1. Artificial black hole singularities M. Cadoni, University of Cagliari COSLAB 2005, Smolenice, August 31 2005 Based on M.C., Class. and Quant. Grav. 22 (2005) 409 M.C and S. Mignemi, gr-qc/0503059, gr-qc/0504143 Abstract We look for acoustic analogues of a spherical symmetric black hole with a pointlike source. We show that the gravitational system has a dynamical counterpart in the constrained, steady motion of a fluid with a planar source. The equations governing the dynamics of the gravitational system can be exactly mapped in those governing the motion of the fluid. The different meaning that singularities and sources have in fluid dynamics and in general relativity is also discussed.

2. COSLAB 2005 Introduction Since the seventies of the last century black hole physics plays a crucial role for fundamental theoretical physics : Since the seventies of the last century black hole physics plays a crucial role for fundamental theoretical physics :  Quantum mechanics and spacetime structure: Hawking radiation, information loss for black holes, meaning of singularities  Thermodynamics and statistical mechanics: Microscopic derivation of the Bekenstein-Hawking area law S=A/4  Non perturbative solutions of string theory: Branes, AdS/CFT correspondence  Holographic principle : Fundamental or Emergent ?

3. COSLAB 2005 Problem: Present (and near-future) astrophysical observations can not test most of theoretical predictions Recently has become increasingly clear that condensed matter system (e.g. fluids) can be used to mimic various kinematical aspects of general relativity Condensed matter analogues have been used to mimic Black holes and event horizons Cosmological solutions Field theory in curved spacetime Hawking radiation

4. COSLAB 2005  Condensed matter system are experimentally testable in laboratory in the near future we could have a “ Black hole phenomenology” based not on astrophysical but condensed matter experiments  Condensed matter system are experimentally testable in laboratory  in the near future we could have a “ Black hole phenomenology” based not on astrophysical but condensed matter experiments  Can we also describe in this way spacetime singularities?  Main obstruction: usual approach works for kinematical but not dynamical aspects of gravitation   To extend the analogy at a dynamical level is a very difficult task: Gravitational systems: Gravitational systems: 1)Huge redundancy of gauge degrees of freedom 2) Separation between gravitational field and sources

5. COSLAB 2005 Fluids: 1) Few physical parameters (pressure, velocity, density) 2)Do not seem to allow for a source-field description  We will show that an analogy gravitational dynamics  (constrained) fluid dynamics can be found at least for spherically symmetric black holes with ( or without) sources with gauge degrees of freedom completely fixed

6. COSLAB 2005 Summary Summary 1. Gravitational dynamics and fluid dynamics A. Gravitational dynamics without sources B. Fluid dynamics C. Gravitational dynamics with pointlike sources 2. Black hole thermodynamics and fluid dynamics 3. Solutions of the constrained fluid dynamics 4. Conclusions

7. COSLAB 2005 1.Gravitational dynamics and fluid dynamics We only consider spherically symmetric solutions (Topological theory, no propagating gravitational degrees of freedom). They can be described by an effective 2D (dilaton) gravity model obtained retaining only the radial modes of Einstein gravity ( G= -2 ) ds 2 (4) = ds 2 (2) +(2/ 2 )  d  2 (2) We consider Einstein gravity coupled to matter fields and a pointlike source of mass m We consider Einstein gravity coupled to matter fields and a pointlike source of mass m

8. COSLAB 2005 After a Weyl rescaling of the metric we get a 2D model with potential V(  ) and coupling function W(  )After a Weyl rescaling of the metric we get a 2D model with potential V(  ) and coupling function W(  ) For static solutions in the Schwarzschild gauge (ds 2 =Ud  2 + U - 1 dx 2 ) and source at rest in the origin the field equations areFor static solutions in the Schwarzschild gauge (ds 2 =Ud  2 + U - 1 dx 2 ) and source at rest in the origin the field equations are

9. COSLAB 2005 A. Gravitational dynamics without sources 2D dilaton gravity without matter allows for the definition of a scalar mass function, which on shell is constant (F 0 is a normalization constant) The gravitational field equation take the simple form

10. COSLAB 2005 where J=  Vd , the event horizon is located at r=r h, with J(r h )=2M/, in all physically interesting situations the curvature singularity is located at r=0 The general solutions of the field equations, describing BH are B. Fluid dynamics Fluid dynamics is a classical field theory which can be completely understood in terms of Newtonian physics. The fundamental equations are the equation of continuity and Euler equationFluid dynamics is a classical field theory which can be completely understood in terms of Newtonian physics. The fundamental equations are the equation of continuity and Euler equation

11. COSLAB 2005 , v and P are the fluid density, velocity and pressure,  is the potential for external forces. We will consider zero inviscid, irrotational and barotropic fluids In particular this means that P is a function of  only so that we can define a specific entalpy h and an equation of state and that v can be derived from a potential

12. COSLAB 2005 The notion of acoustic horizon (and acoustic black hole) arises by considering perturbations (sound waves) around some fixed background at the linearized level Let us consider the steady flow of a fluid whose motion is essentially one-dimensional  transverse velocities (along y,z) are small with respect to that along x The flux tube has a slab geometry with v,  and the flux tube profile A depending only on x In this case the fluid dynamics equations become ( F is the flux of matter fluid)

13. COSLAB 2005 Unruh has shown that the equation of motion for the sound wave    is that of a scalar massless field propagating in a Lorentzian manifold, Where g  is the acoustic metric Where c is the local speed of sound

14. COSLAB 2005 As long as the propagation of sound waves is concerned we can use the usual (kinematical) spacetime notions of general relativity Ergo-regions: K  =(1,0), K 2 = g tt = -(    c) (c 2 -v 2 ). Ergo-regions are regions where K 2 >0. Supersonic regions, v 2 >c 2 are ergo-regions Trapped surface: region where the normal component of v is always greater then c Apparent Horizon: the boundary of a trapped region Event horizon: the boundary of a region from where phonons cannot escape

15. COSLAB 2005 At the dynamical level the gravity/fluid correspondence can be realized defining the new variables The causal structure of the gravitational black hole can be put in correspondence with that of the acoustic black hole using a Painlevè Gullstrand-like coordinate transformation

16. COSLAB 2005 They can be put in the gravitational form by the identification So that the fluid equations become and imposing the constraint

17. COSLAB 2005 Dynamical equivalence of gauge fixed spherical symmetric gravity with a constrained fluid dynamics If the external parameters for the fluid (A(x),  (x)) are given the constraint is equivalent to an equation of state for the fluid If the equation of state for the fluid is given (usual physical situation) the constraint determines the external parameters ( ,A) for the fluid. The conserved quantity of fluid dynamics ( the flux F ) is identified with the conserved quantity of the gravitational dynamics ( the black hole mass M)

18. COSLAB 2005 C. Gravitational dynamics with pointlike sources In order to give an acoustic meaning to the curvature singularity the discussion has to be generalized to the case m  0 In order to give an acoustic meaning to the curvature singularity the discussion has to be generalized to the case m  0 In the presence of the pointlike source the gravitational equations can be written as In the presence of the pointlike source the gravitational equations can be written as

19. COSLAB 2005 Everything works as in the previous case, the only difference is that now the continuity equation for the fluid acquires a source term proportional to m Everything works as in the previous case, the only difference is that now the continuity equation for the fluid acquires a source term proportional to m The acoustic analogue of the gravitational curvature singularity is a planar source for the fluid flux The acoustic analogue of the gravitational curvature singularity is a planar source for the fluid flux In the acoustic description the delta function singularity is not associated with a singularity of the dynamics ( the Euler equation is not singular) In the acoustic description the delta function singularity is not associated with a singularity of the dynamics ( the Euler equation is not singular)

20. COSLAB 2005 Explicit solutions of the dynamical equations can be found taking the solutions as function of |r| ( to generate the delta function singularity at r=0) Explicit solutions of the dynamical equations can be found taking the solutions as function of |r| ( to generate the delta function singularity at r=0)

21. COSLAB 2005 2. Black hole thermodynamics and fluid dynamics Considering the acceleration of fiducial observers or the periodicity of the Euclidean section at the horizon one can define also for acoustic black holes the notion of Surface gravity and Hawking temperature: However, usually it is very difficult to find the acoustic counterpart of the other BH thermodynamical parameters mass M and entropy S Our approach allows us to give a natural acoustic meaning to M and S

22. COSLAB 2005 In fact it is well known that black hole thermodynamics follows directly from the gravitational field equations Having found a dynamical correspondence gravity/fluid we just need to express the BH mass and entropy in terms of the fluid parameters We find acoustic analogues M a,S a satisfying automatically the first principle dM a =T a dS a The explicit expression are simple and transparent in the case of a flux tube of constant section ( M (r h ) is the fluid mass inside the horizon)

23. COSLAB 2005 The entropy has a simple physical meaning is proportional to the total mass of the fluid outside the horizon, whereas the first principle gives The temperature measures the rate of change of the flux of fluid mass when the horizon position is changed

24. COSLAB 2005 3. Solutions of the constrained fluid dynamics Let us now solve the equations of the constrained fluid dynamics Let us now solve the equations of the constrained fluid dynamics  find the form of v 0,  0, c, A,  Let us consider a generic power law equation of state for the fluid ( a arbitrary real constant), Let us consider a generic power law equation of state for the fluid ( a arbitrary real constant), This Equation describes almost all physically interesting fluids: perfect fluid (n=1), Bose-Einstein condensates (n=2), Chaplygin gas (n= - 1)

25. COSLAB 2005 To solve completely the dynamics we have to consider separately two cases 1) Flux tube with constant section 2) Flux tube with non-constant section and homogeneous external potential In terms of the new variables the equation of state reads Using this equation. one has

26. COSLAB 2005 Flux tube with constant section In this case we can use the continuity equation to solve for X=X(Y). The constraints determines the form of the external potential. The solution can be given as function of Y. Where  is the (constant) flux of mass in the tube. The acoustic horizon is located at Y=Y h =  (X=0), the subsonic region Y>  (X>0), the black hole singularity at Y=Y s =  [(A 2 +1) 1/2 -A].

27. COSLAB 2005 The external potential becomes extremal on the horizon (null force condition) All the fluid parameters remain finite both at the horizon and the singularity Behaviour of the fluid parameters ( External potential bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of Y, for n=2,  =1, M=(3/4). The acoustic horizon is located at Y=1, the singularity at Y=1/2

28. COSLAB 2005 Behaviour of the fluid parameters ( External potential bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of the Y, for n=1/2,  =1, M=(3/4). The acoustic horizon is located at Y=1, the singularity at Y=1/2 Behaviour of the fluid parameters ( External potential bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of the Y, for n=1/2,  =1, M=(3/4). The acoustic horizon is located at Y=1, the singularity at Y=1/2

29. COSLAB 2005 Flux tube with non-constant section In this case we take the external potential constant whereas A changes along the flux tube. We can use the continuity equation to find A(X,Y) The constraint can be now easily solved to give (  integration constant)

30. COSLAB 2005 Again the fluid parameters can be given (for n generic in implicit form) as function of the variable Y The acoustic horizon is located at Y=Y h =  (n+1)/(n-1), the black hole singularity at Y=Y s, the supersonic region at Y s <Y<Y h All the fluid parameters remain finite both at the horizon and the singularity The horizon must forms at a minimum of the section A  the flux tube has the form of a Laval nozzle

31. COSLAB 2005 This means that the constraint, which is necessary to have a correspondence between gravitational and fluid dynamics takes the form of a “geometrical constraint” on the form of the flux tube. This “geometrical constraint” forces the fluid to develop an acoustic horizon This is an well-known fact from hydrodynamics: the acoustic horizon must form at the narrowest cross-section of the nozzle

32. COSLAB 2005 Behaviour of the fluid parameters ( Tube cross- section bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of Y, for n=2,  =1, M=(9/16). The acoustic horizon is located at Y=1, the singularity at Y=1/8. The range of the coordinate Y is 1/8<Y<(27/8) 1/2. For this value of Y the tube cross- section diverges. Behaviour of the fluid parameters ( Tube cross- section bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of Y, for n=2,  =1, M=(9/16). The acoustic horizon is located at Y=1, the singularity at Y=1/8. The range of the coordinate Y is 1/8<Y<(27/8) 1/2. For this value of Y the tube cross- section diverges.

33. COSLAB 2005 Behaviour of the fluid parameters ( Tube cross- section bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of the Y, for n=1/2,  =1, M=(9/16). The acoustic horizon is located at Y=1, the singularity at Y=1/8. The range of the coordinate Y is 1/8<Y<(8/3) 3/2. For this value of Y the tube cross- section diverges. Behaviour of the fluid parameters ( Tube cross- section bold black line, velocity black line, density dashed line, speed of sound grey line) normalized at their horizon values as a function of the Y, for n=1/2,  =1, M=(9/16). The acoustic horizon is located at Y=1, the singularity at Y=1/8. The range of the coordinate Y is 1/8<Y<(8/3) 3/2. For this value of Y the tube cross- section diverges.

34. COSLAB 2005 4. Conclusions  Main results:  The gravity/fluid analogy can be promoted from the pure kinematical to a full dynamical level  It is possible to find a dynamical equivalence between high symmetric gauge-fixed gravitational systems and a fluid  We can construct artificial BH singularities and give a meaning to artificial BH thermodynamics  Singularities have different meaning in general relativity and fluid dynamics

35. COSLAB 2005 In the former case they are related to true singularities of the dynamics in the latter they appear as mere source terms for the matter (cusp terms in the fiels)  Open questions How far reaching is the analogy? How far reaching is the analogy? It seems to hold for gravitational systems with high symmetry and with no propagating gravitational degrees of freedom (topological gravity). It is therefore likely that it can be extended to cosmological solutions (Cosmological singularity) but extremely unlikely that it can be used to describe gravitational systems with propagating degrees of freedom (solutions depending on both the timelike and spacelike coordinate, gravitational waves), backreaction etc Role of the Weyl transformation Role of the Weyl transformation

36. COSLAB 2005