on general solution in topological massive gravity and ricci flows
DESCRIPTION
Seminar at Department of Mathematics (host prof. Metin Gurses) Bilkent University, July 22, 2010; Ankara, TurkeyTRANSCRIPT
On General Solutions in Topological Massive Gravity and Ricci Flows
On General Solutions in TopologicalMassive Gravity and Ricci Flows
Research Seminar
Sergiu I. Vacaru
Science DepartmentUniversity Al. I. Cuza, Iasi, Romania
Seminar at Department of Mathematics(host prof. Metin Gurses) Bilkent University
July 22, 2010; Ankara, Turkey
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On General Solutions in Topological Massive Gravity and Ricci Flows
Subjects
I. Motivations: Topological Massive Gravity andNonholonomic Structures
II. Nonholonomic Ricci flow evolution of topologicalmassive gravity solutions
III. Nonholonomic Distributions and Massive Gravity
IV. General Solutions in Massive Gravity
a. Separation of Einstein eqs fordistinguished connections
b. Integration of (non) holonomic Einstein eqs
V. Extracting topological massive black ellipsoids
VI. Conclusions and Perspectives
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On General Solutions in Topological Massive Gravity and Ricci Flows
Motivations:
Concept/ coefficient form of N–connections: E. Cartan, A. Kawaguchi, C. Ehresmann ....
1. M. Gurses, Killing Vector Fields in Three Dimensions: A Method to Solve MassiveGravity Field Equations, arXiv: 1001.10392. A. Ulas Ozgur Kisised, O. Sarioglu and B. Tekin, Cotton Flow, arXiv: 0803.1603v23. David D. K. Chow, C. N. Pope and E. Sezgin, Classification of solutions in topologicallymassive gravity, CQG 27 (2010) 1050014. David D. K. Chow, C. N. Pope and E. Sezgin, Kundt spacetimes as solutions of topolog-ically massive gravity, CQG 27 (2010) 105002
5. S. Vacaru, On General Solutions in Einstein Gravity, accepted: Int. J. Geom. Meth.Mod. Phys. (IJGMMP) 8 N1 (2011); arXiv: 0909.3949v16. S. Vacaru, On General Solutions in Einstein and High Dimensional Gravity, Int. J. Theor.Phys. 49 (2010) 884-913; arXiv: 0909.3949v47. S. Vacaru, Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics, ActaApplicandae Mathematicae [ACAP] 110 (2010) 73-107; arXiv: 0810.07078. S. Vacaru, Parametric Nonholonomic Frame Transforms and Exact Solutions in Gravity,Int. J. Geom. Meth. Mod. Phys. (IJGMMP) 4 (2007) 1285-1334; arXiv: 0704.3986
Topological Massive Gravity and Nonholonomic Strs
TMG equations found by Deser, Jackiw and Templeton(DJT). For nontrivial cosmological constant,
Eαβ + µ−1Cα
β = λδαβ
The Bergshoeff–Hohm–Townsend (BHT) theorypreserving parity (more sophisticate) can be also solved”almost” in general form.
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On General Solutions in Topological Massive Gravity and Ricci Flows
Nonholonomic Structures and Exact Solutions inTopological Massive Gravity
3-d Levi–Civita topological massive configs
Eαβ
+ µ−1C αβ
= λδαβ,
Eαβ
= Rαβ− 1
2δα
β[3]R,
C αβ =1
2√
g
(εατ γ∇τR
βγ + εβτ γ∇τR
αγ
)
for the Levi–Civita connection ∇ = pΓγ
αβ.
For metrics g,
ds2 = ± (dx1
)2+ gαβ(x2, y3, y4)duαduβ
= gαβ(uα)duαduβ
coordinates uα = (xi, ya) = (x1, x2, y3, y4) = (x1, uα);
i, j, ...k = 1, 2; a, b... = 3, 4; α, β = 2, 3, 4
4-d nonholonomic gravitational configurations
∇g = 0, torsion ∇T = 0.
Another class of linear connections gD = g Γγαβ,
gD = g∇ + gZ
with distortion/torsion gZ completely determined by g,when gDg = 0.
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On General Solutions in Topological Massive Gravity and Ricci Flows
Consider metrics g of type
ds2 = gi(xk)(dxi)2 + ω2(xk, yb)ha(x
k, y3)(dya + Na
j (xk, y3))2
(1)
= g1(x1, x2)(dx1)2 + g2(x
1, x2)(dx2)2 + ω2(x1, x2, y3, y4)×[h3(x
1, x2, y3)(dy3 + w1(x
1, x2, y3)dx1 + w2(x1, x2, y3)dx2)2
+h4(x1, x2, y3)
(dy4 + n1(x
1, x2, y3)dx1 + n2(x1, x2, y3)dx2)2
]
for N 3i = wi and N 4
i = ni.
Remark: Reduction 4-d →3-d
Source: Υαβ =
[vΛ(xk, y3)δi
k,hΛ(xk)δa
v
]
Theorem 1: The Einstein equations for gD,
gRαβ − 1
2gαβ
gsR = Υα
β
for the metrics g (1) transform into an exactly integrablesystem of partial differential equations
gR11 = gR2
2 = hΛ(xk) (2)gR3
3 = gR44 = vΛ(xk, y3)
Einstein gravity solutions with g∇ are extracted byconstraining gZ = 0.
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On General Solutions in Topological Massive Gravity and Ricci Flows
Effective 4–d topological theory
Eαβ + µ−1Cα
β = λδαβ, (3)
Gαβ = Rα
β − 1
2δα
β sR,
Cαβ =1
2√
g
(εατγDτR
βγ + εβτγDτR
αγ
)
Lemma: Cαβ(uγ) = ( hC(uγ) δi
j,vC (uγ)δa
b)
Theorem 2: The solutions of (3) can be extracted from(2) for
hΛ(xk) = λ− µ−1 hC(xk),vΛ(xk) = λ− µ−1 vC(xk).
Theorem: Nonholonomic Cotton flows are determined bysolutions:
∂χgij(χ) =
(1− 1
4δl
l
)εikj√gh
Dk( hΛ(χ)) = Cij(χ),
∂χhab(χ) =
(1− 1
4δe
e
)εacb√hg
Dc( vΛ(χ)) = Cab(χ)
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On General Solutions in Topological Massive Gravity and Ricci Flows
II. Black Ellipses in TMG
Nontrivial effective cosmological source
Consider a diagonal 4-d metricελg = −dξ ⊗ dξ − r2(ξ) dϑ⊗ dϑ−
r2(ξ) sin2 ϑ dϕ⊗ dϕ + λ$2(ξ) dt⊗ dt,
where coordinates and nontrivial metric coefficients
x1 = ξ, x2 = ϑ, y3 = ϕ, y4 = t,
g1 = −1, g2 = −r2(ξ), h3 = −r2(ξ) sin2 ϑ, h4 = λ$2(ξ)
forξ =
∫dr
∣∣∣∣1−2µ
r+ ε
(1
r2+
λ
34κ
2 r2
)∣∣∣∣1/2
,
λ$2(r) = 1− 2µ
r+ ε
(1
r2− λ
34κ
2 r2
),
where 4κ2 = 1/M 2
∗ is the 4–dimensional Newton’sconstant, λ = ε λ is a positive cosmological constant andµ1 is the ADM mass (review on Schwarzschild–de Sitterblack holes in (4 + n1)–dimensions, for n1 = 1, 2, ...)
For ε → 0 and µ taken to be a point mass (for astationary locally anisotropic model, µ = µ
0+ ε
µ1(ξ, ϑ, ϕ) , for µ
0= const and function µ
1(ξ, ϑ, ϕ)
taken from phenomenological considerations).
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On General Solutions in Topological Massive Gravity and Ricci Flows
The metric εg has a true singularity at r = 0 and theequation
1− 2µ0
r+
1
3λ 4κ
2 r2 = 0
has three solutions for not small r (when we can neglectthe term 1/r2) corresponding to three horizons.
3–d configurations are generated if we do notconsider coordinate ϑ
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On General Solutions in Topological Massive Gravity and Ricci Flows
Off–diagonal Configurations
We consider the ansatzλg = −eφ(ξ,ϑ) dξ ⊗ dξ − eφ(ξ,ϑ) dϑ⊗ dϑ
+h3 (ξ, ϑ, ϕ) δϕ⊗ δϕ + h4 (ξ, ϑ, ϕ) δt⊗ δt,
δϕ = dϕ + w1 (ξ, ϑ, ϕ) dξ + w2 (ξ, ϑ, ϕ) dϑ,
δt = dt + n1 (ξ, ϑ, ϕ) dξ + n2 (ξ, ϑ, ϕ) dϑ,
for h3 = η3(ξ, ϑ, ϕ)r2(ξ) sin2 ϑ, h4 = η4(ξ, ϑ, ϕ) λ$2(ξ),
where the coefficients (solutions of TMG) satisfy
φ••(ξ, ϑ) + φ′′(ξ, ϑ) = 2 hΛ(ξ, ϑ);
h3 = ± (φ∗)2
4 vΛ(ξ, ϑ)e−2 0φ(ξ,ϑ),
h4 = ∓ 1
4 vΛ(ξ, ϑ)e2(φ− 0φ(ξ,ϑ));
wi = −∂iφ/φ∗;
ni = 1ni(ξ, ϑ) + 2ni(ξ, ϑ)
∫(φ∗)2 e−2(φ− 0φ(ξ,ϑ))dϕ,
= 1ni(ξ, ϑ) + 2ni(ξ, ϑ)
∫e−4φ(h∗4)
2
h4dϕ, n∗i 6= 0;
= 1ni(ξ, ϑ), n∗i = 0;
for any nonzero ha and h∗a and (integrating) functions1ni(ξ, ϑ), 2ni(ξ, ϑ), generating function φ(ξ, ϑ, ϕ), and0φ(ξ, ϑ) to be determined from boundary cond.
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On General Solutions in Topological Massive Gravity and Ricci Flows
Ellipsoid de Sitter configurations and TMG
rotλ g = −eφ(ξ,ϑ) (dξ ⊗ dξ + dϑ⊗ dϑ)
−h20
[(√|q|)∗
]2
1 + ε
1
(√|q|)∗
s√
|q|
∗ δϕ⊗ δϕ
+(q + εs
)δt⊗ δt,
δϕ = dϕ + w1dξ + w2dϑ, δt = dt + n1dξ + n2dϑ.
q = 1− 2 1µ(r, ϑ, ϕ)
r, s =
q0(r)
4µ20
sin(ω0ϕ + ϕ0),
with 1µ(r, ϑ, ϕ) = µ + ε(r−2 − λ 4κ
2 r2/3)/2, chosen
to generate an anisotropic rotoid configuration for thesmaller ”horizon” (when h4 = 0), for a q
0(r),
r+ '2 1µ
1 + εq0(r)
4µ20
sin(ω0ϕ + ϕ0).
Levi–Civita: φ(r, ϕ, ϑ) = ln |h∗4/√|h3h4|| must be any
function 2e2φφ = vΛ.
N–connection coefficients: w1w2
(ln |w1
w2|)∗
= w•2 − w′
1 for
w∗i 6= 0; w•
2 − w′1 = 0 for w∗
i = 0; and take ni = 1ni(xk)
for 1n′1(xk)− 1n•2(x
k) = 0.
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On General Solutions in Topological Massive Gravity and Ricci Flows
Rotoids and solitonic distributions
On a N–anholonomic spacetime V defined by a rotoidd–metric rotg, we can consider a static 3–d solitonicdistribution η(ξ, ϑ, ϕ) as a solution of solitonic equation1
η•• + ε(η′ + 6η η∗ + η∗∗∗)∗ = 0, ε = ±1.
It is possible to define a nonholonomic transform from rotgto a d–metric rot
st g determining a stationary metric for arotoid in solitonic background to be reduces for 3–d TMG:
rotst g = −eψ (dξ ⊗ dξ + dϑ⊗ dϑ)
−4[(√|ηq|)∗
]2[1 + ε
1
(√|ηq|)∗
(s√|ηq|
)∗]δϕ⊗ δϕ
+η (q + εs) δt⊗ δt,
δϕ = dϕ + w1dξ + w2dϑ, δt = dt + 1n1dξ + 1n2dϑ,
N–connection coefficients are taken the same as inprevious solution.
Limit ε → 0, a nonholonomic embedding of theSchwarzschild solution into a solitonic vacuum, whichresults in a vacuum solution of the Einstein gravity definedby a stationary generic off–diagonal metric and datainduced by TMG.
1as a matter of principle, we can consider that η is a solution of any three dimensional solitonic and/ or other nonlinear wave equations
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”Fictive” Vacuum Solutions
Solutions of Rαβ = 0 by d-metrics with stationarycoefficients subjected to conditions
ψ••(ξ, ϑ) + ψ′′(ξ, ϑ) = 0;
h3 = ±e−2 0φ(h∗4)2
h4for given h4(ξ, ϑ, ϕ), φ = 0φ = const;
wi = wi(ξ, ϑ, ϕ) are any functions if vΛ = 0;
ni = 1ni(ξ, ϑ) + 2ni(ξ, ϑ)
∫(h∗4)
2 |h4|−5/2dv, n∗i 6= 0;
= 1ni(ξ, ϑ), n∗i = 0,
for h4 = η(ξ, ϑ, ϕ) [q(ξ, ϑ, ϕ) + εs(ξ, ϑ, ϕ)] . In the limitε → 0, we get a so–called Schwarzschild black holesolution mapped nonholonomically on a N–anholonomic(pseudo) Riemannian spacetime, or constrained to TMG.
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III. Nonholonom. Distr. & Gravity
Nonholonomic Manifolds/Bundles
Geometrization of nonholonomic mechanics =⇒concept of nonholonomic manifold V = (M,D)smooth & orientable M , non–integrable distribution DN–anholonomic manifold enabled with nonlinearconnection (N–connection) structure
N : TV = hV ⊕ vV, D = hV, integrable vV
Examples: 1) V (semi/pseudo) Riemannian manifold2) V = E(M), or = TM, is a vector, or tangent, bundle
Local coordinates u = (x, y), or uα = (xi, ya)
h–indices: i, j, ... = 1, 2, 3 and v–indices: a, b, ... = 4, 5
Local coefficients N = Nai (u) dxi ⊗ ∂
∂ya
Particular case: Nai (u) = Γa
bi(x)yb
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∃ N–adapted (co) frames (vielbeins)
eν =
(ei =
∂
∂xi−Na
i (u)∂
∂ya, ea =
∂
∂ya
),
eµ =(ei = dxi, ea = dya + Na
i (u)dxi).
Nonholonomy: [eα, eβ] = eαeβ − eβeα = W γαβeγ
Anholonomy coefficients W bia = ∂aN
bi and W a
ji = Ωaij,
N–connection curvature Ωaij = ej (Na
i )− ei
(Na
j
)
holonomic/ integrable case W γαβ = 0.
Distinguished Connections:A d–connection D on V is a linear connection preservingunder parallelism the N–connection splitting.
Locally, D ⇒ Γγαβ =
(Li
jk, Labk, C
ijc, C
abc
),
hD = (Lijk, L
abk) and vD = (C i
jc, Cabc)
Distinguished objects: d–objects, d–tensors
d–vectors X = hX + vX = hX + vX
N–adapted geometric constructions.
Levi–Civita connection ∇ is not N–adapted
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On General Solutions in Topological Massive Gravity and Ricci Flows
Torsion and Curvature d–tensors
Torsion of D =( hD, vD), by the d–tensor field
T(X,Y) + DXY −DYX− [X,Y],
T(X,Y) = hT(hX, hY), hT(hX, vY), ..., vTvX, vY)N–adapted: T = Tα
βγ = (T ijk, T
ija, T
ajk, T
bja, T
bca)
Curvature: R(X,Y) + DXDY −DYDX−D[X,Y],
R = Rαβγδ =
(Ri
hjk,Rabjk,R
ihja,R
cbja,R
ihba,R
cbea
)Ricci: Ric + Rβγ = Rα
βγα = (Rij, Ria, Rai, Rab)Distinguished metric (d–metric)
g = hg⊕Nvg = [ hg, vg]
g = gij(x, y) ei ⊗ ej + hab(x, y) ea ⊗ eb
Coordinate co–frames: g = gαβ
(u) duα ⊗ duβ
gαβ
=
[gij + Na
i N bjhab N e
j hae
N ei hbe hab
]
for N ej (u) = N e
j (x, y)
Data for generic off–diagonal metrics and N–connections
g = gαβ =[gij, hab, N
bj
]
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Levi–Civita and canonical d–connection
Levi–Civita connection ∇ = gΓγαβ,
T αβγ = 0 and ∇g =0
Canonical d–connection D = gΓγαβ
Dg =0 and hT(hX, hY ) = 0, vT(vX, vY ) = 0
gΓγαβ = gΓγ
αβ + gZγαβ
Distortion tensor gZγαβ completely defined by g
N–adapted coefficients Γγαβ =
(Li
jk, Labk, C
ijc, C
abc
),
Lijk =
1
2gir (ekgjr + ejgkr − ergjk) ,
Labk = eb(N
ak ) +
1
2hac
(ekhbc − hdc ebN
dk − hdb ecN
dk
),
C ijc =
1
2gikecgjk, Ca
bc =1
2had (echbd + echcd − edhbc) .
Nontrivial torsion Tγαβ ∼ gZγ
αβ
T ija = C i
jb, Taji = −Ωa
ji, Tcaj = Lc
aj − ea(Ncj )
If Tγαβ = 0
gΓγαβ = gΓγ
αβ
even, in general, ∇ 6= D
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IV. General Sols in Topolog. Massive Grav.
a. Separation of Einstein eqs for d–connections
Einstein eqs for gβδ rewritten equivalently using thecanonical d–connection,
R βδ − 1
2gβδ
sR = Υβδ,
Lcaj = ea(N
cj ), C i
jb = 0, Ωaji = 0,
Ricci tensor R βδ is for Γγαβ,
sR = gβδR βδ for D → ∇,also contributions of Cotton tensor.
For instance, (2+2) splitting, (uα = (xk, t, y4 = y),
ansatz with Killing symmetry ∂/∂y4,
Kg = g1(xk)dx1 ⊗ dx1 + g2(x
k)dx2 ⊗ dx2
+h3(xk, t)e3⊗e3 + h4(x
k, t)e4⊗e4
where for N 3i = wi(x
k, t), N 4i = ni(x
k, t),
e3 = dt + wi(xk, t)dxi,
e4 = dy4 + ni(xk, t)dxi
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Theorem 1 (Separation of Eqs)
The Einstein eqs for ansatz Kg and D are:
−R11 = −R2
2 =1
2g1g2(4)
[g••2 − g•1g
•2
2g1− (g•2)
2
2g2+ g′′1 −
g′1g′2
2g2− (g′1)
2
2g1
]= Υ4(x
k),
−R33 = −R4
4 =
1
2h3h4
[h∗∗4 − (h∗4)
2
2h4− h∗3h
∗4
2h3
]= Υ2(x
k, t), (5)
R3k =wk
2h4
[h∗∗4 − (h∗4)
2
2h4− h∗3h
∗4
2h3
](6)
+h∗44h4
(∂kh3
h3+
∂kh4
h4
)− ∂kh
∗4
2h4= 0,
R4k =h4
2h3n∗∗k +
(h4
h3h∗3 −
3
2h∗4
)n∗k2h3
= 0, (7)
where a• = ∂a/∂x1, a′ = ∂a/∂x2, a∗ = ∂a/∂t.
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On General Solutions in Topological Massive Gravity and Ricci Flows
b)Integration of (non)holonomic Einstein eq
Theorem 2 (Integral Varieties)
Eqs(4)–(7), written for h∗3,4 6= 0 and Υ2,4 6= 0,
ψ + ψ′′ = 2Υ4(xk) (8)
h∗4 = 2h3h4Υ2(xi, t)/φ∗ (9)
βwi + αi = 0 (10)
n∗∗i + γn∗i = 0 (11)
αi = h∗4∂iφ, β = h∗4 φ∗, φ = ln | h∗4√|h3h4|
|, γ =(ln |h4|3/2
|h3|)∗
General solution: g1 = g2 = eψ(xk)
h4 = 0h4(xk)± 2
∫(exp[2 φ(xk, t)])∗
Υ2dt,
h3 = ± 1
4
[√|h∗4(xi, t)|
]2
exp[−2 φ(xk, t)]
wi = −∂iφ/φ∗
nk = 1nk
(xi
)+ 2nk
(xi
) ∫[h3/(
√|h4|)3]dt
Levi–Civita (LC) conditions:
w∗i = ei ln |h4|, ekwi = eiwk, n∗i = 0, ∂ink = ∂kni
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On General Solutions in Topological Massive Gravity and Ricci Flows
General Commutative Solutions
Non–Killing metrics
Dependence on 4th coordinate via ω2(xj, y3 = t, y4 = y)
g = gi(xk)dxi ⊗ dxi + ω2(xj, t, y)ha(x
k, t)ea⊗ea,
e3 = dy3 + wi(xk, t)dxi, e4 = dy4 + ni(x
k, t)dxi,
ekω = ∂kω + wkω∗ + nk∂ω/∂y = 0,
ω2 = 1 results in solutions with Killing symmetry.
Sketch proof: Recompute the Ricci h–v and v–componentsand get zero distortion contribution for v– d’Allambertconditions, and/or additional sources determined by ω.
N–deformations and exact solutions
’Polarizations’ ηα and ηai , nonholonomic deformations,
g = [ gi,ha,
Nak ] → ηg = [ gi, ha, N
ak ].
Deformations of frame/metric/fundamental geometricstructures are more general than moving frame method:
ηg = ηi(xk, t) gi(x
k, t)dxi ⊗ dxi
+ ηa(xk, t) ha(x
k, t)ea⊗ea,
e3 = dt + η3i (x
k, t) wi(xk, t)dxi,
e4 = dy4 + η4i (x
k, t) ni(xk, t)dxi.
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V. Eqs for Nonhol. Ricci Flows
1. Families fundamental Einstin/ Cotton/ Lagrange/Finsler L(x, y, χ)/ F (x, y, χ), on TM, or V, real χ,
g = gi(xk, χ)dxi ⊗ dxi
+ω2(xj, t, y, χ)ha(xk, t, χ)ea⊗ea,
e3 = dy3 + wi(xk, t, χ)dxi, e4 = dy4 + ni(x
k, t, χ)dxi,
2. For the previous works, gαβ are solutions of Einsteineqs for a d–connection, Rαβ = λgαβ, for instance,
R βδ − 1
2gβδ
sR = Υβδ,
LC–cond. Lcaj = ea(N
cj ), C i
jb = 0, Ωaji = 0
3. This seminar, gαβ(χ) are solutions of∂gαβ
∂χ = −2Cαβ
4. Canonical d–connection evolution (postulates)∂
∂χgi = −2
(Rii − λgi
)− hc
∂
∂χ(N c
i )2,
∂
∂χha = −2
(Raa − λha
),
Rαβ = 0, for α 6= β, (R− λg) → µ−1C
In general, Rαβ 6= Rβα resulting in nonsymmetric metrics.
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On General Solutions in Topological Massive Gravity and Ricci Flows
N–connection evolution of massive gravity
Include exact solutions for Einstein (non) holonomicmanifolds into nonholonomic Ricci/ -Cotton flow eqs?
Examples with nontrivial N ci (χ):
a) ∂∂χgi = ∂
∂χha = 0, Υ2 = Υ4 = 0
constraints:∂
∂χ[hc (N c
i )2] = 0, (12)
gij and hab are d–metrics not depending on χfamilies of N 3
i = wi(xk, t, χ), N 4
i = ni(xk, t, χ) with (12)
g = gi(xk)dxi ⊗ dxi + ω2(xj, t, y, χ)ha(x
k, t)ea⊗ea,
e3 = dy3 + wi(xk, t, χ)dxi, wi(x
k, t, χ) → wi(xk, t), (15)
e4 = dy4 + ni(xk, t, χ)dxi,
Use Theorem 2, h∗3,4 6= 0, Υ2,4 → λ,
ψ + ψ′′ = 2λ (13)
h∗4 = 2h3h4λ/φ∗ (14)
βwi + αi = 0 (15)
n∗∗i (λ) + γ n∗i (λ) = 0 (16)
αi = h∗4∂iφ, β = h∗4 φ∗, φ = ln | h∗4√|h3h4|
|,
γ =
(ln|h4|3/2|h3|
)∗
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On General Solutions in Topological Massive Gravity and Ricci Flows
Solutions for N–con. Ricci flows
g1 = g2 = eψ(xk), wi = −∂iφ/φ∗,
h4 = 0h4(xk)± 2
λexp[2 φ(xk, t)],
h3 = ± 1
4
[√|h∗4(xi, t)|
]2
exp[−2 φ(xk, t)]
nk(χ) = 1nk(xi, χ) + 2nk(x
i, χ)
∫[h3/(
√|h4|)3]dt
ekω(χ) = ∂kω(χ) + wkω∗(χ) + nk(χ)∂ω(χ)/∂y = 0
LC–conditions: w∗i = ei ln |h4|, ekwi = eiwk,
n∗i (χ) = 0, ∂ink(χ) = ∂kni(χ)
b) We can consider nontrivial Υ2, Υ4 imposing constr.(12)
R11 = R2
2 = −Υ4(xk, χ), R3
3 = R44 = −Υ2(x
k, t, χ)∂
∂χgi = 2(Υ4 + λ)gi,
∂
∂χha = 2(Υ2 + λ)ha
ekω(χ) = ∂kω(χ) + wk(χ)ω∗(χ) + nk(χ)∂ω(χ)/∂y = 0
Conclusion: Exact solutions for Einstein–Lagrange/–Finsler, massive gravity models, both with trivial andnontrivial matter sources, can be generalized tononholonomic Ricci flows, and re–defined for Cotton flows.
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On General Solutions in Topological Massive Gravity and Ricci Flows
VI. Conclusions and Perspectives
1. ”Almost” any solution of Generalized Einstein eqs withsources, for gα′β′, via frame/ coordinate transform
eα = eα′α(xi, ya)eα′, gαβ = eα′
αeβ′βgα′β′,
can be expressed in a form gαβ =∣∣∣∣∣∣∣∣
g1 + ω2(w 21 h3 + ω2(n 2
1 h4) ω2(w1w2h3 + n1n2h4) ω2 w1h3 ω2 n1h4
ω2(w1w2h3 + n1n2h4) g2 + ω2(w 22 h3 + n 2
2 h4) ω2 w2h3 ω2 n2h4
ω2 w1h3 ω2 w2h3 h3 0ω2 n1h4 ω2 n2h4 0 h4
∣∣∣∣∣∣∣∣
2. Exact solutions Related to Nonholonomic Ricci/ – Cottonflows: physical applications in classical and quantumgravity.
3. Black ellipsoid/torus/solitonic/ pp–waves in massivetopological gravity (TMG)
4. Cosmological solutions in TMG
5. Imbedding of M. Gurses solutions into nonholonomicconfigurations and generalizations
6. Solitonic hierarchies in TMG
7. Algebraic classification and nonholonomic deforms TMG
8. Deformation quantization of TMG
9. Quantum Einstein–Finsler gravity and TMG
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