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]
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
ææ
æ
æ
æ
æ
æ
æ
æ
æ
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
rvat
ure
radiu
sL
ææ
æ
æ
æ
æ
æ
æ
æ
æ
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
rvat
ure
radiu
sL
ææ
æ
æ
æ
æ
æ
æ
æ
æ
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
�r3L1
�2@k
m-
1D
Hinv.cu
rvat
ure
radiu
sL
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
ææ
æ
æ
æ
æ
æ
æ
æ
æ
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
rvat
ure
radiu
sL
ææ
æ
æ
æ
æ
æ
æ
æ
æ
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
�r3L1
�2@k
m-
1D
Hinv.cu
rvat
ure
radiu
sL
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