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
of
doubly spinning black objects
&
1003.2421[hep

th]
1
Motivation
Black holes are the most elementary and fascinating
objects in General Relativity
In their presence the effects of the space

time curvature are dramatic
In string theory, mathematics and recent cutting edge
experiments black objects are also relevant.
The study of the
properties of higher
dimensional black holes
is essential to
understand the
dynamics of space

time
2
Asymptotically flat
On the BH species
(by means of natural selection)
Vacuum Einstein
´
s equation
Boundary conditions
Equilibrium
Stationary
–
no time dependence
Regular solutions on and outside the event horizon
R
µ
ν
=0
[*]
[*] We start from a five dimensional continuum which is x
1
,x
2
,x
3
,x
4
,x
0
.
In it there exists a Riemannian metric with a line element ds
2
= g
μν
dx
μ
dx
ν
μ
,
ν
= 1,2,…,D
3
On the BH species
(by means of natural selection)
S
5
S
D

2
S
4
S
3
S
2
The boundary stress tensor satisfies
a local conservation law
Summary of neutral
D

dim BHs classified
by its horizon
topologies
4
In D>4 many black holes have been found:
Motivation
Galloway+Schoen
Topology:
Rigidity :
Hollands
et al
Stationary &
axisymmetric
In D=4 stationary black holes are spherical and unique
The main feature of high

D is the richer rotation dynamics
Study thermodynamical properties of spinning BH in D

dim to learn
how these solutions connect.
Our goal:
5
One&Two angular momenta +
Vacuum + Asymtotically flat
Phase diagram of black objects
6
What do we know
about black objects?
In D=4 dimensions
In D=5 dimensions

Kerr black hole
In D>5 dimensions
BH w/ one J in D

dim

the Myers

Perry black hole

the Myers

Perry black hole
a
H
j
It seems that there is an infinite number of BHs.
a
H
j
2
1
a
H
j
1

black ring

thin
black ring and
black saturn
j
1
2
7
The generalization of the black hole solution with ANY #
of angular
momenta
is the Myers

Perry (MP) solution.
singular
The gray curve is the phase of zero temperature BH’s
Representative phase of MP

BHs with one of the two
angular momenta
fixed
BH w/ two J in D

dim
D=5
The dashed lines show MP for fixed values of =0.1,0.3,0.5 right to left
j
2
8
The generalization of the black ring with TWO
angular
momenta
is known.
BR w/ two J in D

dim
D=5
The dark gray curve is the phase of
zero temperature BR’s
Representative phase of the doubly spinning BR
with the S
2
angular momenta
fixed
The dashed lines show BR for fixed values of
(right towards left)
The black dashed curve is the phase of
zero temperature MP BH’s
The angular
momenta
are bounded
The fat ring branch disappears for
9
What do we know about
these black objects?
In D=5 dimensions
In D>5 dimensions

Myers

Perry Black Hole (BH)

Black Ring (BR)

Myers

Perry Black Hole

Black Ring (BR)
BH w/ two J in D

dim

Helical BH

Black Saturn

Bicycling BR
j
2
is fixed
Not shown here

Helical BH

Black Saturn

Bicycling BR

Blackfolds
Not shown here
a
H
j
1
j
2
is fixed
a
H
j
1
1
10
Why are we interested doubly spinning solutions?
Black Holes with T=0 are interesting because they can teach us
about the microscopic origin of their physical properties
SUSY
Asymptotically flat
Non SUSY
T=0
BH w/ two J in D

dim
Doubly spinning black rings, in contrast to the singly spinning
black rings, can be
extremal
.
11
One&Two angular momenta +
Vacuum + Asymtotically flat
Ultra

spinning black objects
12
Ultra

spinning black objects
R
Balance condition
Parameters in the solution
where
S
1
x S
D

3
R
Thin Black ring
R
∞
S
D

2
+

+

13
Thin Blackring/fold
Recently the matched asymptotic expansion has been applied to solve
Einstein
´
s equations to find thin Black ring/folds in D>4 dimensions
The basic idea
Black ring
Black string
≡
R
∞
R
ro
ro
ro
Emparan + Harmark + Niarchos + Obers + MJR 0708.2181
[hep

th]
Having a better
understanding of the
properties of BO may
be useful to construct
new solutions
Thin
14
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
+
Wiseman
Branch
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
15
Q1: If black objects are thermally unstable in the grand

canonical ensemble for
j
th
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 j
m
?
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, j
m
?
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?
16
Thermodynamics of black objects
17
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
18
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.
Brown
+
York gr

qc/9209012
Mann
+
Marolf
Quasilocal thermodynamics
Compute directly the Gibbs

Duhem relation
by integrating the action supported with
counterterms.
19
Instabilities from Thermodynamics
20
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?
21
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.
Compressibility
Heat 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.
22
Black ring thermal stability
The black ring is thermally unstable, both in the canonical and grand

canonical ensembles.
The C
Ω
→0 as T→0 which is
expected and can be drawn
from Nernst theorem.
Heat capacity
Compressibility
Singly spinning black ring
23
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?
24
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
25
Critical points & turning points
26
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=5
D=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.
27
Indicates where the transition
to
the
black membrane phase.
More general black holes with N spins ultra

spin iff
Critical points: MP BH
j
m
Black holes with one spin
0
where
Are there other ultra spinning MP black holes?
And for
and
28
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 holes
is 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.
29
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 the
zero 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.
30
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 role
as the minimum of the temperature for the BH
The Ruppeiner curvature diverges.
31
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
j
ψ
> 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
32
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 unstable
Found 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.
33
34
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
35
Black String
=0
The stress energy tensor is conserved
for any value of the parameters
Observe that
=0
Corresponds to the
thin
black ring limit
Boundary stress energy tensor for black strings
36
To 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
a
H
j
2
ω
…
Disconnected compact horizons: multi horizon black hole solutions
One compact horizons:
uni
horizon black hole solutions
On the number of angular
momenta
On the number of horizons
Maximum # angular
momenta
:
On how we compare solutions
Compare by drawing diagrams
i.e.
a phase diagram
a
H
j
2
j
1
j
2
Jargon and reminder
j
[(D

1)/2]
. . .
37
Outline
Introduction
Thermodynamics
Instabilities from Thermodyn.
Summary and outlook
Motivation

Ultra

spinning BH

BH solutions in D

dim

Membrane phase

Critical points & turning points

Thermal ensembles

Thermodynamic stability

Grand

canonical ensemble
38
39
Ultra spinning multi BH
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