Cosmological constant is a conserved charge

Transcription

1 Cosmological constant is a conserved Kamal Hajian Institute for Research in Fundamental Sciences (IPM) In collaboration with Dmitry Chernyavsky (Tomsk Polytechnic U.) arxiv: , to appear in Classical & Quantum Gravity Recent Trends in String Theory and Related Topics IPM, May 8 th 2018

2 Outline 1 History and motivations 2 Λ is a conserved 3 Conjugate chemical potential and first law of thermodynamics 2 / 24

3 3 / 24 History and motivations

4 4 / years ago: introduction of Λ

5 Why Einstein introduced Λ? Einstein static universe Einstein expected/assumed the universe to be: spatially homogenious and isotropic static ρ with some matter T µ ν = 0 P P 0, ρ > 0, P P 5 / 24

6 Why Einstein introduced Λ? Einstein static universe Einstein expected/assumed the universe to be: spatially homogenious and isotropic static ρ with some matter T µ ν = 0 P P 0, ρ > 0, P P Metric ds 2 = dt 2 + a 2 dr 2 ( 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdφ 2 ) { { Λ Equation of motion: 3 = 4πG 3 (ρ + 3P ) Λ > 0 k a = Λ πG 3 ρ k > 0 5 / 24 Einstein universe is spatially closed with positive Λ.

7 Instability Problems with Einstein universe Einstein universe is unstable: Λ Expansion, Λ Collapse 6 / 24

8 Problems with Einstein universe Instability Einstein universe is unstable: Λ Expansion, Λ Collapse Observation: universe is not static! In 1929 Hubble discovered that the universe is expanding. Λ is not needed anymore. Λ: Biggest Einstein s mistake. 6 / 24

9 Revival of Λ in 1997: Λ as dark energy ΛCDM: a successful model for the universe 7 / 24 About 68 percent of the universe is Λ!

10 AdS/CFT correspondence J. Maldacena 1997 IIb supergravity on AdS 5 S dim N = 4 SYM in large N 8 / 24 AdS is a solution to G µν + Λg µν = 0 for Λ < 0

11 1980 s: Λ as pressure Λ can be considered as a pressure term G µν = 8πG T µν ρ {}}{ Λ πG µ 0 P 8πGT µν Λg µν, T ν = Λ 0 0 8πG 0 0 P Λ 0 8πG P Λ 8πG ρ ρ + Λ 8πG, P P Λ 8πG Generalized first law of thermodynamics δq V δp 9 / 24

12 1980 s: Λ as pressure Λ can be considered as a pressure term G µν = 8πG T µν ρ {}}{ Λ πG µ 0 P 8πGT µν Λg µν, T ν = Λ 0 0 8πG 0 0 P Λ 0 8πG P Λ 8πG ρ ρ + Λ 8πG, P P Λ 8πG Generalized first law of thermodynamics δq V δp 9 / 24 S M Ω H J Φ H Q Hawking temperature Entropy Mass Horizon angular velocity Angular momentum Horizon electric potential Electric

13 Some questions about V δp 1) Λ is a parameter in the Lagrangian. What does δλ mean? δq V δp 2) All chemical potentials can be found around horizon. Why V can not be? on the horizon on the horizon on the horizon δq V δp 3) Why the order of Intensive/Extensive is reversed in V δp? δq V δp 10 / 24

14 What is V? (A)dS-Schwarzschild black hole V = r H A H 3 = 4πr3 H 3, (A)dS-Kerr black hole V = 4π(r 3 +r H H a2 + Gma2 ) 1+ Λa2 3, 3(1+ Λa2 3 ) In general, one can find V from the first law V : Thermodynamic volume (not a geometric volume) 11 / 24

15 Our motivations Is Λ a conserved? δq V δp How to define an alternative to V on the horizon? on the horizon Can we prove the first law? on the horizon on the horizon? on the horizon? δq V δp δs? = δm Ω H δq V δp 12 / 24

16 13 / 24 Λ is a conserved

17 Λ from a top form Introducing a top form F to the theory Action: d D x (L g + L M ) d D x (L g + L M ) 1 d D x 2 16πG D! F 2 F = da, F 2 = F µ1...µ D F µ 1...µ D, gauge symmetry { A A + dλ F F 14 / 24

18 Λ from a top form Introducing a top form F to the theory Action: d D x (L g + L M ) d D x (L g + L M ) 1 d D x 2 16πG D! F 2 F = da, F 2 = F µ1...µ D F µ1...µ D, gauge symmetry { A A + dλ F F Field equations (equations of motion) E g µν ± 2 (D 1)! (F ρ µρ 2...ρ D F 2...ρ D ν 1 2D F 2 g µν ) = 8πGT µν, µ F µµ2...µ D = 0 Field equations for the matter field 14 / 24

19 Λ from a top form Introducing a top form F to the theory Action: d D x (L g + L M ) d D x (L g + L M ) 1 d D x 2 16πG D! F 2 F = da, F 2 = F µ1...µ D F µ 1...µ D, gauge symmetry { A A + dλ F F Field equations (equations of motion) E g µν ± 2 (D 1)! (F ρ µρ 2...ρ D F 2...ρ D ν 1 2D F 2 g µν ) = 8πGT µν, µ F µµ 2...µ D = 0 F µ1...µ D = c g ϵ µ1...µ D, c = const. 0 Field equations for the matter field 14 / 24

20 Λ from a top form Introducing a top form F to the theory Action: d D x (L g + L M ) d D x (L g + L M ) 1 d D x 2 16πG D! F 2 F = da, F 2 = F µ1...µ D F µ1...µ D, gauge symmetry { A A + dλ F F Field equations (equations of motion) Replacing on-shell F µ1...µ D { E g µν + Λg µν = 8πGT µν, Λ = ±c 2 Field equations for the matter field A. Aurilia et al, 1980, M. J. Duff et al, 1980, S. Hawking 1984, M. Henneaux et al, / 24

21 In brief Λ: parameter of theory parameter of solution Λ = ±c 2, F µ1...µ D = c g ϵ µ1...µ D 15 / 24

22 Λ: conserved of global part of the gauge symmetry Noether current and for arbitrary λ J µ λ = µνρ3...ρd L ( µ A νρ3...ρ D ) δ F λa νρ3...ρ D = 4πG(D 2)! νλ ρ3...ρ D = ν Q µν Surface density Q µν λ = F µνρ 3...ρ D 4πG(D 2)! λ ρ 3...ρ D 16 / 24

23 Λ: conserved of global part of the gauge symmetry Noether current and for arbitrary λ J µ λ = µνρ3...ρd L ( µ A νρ3...ρ D ) δ F λa νρ3...ρ D = 4πG(D 2)! νλ ρ3...ρ D = ν Q µν Surface density Q µν λ = F µνρ3...ρ D 4πG(D 2)! λ ρ 3...ρ D ˆλ: global part of the gauge symmetry Considering the set of closed gauge transformation generators dλ = 0, global part of the gauge transformation is defined to be ˆλ = λ S λ. C: conserved of ˆλ 16 / 24 { λ ˆλ F c C = Qˆλ = ±c ˆλ = ±c g ϵ µ1...µ D S 4πG S 4πG.

24 Λ: conserved of global part of the gauge symmetry Noether current and for arbitrary λ J µ λ = µνρ3...ρd L ( µ A νρ3...ρ D ) δ F λa νρ3...ρ D = 4πG(D 2)! νλ ρ3...ρ D = ν Q µν Surface density Q µν λ = F µνρ3...ρ D 4πG(D 2)! λ ρ 3...ρ D ˆλ: global part of the gauge symmetry Considering the set of closed gauge transformation generators dλ = 0, global part of the gauge transformation is defined to be ˆλ = λ S λ. C: conserved of ˆλ 16 / 24 C = ± Λ 4πG, (or) Λ = ±(4πG)2 C 2

25 In brief Λ: parameter of theory parameter of solution conserved Λ = ±(4πG) 2 C 2 C : conserved of ˆλ 17 / 24

26 18 / 24 Conjugate chemical potential of Λ and generalized first law of thermodynamics

27 Conjugate chemical potential Inspiration: electric potential on the horizon Maxwell field: L = 1 4 F 2, F = da, A A + dλ, Φ H ζ H A H 19 / 24

28 Conjugate chemical potential Inspiration: electric potential on the horizon Maxwell field: L = 1 4 F 2, F = da, A A + dλ, Φ H ζ H A H Θ H : conjugate chemical potential for C ζ H A Θ H H 19 / 24

29 Conjugate chemical potential Inspiration: electric potential on the horizon Maxwell field: L = 1 4 F 2, F = da, A A + dλ, Φ H ζ H A H Θ H : conjugate chemical potential for C ζ H A Generalized first law of thermodynamics δq Θ H δc Θ H H 19 / 24

30 Conjugate chemical potential Inspiration: electric potential on the horizon Maxwell field: L = 1 4 F 2, F = da, A A + dλ, Φ H ζ H A H Θ H : conjugate chemical potential for C ζ H A Θ H Generalized first law of thermodynamics δq Θ H δc δp = Λ δc V Θ T H H δq V δp Λ 19 / 24 H

31 Proof of the generalized first law Based on K.H and M. M. Sheikh-Jabbari, Phys. Rev. D 93, no. 4, (2016). Basics: Focusing on backgrounds with symplectic symmetry generators Diffs + gauge transformations η = {ξ µ, λ, λ} 20 / 24

32 Proof of the generalized first law Based on K.H and M. M. Sheikh-Jabbari, Phys. Rev. D 93, no. 4, (2016). Basics: Focusing on backgrounds with symplectic symmetry generators Diffs + gauge transformations η = {ξ µ, λ, λ} Symplectic symmetry generators of s entropy: η S = 1 {ζ µ H, Φ H1, Θ H ˆλ}, angular momentum: ηj ={ φ, 0, 0} mass: η M ={ t, 0, 0}, electric : η Q ={0, 1, 0}, C : η C ={0, 0, ˆλ} Linearity of conserved s w.r.t generators: η S = η M Ω H η J Φ H η Q Θ H η C L.o.c.c.w.r.t.g δq Θ H δc 20 / 24

33 In brief Θ H is defined independent of the first law, and on the horizon ζ H A Θ H The generalized first law is proved H δq Θ H δc 21 / 24

34 Example: Kerr-AdS black hole ds 2 = θ ( 1 Λr 2 3 Ξ + ρ2 r dr 2 + ρ2 θ dθ 2, ρ 2 r 2 + a 2 cos 2 θ, θ f)dt 2 2 θ fa sin 2 θ dtdφ + ( r 2 +a 2 + fa 2 sin 2 θ ) sin 2 θ dφ 2 Ξ Λ (r 3 + 3ra 2 cos 2 θ+ Gma2 Ξ A= ) sin θ dt dθ dφ 3Ξ r (r 2 + a 2 )(1 Λr2 3 ) 2Gmr, θ 1 + Λa2 3 cos2 θ, Ξ 1 + Λa2 3, f 2Gmr ρ 2 Ξ 2 r H (1 Λa2 3 Λr2 a2 H r κ H = 2 H 2(r 2 + H a2 ) ), Ω H = 2 Λr H a(1 ) 3 Λ 4π(r 3 r 2 +, Θ H a2 H = Ξ ) +r H H a2 + Gma2 3Ξ M = m Ξ 2, J = ma Ξ, S = π(r2 + H a2 ), C = ± Λ 2 GΞ 4πG 22 / 24 δq Θ H δc

35 Conclusion We showed that Λ is a conserved δq Θ H δc, Λ = ±(4πG) 2 C 2 We defined Θ H on the horizon, and independent of first law on the horizon on the horizon on the horizon on the horizon δq Θ H δc, Θ H = We proved the generalized first law H ζ H A δq Θ H δc 23 / 24

36 Conclusion We showed that Λ is a conserved δq V δp, (8πG)P = Λ = ±(4πG) 2 C 2 We defined Θ H on the horizon, and independent of first law on the horizon on the horizon on the horizon on the horizon δq V δp, ΛV Θ H = We proved the generalized first law H ζ H A δq V δp 23 / 24

37 THANK YOU