Geons in Asymptotically Anti-de Sitter spacetimes Grgoire Martinon - - PowerPoint PPT Presentation

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Geons in Asymptotically Anti-de Sitter spacetimes Grgoire Martinon Observatoire de Paris Universit Paris Diderot 6 Octobre 2015 Grgoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 1 / 24 AdS/CFT correspondance AdS/CFT

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SLIDE 1

Geons in Asymptotically Anti-de Sitter spacetimes

Grégoire Martinon

Observatoire de Paris Université Paris Diderot

6 Octobre 2015

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 1 / 24

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SLIDE 2

AdS/CFT correspondance

AdS/CFT correspondance

(Anti-de Sitter/Conformal Field Theory)

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 2 / 24

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SLIDE 3

AdS/CFT correspondance AdS/CFT

AdS/CFT

Seminal paper The Large N Limit of Superconformal Field Theories and Supergravity

  • J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231

More than 11 000 citations !

E

Mc2

Casimir

AdS5

=

Schwarzschild Quark-gluon plasma

AdS to CFT CFT to AdS

=

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 3 / 24

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SLIDE 4

AdS/CFT correspondance AdS/CFT

Holographic principle

Strongly coupled 4D gauge theory = Grav. theory in 5D AAdS QCD, QED at strong coupling is hard Super gravity in AAdS much easier AdS5 dual to N = 4 Super Yang-Mills Black hole thermodynamics AAdS without Black Hole : T = 0 AAdS with Black Hole : T = 0

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 4 / 24

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SLIDE 5

Anti-de Sitter spacetime

Anti-de Sitter (AdS) spacetime

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 5 / 24

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SLIDE 6

Anti-de Sitter spacetime Negative cosmological constant

Negative cosmological constant

AdS = unique maximally symmetric solution of Einstein with Λ < 0 Rµν − R 2 gµν + Λgµν = 0 AdS 10 Killing vectors AdS length L : Λ = − 3 L2 Constant curvature : R = −12 L2 Positive ! Negative !

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 6 / 24

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SLIDE 7

Anti-de Sitter spacetime Radial Geodesics

Radial geodesics

t r

null

timelike

πL

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 7 / 24

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SLIDE 8

Anti-de Sitter spacetime Conformal representation

Conformal representation

ds2 = − 1 + ρ2 1 − ρ2 2 dt2+ 4 (1 − ρ2)2

  • dr 2 + r 2(dθ2 + sin2 θdϕ2)
  • ρ = r

L

  • Properties

compactified r ∈ [0, L] boundary at r = L ε = 1 − ρ2 1 + ρ2 gαβ = O 1

ε2

  • ˆ

gαβ = ε2gαβ = O(1)

L

2

ϵ = 0

Conformal metric : dˆ s2 = −dt2 + 4 (1 + ρ2)2

  • dr 2 + r 2(dθ2 + sin2 θdϕ2)
  • Grégoire Martinon (LUTH)

Geons in AAdS spacetimes 6 Octobre 2015 8 / 24

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SLIDE 9

Asymptotically AdS spacetime

Asymptotically AdS (AAdS) spacetime

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 9 / 24

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SLIDE 10

Asymptotically AdS spacetime AdS + something

AdS + something

AdS

AAdS

1 1 1 1

Problem : gαβ diverges like O 1

ε2

  • at the boundary !

Cure : conformal structure at the boundary AAdS boundary conditions (Ashtekar and Magnon 1984) ε2gαβ =

r=L diag(−1, 1, 1, 1)

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 10 / 24

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SLIDE 11

Asymptotically AdS spacetime Global quantities

Global quantities

Weyl tensor Definition : Cαβµν = Rαβµν − [trace part] Cα

βαν = Cα αµν = Cα βµα = 0

Conformal invariance : ˆ gαβ = ε2gαβ ⇒ ˆ Cαβµν = Cαβµν AdS boundary : ⇒ Cαβµν = O(ε) Global quantities (Ashtekar and Das 2000)

Define : ˆ Kαβµν = ˆ Cαβµν ε = O(1), nα = ∇αε and nα = ˆ gαβ∇βε

M = L3 8π

  • S∞

ˆ Kαβµνnβnν(∂t)αuµ√ ˆ σd2y J = L3 8π

  • S∞

ˆ Kαβµνnβnν(∂ϕ)αuµ√ ˆ σd2y

n u

t

∂ Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 11 / 24

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SLIDE 12

Geons in AAdS spacetime

Geons in AAdS spacetime

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 12 / 24

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SLIDE 13

Geons in AAdS spacetime What’s a geon ?

What’s a geon ?

GEON = Electro-Gravitational Entity Seminal papers : 50’s and 60’s Wheeler, Power, Brill, Ernst, Melvin, Hartle, Thorne, Kaup At the beginning : asymptotically flat cylindrical, toroidal, spherical EM/ν/GW/φ wave packet Geon properties need rotation to avoid collapse photon self attraction Black body radiation

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 13 / 24

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SLIDE 14

Geons in AAdS spacetime Linear geon in AAdS

Linear geon in AAdS

Seminal papers (AAdS) : 2010’s Bizon, Rostworowski, Maliborski, Dias, Horowitz, Santos, Kodama, Ishibashi, Seto, Wald Properties vaccum solution need rotation to avoid collapse stationnary in corotating frame ⇒ Helical Killing vector Mathematics gαβ = ¯ gαβ + Ahij with A ≪ 1 Rµν − R

2 gµν + Λgµν = O(A2)

Tensor spherical harmonics

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 14 / 24

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SLIDE 15

Geons in AAdS spacetime Linear geon in AAdS

  • 0.02
  • 0.01

0.01 0.02

  • 0.04
  • 0.03
  • 0.02
  • 0.01

0.01 0.02 0.03 0.04

  • 0.04
  • 0.02

0.02 0.04

ˆ n in z = 0 plane By in z = 0 plane ˆ hxy in z = 0 plane

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15

ˆ hxx in x = 0 plane ˆ hyy in x = 0 plane ˆ hzz in z = 0 plane

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 15 / 24

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SLIDE 16

Geons in AAdS spacetime What is a geon useful for ?

What is a geon useful for ?

AdS/CFT dual representation : No black hole ⇒ T = 0

Geon in AAdS Spin-2 Bose-Einstein (glueball)

Explore a non-linear stability island of AdS spacetime 1, 2

  • 1. P

. Bizo´ n and A. Rostworowski. Physical Review Letters, 107, July 2011

  • 2. P

. Bizo´ n, M. Maliborski, and A. Rostworowski. ArXiv, June 2015, 1506.03519

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 16 / 24

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Geons in AAdS spacetime Non-linear geon

Non-linear geon

Step by step

  • 1. Find linear geon
  • 2. AAdS-AdS formulation
  • 3. Choose a gauge
  • 4. Invert Einstein system

What you need : Kodama-Ishibashi formalism Spectral method Numerical library : KADATH Newton-Raphson in dimension 104 Horowitz and Santos 2014

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 17 / 24

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Conclusion

Conclusion

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 18 / 24

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Conclusion

Conclusion

State of the art : small sequences of geons computed with central amplitude ∼ 10% of AdS at several resolutions TO-DO list : compute larger geon sequences maximum mass and maximum angular momentum of a geon ? check stability with evolution code AdS/CFT interpretation add ingredients (black holes, boson stars. . .)

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 19 / 24

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Conclusion

Holy grail of AdS gravitational systems : Exact non-coalescing binaries with exact helical symmetry

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 20 / 24

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SLIDE 21

Conclusion

Thank you for your attention

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 21 / 24

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SLIDE 22

References

References

P . Bizo´ n and A. Rostworowski. Physical Review Letters, 107, July 2011. P . Bizo´ n, M. Maliborski, and A. Rostworowski. ArXiv, June 2015, 1506.03519.

  • L. Andersson and V. Moncrief.

Elliptic-hyperbolic systems and the einstein equations. Annales Henri Poincaré, 4(1) :1–34, 2003. Dennis M. DeTurck. Deforming metrics in the direction of their ricci tensors.

  • J. Differential Geom., 18(1) :157–162, 1983.

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 22 / 24

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SLIDE 23

References

References (cont.)

  • M. Headrick, S. Kitchen, and T. Wiseman.

A new approach to static numerical relativity and its application to Kaluza-Klein black holes. Classical and Quantum Gravity, 27(3) :035002, February 2010, 0905.1822. P . Figueras, J. Lucietti, and T. Wiseman. Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua. Classical and Quantum Gravity, 28(21) :215018, November 2011, 1104.4489. Juan Maldacena. The large-n limit of superconformal field theories and supergravity. International Journal of Theoretical Physics, 38(4) :1113–1133, 1999.

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 23 / 24

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SLIDE 24

References

References (cont.)

  • H. Kodama, A. Ishibashi, and O. Seto.
  • Phys. Rev. D, 62(6) :064022, September 2000, hep-th/0004160.
  • A. Ishibashi and R. M. Wald.

Classical and Quantum Gravity, 21 :2981–3013, June 2004, hep-th/0402184.

Grégoire Martinon (LUTH) Geons in AAdS spacetimes 6 Octobre 2015 24 / 24