Black holes and the leaking faucet in your bathroom

Transcription

1 Black holes and the leaking faucet in your bathroom Nicolas Vasset Journal club May 5th, 2011 Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 1 / 17

2 Based on the following articles [Lehner & Pretorius, Phys. Rev. Lett. 105, (2010)]: Black strings, low viscosity fluids and violation of cosmic censorship. [Camps et al., JHEP 05, 042 (2010)]: Black brane viscosity and the Gregory-Laflamme instability. [Cardoso & Dias, Phys. Rev. Lett. 96, (2006)]: Rayleigh-Plateau and Gregory-Laflamme instabilities of black strings. [Gregory & Laflamme, Phys. Rev. Lett. 70, 19 (1993)] Black Strings and p-branes are unstable. Your favorite hydrodynamics book... or if you don t have one, [Drazin, Cambridge U.P (2002)], An introduction to hydrodynamics stability. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 2 / 17

3 Black holes in 5 dimensions!! Why would anyone study this? Because it s fun! The world is more than 4-dimensional in string theory, quantum gravity, AdS-CFT (5D), brane cosmology...you name it! Black hole physics is much more rich in 5D than 4D (see later). It could teach us a lot about GR theory itself (see later about the cosmic censorship). Very unexpected features of black hole dynamics in 5D (point of this talk). Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 3 / 17

4 Black holes in 5 dimensions!! Why would anyone study this? Because it s fun! The world is more than 4-dimensional in string theory, quantum gravity, AdS-CFT (5D), brane cosmology...you name it! Black hole physics is much more rich in 5D than 4D (see later). It could teach us a lot about GR theory itself (see later about the cosmic censorship). Very unexpected features of black hole dynamics in 5D (point of this talk). Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 3 / 17

5 Black holes in 5 dimensions!! Why would anyone study this? Because it s fun! The world is more than 4-dimensional in string theory, quantum gravity, AdS-CFT (5D), brane cosmology...you name it! Black hole physics is much more rich in 5D than 4D (see later). It could teach us a lot about GR theory itself (see later about the cosmic censorship). Very unexpected features of black hole dynamics in 5D (point of this talk). Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 3 / 17

6 5D Kaluza-Klein spacetime Context of Kaluza-Klein theory (1921) for gravitation and elecromagnetism Topology: 4D spacetime a circle (wrapped, finite dimension) of length a. Assume 5D Einstein equations hold on this spacetime. Are there any black hole solutions on this? Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 4 / 17

7 5-dimensional black objects In 4 dimensions, stationary vacuum black hole solutions are exactly known: Schwarzschild, Kerr... No hair theorem! They all have spherical topology [Israel, 1968]. In KK, this is not true anymore... (Restricted to spherical symmetry in the 4D spacetime). All are stationary solutions of vacuum Einstein equations in this topology : very rich space of solutions, even in spherical symmetry! Objects fully characterized by mass M and tension T (see later). Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 5 / 17

8 Stability of the black holes/strings in 4D and 5D 4D black holes We know the static Schwarzschild solution is stable against linear/nonlinear perturbations. We almost know it is also true for the Kerr spacetimes... 5D black objects spheroidal black hole: the 4D theorem generalizes black string: doubtful... Entropy argument: a spherical solution in 5D has more entropy than the corresponding black string with the same mass: suspicion of instability. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 6 / 17

9 Black strings are linearly unstable [Gregory & Laflamme, 1993]: Study of linear stability of the black string solution of radius R 0 against perturbations. (D-dimensional Kaluza Klein spacetimes). (Linear) perturbations of the form P ρσ = h ρσ e Ωt+ikz. µ = k.r 0 2π. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 7 / 17

10 Generic axisymmetric instability of black strings, for modes with a small wavenumber. Modes only present if perturbations have a long enough wavelength! (impossible if the string length a is too small: parameter of the system!) one characteristic mode for each D, that will grow faster than the others. End state of the instability? A linear study cannot say anything about it... numerical simulations to the rescue! But hold on... Long wavelength unstabilities on a cylinder? Where have I seen this before? Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 8 / 17

11 Generic axisymmetric instability of black strings, for modes with a small wavenumber. Modes only present if perturbations have a long enough wavelength! (impossible if the string length a is too small: parameter of the system!) one characteristic mode for each D, that will grow faster than the others. End state of the instability? A linear study cannot say anything about it... numerical simulations to the rescue! But hold on... Long wavelength unstabilities on a cylinder? Where have I seen this before? Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 8 / 17

12 The Rayleigh-Plateau instability Take a uniform incompressible fluid flowing in a cylinder of length a, radius R 0. Euler equations are verified, gravity is neglected. Laplace-Young Surface tension T on the cylinder surface corresponds to a pressure difference p = T n = T/R 0, and corresponding potential energy. Consider a set of mode perturbations for speed, pressure, displacement: ( u, p, ζ) = (û(r), ˆp(r), ˆζ(r))e st+i(kx+nθ) (1) Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 9 / 17

13 Perturbed surface energy: δu T (k 2 R 2 0 1). unstable modes growth as a function of wavenumber (plus experimental points). modes unstable for wavelength λ = 2π/k 2πR 0. Only axisymmetric modes are unstable. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 10 / 17

14 Rayleigh and Plateau in your bathroom End state: the fluid cylinder breaks out into drops. timescale and size depend on the original radius (fluid properties) Instability also present when adding viscosity effects (see Eggers(1997)) [Eggers, 1997] Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 11 / 17

15 Rayleigh and Plateau on a black string [Cardoso and Dias, 2006]: Strong analogy between both instability problems: timescales, typical lengthes... Quantitative agreement in very high dimensions Unstable only in axisymmetry! Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 12 / 17

16 Going further in the analogy [Camps et al., 2010]: Perturbative nonlinear description of black strings in full general relativity. String dynamics are perturbatively described by a viscous hydrodynamics equation in a 3D cylinder. Gregory-Laflamme dispersion relation is derived using this description! (instability of sound modes). Can we extend the analogy beyond the linear part? End state of the system? Need of simulations: [Choptuik et al., 2004],[Garfinkle et al., 2004]. [Lehner & Pretorius, 2010]: first long-time simulation in 5D. Adaptive mesh refinement (ratio of 2) in time and space throughout evolution. Harmonic gauge for General relativity, cartesian coordinates 4th order explicit time evolution. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 13 / 17

17 Going further in the analogy [Camps et al., 2010]: Perturbative nonlinear description of black strings in full general relativity. String dynamics are perturbatively described by a viscous hydrodynamics equation in a 3D cylinder. Gregory-Laflamme dispersion relation is derived using this description! (instability of sound modes). Can we extend the analogy beyond the linear part? End state of the system? Need of simulations: [Choptuik et al., 2004],[Garfinkle et al., 2004]. [Lehner & Pretorius, 2010]: first long-time simulation in 5D. Adaptive mesh refinement (ratio of 2) in time and space throughout evolution. Harmonic gauge for General relativity, cartesian coordinates 4th order explicit time evolution. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 13 / 17

18 generation of satellites and ministrings Successively forming strings experience unstability, and generate smaller satellites. Smaller strings then form. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 14 / 17

19 Rayleigh Plateau for low-viscosity fluids Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 15 / 17

20 Self-similar generation of satellites Self similar dynamics in low-viscosity fluids: radius of filaments follow r(t) (t 0 t), where at t 0 filaments break up. Close to the same law for string radius! (self-similarity assumed). Filament break up at finite time. Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 16 / 17

21 So what s up? A strong analogy has been built between low viscosity fluid dynamics and 5D strings. Linear and non-linear regimes present strong analogies. However, hard to assess analytically in the non-linear case. Why is it interesting? Study of analogue systems for better understanding... Fluid on the boundary in AdS/CFT theories? Breakup of the string in finite time: violation of cosmic censorship!. A naked singularity forms classically (horizon bifurcation). First example in 5 dimensions. On this system, quantum gravity effects should appear at this point (formation of droplets). Further studies expected... Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 17 / 17

22 So what s up? A strong analogy has been built between low viscosity fluid dynamics and 5D strings. Linear and non-linear regimes present strong analogies. However, hard to assess analytically in the non-linear case. Why is it interesting? Study of analogue systems for better understanding... Fluid on the boundary in AdS/CFT theories? Breakup of the string in finite time: violation of cosmic censorship!. A naked singularity forms classically (horizon bifurcation). First example in 5 dimensions. On this system, quantum gravity effects should appear at this point (formation of droplets). Further studies expected... Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 17 / 17