Rapidly rotating black holes in Chern-Simons gravity ...cxc.harvard.edu/fellows/symp_presentations/2014/Stein_L.pdfRapidly rotating black holes in Chern-Simons gravity: Decoupling

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]

http://arxiv.org/abs/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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NS merger

BH merger

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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?


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


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


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


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


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?


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


ææ

æ

æ

æ

æ

æ

æ

æ

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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


Ξ=

HGm

�r3L1

�2@k

m-

1D

Hinv.cu

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ure

radiu

sL

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æ

æ

æ

æ

æ

æ

æ

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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


Ξ=

HGm

�r3L1

�2@k

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1D

Hinv.cu

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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


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


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]



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


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


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


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


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!



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”)


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



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


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



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



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


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


01

23

r�

sin Θ

-20

2r�

cos Θ

0.2

0.4


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




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



=⇒ ` . 22km

• Better by 107 than Solar System bounds


Future work

• a→ GM limit analytically?

• All a analytically?

• Rest of the metric

• Accretion disk modeling


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



• 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]



Documents

Rapidly rotating black holes in Chern-Simons gravity ...cxc.harvard.edu/fellows/symp_presentations/2014/Stein_L.pdfRapidly rotating black holes in Chern-Simons gravity: Decoupling