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Structure of black holes in theories beyondgeneral relativity
Weiming Wayne Zhao
LIGO SURF ProjectCaltech TAPIR
August 18, 2016
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 1 / 16
Acknowledgments
Leo C. SteinVijay VarmaMark ScheelKevin Barkett and other people in TAPIRAlan and other people in LIGOSURF, TAPIR, LSC, Caltech, NSF
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 1 / 16
Table of Contents
1 Introduction to beyond GR theories
2 Perturbation theory
3 Gauge for black holes
4 Numerics
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 1 / 16
Why corrections to GR?
• No known experiment has been directly inconsistent with GR• GR + QFT is not renormalizable at high energies• The full gravity theory has an effective field theory (EFT) with
GR as its zeroth order term
General relativity
Special relativity
post-NewtonianG→0
v/c→0Standard Model
QED
Maxwellh→0
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 2 / 16
Tests of GR
Two ways to study theories of gravity
• Theory independent - framework for calculations
• Theory dependent - statistical tests for detections
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 3 / 16
Why numerics?
• In principle, LIGO and descendants can detect deviations.
• We need waveform templates for matched filtering,
• generated from new types of numerical simulations of gravity.
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 4 / 16
Perturbations of GR
EFT =⇒ alternative theories of gravity are corrections to GR.• We can do perturbation theory!
We develop a formalism that can represent• corrections to GR• matter near the horizons of black holes
(i.e. “bumpy black holes”)Note that the spacetime does not have to be Ricci-flat.
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 5 / 16
Table of Contents
1 Introduction to beyond GR theories
2 Perturbation theory
3 Gauge for black holes
4 Numerics
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 5 / 16
Perturbation Theory
We have a 1-parameter family of geometries, described by themetric gab with the parameter ε. Expanding around ε = 0,
gab(ε) = g (ε=0)ab + ε
dgabdε
∣∣∣∣ε=0︸ ︷︷ ︸
h(1)ab
+O(ε2)
Let g (ε=0)ab be a Ricci-flat geometry like Schwarzschild or Kerr.
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 6 / 16
Theory independent Lagrangian
(Now to the physics)The action for a general interacting scalar θ
I =
∫d4x√−g
[R − 1
2∂aθ∂
aθ + εLint
]
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 7 / 16
Theory independent Equations of Motion
After making various calculations and arguments, the gist is that
Gab = T effab
where T effab = O(ε2) and ∇aT eff
ab = 0
�θ = S
and S = O(ε1)
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 8 / 16
Table of Contents
1 Introduction to beyond GR theories
2 Perturbation theory
3 Gauge for black holes
4 Numerics
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 8 / 16
Black hole spacetimes
In this project,• Conservation of angular momentum =⇒ axisymmetric• Gravitation radiation + late times =⇒ stationary
We can always∗ put the stationary, axisymmetric metrics into theWeyl-Lewis-Papapetrou (WLP) form:
ds2 = −V (dt −wdφ)2 + V−1(ρ2dφ2 + e2γ(dρ2 + e2λdz2)
)Unlike Kerr, this metric doesn’t have to be Ricci-flat.
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 9 / 16
Weyl-Lewis-Papapetrou (WLP) Gauge
Spin-2 gauge in gravity = choice of coordinatesWe see manifestly four metric degrees of freedom ~vIn perturbation, we have
V (ρ, z)w(ρ, z)γ(ρ, z)λ(ρ, z)
=
V0w0γ0λ0
+12ε2
δVδwδγδλ
︸ ︷︷ ︸
~v
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 10 / 16
WLP Gauge
The WLP metric:
ds2 = −V (dt −wdφ)2 + V−1(ρ2dφ2 + e2γ(dρ2 + e2λdz2)
)One can always fix λ = 0 in the Ricci-flat case.
For the non-Ricci-flat spacetimes, z 7→ f (z) keeps the WLP gauge,i.e. λ 7→ λ+ ∂z log f .
λ is ambiguous
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 11 / 16
z gauge fixing
I proved that the gauge freedom exactly corresponds to aconstant of integration C(z)
∂zδλ(ρ, z) =
∫ ρ
∞ρ′∂z
(T effρρ (ρ′, z)− T eff
zz (ρ′, z))
dρ′ + C(z)
And therefore we can choose C(z) = 0 to fully fix our gauge.
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 12 / 16
Table of Contents
1 Introduction to beyond GR theories
2 Perturbation theory
3 Gauge for black holes
4 Numerics
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 12 / 16
Linearized equations
Despite the redundancies in Gab = T effab, we can now cast the
linearized equations into the form
∆~v ≡ ∆0
δVδwδγδλ
+ lower order terms = source
where ∆0 is the induced 3-Laplacian of the Kerr background
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 13 / 16
Newton-Raphson method
Given this form, we use an iterative scheme to solve for thesefour metric variables in ∆~v = ~S
initial guess: ∆0~v0 = ~S0
∆(~v0 + δ~v) = ~S
∆~v0 + ∆0δ~v ≈ ~S
=⇒ ∆0δ~v ≈ ~S −∆~v0
iteratively solve: δ~v ≈ ∆−10
(~S −∆~v0
)
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 14 / 16
Current goals
• Simulating over a Kerr background with dynamicalChern-Simons
• Source is in rational polynomial Boyer-Lindquist coordinates
• Implementing the formalism directly (in SpEC)• Physically interesting quantities of these BHs
• thermodynamic entropy• ISCO• orbital frequencies• locations of the new horizons
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 15 / 16
Long Term Goals
• Non-stationary perturbations• Quasi-normal modes of these black holes
• Binary black holes• Gravitational waveforms at infinity• Non-perturbative solutions[1]• Extremal-Kerr solutions and relate to the Weak Gravity
Conjecture in string theory
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 16 / 16
Summary
• Tests of GR are important• Insight into developing theory-independent frameworks• Choosing gauge is important for numerical calculations• Numerical simulations =⇒ LIGO waveforms
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 17 / 16
Works cited
A. I. Harte and J. Vines, “Generating exact solutions to Einstein’s equation using linearized approximations,”arXiv:1608.04359 [gr-qc].
D. Hilditch, “An Introduction to Well-posedness and Free-evolution,” Int. J. Mod. Phys. A28 (2013) 1340015,arXiv:1309.2012 [gr-qc].
L. C. Stein, “Rapidly rotating black holes in dynamical Chern-Simons gravity: Decoupling limit solutions andbreakdown,” Phys. Rev. D90 no. 4, (2014) 044061, arXiv:1407.2350 [gr-qc].
A. Maselli, P. Pani, L. Gualtieri, and V. Ferrari, “Rotating black holes in Einstein-Dilaton-Gauss-Bonnet gravity withfinite coupling,” Phys. Rev. D92 no. 8, (2015) 083014, arXiv:1507.00680 [gr-qc].
K. Yagi and L. C. Stein, “Black Hole Based Tests of General Relativity,” Class. Quant. Grav. 33 no. 5, (2016) 054001,arXiv:1602.02413 [gr-qc].
M. Visser, “The Kerr spacetime: A Brief introduction,” in Kerr Fest: Black Holes in Astrophysics, General Relativityand Quantum Gravity.2007.arXiv:0706.0622 [gr-qc].
J. Boyd, Chebyshev and Fourier Spectral Methods: Second Revised Edition.Dover Books on Mathematics. Dover Publications, 2013.https://books.google.com/books?id=b4TCAgAAQBAJ.
G. B. Cook and M. A. Scheel, “Well-behaved harmonic time slices of a charged, rotating, boosted black hole,”Phys. Rev. D 56 (Oct, 1997) 4775–4781. http://link.aps.org/doi/10.1103/PhysRevD.56.4775.
Wayne Zhao (LIGO SURF) Structure of BHs beyond GR August 18, 2016 17 / 16
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