij (τ) = e i i (τ, u) e j j (τ, u)η ij subjected to eqs τ e α α = L g αβ R βγ e γ α, τ e α α = L g αβ Rβγ e γ α, for the Levi-Civita connection; for the canonical d connection. Principles of

27 Nonholonomic Perelman s functionals Finsler branes & cosmological solutions Finsler branes & cosmological solutions Nonholon. trapping solutions (cosmology, with h 3 (x i, y 3 = t)) : g = g 1 dx 1 dx 1 + g 2 dx 2 dx 2 + h 3 e 3 e 3 + h 4 e 4 e 4 + (l P ) 2 h φ 2 [ q h 5 e 5 e 5 + q h 6 e 6 e 6 + q h 7 e 7 e 7 + q h 8 e 8 e 8 ] e 3 = dy 3 + w i dx i, e 4 = dy 4 + n i dx i, e 5 = dy w i dx i, e 6 = dy n i dx i, e 7 = dy w i dx i, e 8 = dy n i dx i. φ 2 (y 5 ) = 3ɛ2 + a(y 5 ) 2 3ɛ 2 + (y 5 ) 2 and l P h(y 5 9ɛ 4 ) = [ 3ɛ 2 + (y 5 ) 2] 2, N connection coefficients determined by sources h Λ(x i ) = Υ 4 + Υ 6 + Υ 8, v Λ(x i, v) = Υ 2 + Υ 6 + Υ 8, 5 Λ(x i, y 5 ) = Υ 2 + Υ 4 + Υ 8, 7 Λ(x i, y 5, y 7 ) = Υ 2 + Υ 4 + Υ 6. Principles of

28 Almost all models of QG with nonlinear dispersions can be geometrized as certain Finsler spacetimes. Natural/ Canonical Principles for metric compatible EFG generalizing the GR on TV, Cart D. Finsler branes, trapping: "new" QG/ LV phenomenology. Outlook (recently developed, under elaboration): EFG is almost completely integrable, can be quantized as almost Kähler Fedosov/ A brane geometries, and renormalizable for bi connection/gauge gravity models. Finsler for black holes (ellipsoids, toruses, holes, wormholes, solitons); anisotropic cosmological models (off diagonal inflation, dark energy/matter etc). Noncommutative/ Ricci Finsler flows, emergent (non) commutative Lagrange Finsler analogous gravity and quantization, Clifford Finsler algebroids etc. Principles of