XVI European Workshop on Strings Theory Madrid– 14 June 2010 arXiv:0909.0008 [hep-th] arXiv:0909.3852 [hep-th] Dumitru Astefanesei, MJR, Stefan Theisen Dumitru Astefanesei, Robert B. Mann, MJR, Cristian Stelea Maria J. Rodriguez Thermodynamic instability Thermodynamic instability of of doubly spinning black objects doubly spinning black objects &1003.2421[hep-th]
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XVI European Workshop on Strings Theory Madrid– 14 June 2010
Thermodynamic instability of doubly spinning black objects. Maria J. Rodriguez. XVI European Workshop on Strings Theory Madrid– 14 June 2010. Dumitru Astefanesei, Robert B. Mann, MJR, Cristian Stelea. arXiv:0909.3852 [hep-th]. &. 1003.2421[hep-th]. - PowerPoint PPT Presentation
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XVI European Workshop on Strings Theory
Madrid– 14 June 2010
arXiv:0909.0008 [hep-th]
arXiv:0909.3852 [hep-th]
Dumitru Astefanesei, MJR, Stefan Theisen
Dumitru Astefanesei, Robert B. Mann, MJR, Cristian Stelea
Having a better understanding of the properties of BO may be useful to construct
new solutions
Thin
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Black Holes and black rings in ultra-spinning regime will inherit the instabilities.
In certain regimes black holes and black rings behave like black strings and black p-branes.
Ultra-spinning black objects
Black strings and branes exhibit Gregory-Laflamme instability
Gubser + WisemanBranch of static lumpy black strings
A black hole solution which is thermally unstable in the grand-canonical ensemble will develop a classical instability.Gubser + Mitra
Emparan + Myers
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Q1: If black objects are thermally unstable in the grand-canonical ensemble for jth does this imply that there they are classically unstable?
Instabilities from thermodynamics
But to investigate this and where the threshold of the classical instability is one has to perform a linearized analysis of the perturbations.
Q2c: Is there any relation between zeros of eigenvalues of Hess(G) and jm?
Q2: What information can we get from the study of the thermodynamical instabilities?
We can establish a membrane phase signaled by the change in its thermodynamical behavior which could imply the classical instability.
Q2b: Which is the threshold of the membrane phase, jm?
We can study the zeros of the Hess(G) which seem to be linked to the classical instabilities
Q2a: How to establish the membrane phase?
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Thermodynamics of black objects
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Which ensemble is the most suitable for this analysis?
Entropy – microcanonical ensemble
Thermal ensembles
Gibbs potential – grand canonical ensemble
Enthalpy Helmholtz free energy – canonical ensemble
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Due to the equivalence principle, there is no local definition of the energy in gravitational theories
Basic idea of the quasilocal energy: enclose a region of space-time with some surface and compute the energy with respect to that surface – in fact all thermodynamical quantities can be computed in this way
For asymptotical flat space-time, it is possible to extend the quasilocal surface to spatial infinity provided one incorporates appropriate boundary (counterterms) in the action to remove
divergences from the integration over the infinite volume of space-time.
r =co
nst.
Brown + York gr-qc/9209012
Mann + Marolf
Quasilocal thermodynamics
Compute directly the Gibbs-Duhem relation
by integrating the action supported with counterterms.
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Instabilities from Thermodynamics
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Thermal stability
In analogy with the definitions for thermal expansion in the liquid-gas system, the specific heat at a constant angular velocity, the isothermal compressibility, and the coefficient of thermal expansion can be defined
The conditions for thermal stability in the grand-canonical ensemble
or
What do we know about the thermal stability of black objects?
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Black hole thermal stability
Monterio + Perry + Santos 0903.3256[gr-qc]
The response functions are positive for different values of the parameters implying there is no region in parameters space where both are simultaneously positive.
The black holes is thermally unstable, both in the canonical and grand-canonical ensembles.
CompressibilityHeat capacity
Singly rotating Myers-Perry black hole
For doubly spinning MP-BH the response functions are positive for different (complementary) regions of the parameter space implying its
instability.
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Black ring thermal stability
The black ring is thermally unstable, both in the canonical and grand-canonical ensembles.
The CΩ→0 as T→0 which is expected and can be drawn from Nernst
theorem.
Heat capacity Compressibility
Singly spinning black ring
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We investigated the stability of the doubly spinning black ring
The doubly spinning black hole
and the singly spinning black ring
are thermally unstable in the grand-canonical
ensembles.
A second rotation could help to stabilize the solution
Doubly spinning black ring
What about the thermal stability of the doubly spinning black ring?
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Doubly spinning black ring
The grand canonical potential for doubly spinning black ring (using the quasi local formalism)
The Hessian should be negatively defined
The doubly spinning black ring is local thermally unstable.
where
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Critical points & turning points
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These points should not be considered as a sign for an instability or a new branch but a transition to an infinitesimally nearby solution along the same family of solutions.
Instabilities from thermodynamics
The instabilities and the threshold of the membrane phase of the singly spinning MP BH are 0
D=5
D=10
D=6
D=5D=10
D=6
Numerical evidence supports this connection with the zero-mode perturbation of the solution.
Note that the relation between ensembles is not in general valid.
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Indicates where the transitionto the black membrane phase.
More general black holes with N spins ultra-spin iff
Critical points: MP BH
jm
Black holes with one spin
0 where
Are there other ultra spinning MP black holes?
And for
and
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The existence and location of the threshold of this regime is signaled by the minimum of the temperature and the maximum
angular velocity as functions of the angular momentum.
The transition to a membrane-like phase of the rapidly spinning black holesis established from the study of the thermodynamics of the system.
where for the ultra spinning MP BH
Critical points: MP BH
while the angular velocity reaches its maximum value.
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But let´s take a closer look to the Hessian, which has to be negatively defined,
Do the zeros of the eigenvalues of this Hessian have any physical interpretation?
We´ve checked that at least one of the eigenvalues of the Hess[G] is zero.
Critical points: MP BH
And also checked that the Ruppeiner curvature pinpoints thezero of the determinant of the Gibbs potential’s hessian
These points seem to be related to the classical instabilities.
is the so called Ruppeiner metric
The Ruppeiner metric measures the complexity of the underlying statistical mechanical model
A curvature singularity is a signal of critical behavior.
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Turning points : BR
We´ve checked that at least one of the eigenvalues of the Hess[G] is zero there.
λ=0.5 At the cusp in s vs j
In this case the temperature does not have a minimum, but there exists a turning point and
plays a similar roleas the minimum of the temperature for the BH
The Ruppeiner curvature diverges.
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Indicate where the transition to the thermodynamical black membrane phase.
λ [ν] At the cusp in s vs j
No eigenvalue of the Hess[G] is zero there.
Turning points : BR
Particular BR solutions with jψ > 1/5 fall into the same category as other black holes with no membrane phase as the four dimensional Kerr black
hole and the five dimensional Myers-Perry black hole.
I
III
I
III
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Summary and outlook
It would be interesting to investigate numerically whether these correspond to the zero-mode perturbations.
We showed ,in parameter space, that doubly spinning black rings are thermally unstableFound the thresholds of the transition to the black membrane phase of black holes and black rings with at least two spins.Identified particular cases of doubly spinning BR with no membrane phase
Study the ultraspinning behavior of multi black holes, such as the bicycling black ring and saturn, which can be relevant in finding new higher dimensional multi
black hole solutions.
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In five dimensions stationary implies axisymmetric
To calculate the physical quantities we employ the complex instanton method
The ADM decomposition of the full spacetime
We can write (B) in the (A) form
The Wick transformation changes the intensive variables
but not the extensive ones
(B)
(A)
Lapse function Shift function
Angular velocity Temperature
Quasilocal thermodynamics
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Black String
=0The stress energy tensor is
conserved for any value of the parameters
Observe that
=0Corresponds to the thin black ring limit
Boundary stress energy tensor for black strings
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To compare solutions we need to fix a common scale Classical GRTo compare solutions we need to fix a common scale Classical GR
We'll fix the mass M equivalently and factor it out to get dimensionless quantities
We'll fix the mass M equivalently and factor it out to get dimensionless quantities
aHaH j2j2 ωω ……
Disconnected compact horizons: multi horizon black hole solutions
One compact horizons: uni horizon black hole solutions