Thermodynamic instability of doubly spinning black objects

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