Dilatonic Black Saturn

Transcription

1 Dilatonic Black Saturn Saskia Grunau Carl von Ossietzky Universität Oldenburg

2 Introduction In higher dimensions black holes can have various forms: Black rings Black di-rings Black saturns... Picture: Max Planck Institute for Gravitational Physics

3 Outline 1. The Black Saturn Solution 2. Adding the Dilaton 3. Analysis of the Dilatonic Black Saturn 4. Conclusion

4 Outline The Black Saturn Solution 1. The Black Saturn Solution 2. Adding the Dilaton 3. Analysis of the Dilatonic Black Saturn 4. Conclusion

5 The black saturn The Black Saturn Solution Metric of a black Saturn: ds 2 = H [ ( y ωψ ) ] 2 dt + + q dψ H x H y + H x {k 2 P ( dρ 2 + dz 2) + G y H y dψ 2 + G x H x dφ 2 } Stationary axisymmetric metric in canonical coordinates: ds 2 = G ab dx a dx b + e 2ν( dρ 2 + dz 2) The metric functions depend onρand z only: G ab = G ab (ρ, z) andν=ν(ρ, z). [ H. Elvang and P. Figueras, JHEP 0705, 050 (2007) ]

6 The Black Saturn Solution Metric functions of the black saturn Metric functions: H x = F [M c 2 1 M 1 + c 2 2 M 2 + c 1 c 2 M 3 + c 2 1 c2 2 M 4 [ ] H y = F 1µ 3 µ 1 M 0 c 2 1 µ 4 µ M ρ 2 1 c µ 1 µ M µ 1 µ ρ 2 + c 1 c 2 M 3 + c 2 1 c2 2 M µ 2 4 µ 1 c 1 R 1 M 0 M 1 c 2 R 2 ω ψ = 2 P = (µ 3 µ 4 +ρ 2 ) 2 (µ 1 µ 5 +ρ 2 )(µ 4 µ 5 +ρ 2 ) ], G x = ρ2 µ 4 µ 3 µ 5, M 0 M 2 + c 2 1 c 2 R 2 M 1 M 4 c 1 c 2 2 R 1 M 2 M 4, F G x F =µ 1 µ 5 (µ 1 µ 3 ) 2 (µ 2 µ 4 ) 2 (ρ 2 +µ 1 µ 3 )(ρ 2 +µ 2 µ 3 )(ρ 2 +µ 1 µ 4 )(ρ 2 +µ 2 µ 4 ) (ρ 2 +µ 2 µ 5 )(ρ 2 +µ 3 µ 5 )(ρ 2 +µ 2 1 )(ρ2 +µ 2 2 )(ρ2 +µ 2 3 )(ρ2 +µ 2 4 )(ρ2 +µ 2 5 ) Whereµ i = ρ 2 + (z a i ) 2 (z a i ), R i = ρ 2 + (z a i ) 2 and, G y = µ 3µ 5 µ 4, M 0 =µ 2 µ 2 5 (µ 1 µ 3 ) 2 (µ 2 µ 4 ) 2 (ρ 2 +µ 1 µ 2 ) 2 (ρ 2 +µ 1 µ 4 ) 2 (ρ 2 +µ 2 µ 3 ) 2, M 1 =µ 2 1 µ 2µ 3 µ 4 µ 5 ρ 2 (µ 1 µ 2 ) 2 (µ 2 µ 4 ) 2 (µ 1 µ 5 ) 2 (ρ 2 +µ 2 µ 3 ) 2, M 2 =µ 2 µ 3 µ 4 µ 5 ρ 2 (µ 1 µ 2 ) 2 (µ 1 µ 3 ) 2 (ρ 2 +µ 1 µ 4 ) 2 (ρ 2 +µ 2 µ 5 ) 2, M 3 = 2µ 1 µ 2 µ 3 µ 4 µ 5 (µ 1 µ 3 )(µ 1 µ 5 )(µ 2 µ 4 )(ρ 2 +µ 2 1 )(ρ2 +µ 2 2 )(ρ2 +µ 1 µ 4 )(ρ 2 +µ 2 µ 3 )(ρ 2 +µ 2 µ 5 ), M 4 =µ 2 1 µ 2µ 2 3 µ2 4 (µ 1 µ 5 ) 2 (ρ 2 +µ 1 µ 2 ) 2 (ρ 2 +µ 2 µ 5 ) 2.

7 The Black Saturn Solution Rod structure z-axis:ρ = 0 The z-axis is divided into intervalls, called rods. On the rods we have dim(ker[g(ρ = 0, z)]) = 1 Kernel: kerg ={ v G v = 0} t (1, 0, Ω BR ψ ) (1, 0, Ω BH ψ ) φ ψ (0, 1, 0) (0, 1, 0) 0 κ 3 κ 2 κ 1 1 (0, 0, 1) [ H. Elvang and P. Figueras, JHEP 0705, 050 (2007) ]

8 Outline Adding the Dilaton 1. The Black Saturn Solution 2. Adding the Dilaton 3. Analysis of the Dilatonic Black Saturn 4. Conclusion

9 The dilaton Adding the Dilaton Gravitational scalar field Φ Coupling: interaction of gravitational and electromagnetic field Action in 5-D EMD-theory: S = 1 d 5 x g (R 12 16πG Φ,µΦ,µ 14 ) e 2hΦ F µν F µν Kaluza-Klein value of the dilaton coupling constant: h = 2 6

10 Adding the Dilaton Construction of a Kaluza-Klein black hole 1. Add an extra coordinate U to any (D-dimensional) black hole metric: 2. Perfom a boost in the t-u plane: ds 2 D+1 = ds2 D + du t t cosh(β) + U sinh(β) U t sinh(β) + U cosh(β) 3. Compare the boosted metric to the Kaluza-Klein parametrization of a (D + 1)-dimensional metric ds 2 D+1 = e2ιφ g µν dx µ dx ν + e 2(D 2)ιΦ (du + A µ dx µ ) 2 and read off the metric g µν of the new D-dimensional Kaluza-Klein black hole, the vector potential A µ and the scalar potential Φ.

11 Adding the Dilaton Kaluza-Klein black hole in 5 dimensions Let ds 2 5 be the metric of a 5-dimensional rotating black hole: ds 2 5 = g ttdt 2 + 2g tψ dtdψ + g ψψ dψ 2 + g φφ dφ 2 + g ρρ dρ 2 + g zz dz 2. Then the corresponding Kaluza-Klein black hole has the metric dskk 2 = 1 V 2/3 g ttdt cosh(β) V 2/3 g tψdtdψ sinh(β)2 V 2/3 gtψdψ V 1/3 g ψψ dψ 2 + V 1/3 g φφ dφ 2 + V 1/3 g ρρ dρ 2 + V 1/3 g zz dz 2, where V = cosh(β) 2 + g tt sinh(β) 2.

12 Outline Analysis of the Dilatonic Black Saturn 1. The Black Saturn Solution 2. Adding the Dilaton 3. Analysis of the Dilatonic Black Saturn 4. Conclusion

13 Analysis of the Dilatonic Black Saturn The dilatonic black saturn The metric: ds 2 = 1 H [ y V 2/3 dt + cosh(β) 2( ω ) ] ψ 2 + q dψ H x H y + V 1/3 H x {k 2 P where V = cosh(β) 2 sinh(β) 2 H y H x. ( dρ 2 + dz 2) + G y H y dψ 2 + G x H x dφ 2 }

14 Analysis of the Dilatonic Black Saturn Saturn in equilibrium The black saturn consists of 2 black objects: a sphere and a ring. Conical singularity between the two objects The sphere and the ring can rotate differently, can have different Hawking temperatures,... Conditions for equilibrium 1. Mechanical equilibrium No conical singulariy:δ = 0 2. Thermodynamical equilibrium Same angular momentum: Ω BH ψ = ΩBR ψ Same Hawking temperature: T BH H Same electrostatic potential: Ψ BH el = T BR H = Ψ BR el

15 Analysis of the Dilatonic Black Saturn Parameter space: Concial singularity The dilaton parameter β has no influence on the conical singularity. Coloured area: thermodynamical equilibrium δ< 0: conical deficit δ> 0: conical excess Curved black line: mechanical equilibrium

16 Analysis of the Dilatonic Black Saturn Mechanical moment of inertia β has no influence on the mechanical moment of inertia. 1 I = Ω J M,δ I< 0: thin solutions I> 0: fat solutions : regular solutions (δ = 0)

17 Analysis of the Dilatonic Black Saturn Physical quantities of the dilatonic black saturn Physical quantities of the dilatonic solution in relation to the physical quantities without the dilaton (β = 0). Area of the horizon: A β = cosh(β) A 0 Hawking temperature: T β = Mass: M β = ( 2 3 sinh(β)2 + 1) M 0 1 cosh(β) T 0 Angular momentum: J β = cosh(β) J 0 Charge: Q = 2 3 sinh(β) cosh(β) M 0 Electrostatic potential: Ψ el = sinh(β) cosh(β)

18 Phase diagram Analysis of the Dilatonic Black Saturn Phase diagram of the dilatonic black saturn in equilibrium

19 Phase diagram Analysis of the Dilatonic Black Saturn j q-diagram a H q-diagrams

20 Analysis of the Dilatonic Black Saturn Temperature Temperature of the dilatonic black saturn in equilibrium S. Grunau (Uni Oldenburg) Dilatonic Black Saturn

21 Analysis of the Dilatonic Black Saturn The first law of thermodynamics First law of thermodynamics for vacuum, stationary, asymptotically flat spacetimes with conical singularities: dm = T H ds + ΩdJ + Ψ el dq AdP P: Pressure by conical singularity A T H : Area spanned by the conical singularity [C. Herdeiro, B. Kleihaus, J. Kunz and E. Radu, Phys. Rev. D. 81, (2010)] Analysis of thermodynamical stability: S Specific heat at constant angular momentum C J = T H T H Isothermal moment of inertiaǫ = J Ω TH,A J,A

22 Analysis of the Dilatonic Black Saturn Specific heat at constant J (fixed Q ensemble) S C J = T H T H J,A,Q : regular solutions β = 0 β = 1 β = 5

23 Analysis of the Dilatonic Black Saturn Isothermal moment of inertia (fixed Q ensemble) ǫ = J Ω TH,A,Q : regular solutions β = 0 β = 0.5 β = 1 It is not possible that C J andǫare positive at the same time. thermodynamically instable

24 Analysis of the Dilatonic Black Saturn Fixed Φ el ensemble specific heat at constant J isothermal moment of inertia Hereβ has no influence on C J andǫ. It is not possible that C J andǫare positive at the same time.

25 Outline Conclusion 1. The Black Saturn Solution 2. Adding the Dilaton 3. Analysis of the Dilatonic Black Saturn 4. Conclusion

26 Conclusion Summary Dilatonic black saturn: 1. Start with neutral solution and add new dimension 2. Boost with respect to time and the new dimension 3. Obtain dilatonic solution by dimensional reduction Analysis: The dilatonic black saturn is not thermodynamically stable.