Modified Dispersion Relations in Horava-Lifshitz Gravity and Finsler Brane Models

8/8/2019 Modified Dispersion Relations in Horava-Lifshitz Gravity and Finsler Brane Models

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Modied Dispersion Relations in Ho rava–Lifshitz

Gravity and Finsler Brane Models

Sergiu I. Vacaru

Department of Science University Al. I. Cuza (UAIC), Ia¸ si, Romania

Research Seminar at

Albert–Einstein–Institut, Potsdam–Golm,Max–Planck–Institut für Gravitationsphysik

October 29, 2010

Sergiu I. Vacaru (UAIC, Ia¸ si, Romania) MDR in HL gravity & Finsler brane models October 29, 2010 1 / 19

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Differential & Finsler Geometry, Ia¸ si, RomaniaResearch group "Geometry & Applications in Physics"

100 years traditions on math & applications; supervision/ collaborations by/with D. Hilbert, T. Levi–Civita and E. Cartan of PhDof prominent members of Romanian Academy.

E. Cartan visit at Ia¸ si in 1931 induced 80 years of research onFinsler/integral geometry etc, "isolation" after 1944;"Japanese–Finsler geometry orientation" after 1968Alexandru Myller (1879–1965), PhD–1906: D. Hilbert(chair/adviser) and F. Klein, H. Minkowski (commission).

Gheorghe Vranceanu (1900–1979), PhD-1924, from Levi–Civita,commission head: Volterra; 1927-28, Rockefeller scholarship forFrance, E. Cartan, and USA at Harvard & Princeton (Morse,Birkhoff, Veblen)


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Differential & Finsler Geometry, Ia¸ si, Romania(prolongation)

Mendel Haimovici (1906–1973); PhD-1933- Levi–Civita.Radu Miron (1927 - ); 28 monogr., 240 rev. MathSciNetLagrange–Finsler, Hamilton–Cartan & higher order, applications tomechanics and relativity etc.Iasi team and "Romanian Finsler diaspora": M. Anastasiei, D.Buc ataru and M. Crâsmâreanu (Ia¸ si);A.Bejancu(Kuwait);D.Hrimiuc(Canada);V.Sabau(Japan);

S. Vacaru (Cernâuti/Chernivtsy, Chi¸ sinâu/ Kishinev, Tomsk,Dubna, Moscow, Kyiv, Bucharest–M agurele, Lisbon, Madrid,Toronto, Ia¸si)


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Outline

1 Goals and Motivation

2 Finsler Geometry from MDR and HL GravityPreliminaries on HL and GR theoriesMDR and HL gravityFundamental Finsler functions from HL gravity

3 Horava–Finsler GravityFundamental geometric objects for HL gravityField equations in HF gravityMagic splitting of gravitational HF led equations

4 Finsler Branes and Trapping to HL and GRAnsatz for HF–brane solutionsFinsler brane solutions

5 Summary & Conclusions


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Goals and Motivation

Motivation and Goals

Motivation1 QG–LV–MDR–Finsler geometry scheme. HL theory with scaling

anisotropy and MDR can be included.2 Finsler in analogous gravity, geometric mechanics, condensed

matter, phase transitions, QG and phenomelogy.3 Restricted special relativity, modied Lorentz geoms etc4 Modern cosmology and dark energy/dark matter5 Non–integrable Ricci ows from Rimenann to Finsler geoms6 Noncommutative generaliz. of gravity as complex Finsler geoms7 Finsler congurations as exact solutns in GR, string gravity etc8 Finsler methods of constructing exact solutions9 almost Kähler – Finsler variables in GR and modications -

deformation quantization, A–brane, gauge like, nonholonomic

canonical quantization etc.Sergiu I. Vacaru (UAIC, Ia¸ si, Romania) MDR in HL gravity & Finsler brane models October 29, 2010 5 / 19

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Goals and Motivation

Motivation and Goals

Goals:HL and MDR result in Finsler generating functionsFor GR community: Metric compatible Finsler geometry andgravityA model of Ho rava–Finsler gravity for HL gravity nonholonomicOn general solutions in GR, HL and Finsler gravity theoriesFinsler Brane solutions and trapping HF gravity

Reviews new results:S. Vacaru, in: NPB, PLB, AP NY, JHEP, IJGMMP, JMP, JGP, CQG,1995–2010Details in: S. Vacaru, arXiv: 1010.5457Partner works: arXiv: 1008.4912, 1004.3007, 1003.0044


l f d l d h

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Finsler Geometry from MDR and HL Gravity Preliminaries on HL and GR theories

Preliminaries on HL and GR theories

HL theory

ADM: ds 2 = g ij dx i dx j = −N 2dt 2 + g i j (dx i + N i dt )(dx j + N j dt )foliation–preserving t = t (t ) and x

i = x

i (t , x

k ), x i = ( x 1 = t , x

i )

Anisotr. t →l z t , x i →lx i ; N →l − 2 N , N i →l − 2 N i , g i j →g i j ,z = 3 , power–counting renormalizable gravity in 4–d,Action for HL gravity HLS = K S + V S ,

K S =2

κ 2 dtd 3 x | g |N K

i

j K

i

j − λ K 2

V S = dtd 3

x | g |N [κ 2 µ2 2

i

j

k R

i

l

j R

l

k −κ 2 µ

8 R

i

j R

i

j

+κ 2 µ

8(1 − 3λ )1 − 4λ

4R 2 + ΛR − 3Λ2 −

κ 2

2 2 C

i

j C

i

j ],

κ,µ, , Λ = const , dynamical constant λ. In GR λ = 1.Sergiu I. Vacaru (UAIC, Ia¸ si, Romania) MDR in HL gravity & Finsler brane models October 29, 2010 7 / 19

Fi l G f MDR d HL G i MDR d HL i

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Finsler Geometry from MDR and HL Gravity MDR and HL gravity

MDR and HL gravity

Stability analysis of HL, possible instabilities, dispersions etc,Fourier transforms ψ(t , x i ) = d 3 k

(2π )3/ 2 ψp (t )e ip i x i )

Detailed balance conditionsscalar perturbations and low– p , ω2 = − 9 κ 4 µ 2 Λ2

32 (1 − 3 λ )2 < 0, inducesinstabilities at the IR for all values of λ and both sings of

Λhigh– p , ω2 = κ 4 µ 2

161 − λ

1 − 3 λ

2

p 4

tensor perturbations, ω2 = c 2 p 2 + κ 4 µ 2

16 p 4 ± κ 4 µ4 2 p 5 + κ 4

4 4 p 6 .Perturbative analysis extended beyond detailed balance

scalar UV–behavior, ω2

=κ 2 (1 − λ )2

16 (1 − 3 λ )2 p 4

−3 κ 2 (1 − λ )2 (1 − 3 λ ) ηp

6

.tensor perturbations:

ω2 = c 2 p 2 +κ 4 µ2

16p 4 ±

κ 4 µ4 2 p 5 +

κ 4

4 4 −κ 2 η

2p 6 .


Finsler Geometry from MDR and HL Gravity Fundamental Finsler functions from HL gravity

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Finsler Geometry from MDR and HL Gravity Fundamental Finsler functions from HL gravity

Fundamental Finsler functions from HL gravity

Stability analysis, light propagation, k i →p i ∼

y a ,

ω2 = c 2 g i j k i k j 2

1 −1r

q i 1 i 2 ... i 2r y i 1 ...y i 2r

g

i

j k i k j

2r .

nonlinear homogeneous quadratic elements (withF (x i , β y j ) = β F (x i , y j ), for any β > 0), when ds 2 = F 2 (x i , y j )

≈ −(cdt )2 + g

i

j (x k )y i y j 1 + 1

r

q i 1 i 2 ... i 2r (x k )y i 1 ... y i 2r

g

i j (x k

)y

i y

j

r + O (q 2 ) F is a

fundamental Finsler function, Hessian F g ij (x i , y j ) = 12

∂ F 2∂ y i ∂ y j

HL theory is with generic anisotropy and LV characterized bydispersion relations ω(p i ,κ,µ, Λ, , c ,λ ,η ) and associated F .


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Horava–Finsler Gravity Fundamental geometric objects for HL gravity




Canonical N–connections and d–metrics for HF gravity

Canonical N–connections and d–metrics for HF gravity

L = H F 2 regular Lagrangian, S (τ ) =1

0L(x (τ ), y (τ ))d τ with

y k (τ ) = dx k (τ )/ d τ, for x (τ ) smooth curves on V , τ ∈[0 , 1].Euler–Lagrange eqs for S (τ ), d

d τ ∂ L∂ y i

−∂ L∂ x i = 0 , equivalent eqs

d 2 x k

d τ 2 + 2G k (x , y ) = 0 , G k = 14 g kj y i ∂ 2 L

∂ y j ∂ x i − ∂ L∂ x j

induces c N = c N a j = ∂ G a /∂ y j .

Sasaki lift e i = dx i and F e a = dy a + F N a i (u )dx i ,

F g =

F g ij (x , y ) e

i

⊗e

j + (

∗

l P )2 F

g ij (x , y )F

ei

⊗

F e

j ,

off–diagonal metrics e α = e αα e α , H g = H g αβ (u ) du α⊗du β ,

H g αβ =g ij + ( ∗l P )2 h ab N a

i N b j ( ∗l P )2 h ae N e

j

( ∗l P )2 h be N e i ( ∗l P )2 h ab



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Horava Finsler Gravity Fundamental geometric objects for HL gravity

N–adapted d–connections

Canonical d–connection & Levi–Civita connectionA d–connection D = ( hD , vD ) preserves under parallelism the h– v–spitting. Metric compatible: D g = 0

Canonic. d–con. D : Γγ αβ = (

Li

jk ,

La

bk , C i jc , C a bc ); T i jk = 0 , T a

bc = 0 .

Li

jk =12 g ir e k g jr + e j g kr

−e r g jk ,

La bk = e b (N a

k ) +12 h ac e k h bc −h dc e b N d

k −h db e c N d k ,

C i jc =12 g ik e c g jk , C a

bc = 12 h ad (e c h bd + e c h cd −e d h bc ) .

Levi–Civita = Γγ αβ distortion: g Γγ

αβ = g

Γγ

αβ + g Z γ αβ ,

all components dened by metric. Cartan d–con. canonicalalmost symplectic. Chern d–con. not metric compatible!

Denition:A nonholonomic manifold is a pair (V,

N)


Horava–Finsler Gravity Field equations in HF gravity

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y q g y

The Einstein eqs on nonholonomic manifolds

Two equivalent representations of Einstein eqs

Levi–Civita , R βδ − 12 g βδ R = κ T βδ ,

canonic d–connect D, R βδ −12

g βδs R = Υ βδ ,

Lc aj = e a (N

c j ), C

i jb = 0 , Ω

a ji = 0 .

Ansatz for solutions:

η g = ηi (x k , v ) g i (x k , v )dx i ⊗ dx i + ηa (x k , v ) h a (x k , v )e a ⊗e a ,

e3 = dv + η3

i (x k , v ) w i (x k , v )dx i , e

4 = dy 4 + η4

i (x k , v ) n i (x k , v )dx i

g ij = diag [g i = ηi g i ] and h ab = diag [h a = ηa

h a ] andN 3k = w i = η3

i w i and N 4k = n i = η4

i n i ; Gravit.’polarizations’ ηα

and ηa i ,

g = [ g i , h a , N a k ] →ηg = [ g i , h a , N a

k ], functions ofnecessary smooth class and/or any rando m (s toc ha st ic) va riables .


Horava–Finsler Gravity Magic splitting of gravitational HF led equations

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y g p g g q

Nonholonomic separation of Einstein eqs

MAGIC d–connection SPLITTING for one Killing ansatz:

ηg = g i (x k )dx i ⊗

dx i + h 3 (x k , t )e 3⊗

e 3 + h 4 (x k , t )e 4⊗

e 4 ,e 3 = dt + w i (x k , t )dx i , e 4 = dy 4 + n i (x k , t )dx i

a • = ∂ a /∂ x 1 , a = ∂ a /∂ x 2 , a ∗= ∂ a /∂ v ; v = t ;

R 11 = R

22 =

− 12g 1 g 2 [g

••

2 −g •1 g •22g 1 −

(g •2 )2

2g 2 + g 1 −g 1 g 22g 2 −

(g 1 )2

2g 1 ] = − Υ4 (x k ),

R 33 = R 44 = −1

2h 3 h 4[h ∗∗4 −

(h ∗4 )2

2h 4−

h ∗3 h ∗42h 3

] = − Υ2 (x k , v ),

R 3

k =

w k

2h 4[h ∗∗4 −

(h ∗4 )2

2h 4−

h ∗3 h ∗42h 3

] +h ∗4

4h 4∂ k h 3

h 3+

∂ k h 4

h 4−

∂ k h ∗42h 4

= 0 ,

R 4 k =h 4

2h 3n ∗∗k +

h 4h 3

h ∗3 −32

h ∗4n ∗k

2h 3= 0 ,

w ∗i = e i ln |h 4 | , e k w i = e i w k , n ∗i = 0 , ∂ i n k = ∂ k n i


Horava–Finsler Gravity Magic splitting of gravitational HF led equations

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Exact solutions with h ∗3 ,4

= 0 and Υ 2,4 = 0

Formal integrationh–metric: g 1 = g 2 = e ψ(x k ) , ψ + ψ = 2Υ4 (x k )coefcients:φ = ln |

h ∗4√|h 3h 4 | |, α i = h ∗4∂ i φ, β = h ∗4 φ∗, γ = ln |h 4 |3 / 2 / |h 3 |∗

N–connection eqs: β w i + α i = 0 , n ∗∗i + γ n ∗i = 0v–metric, if h ∗4 = 0; Υ2 = 0 , we get φ∗= 0 .∀φ = φ(x i , t ) = const isa solution generating function, h ∗4 = 2h 3h 4 Υ2 (x i , t )/φ ∗.

solution: h 3 =

±|φ∗ (x i ,t ) |

Υ2

, h 4 = 0 h 4 (x k )

±2

(exp [2 φ(x k ,t )]) ∗

Υ2

dt ,w i = −∂ i φ/φ ∗, n i = 1 n k x i + 2 n k x i [h 3 / ( |h 4 |)3 ]dt ,integration functions 0 h 4 (x k ), 1 n k x i and 2 n k x i

Υi = λ, λ →h λ(x k ) = Υ 4 (x k ) and λ →v λ(x k , t ) = Υ 2 (x k , t ).


Finsler Branes and Trapping to HL and GR Finsler brane solutions

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Holonomic/ Diagonal Conf. for HF–branes

Trapping scenario with diagonal HF–solutionsSimilarly to Gogberashvili et all, but on TM , whereηαβ = diag [1, −1 , −1−, 1], g = φ2 (y 5 )ηαβ du α⊗du β −( ∗l P )2 h (y 5 )[ dy 5⊗dy 5 + dy 6⊗dy 6 ±dy 7⊗dy 7 ±dy 8⊗dy 8 ],

Sources: cosm. const. Λ and stress–energy tensor, fundamentalmass scale M on T V, dim T V = 8 .

Υβ δ = δβ

δ[Λ−M − (m + 2)K 1 (y 5 )], Υ55 = Υ 6

6 = Λ −M − (m + 2) K 2 (y 5 ),

Solution, width 2 = 40 M 4 / 3Λ:

φ2 (y 5 ) =3 2 + a (y 5 )2

3 2 + ( y 5 )2 and ∗l P |h (y 5 )| =9 4

3 2 + ( y 5 )2 2 .



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Off–Diagonal Solutions for HF–branes

Off–Diagonal Solutions for HF–branes

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 φ2 [ q h 5 e 5

⊗e 5 + q h 6 e 6⊗e 6 + q h 7e 7

⊗e 7 + q h 8e 8⊗e 8 ]

e 3 = dy 3 + w i dx i , e 4 = dy 4 + n i dx i ,e 5 = dy 5 + 1 w i dx i , e 6 = dy 6 + 1 n i dx i ,

e 7 = dy 7 + 2 w i dx i , e 8 = dy 8 + 2 n i dx i .

h 5 (x i , y 5 ) = ∗l P h (y 5)

φ2 (y 5 )q h 5 (x i , y 5 ), h 6 (x i , y 5 ) = ∗l P h (y

5)

φ2 (y 5 )q h 6 (x i , y 5 ),

h 7 (x i , y 5 , y 7 ) = ∗l P h (y 5 )

φ2 (y 5 )q h 7 (x i , y 5 , y 7 ),

h 8 (x i , y 5 , y 7 ) = ∗l P h (y 5 )

φ2 (y 5 )q h 8 (x i , y 5 , y 7 ).



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with respect to coordinate co–framesg 2 α 2 β =

B 11 B 12 w 1 h 3 n 1 h 4 1 w 1 h 5 1 n 1 h 6 2 w 1 h 7 2 n 1 h 8B 21 B 22 w 2 h 3 n 2 h 4 1 w 2 h 5 1 n 2 h 6 2 w 2 h 7 2 n 2 h 8

w 1 h 3 w 2 h 3 h 3 0 0 0 0 0n 1 h 4 n 2 h 4 0 h 4 0 0 0 0

1 w 1 h 5 1 w 2 h 5 0 0 h 5 0 0 01 n 1 h 6 1 n 2 h 6 0 0 0 h 6 0 02 w 1 h 7 2 w 2 h 7 0 0 0 0 h 7 02 n 1 h 8 2 n 2 h 8 0 0 0 0 0 h 8

observable Finsler brane and LV contributions by terms proportional to ( ∗ l P )2

B 11 = g 1 + w 21 h 3 + n

21 h 4 +

∗

l P 2 h

φ 2 (1

w 1 )2 q h 5 + (

1n 1 )

2 q h 6 + (2

w 1 )2 q h 7 + (

2n 1 )

2 q h 8 ,

B 12 = B 21 = w 1 w 2 h 3 + n 1 n 2 h 4 +∗

l P 2 h

φ 2

1w 1

1w 2

q h 5 +1

n 11

n 2q h 6 +

2w 1

2w 2

q h 7 +2

n 12

n 2q h 8 ,

B 22 = g 2 + w 22 h 3 + n

22 h 4 +

∗

l P 2 h

φ 2t (

1w 2 )

2 q h 5 + (1

n 2 )2 q h 6 + (

2w 2 )

2 q h 7 + (2

n 2 )2 q h 8 .


Summary & Conclusions

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Summary & Conclusions

Gravity theories with LV & MDR -> fundamental Finsler functs.

HF gravity: Finsler geometry models derived in a metriccompatible theory on tangent bundle.HF gravity can be integrated in very general forms. The methodcan be applied in GR and modications.

Finsler brane models with warping/trapping for nonholonomictransforms HF-> HL / GR without "velocity coordinates": genericoff–diagonal solutions with N–connection structure.

Outlook (recently developed, under elaboration):Deformation, A–brane, bi–connection quantization of HF and HLgravity models.Exact solutions in HF and HL. Applications in cosmology andastrophysics. Exotic solutions with diffusion, fractional derivatives,chaos, solitons etc.Nonholonomic Ricci ows and HF gravity.


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Documents

Modified Dispersion Relations in Horava-Lifshitz Gravity and Finsler Brane Models