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Linear dilaton for asymptotically Lifshitz-like spacetimes Anastasia Golubtsova 1 based on a collabotation with Irina Ia. Arefeva and Eric Gourgoulhon JHEP 1504 (2015) (011),arXiv:1410.4595, 1511.XXXXX 1 BLTP JINR LUTh, Meudon, 2015

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Linear dilaton for asymptotically Lifshitz-like spacetimes

Anastasia Golubtsova1

based on a collabotation with Irina Ia. Aref’eva and Eric Gourgoulhon JHEP 1504 (2015) (011),arXiv:1410.4595, 1511.XXXXX

1BLTP JINR

LUTh, Meudon, 2015

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Outline

1

Motivation

2

Asymptotycally Lifshitz backgrouds

3

Linear dilaton

4

Out of equillibrium

5

Summary and Outlook

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 2

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Motivation

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 3

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Strongly coupled systems

Ultra-cold atoms High temperature conductors Quantum liquids QUARK-GLUON PLASMA THE BIG BANG

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 4

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Strongly coupled systems

Ultra-cold atoms High temperature conductors Quantum liquids QUARK-GLUON PLASMA THE BIG BANG

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 4

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Strongly coupled systems

Ultra-cold atoms High temperature conductors Quantum liquids QUARK-GLUON PLASMA THE BIG BANG

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 4

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Strongly coupled systems

Ultra-cold atoms High temperature conductors Quantum liquids QUARK-GLUON PLASMA THE BIG BANG

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 4

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

The quark-gluon plasma (2005)

Experiments on Heavy Ion Collisions at RHIC and LHC: A new state of matter: deconfined quarks, antiquarks, and gluons at high temperature. QGP does not behave like a weakly coupled gas of quarks and gluons, but a strongly coupled fluid. τtherm(0.1fm/c) < τhydro < τhard(10fm/c) < τf (20fm/c)

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 5

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Difficulties and solution

Quantum field theories with large coupling constant: long distances, strong forces Perturbative methods are inapplicable No consistent quantum field theory at strong coupling SOLUTION ? GAUGE/GRAVITY DUALITY A correspondence between the gauge theory in D Minkowski spacetime and supergravity in (D + 1) AAdS ’t Hooft’ 93, Susskind’94. Example: The AdS/CFT correspondence J.M. Maldacena, Adv.Theor.Math.Phys. 2, (1998). Supergravity theories in AdS-backgrounds Gravity theories with scalar fields, form fields in AdS. etc.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 6

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Difficulties and solution

Quantum field theories with large coupling constant: long distances, strong forces Perturbative methods are inapplicable No consistent quantum field theory at strong coupling SOLUTION ? GAUGE/GRAVITY DUALITY A correspondence between the gauge theory in D Minkowski spacetime and supergravity in (D + 1) AAdS ’t Hooft’ 93, Susskind’94. Example: The AdS/CFT correspondence J.M. Maldacena, Adv.Theor.Math.Phys. 2, (1998). Supergravity theories in AdS-backgrounds Gravity theories with scalar fields, form fields in AdS. etc.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 6

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Difficulties and solution

Quantum field theories with large coupling constant: long distances, strong forces Perturbative methods are inapplicable No consistent quantum field theory at strong coupling SOLUTION ? GAUGE/GRAVITY DUALITY A correspondence between the gauge theory in D Minkowski spacetime and supergravity in (D + 1) AAdS ’t Hooft’ 93, Susskind’94. Example: The AdS/CFT correspondence J.M. Maldacena, Adv.Theor.Math.Phys. 2, (1998). Supergravity theories in AdS-backgrounds Gravity theories with scalar fields, form fields in AdS. etc.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 6

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Difficulties and solution

Quantum field theories with large coupling constant: long distances, strong forces Perturbative methods are inapplicable No consistent quantum field theory at strong coupling SOLUTION ? GAUGE/GRAVITY DUALITY A correspondence between the gauge theory in D Minkowski spacetime and supergravity in (D + 1) AAdS ’t Hooft’ 93, Susskind’94. Example: The AdS/CFT correspondence J.M. Maldacena, Adv.Theor.Math.Phys. 2, (1998). Supergravity theories in AdS-backgrounds Gravity theories with scalar fields, form fields in AdS. etc.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 6

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Difficulties and solution

Quantum field theories with large coupling constant: long distances, strong forces Perturbative methods are inapplicable No consistent quantum field theory at strong coupling SOLUTION ? GAUGE/GRAVITY DUALITY A correspondence between the gauge theory in D Minkowski spacetime and supergravity in (D + 1) AAdS ’t Hooft’ 93, Susskind’94. Example: The AdS/CFT correspondence J.M. Maldacena, Adv.Theor.Math.Phys. 2, (1998). Supergravity theories in AdS-backgrounds Gravity theories with scalar fields, form fields in AdS. etc.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 6

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Holographic dictionary

d gravity on AdS = the (d − 1) strongly coupled theory T = 0 : AdS vacuum, T = 0: black-hole solutions in AdS. 4d Multiplicity in HIC = BH entropy in AdS5 Gubster et al.’08 Thermalization time in M1,3 = BH formation time in AdS5 Non-local observables: Wilson loops, Entarglement entropy, two point correlators. PROFIT?

  • Calculations in gravitational backgrounds with certain

asymptotics

  • Reduciton to classical mechanics

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 7

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Multiplicity: experimental data, theoretical estimation

D = 4 Multiplicity = Area of trapped surface in D = 5

Experiment:

Sdata = s0.15

NN

ALICE collaboration’10 Modified AdS:

Sdata = s0.12

NN

Kiritis & Taliotis’11

Modified AdS+ ghosts: Sdata = s0.16

NN

Aref’eva et al.’14 ALICE collaboration’10

  • The QGP is spartially anisotropic

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 8

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Multiplicity: experimental data, theoretical estimation

D = 4 Multiplicity = Area of trapped surface in D = 5

Experiment:

Sdata = s0.15

NN

ALICE collaboration’10 Modified AdS:

Sdata = s0.12

NN

Kiritis & Taliotis’11

Modified AdS+ ghosts: Sdata = s0.16

NN

Aref’eva et al.’14 ALICE collaboration’10

  • The QGP is spartially anisotropic

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 8

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Asymptotycally Lifshitz-like backgrouds and holography

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 9

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Outline

1 Spacetimes with Lifshitz scaling 2 Lifshitz-like backgrounds for holography 1 Lifshitz-like metrics 2 Shock waves in Lishitz spacetimes 3 Lifshitz-like backgrounds with spherical symmetry 1 Lifshitz black holes 2 Lifshitz-Vadya solutions 3 Boson stars in Lifshitz-like backgrounds

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 10

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Lifshitz scaling

The AdS/CFT correspondence: The Field Theory The Gravitational Background

  • the conformal group SO(D, 2)
  • the group of isometries
  • f a D-dimensional CFT
  • f AdSD+1

(t, xi) → (λt, λxi) , i = 1, .., d − 1 ds2 = r2 −dt2 + d x2

d−1

  • + dr2

r2

Generalizations? Lifshitz scaling: t → λνt,

  • x → λ

x, r → 1

λr,

where ν is the Lifshitz dynamical exponent Lifshitz metric: ds2 = −r2νdt2 + dr2 r2 + r2d x2

d−1

Kachru, Liu, Millgan ’08

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 11

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Lifshitz scaling

The AdS/CFT correspondence: The Field Theory The Gravitational Background

  • the conformal group SO(D, 2)
  • the group of isometries
  • f a D-dimensional CFT
  • f AdSD+1

(t, xi) → (λt, λxi) , i = 1, .., d − 1 ds2 = r2 −dt2 + d x2

d−1

  • + dr2

r2

Generalizations? Lifshitz scaling: t → λνt,

  • x → λ

x, r → 1

λr,

where ν is the Lifshitz dynamical exponent Lifshitz metric: ds2 = −r2νdt2 + dr2 r2 + r2d x2

d−1

Kachru, Liu, Millgan ’08

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 11

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Lifshitz scaling

The AdS/CFT correspondence: The Field Theory The Gravitational Background

  • the conformal group SO(D, 2)
  • the group of isometries
  • f a D-dimensional CFT
  • f AdSD+1

(t, xi) → (λt, λxi) , i = 1, .., d − 1 ds2 = r2 −dt2 + d x2

d−1

  • + dr2

r2

Generalizations? Lifshitz scaling: t → λνt,

  • x → λ

x, r → 1

λr,

where ν is the Lifshitz dynamical exponent Lifshitz metric: ds2 = −r2νdt2 + dr2 r2 + r2d x2

d−1

Kachru, Liu, Millgan ’08

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 11

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Lifshitz scaling

The AdS/CFT correspondence: The Field Theory The Gravitational Background

  • the conformal group SO(D, 2)
  • the group of isometries
  • f a D-dimensional CFT
  • f AdSD+1

(t, xi) → (λt, λxi) , i = 1, .., d − 1 ds2 = r2 −dt2 + d x2

d−1

  • + dr2

r2

Generalizations? Lifshitz scaling: t → λνt,

  • x → λ

x, r → 1

λr,

where ν is the Lifshitz dynamical exponent Lifshitz metric: ds2 = −r2νdt2 + dr2 r2 + r2d x2

d−1

Kachru, Liu, Millgan ’08

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 11

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Lifshitz-like spacetimes for holography

  • Lifshitz-like metrics

ds2 = r2ν −dt2 + dx2 + r2dy2

1 + r2dy2 2 + dr2

r2 , (t, x, y, r) → (λνt, λνx, λy1, λy2, r

λ), M. Taylor’08, Pal’09.

Gauge/gravity duality: theory with T = 0.

  • Shock-waves in Lifshitz-spacetimes

ds2 = φ(y1, y2, z)δ(u) z2 du2 − 1 z2 dudv + 1 z2/ν

  • dy2

1 + dy2 2

  • + dz2

z2 ,

u = t − x and v = t + x, z = 1/rν, I.Ya.Aref’eva, AG’14. Gauge/gravity duality: Multiplicity in HIC in D = 4 can be estimated by the area of trapped surface in AdS5 formed in collision of shock waves. Gubster, Pufu, Yarom’09

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 12

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Multiplicity: experimental data, theoretical estimation

D = 4 Multiplicity = Area of trapped surface in D = 5

Experiment:

Sdata = s0.15

NN

ALICE collaboration’10 Modified AdS:

Sdata = s0.12

NN

Kiritis & Taliotis’11

Modified AdS+ ghosts: Sdata = s0.16

NN

Aref’eva et al.’14 ALICE collaboration’10

5d Lifshitz spacetimes ν = 4, ν = −2 Sdata = s0.16

NN

Aref’eva & A.G.’14

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 13

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Multiplicity: experimental data, theoretical estimation

D = 4 Multiplicity = Area of trapped surface in D = 5

Experiment:

Sdata = s0.15

NN

ALICE collaboration’10 Modified AdS:

Sdata = s0.12

NN

Kiritis & Taliotis’11

Modified AdS+ ghosts: Sdata = s0.16

NN

Aref’eva et al.’14 ALICE collaboration’10

5d Lifshitz spacetimes ν = 4, ν = −2 Sdata = s0.16

NN

Aref’eva & A.G.’14

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 13

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Linear dilaton and asymptotycally Lifshitz-like metric

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 14

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Possible models

A massive form field S =

1 16πG5

  • d5x
  • |g|
  • R + Λ − 1

6

  • H2

(3) + m2 0B2 (2)

  • ,

with H(3) = dB(2), Λ is negative cosmological constant B(2) =

  • ν−1

ν L2r2νdt ∧ dx,

H(3) = 2ν

  • ν−1

ν L2r2ν−1dr ∧ dt ∧ dx

m0 = ν

L2 ,

c2 = (ν+1)ν

16L2 ,

Λ = − 4ν2+ν+1

2L2

.

  • M. Taylor’08

Lifshitz-metrics Shock waves and its collision No analytic black hole solutions

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 15

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Lifshitz black holes, Lifshitz-Vaidya, etc.

  • Black holes in Lifshitz background

ds2 = r2ν −f(r)dt2 + dx2 + r2 dy2

1 + dy2 2

  • +

dr2 r2f(r), where f(r) = 1 − m r2ν+2 .

Gravity/gauge duality: T = 0, non-local observables in equillibrium.

  • Lifshitz-Vaidya metrics, a shell falling along v = 0.

ds2 = −f(v, z) z2 dv2 − 2dvdz z2 + dx2 z2 + (dy2

1 + dy2 3)

z2/ν , f = 1 − m(v)z2ν+2, m(v)defines the thickness of the shell.

Gravity/gauge duality: non-local observables out of equillibrium.

  • Boson stars in Lifshitz background

Gravity/gauge duality: condensed matter S.A Hartnoll’11

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 16

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Linear dilaton

Linear dilaton field and 2-form fields S = 1 16πG5

  • d5x
  • |g|
  • R[g] + Λ − 1

2∂Mφ∂Mφ − 1 4eλφF2

(2)

  • .

The Einstein equations are Rmn = −Λ 3 gmn + 1 2(∂mφ)(∂nφ) + 1 4eλφ (2FmpFp

n) − 1

12eλφF2gmn. The scalar field equation φ = 1 4λeλφF2, with φ = 1

  • |g|

∂m(gmn |g|∂nφ). The gauge field obeys the following equation Dm

  • eλφFmn

= 0.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 17

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Black hole (brane) solutions

ds2 = e2νr −f(r)dt2 + dx2 + e2r dy2

1 + dy2 2

  • + dr2

f(r),

where f(r) = 1 − me−(2ν+2)r. Aref’eva,AG, Gourgoulhon’15 F(2) = 1

2qdy1 ∧ dy2,

φ = φ(r), eλφ = µe4r. ν = 4, λ = ± 2 √ 3 , Λ = 90, µq2 = 240. SUGRA IIA on M = X(1)5 × X(2)5,: F(2), F(4), H(3). with F(2) = 1 2qdy1 ∧ dy2, F(4) ∼ const, H3 = 0, Azeyanagi et al. ’09 5d GAUGED U(1)3 SUGRA ⇒ SUGRA IIB Gauntlett et al.’11

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 18

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Black hole (brane) solutions

ds2 = e2νr −f(r)dt2 + dx2 + e2r dy2

1 + dy2 2

  • + dr2

f(r),

where f(r) = 1 − me−(2ν+2)r. Aref’eva,AG, Gourgoulhon’15 F(2) = 1

2qdy1 ∧ dy2,

φ = φ(r), eλφ = µe4r. ν = 4, λ = ± 2 √ 3 , Λ = 90, µq2 = 240. SUGRA IIA on M = X(1)5 × X(2)5,: F(2), F(4), H(3). with F(2) = 1 2qdy1 ∧ dy2, F(4) ∼ const, H3 = 0, Azeyanagi et al. ’09 5d GAUGED U(1)3 SUGRA ⇒ SUGRA IIB Gauntlett et al.’11

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 18

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

From Lifshitz to AdS asymptotycs

  • Let φ = const and F2 = 0

Black hole solutions with AdS asymptotics ds2 = e2r −f(r)dt2 + dx2 + e2r dy2

1 + dy2 2

  • + dr2

f(r), where f(r) = 1 − me−4r. ds2 = ˜ r2 −f(˜ r)dt2 + dx2 + ˜ r2(dy2

1 + dy2 2) +

d˜ r2 f(˜ r)˜ r2 , f(˜ r) = 1 − m

˜ r4 .

Corresponds to the UV limit.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 19

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

From Lifshitz to AdS asymptotycs

  • Let φ = const and F2 = 0

Black hole solutions with AdS asymptotics ds2 = e2r −f(r)dt2 + dx2 + e2r dy2

1 + dy2 2

  • + dr2

f(r), where f(r) = 1 − me−4r. ds2 = ˜ r2 −f(˜ r)dt2 + dx2 + ˜ r2(dy2

1 + dy2 2) +

d˜ r2 f(˜ r)˜ r2 , f(˜ r) = 1 − m

˜ r4 .

Corresponds to the UV limit.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 19

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

From Lifshitz to AdS asymptotycs

  • Let φ = const and F2 = 0

Black hole solutions with AdS asymptotics ds2 = e2r −f(r)dt2 + dx2 + e2r dy2

1 + dy2 2

  • + dr2

f(r), where f(r) = 1 − me−4r. ds2 = ˜ r2 −f(˜ r)dt2 + dx2 + ˜ r2(dy2

1 + dy2 2) +

d˜ r2 f(˜ r)˜ r2 , f(˜ r) = 1 − m

˜ r4 .

Corresponds to the UV limit.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 19

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Out of equillibrium

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 20

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Construction of Lifshitz-Vaidya spacetimes

The Lifshitz-like metric z = 1 rν ds2 = z−2 −f(z)dt2 + dx2 + z−2/ν(dy2

1 + dy2 2) + dz2 z2f(z),

f = 1 − mz2/ν+2. The Eddington-Finkelstein coordinates: dv = dt + dz f(z). A matter shell infalling in Lifshitz background ds2 = −z−2f(z)dv2 − 2z−2dvdz + z−2dx2 + z−2/ν(dy2

1 + dy2 2),

f = 1 − m(v)z2/ν+2, v < 0 − inside the shell, v > 0 − outside, F2 = 1

2qdy1 ∧ dy2,

λφ = 4r + r0.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 21

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Thermalization

Def. Thermalization time at scale l is the time at which the tip of the geodesic with endpoints (−l/2) and (l/2) grazes the middle of the shell. The Lagrangian of the pointlike probe L =

  • −f(z, v)

z2 dv dτ dv dτ − 2 z2 dv dτ dz dτ + 1 z2 dx dτ dx dτ + 1 z2/ν dyi dτ dyi dτ

  • τ = x or τ = yi, i = 1, 2.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 22

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Thermalization time

Let τ = x, the Lagrangian L = √ R/z

The integrals of motion

J = − 1 z √ R = − 1 z2L I1 = f(z)v

′

x + z

′

x

z √ R , I2 = z−2/νy

′

1,x

√ R , I3 = z−2/νy

′

2,x

√ R , where R = 1 − f(z)(v

′

x)2 − 2v

′

xz

′

x + z2−2/ν((y

′

1,x)2 + (y

′

2,x)2).

z

′

x = ±

  • f(z)
  • 1

z2J 2 − z2/ν I2

2

J 2 + I2

3

J 2

  • − 1
  • + I2

1

J 2 , x = ±

  • dz
  • f(z)
  • 1

z2J 2 − z2/ν

I2

2

J 2 + I2

3

J 2

  • − 1
  • + I2

1

J 2

.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 23

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Thermalization time

The turning point can be found from f(z∗) 1 z2

∗

− z2/ν

∗

  • I2

2 + I2 3

  • − J 2
  • + I2

1 = 0.

xI1=I2=I3=0 = ± z∗

ǫ

dz

  • f(z)
  • 1

z2 1

z2 ∗

− 1 . The thermalization time ttherm at scale l ttherm = z∗

ε

dz f(z), ℓ = 2 z∗

ǫ

dz

  • f(z)
  • 1

z2 1

z2 ∗

− 1 .

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Thermalization in the x-direction

0.5 1.0 1.5 2.0 2.5 [] 0.2 0.4 0.6 0.8 1.0 1.2

τ[]

Figure : Dependencies of τ on ℓ for the 5-dimensional Lifshitz metric for ν = 2, ν = 3, ν = 4 with m = 0.1 and m = 0, 5.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 25

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

τ = yi

L = √ R z , R = 2z2−2/ν − f(z)(˙ vy)2 − 2˙ vy˙ zy + (˙ x)2. The integrals of motion read J = −2z1−2/ν √ R = − 2 z2/νL, I1 = f(z)˙ vy + ˙ zy z √ R , I2 = − ˙ xy z √ R . (5.1) ℓ = 2 z∗

ǫ

dz

  • 2z2−2/νf(z)
  • z2/ν

∗

z2/ν − 1

, ttherm = z∗

ǫ

dz f(r).

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 26

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Thermalization in the yi-direction

1 2 3 4 5 6

[]

0.2 0.4 0.6 0.8 1.0 1.2

τ[]

Figure : Dependencies of the thermalization times τ on ℓ for the Lifshitz metric for ν = 2, ν = 3 and ν = 4 (left to right). The solid and dotted curves correspond to m = 0.5 and m = 0.1, respectively

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Summary and Outlook

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Summary and Outloook

Done

1 The shock waves and estimations of multiplicity 2 Black brane and Vaidya solutions in Lifshitz-like

backgrounds

3 Computation of thermalization time 4 Wilson loops and Entanglement entropy in black brane

background IN PROGRESS

1 Non-local observables in Lifshitz-Vaidya background 2 Lifshitz boson stars and interpretation condensed matter ? 3 The underlying theory ??? 4 Interpolating solutions Lif5 ⇒ AdS5,

AdS2/CFT1, 1d CFT = SQM.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 29

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Summary and Outloook

Done

1 The shock waves and estimations of multiplicity 2 Black brane and Vaidya solutions in Lifshitz-like

backgrounds

3 Computation of thermalization time 4 Wilson loops and Entanglement entropy in black brane

background IN PROGRESS

1 Non-local observables in Lifshitz-Vaidya background 2 Lifshitz boson stars and interpretation condensed matter ? 3 The underlying theory ??? 4 Interpolating solutions Lif5 ⇒ AdS5,

AdS2/CFT1, 1d CFT = SQM.

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 29

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Motivation Asymptotycally Lifshitz backgrouds Linear dilaton Out of equillibrium Summary and Outlook

Thank you for your attention!

Anastasia Golubtsova — Linear dilaton for asymptotically Lifshitz-like spacetimes 30