oscillations and stability of black holes in dynamical ...bhw/gualtieri.pdfchern-simons modiﬁed...

Oscillations and stability of black holesin dynamical Chern-Simons gravity

V. Cardoso, L.G., Phys. Rev. D80 064008 (2009)

V. Cardoso, L.G., C. M. Mendes, P. Pani, in preparation

Leonardo Gualtieri II Workshop on Black Holes IST, Lisbon, December 2009


Chern-Simons modified gravity is one of the most interesting extensions of General Relativity

The gravitational field is coupled with a scalar field through a parity-violating Chern-Simons term:

∗RR =12Rabcd�

abefRcdef

S = κ

�d4x√−gR+

α

4

�d4x√−gϑ ∗RR

Where is a scalar field andϑ



The gravitational field is coupled with a scalar field through a parity-violating Chern-Simons term:

∗RR =12Rabcd�

abefRcdef

S = κ

�d4x√−gR+

α

4

�d4x√−gϑ ∗RR

Where is a scalar field andϑ

This theory has first been introduced in a non-dynamical version, in which the scalar field is given a priori, like an external field.

Later, a dynamical version of the theory has been proposed, in which the scalar field is treated as a dynamical field:

−β

2

�d4x√−g

�gab∇aϑ∇bϑ + V (ϑ)

�

We have been studying dynamical Chern-Simons (DCS) gravity.



Indeed:



Indeed:

- The CS term violates parity, thus it has a characteristic observational signature which could allow to discriminate an effect of this theory from other phenomena (for instance, the two polarizations of gravitational waves propagate differently)



Indeed:


- Could help to explain several problems of cosmology, from inflation to baryon asymmetry



Indeed:



- It is predicted from String Theory: a CS term is required in most of its solutions to preserve unitarity, and it is induced in most string theories by duality



Indeed:



- It is predicted from String Theory: a CS term is required in most of its solutions to preserve unitarity, and it is induced in most string theories by duality

- It is predicted from Loop Quantum Gravity: it is required to ensure gauge invariance of the Ashtekar variables


Much work has been done to study possible observational signatures from DCS gravity(see for instance the review Alexanded & Yunes, Phys. Rept. 480, 1 (2009) and references therein)

and it was found that astronomical observations pose constraints on the parameters od DCS gravity.

In particular, it has been found that observations from the double pulsar PSR J0737-3039 imply

β

α2� 10−15km−4





β

α2� 10−15km−4

We have studied perturbations of black holes in DCS gravityWe have found that if a spherically symmetric black hole is perturbed

and we assume that the scalar field is , i.e. vanishing scalar field in the background,then the BH is unstable unless

gµν = ηµν + hµν

O(h)

β

α2� 10−2km−4





β

α2� 10−15km−4

We have studied perturbations of black holes in DCS gravityWe have found that if a spherically symmetric black hole is perturbed

and we assume that the scalar field is , i.e. vanishing scalar field in the background,then the BH is unstable unless

gµν = ηµν + hµν

O(h)

β

α2� 10−2km−4

As I will argue, this should be considered as a strong indication of the range of valueswe could expect for the parameters of DCS gravity


Spherically symmetric BH in DCS gravity are described by the Schwarzschild metric.

In previous work on this subject (Yunes & Sopuerta, PRD 77, 064007, 2008) is was found thatin presence of a background scalar field,

perturbations of the spacetime metric with polar parity and axial parity are mixed,and their equations are very involved.

We have found that if , polar and axial metric perturbations decouple.Only axial perturbations are coupled with the scalar field; polar perturbations

are not affected by the scalar field, and satisfy the equations of General Relativity (the Zerilli equation).

gµν = g(Schw)µν + hµν ϑ = ϑ(0) + δϑ

ϑ(0) = 0


Spherically symmetric BH in DCS gravity are described by the Schwarzschild metric.

In previous work on this subject (Yunes & Sopuerta, PRD 77, 064007, 2008) is was found thatin presence of a background scalar field,

perturbations of the spacetime metric with polar parity and axial parity are mixed,and their equations are very involved.

We have found that if , polar and axial metric perturbations decouple.Only axial perturbations are coupled with the scalar field; polar perturbations

are not affected by the scalar field, and satisfy the equations of General Relativity (the Zerilli equation).

gµν = g(Schw)µν + hµν ϑ = ϑ(0) + δϑ

ϑ(0) = 0

We expand the scalar field in scalar spherical harmonics:

We expand the metric perturbations in tensor spherical harmonics, defining the Regge-Wheeler master function Q(r), which describes axial perturbations,

as in General Relativity

ϑ =Θr

Y lme−iωt


The functions satisfy the following equations:Q(r) , Θ(r)d2

dr2∗Q +

�ω2 − f

�l(l + 1)

r2− 6M

r3

��Q = −96π iMfω

r5Θ

d2

dr2∗Θ +

�ω2 − f

�l(l + 1)

r2

�1− 576πM2

r6β

�+

2M

r3

��Θ = −f

(l + 2)!(l − 2)!

6iMωr5β

Q

(Here for simplicity we have posed , which is always possible by choosing the arbitrary normalization of the scalar field)

α = 1


The functions satisfy the following equations:Q(r) , Θ(r)d2

dr2∗Q +

�ω2 − f

�l(l + 1)

r2− 6M

r3


r5Θ

d2

dr2∗Θ +

�ω2 − f

�l(l + 1)

r2

�1− 576πM2

r6β

�+

2M

r3

��Θ = −f

(l + 2)!(l − 2)!

6iMωr5β

Q

(Here for simplicity we have posed , which is always possible by choosing the arbitrary normalization of the scalar field)

α = 1

We have tried so solve these equations with Sommerfeld boundary conditions (ingoing waves at the horizon, outgoing waves at infinity)

to find quasi-normal modes (QNM) of Schwarzschild-DCS black holes,but in these equations the well-known numerical problems

related to the divergence of stable solutions far away from the BH (which also plague perturbations of Schwarzschild BH) are much more severe.


In the work I am now reporting, we did not find stable QNM of Schwarzschild-DCS BH,but we found that, for , there exist unstable, non-oscillating modes (with l=2)!β � 10−2 km−4



We know that BH exist in nature.

Actually, observed BH are very likely to be rotating, and rotating BH in DCS gravity require a non-vanishing background scalar field,

so, strictly speaking, our analysis does not apply to them.Therefore, in principle, we could not definitely exclude DCS gravity theory with and then that non-rotating BH are unstable, but (maybe) rotating BH are stable,

and this would explain why we see them.

β � 10−2 km−4



We know that BH exist in nature.

Actually, observed BH are very likely to be rotating, and rotating BH in DCS gravity require a non-vanishing background scalar field,

so, strictly speaking, our analysis does not apply to them.Therefore, in principle, we could not definitely exclude DCS gravity theory with and then that non-rotating BH are unstable, but (maybe) rotating BH are stable,

and this would explain why we see them.

β � 10−2 km−4

β � 10−2 km−4

β � 10−2 km−4

But this explanation seems to me very involved; I think that the fact that Schwarzschild BH (with vanishing background scalar field) are unstable if gives us a strong indication against the possibility of a DCS theory with


These results are described in Phys. Rev. D80 064008 (2009)I shall mention now preliminary results of a new work we are preparing

(V. Cardoso, L.G., C. M. Mendes, P. Pani,) in which we try to solve the equations

d2

dr2∗Q +

�ω2 − f

�l(l + 1)

r2− 6M

r3


r5Θ

d2

dr2∗Θ +

�ω2 − f

�l(l + 1)

r2

�1− 576πM2

r6β

�+

2M

r3

��Θ = −f

(l + 2)!(l − 2)!

6iMωr5β

Q

for QNM of Schwarzschild-DCS black holes.

We are using two approaches: time-evolution simulations and WKB approach.


These results are described in Phys. Rev. D80 064008 (2009)I shall mention now preliminary results of a new work we are preparing

(V. Cardoso, L.G., C. M. Mendes, P. Pani,) in which we try to solve the equations

d2

dr2∗Q +

�ω2 − f

�l(l + 1)

r2− 6M

r3


r5Θ

d2

dr2∗Θ +

�ω2 − f

�l(l + 1)

r2

�1− 576πM2

r6β

�+

2M

r3

��Θ = −f

(l + 2)!(l − 2)!

6iMωr5β

Q

for QNM of Schwarzschild-DCS black holes.

We are using two approaches: time-evolution simulations and WKB approach.

βcrit

Our results (very preliminary, and still to be confirmed) look very interesting, and, if confirmed, will change our perspective of DCS gravity.

It seems that, for any value , there exists an such that, for , the BH in unstable

(and, as increases, the growth time of the corresponding instability decreases).This would mean that Schwarzschild BH in DCS theory are unstable

(at least, if the background scalar field is vanishing)

l

lβ < βcrit


Conclusions:

- DCS gravity could be a promising extension of General Relativity, for many reasons

- perturbations of spherically symmetric BH in DCS gravity satisfy relatively simple equations: the Regge-Wheeler equation for axial metric perturbations couples with the scalar field equation

- numerical problems make difficult to find QNM of Schwarzschild-DCS BH from these equations

- we found that for the BH is unstable to l=2 perturbations; β

α2� 10−2km−4

this limiting value is many orders of magnitude larger than the bound obtained fromastrophysical observations

- preliminary results (to be confirmed) derived by solving the QNM equations with time-evolution integrations and WKB approximation suggest that for all values of the Schwarzschild-DCS BH are unstable to perturbations with enough large

βl

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