Rapidly rotating black holes in Chern-Simons gravity...

Rapidly rotating black holes in Chern-Simons gravity:Decoupling limit solutions and breakdown

Leo C. Stein

Cornell University

Einstein Fellows Symposium, Cambridge, MA, 29 Oct. 2014

Phys. Rev. D 90, 044061 (2014) [arXiv:1407.2350]

Motivation

1 GR successful but incomplete• GR+QM=new physics (e.g. BH thermo)• Planck scale phenomena? Other scales?• Expect GR is low-energy EFT

2 Ask nature• So far, only weak-field tests• Lots of theories ≈ GR• Need to explore strong-field• Strong curvature • non-linear

What phenomena come from UV completions?

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 2

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Earth's

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NS merger

BH merger

SMBH

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NS

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Motivation

1 GR successful but incomplete• GR+QM=new physics (e.g. BH thermo)• Planck scale phenomena? Other scales?• Expect GR is low-energy EFT

2 Ask nature• So far, only weak-field tests• Lots of theories ≈ GR• Need to explore strong-field• Strong curvature • non-linear

What phenomena come from UV completions?

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 4

Theories

Fundamental approach:• String theory • Loop quantum gravity• TeVeS • Einstein-Æther • Horava• Massive gravity • dRGT • bi-metric• . . .

Pedestrian approach: effective field theory• Learned from cond-mat, then nuclear and hep-th• Theory with separation of scales• “Integrate out,” effective theory for long (or short) wavelengths• Works backwards!

General relativity

Special relativity

post-NewtonianG→0

v/c→0

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 5

Theories

Fundamental approach:• String theory • Loop quantum gravity• TeVeS • Einstein-Æther • Horava• Massive gravity • dRGT • bi-metric• . . .

Pedestrian approach: effective field theory• Learned from cond-mat, then nuclear and hep-th• Theory with separation of scales• “Integrate out,” effective theory for long (or short) wavelengths• Works backwards!

General relativity

Special relativity

post-NewtonianG→0

v/c→0

General relativity

Special relativity

post-NewtonianG→0

v/c→0

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 5

EFT works

Worked for describing superconductivity, predicting W, higgs, etc.

Try to build EFT for gravity

• Metric, general covariance, Lorentz invariance

• Lowest order dynamical theory is Λ+GR!

Beyond GR: add new `—want to constrain this

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 6

Dynamical Chern-Simons Gravity

S =

∫d4x√−g[m2

pl

2R− 1

2(∂ϑ)2 +

mpl

8`2ϑ ∗RR

]

• Anomaly cancellation, low-E string theory, . . .

• Lowest-order EFT with parity-odd ϑ, shift symmetry (long range)

• Phenomenology unique from other R2

(e.g. Einstein-dilaton-Gauss-Bonnet)

• Tractable

• Straw-man theory

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 7

Decoupling limit

• Most EFTs don’t make sense as exact theories (seee.g. Delsate+Hilditch+Witek)

• (Almost) All corrections introduce new `

• Can’t be too long

• Expand fields, EOMs in powers of `/RBG, perturbation scheme

Question: What is regime of validity of decoupling limit?

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 8

How do we constrain ` in dCS from astronomicalobservations?

• dCS is higher-curvature and parity-odd

�ϑ = −mpl

8`2 ∗RR ,

• Want ∗RR as large as possible=⇒ smallest M , largest χ = J/M2

• NSs have small M, but BHs have χ→ 1

Want solutions for rapidly rotating black holes in dCS

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 9

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Sun's

surface

Earth's

surface

LAGEOS

J0737

-3039

Mercury

precessionLLR

NS merger

BH merger

SMBH

merger

NS

10-12 10-10 10-8 10-6 10-4 0.01 110-12

10-9

10-6

0.001

1

¶=Gm�r HcompactnessL

Ξ=

HGm

�r3L1

�2@k

m-

1D

Hinv.cu

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ure

radiu

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Sun's

surface

Earth's

surface

LAGEOS

J0737

-3039

Mercury

precessionLLR

NS merger

BH merger

SMBH

merger

NS

Strong

er

10-12 10-10 10-8 10-6 10-4 0.01 110-12

10-9

10-6

0.001

1

¶=Gm�r HcompactnessL

Ξ=

HGm

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Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 10

How do we constrain ` in dCS from astronomicalobservations?

• dCS is higher-curvature and parity-odd

�ϑ = −mpl

8`2 ∗RR ,

• Want ∗RR as large as possible=⇒ smallest M , largest χ = J/M2

• NSs have small M, but BHs have χ→ 1

Want solutions for rapidly rotating black holes in dCS

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 11

Black holes in dCS

• a = 0 (Schwarzschild) is exact solution with ϑ = 0

• Analytically known solutions in decoupling limit• a�M limit up to O(a2), valid ∀r (see Yunes+Pretorius,

Yagi+Yunes+Tanaka)• r �M limit for l = 1, valid ∀a (see Yagi+Yunes+Tanaka)

• I construct numerical solutions ∀r, ∀a

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 12

Takeaway

• I construct numerical solutions ∀r, ∀a for dCS BHs in decoupling limit

• I use solutions to determine the regime of validity of PT

0.0 0.2 0.4 0.6 0.8 1.0

1

2

5

10

20

50

a�GM

È{�G

MÈ

Breakdown of perturbation theory

Decoupling limit valid

• This can be turned around to forecast bounds ` . 22km fromGRO J1655–40 (M = 6.30± 0.27M�, a ≈ 0.65–0.75)

• For details see Phys. Rev. D 90, 044061 (2014) [arXiv:1407.2350]

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 13

Equations to solve

S =

∫d4x√−g[m2

pl

2R− 1

2(∂ϑ)2 +

mpl

8`2ϑ ∗RR

]

�ϑ = −mpl

8`2 ∗RR

m2plGab +mpl`

2Cab = T(m)ab + T

(ϑ)ab

• gabCab = 0

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 14

Decoupling limit

Take `2 → ε`2 and expand in ε,

ϑ = 0 + εϑ(1) +ε2

2ϑ(2) + . . .

g = gGR + ε0 + ε2hdef + . . .

�GRϑ(1) = −mpl

8`2 ∗RR[gGR]

Don’t yet have quantity to test validity of perturbation theory

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 15

Next order

m2plG

(1)ab [hdef] = T

(ϑ)ab [ϑ(1), ϑ(1)]−mpl`

2Cab[ϑ(1)]

• Trace equation

• Lorenz gauge ∇ahab = 0

1

2m2

pl�hdef = −(∇aϑ(1))(∇aϑ

(1)) ,

• Same scalar PDE operator

• Caveat: gauge-dependent but should still capture a-dependence

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 16

Next order

• Now can make comparison:

√−g =

√−gGR

(1 + ε2 12h

def +O(ε3))

• If hdef ∼ O(1), should keep higher O(ε)

• Criterion for validity of PT:

|hdef| . 1 everywhere

• Program: Solve for ϑ(1), hdef as functions of r, θ for all a

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 17

Approach to solving

Symmetry reduced, ϑ = ϑ(r, θ). �→ ∆.

Analytical:

• Static Green’s function ∆−1 known analytically

• Separation of variables

ϑ =∑j

ϑj(r)Pj(cos θ)

• Can do source decomposition (See Konno+Takahashi and[arXiv:1407.0744])

Resort to numerics!

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 18

Numerical approach

• Elliptic PDE. Could solve hyperbolic, parabolic, relaxation scheme

• Numerical separation of variables. Each j mode is an ODE.

• Compactify r

• Pseudospectral collocation method

• Directly solve discrete ODE operator (“numerical Green’s function”)

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 19

Numerical approach

• For each a, find ϑ(r, θ; a), compute (∂ϑ)2, find hdef(r, θ; a)

• Evaluate max |hdef| and find regime of validity

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 20

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 21

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r˜sinθ

r˜cosθ

ϑ˜(a˜=0.979)

-0.2

-0.1

0

0.1

0.2

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 22

Exponential convergence

a=0.01

a=0.35

a=0.65

a=0.85

a=0.999

10 20 30 40 50

10-30

10-24

10-18

10-12

10-6

1

Harmonic coefficient j

Pow

er

inJ�

j

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 23

Exponential convergence

a=0.01

a=0.35

a=0.65

a=0.85

a=0.999

0 10 20 30 40 50

10-30

10-24

10-18

10-12

10-6

1

Harmonic coefficient j

Pow

er

inh�

j

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 23

Exponential convergence

20 30 40 50 6010-12

10-11

10-10

10-9

10-8

10-7

10-6

Nx

FractionalL2errorin

Σh˜

Exponential convergence with Nx in h˜at a=0.999

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 23

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-3

-2

-1

0

1

2

3

r�

sin Θ

r�cos

Θ

h

�

Ha�=0.99L

0.1

0.2

0.3

0.4

0.5

-6-4-20246

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 24

01

23

r�

sin Θ

-20

2r�

cos Θ

0.2

0.4

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 24

Regime of validity

0.0 0.2 0.4 0.6 0.8 1.0

1

2

5

10

20

50

a�GM

È{�G

MÈ

Breakdown of perturbation theory

Decoupling limit valid

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 25

Forecasting bounds

• Observation of BH indistinguishable from GR predictions

• Size of ` correction below breakdown (caveat: cancellation)

• GRO J1655–40: M = 6.30± 0.27M�, a ≈ 0.65–0.75

0.0 0.2 0.4 0.6 0.8 1.0

1

2

5

10

20

50

a�GM

È{�G

MÈ

Breakdown of perturbation theory

Decoupling limit valid

=⇒ ` . 22km

• Better by 107 than Solar System bounds

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 26

Future work

• a→ GM limit analytically?

• All a analytically?

• Rest of the metric

• Accretion disk modeling

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 27

Takeaway

• I construct numerical solutions ∀r, ∀a for dCS BHs in decoupling limit

• I use solutions to determine the regime of validity of PT

0.0 0.2 0.4 0.6 0.8 1.0

1

2

5

10

20

50

a�GM

È{�G

MÈ

Breakdown of perturbation theory

Decoupling limit valid

• This can be turned around to forecast bounds ` . 22km fromGRO J1655–40 (M = 6.30± 0.27M�, a ≈ 0.65–0.75)

• For details see Phys. Rev. D 90, 044061 (2014) [arXiv:1407.2350]

Leo C. Stein Rapidly rotating BHs in dCS Einstein Symposium, 29 Oct. 2014 28