Diffusion Processes on Nonholonomic Manifoldsand Stochastic Solutions for Einstein Spaces
Sergiu I. Vacaru
Department of ScienceUniversity Al. I. Cuza (UAIC), Iasi, Romania
Multidisciplinary Research Seminarat
INSTITUTE OF MATHEMATICS "O. MAYER"ROMANIAN ACADEMY, IASI BRANCH
October 4, 2010
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 1 / 17
Outline
1 Aims and Motivation
2 Stochastic Processes & Nonholonomic ManifoldsGeometry of N–anholonomic manifoldsh– and v–adapted Euclidian diffusionDiffusion on nonholonomic manifolds
3 Nonholonomic Diffusion in General RelativityThe special relativistic nonholonomic diffusionNonholonomic diffusion and gravitational interactions
4 Exact Stochastic Solutions in GravityThe Einstein eqs on nonholonomic manifoldsNonholonomic separation of Einstein eqsStochastic solutions with h∗
3,4 6= 0 and Υ2,4 6= 0
5 Summary & Conclusions
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 2 / 17
Aims and Motivation
Aims and Motivation
Aims:Theory of stochastic processes onnonholonomic Eucliedean and Riemannan manifoldsRelativistic nonholonomic stochastic differential equations anddiffusionExact solutions for stochastic Einstein spacetimes
Definition:A nonholonomic manifold is a pair (V,N )
Review and new results:S. Vacaru, in: IJGMMP, JMP, JGP, CQG, IJTP)Details in: S. Vacaru, arXiv: 1010.0647Diffusion on Curved (Super) Manifolds and Bundle spaces
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 3 / 17
Aims and Motivation
Related directions
Stochastic GravityEinstein–Langeven eqs with additional noise source.Fluctuations of quantum fields in curved spacetime.Semi–classical approximation and renormalizedenergy–momentum tensors.
Diffusion on Curved SpacesRolling Wiener processes on curved manifolds and diffusion inRiemann–Cartan–Weyl spaces.Laplace–Beltrami operators and diffusion.Stochastic processes in Lagrange–Finsler spaces,supersymmetric and higher order generalzitations.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 4 / 17
Stochastic Processes & Nonholonomic Manifolds Geometry of N–anholonomic manifolds
Stochastic Processes & Nonholonomic Manifolds
Geometry of N–anholonomic manifolds
N–connection splitting: T V = hV⊕vV, N = Nak , ex: V = TM
N–adapted d–frames:
eα + (ei = ∂i − Nai ∂a,eb = ∂b =
∂
∂yb ),
eβ + (ei = dx i ,ea = dya + Nai dx i).
Nonholonomic relations: [eα,eβ] = eαeβ − eβeα = wγαβ (u) eγ ,
d–metrics: g = gijdx i ⊗ dx j + hab(dya + Nak dxk )⊗(dyb + Nb
k dxk )
Signature (±,±,±,±), coordinates uα = (x i , ya), x i = (x1, x2)and ya =
(y3 = v , y4 = y
). Indices i , j , k , ... = 1,2 and
a,b, c, ... = 3,4 for (2 + 2)–splitting, when α, β, . . . = 1,2,3,4.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 5 / 17
Stochastic Processes & Nonholonomic Manifolds Geometry of N–anholonomic manifolds
N–adapted d–connections
Canonical d–connection & Levi–Civita connectionA d–connection D = (hD, vD) preserves under parallelism the h–v–spitting. Metric compatible: Dg = 0
Canonic. d–con. D : Γγαβ = (Lijk , L
abk , C
ijc , C
abc); T i
jk = 0, T abc = 0.
Lijk =
12
g ir (ekgjr + ejgkr − er gjk
),
Labk = eb(Na
k ) +12
hac(
ekhbc − hdc ebNdk − hdb ecNd
k
),
C ijc =
12
g ikecgjk , Cabc =
12
had (echbd + echcd − edhbc) .
Levi–Civita con. ∇ = Γγαβ and distortion: Γγαβ = Γγαβ + Z γαβ ,
all components defined by metric and N–connection.Cartan d–connection is canonical almost symplectic.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 6 / 17
Stochastic Processes & Nonholonomic Manifolds h– and v–adapted Euclidian diffusion
h– and v–adapted Euclidian diffusion
Wiener d–processes of dimension n + mLocally a couple of elementary (Wiener) h– and v–processesWα(τ) =
(W i(τ),Wa(τ)
), parameter τ (in particular, τ = t).
A random (stochastic) curve on V is lifted to a horizontal curve onthe frame of orthonormalized bundles O(V) related by transformseα′ = eα
α′(u)∂α = eαα′(u)eα, ∂α = ∂/∂uα = (∂i = ∂/∂x i , ∂a = ∂/∂ya).
Itô d–calculusA diffusion d–process by a couple of h- and v- SDE,dUα = σαα′(τ,U)δWα′
+ bα(τ,U)dτ, where U = (hU, vU) ∈ Rn+m
is a stochastic d–process with U(0) = u, for u = uβ = (x j , yc),Markovian process by Itô stochastic N-adapt. integral (equation)
Uατ = Uα
0 +τ∫0σαα′(ς,U)δWα′
ς +τ∫0
bα(ς,U)dς,
diffusion coeff. σαα′ , drift. coeff. bα.Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 7 / 17
Stochastic Processes & Nonholonomic Manifolds h– and v–adapted Euclidian diffusion
Itô d–processes and associated diffusion d–operator
Associated diffusion d–operatorStochastic N–adapted differential δf = Af with A = hA ⊕ vA,
A =ρ
2
n+m∑
α′=1
12σiα′(τ,U)σ
jα′(τ,U)
(eiej + ejei
)f
+σaα′(τ,U)σb
α′(τ,U)eaebf + bα(τ,U)eαf,
hA =ρ
2
n∑
α′=1
12σiα′(τ,U)σj
α′(τ,U)(eiej + ejei)f + bi(τ,U)ei f.
∀ stochastic N–adapted process Uατ ∈ Rn+m ∃ the probability
density function φ(τ,u), evolution on τ. ∀ f of U, expected valuef (τ,u) := uE [f (Uτ )] :=
∫U
f (u)φ(τ,u)δu1...δun+m.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 8 / 17
Stochastic Processes & Nonholonomic Manifolds h– and v–adapted Euclidian diffusion
Stratonovich d–calculus
Focker–Plank/forward Kolmogorov eq
∂τ f (τ,u) = Af (τ,u), f (0,u) = f (τ,u),
Stratonovich stochastic integral
More convenient for curved spaces, with ”∼ ” and ””,dUα = σαα′(τ,U) δWα′
+ bα(τ,U)dτ. Equivalence:
σαα′(τ,U) = σα
α′(τ,U), bα(τ,U) = bα(τ,U) − ρ2
n+m∑α′=1
σβα′(τ,U)eβσ
αα′(τ,U).
Diffus.oper.: A = ρ2
n+m∑α′=1
Lα′Lα′ + L0; Lα′ = σβα′(τ,u)eβ ,L0 = bα(τ,u)eβ
Associated N–adapted Focker–Plank eq
∂τφ(τ,u) = ρ2
n+m∑α′=1
eβσβα′(τ,u)eγ [σγ
α′(τ,u)φ(τ,u)] − eαbα(τ,u)φ(τ,u)
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 9 / 17
Stochastic Processes & Nonholonomic Manifolds Diffusion on nonholonomic manifolds
Diffusion on nonholonomic manifolds
N–adapted parallel transport r = (u,e) = (uα,eβ′
β) ∈ O(V)
δuα = eαα′(uβ)δγα′ and δeαα′(uµ) = −Γαβν(u
µ)eνα′(uµ)δuβ
Extending on O(V) the fundamental d–vector fieldsLα′ → OLα′ = eαα′e α − Γαβν(u
µ)eβα′eνβ′(∂/∂eββ′),
L0 → OL0 = Aα(τ,U)e α − ΓαβνAβ eνβ′(τ)(∂/∂eββ′)
Proj. f (r) = f (u,0), r = (uα,eβ′
β)),OA = ρ
2
n+m∑α′=1
OLα′OLα′ + OL0
Diffusion and Lapplace–Beltrami d–operators OAf (r) = VAf (u),
VA =ρ
2
∑
α′
eαα′e α(eβα′e β) + Aβ e β =
ρ
24 + Aβ e β
4 =12
gαβ[e αe β + e βe α +
(Γναβ + Γνβα
)e ν
]
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 10 / 17
Nonholonomic Diffusion in General Relativity The special relativistic nonholonomic diffusion
Nonholonomic Diffusion in General Relativity
The special relativistic nonholonomic diffusion
4–d Minkovski spacetime 31M , hyperbolic structure,
−(v1)2 + (v2)2 + (v3)2 + (v4)2 = −1,on T ( 3
1M), uα = (x i , ya = va); i , j , ... = 1, 2, 3, 4; a, b, c... = 5, 6, 7, 8.
hyperboloid metric and Christoffel connectionhab(vc) = δab − v av b/(v1)2, γa
bc(ve) = v ahbc
N–connection and d-metric, N = N ak (uα), u = uα = (x i , v a),
g = ηijdx i ⊗ dx j + hab(dv a + N ak dxk )⊗(dv b + N b
k dxk )
can. d–con. Γγαβ, Eα′ = Eαα′(u)∂α = Eαα′(u)Eα;
gα′β′ = [gi ′j ′ = ηi ′j ′ ,ha′b′ = δa′b′ ] = Eαα′Eβ
β′gαβ, r = uα,Eαα′
N–adapted relativistic stochastic eqs, frame bundle spaceδuα = Eαα′(τ) δW α′
+ Aα(τ)dτ, δEαα′(τ) = −Γαβν
(τ)Eνα′(τ) δuβ
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 11 / 17
Nonholonomic Diffusion in General Relativity Nonholonomic diffusion and gravitational interactions
Nonholonomic diffusion and gravitational interactions
N–adapt. relativ. diffusion & gravit. fields
Local hyperbolic massive particles: gµν(uα)vµvν = −1
N–adapted frames: V, ηα′β′ = gαβeαα′(u) eββ′(u),
Paral. transp. δvα′= −Γα
′
βγ′(uβ)vγ
′δuβ, δuβ = eββ′(uα)vβ
′δτ
N–ad. SDE diffusion δuα = eαα′(uβ)vα′δτ, force exBα = F a/m0,
δv α = E αα′(τ) δW α′ − Γαβγ′(u
γ)eβα′(uγ)vγ′vα′
δτ + ex Bαδτ,
δE αα′(τ) = −γα
βγ(v ϕ)E γ
α′ δv β ,
(A,L)–diff., coord. E r = uα = (x i , ya), v β ,E αα′, diff. F(V)A and
Laplace–Beltrami from F La′ = E αα′
∂∂v α − γαεγ(v
ϕ)E γα′E ε
β′
∂∂E α
β′
,
F L0 = eαα′(uβ)vα′eα − Γαβγ′(u
γ)eβα′(uγ)vγ′vα′ ∂
∂v α + exBα ∂∂v α −
γαεγ(vϕ)E γ
β′Bε ∂
∂E α
β′
; Bε = −Γεβγ′(uγ)eβα′(uγ)vγ
′vα
′+ ex Bε.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 12 / 17
Exact Stochastic Solutions in Gravity The Einstein eqs on nonholonomic manifolds
The Einstein eqs on nonholonomic manifolds
Two equivalent representations of Einstein eqs
Levi–Civita ∇, R βδ − 12gβδR = κTβδ,
canonic d–connect D, R βδ −12
gβδ sR = Υβδ,
Lcaj = ea(Nc
j ), C ijb = 0, Ωa
ji = 0.
Ansatz for solutions:
ηg = ηi (xk , v) gi(xk , v)dx i ⊗ dx i + ηa(xk , v) ha(xk , v)ea⊗ea,
e3 = dv + η3i (xk , v) wi(xk , v)dx i , e4 = dy4 + η4
i (xk , v) ni (xk , v)dx i
gij = diag[gi = ηigi ] and hab = diag[ha = ηa
ha] andN3
k = wi = η3i
wi and N4k = ni = η4
ini ; Gravit.’polarizations’ ηα
and ηai ,
g = [ gi ,ha,
Nak ] → ηg = [ gi ,ha,Na
k ], functions ofnecessary smooth class and/or any random (stochastic) variables.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 13 / 17
Exact Stochastic Solutions in Gravity Nonholonomic separation of Einstein eqs
Nonholonomic separation of Einstein eqs
MAGIC d–connection SPLITTING for one Killing ansatz:ηg = gi(xk )dx i ⊗ dx i + h3(xk , t)e3⊗e3 + h4(xk , t)e4⊗e4,
e3 = dt + wi(xk , t)dx i ,e4 = dy4 + ni(xk , t)dx i
a• = ∂a/∂x1, a′ = ∂a/∂x2, a∗ = ∂a/∂v ; v = t ; τ via δτ =√|h3|δt .
R11 = R2
2 =−1
2g1g2[g••
2 − g•
1 g•
2
2g1− (g•
2 )2
2g2+ g′′
1 − g′
1g′
2
2g2− (g′
1)2
2g1] = −Υ4(xk ),
R33 = R4
4 = − 12h3h4
[h∗∗
4 − (h∗
4)2
2h4− h∗
3h∗
4
2h3] = −Υ2(xk , v),
R3k =wk
2h4[h∗∗
4 − (h∗
4)2
2h4− h∗
3h∗
4
2h3] +
h∗
4
4h4
(∂k h3
h3+∂k h4
h4
)− ∂k h∗
4
2h4= 0,
R4k =h4
2h3n∗∗
k +
(h4
h3h∗
3 − 32
h∗
4
)n∗
k2h3
= 0,
w∗
i = ei ln |h4|, ek wi = eiwk , n∗
i = 0, ∂ink = ∂k ni
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 14 / 17
Exact Stochastic Solutions in Gravity Stochastic solutions with h∗
3,4 6= 0 and Υ2,4 6= 0
Stochastic solutions with h∗3,4 6= 0 and Υ2,4 6= 0
Formal integration, sure solutions
h–metric: g1 = g2 = eψ(xk ), ψ + ψ′′ = 2Υ4(xk )
coefficients:φ = ln | h∗
4√|h3h4|
|, αi = h∗4∂iφ, β = h∗
4 φ∗, γ =
(ln |h4|3/2/|h3|
)∗
N–connection eqs: βwi + αi = 0, n∗∗i + γn∗
i = 0v–metric, if h∗
4 6= 0; Υ2 6= 0, we get φ∗ 6= 0. ∀φ = φ(x i , t) 6= const isa solution generating function, h∗
4 = 2h3h4Υ2(x i , t)/φ∗.
solution: h3 = ± |φ∗(x i ,t)|Υ2
, h4 = 0h4(xk ) ± 2∫ (exp[2 φ(xk ,t)])∗
Υ2dt ,
wi = −∂iφ/φ∗, ni = 1nk
(x i) + 2nk
(x i) ∫
[h3/(√|h4|)3]dt ,
integration functions 0h4(xk ), 1nk(x i) and 2nk
(x i)
Υi = λ, λ→ hλ(xk ) = Υ4(xk ) and λ→ vλ(xk , t) = Υ2(xk , t).
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 15 / 17
Exact Stochastic Solutions in Gravity Stochastic solutions with h∗
3,4 6= 0 and Υ2,4 6= 0
Stochastic solutions with h∗3,4 6= 0 and Υ2,4 6= 0
Metrics and sources with h–diffusion
Random gener. funct. φ(x k , t) → φ(xk , t) = φ(xk , t) +$φ(xk , t),where φ(xk , t) ∼ f (τ,u) = f (τ, x i) diffusion process from h–spaceto v–space; only h–operators and fix Aβ = 0 in VA, τ → t ,v = v0 = const and 4 is computed 4 = h4 for gij = δijeψ(xk ).
The generalized Kolmogorov backward equation∂τ f (τ, x i) = h4f ((τ, x i ) = h4f ((τ, x i), f (0, x i) = f (x i).
The generalized Fokker–Planck eq ∂τz = ρ2h4z,
z = z(τ, 1x i ; 0, 2x i) is the transition probability with the initialcondition z(0, 1x i ; 0, 2x i) = δ( 1x i − 2x i) for any two points1x i , 2x i∈ V and adequate boundary conditions at infinity.Solutions induced by random sources Υ2; similar to stochasticgravity (Einstein–Langeven eq with additional sources due to thenoise kernel).
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 16 / 17
Summary & Conclusions
Summary & Conclusions
We developed the theory of stochastic processed onnonholonomic Euclidean and Riemannian manifolds.Relativistic models of nonholonomic diffusion.Off–diagonal solutions for stochastic Einstein spacetimes.Multidisciplinary character of research: 1) Geometric Methods(nonholonomic, Riemann–Finsler, almost Kähler); 2) NonlinearEvolution/Field Equations (PDE, SDE, evolution eqs); 3)Probability and Stochastics (diffusion, anisotropic kinetics,statistics, thermodynamics); 4) Geometric quantization.
Outlook (recently developed, under elaboration):Gravity and quantum physics, geometric mechanics; variousapplications in modern cosmology and astrophysics, geometricmechanics etc. Generic nonlinear solutions, stochastic evolution,fractional derivatives, solitonics, singularities, memory etc.Diffusion and Porous Media with Self–Organized Criticality in RicciFlow Evolution of Einstein and Finsler Spaces.
Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 17 / 17