On General Solutions in Topological Massive Gravity and Ricci Flows On General Solutions in Topological Massive Gravity and Ricci Flows Research Seminar Sergiu I. Vacaru Science Department University Al. I. Cuza, Ia¸ si, Romania Seminar at Department of Mathematics (host prof. Metin G¨ urses) Bilkent University July 22, 2010; Ankara, Turkey c Sergiu I. Vacaru, July 15, 2010 1/ 24 100722˙trsp˙ank˙bilkent.tex rayslides.sty
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On General Solution in Topological Massive Gravity and Ricci Flows
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
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
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.
∗ 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
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.
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
On General Solutions in Topological Massive Gravity and Ricci Flows
”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.
On General Solutions in Topological Massive Gravity and Ricci Flows
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
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:
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.