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Scalar self-force for highly eccentric orbits
in Kerr spacetime
Jonathan Thornburg
in collaboration with
Barry WardellDepartment of Astronomy and Department of Astronomy
Center for Spacetime Symmetries Cornell University
Indiana University Ithaca, New York, USA
Bloomington, Indiana, USA
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 1 / 18
Goals• Kerr
• highly-eccentric orbits:
• eLISA EMRIs: likely have e up to ∼ 0.8
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 2 / 18
Goals• Kerr
• highly-eccentric orbits:
• eLISA EMRIs: likely have e up to ∼ 0.8• eLISA IMRIs (likely rare, but maybe very strong sources if they exist):
may have e up to ∼ 0.998
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 2 / 18
Goals• Kerr
• highly-eccentric orbits:
• eLISA EMRIs: likely have e up to ∼ 0.8• eLISA IMRIs (likely rare, but maybe very strong sources if they exist):
may have e up to ∼ 0.998
• [now] (bound, geodesic) equatorial orbits: [future] inclined (generic geodesic)
• compute self-force (all around fixed orbit)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 2 / 18
Goals• Kerr
• highly-eccentric orbits:
• eLISA EMRIs: likely have e up to ∼ 0.8• eLISA IMRIs (likely rare, but maybe very strong sources if they exist):
may have e up to ∼ 0.998
• [now] (bound, geodesic) equatorial orbits: [future] inclined (generic geodesic)
• compute self-force (all around fixed orbit)
⇒ easy access to ∆geodesic per orbit, but we haven’t done this yet⇒ [easy access to emitted radiation at J +, but we haven’t done this yet]
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 2 / 18
Goals• Kerr
• highly-eccentric orbits:
• eLISA EMRIs: likely have e up to ∼ 0.8• eLISA IMRIs (likely rare, but maybe very strong sources if they exist):
may have e up to ∼ 0.998
• [now] (bound, geodesic) equatorial orbits: [future] inclined (generic geodesic)
• compute self-force (all around fixed orbit)
⇒ easy access to ∆geodesic per orbit, but we haven’t done this yet⇒ [easy access to emitted radiation at J +, but we haven’t done this yet]
Limitations
• O(µ)
• scalar field; develop techniques for gravitational field (Lorenz-gauge)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 2 / 18
Goals• Kerr
• highly-eccentric orbits:
• eLISA EMRIs: likely have e up to ∼ 0.8• eLISA IMRIs (likely rare, but maybe very strong sources if they exist):
may have e up to ∼ 0.998
• [now] (bound, geodesic) equatorial orbits: [future] inclined (generic geodesic)
• compute self-force (all around fixed orbit)
⇒ easy access to ∆geodesic per orbit, but we haven’t done this yet⇒ [easy access to emitted radiation at J +, but we haven’t done this yet]
Limitations
• O(µ)
• scalar field; develop techniques for gravitational field (Lorenz-gauge)
⋆ may have a solution to Lorenz gauge instabilities [with Sam Dolan]⇒ if this works, then extension to gravitational field looks doable
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 2 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
• 4th order effective source and puncture field
• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
• 4th order effective source and puncture field
• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]
may have a solution to these [joint with Dolan]
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
• 4th order effective source and puncture field
• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]
may have a solution to these [joint with Dolan]
m-mode (e imφ) decomposition, time-domain evolution
• exploit axisymmetry of Kerr background
• can handle (almost) any orbit, including high eccentricity
• separate 2+1-dimensional time-domain (numerical) evolution for each m
• presently equatorial orbits, but no serious obstacles to generic orbits
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
• 4th order effective source and puncture field
• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]
may have a solution to these [joint with Dolan]
m-mode (e imφ) decomposition, time-domain evolution
• exploit axisymmetry of Kerr background
• can handle (almost) any orbit, including high eccentricity
• separate 2+1-dimensional time-domain (numerical) evolution for each m
• presently equatorial orbits, but no serious obstacles to generic orbits
• worldtube scheme
• worldtube moves in (r , θ) to follow the particle around the orbit
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
• 4th order effective source and puncture field
• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]
may have a solution to these [joint with Dolan]
m-mode (e imφ) decomposition, time-domain evolution
• exploit axisymmetry of Kerr background
• can handle (almost) any orbit, including high eccentricity
• separate 2+1-dimensional time-domain (numerical) evolution for each m
• presently equatorial orbits, but no serious obstacles to generic orbits
• worldtube scheme
• worldtube moves in (r , θ) to follow the particle around the orbit
• hyperboloidal slices (reach horizon and J +)[Zenginoglu, arXiv:1008.3809 = J. Comp. Phys. 230, 2286]
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Overall Plan of the Computation
Effective-Source (puncture-field) regularization
• 4th order effective source and puncture field
• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]
may have a solution to these [joint with Dolan]
m-mode (e imφ) decomposition, time-domain evolution
• exploit axisymmetry of Kerr background
• can handle (almost) any orbit, including high eccentricity
• separate 2+1-dimensional time-domain (numerical) evolution for each m
• presently equatorial orbits, but no serious obstacles to generic orbits
• worldtube scheme
• worldtube moves in (r , θ) to follow the particle around the orbit
• hyperboloidal slices (reach horizon and J +)[Zenginoglu, arXiv:1008.3809 = J. Comp. Phys. 230, 2286]
• (fixed) mesh refinement; some (finer) grids follow the worldtube/particle
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 3 / 18
Effective source (puncture field) regularization
Assume a δ-function particle with scalar charge q.
The particle’s physical (retarded) scalar field ϕ satisfies �ϕ = δ(
x − xparticle(t))
.ϕ is singular at the particle.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 4 / 18
Effective source (puncture field) regularization
Assume a δ-function particle with scalar charge q.
The particle’s physical (retarded) scalar field ϕ satisfies �ϕ = δ(
x − xparticle(t))
.ϕ is singular at the particle.
If we knew the Detwiler-Whiting decomposition ϕ = ϕsingular + ϕregular explicitly,we could compute the self-force via Fa = q(∇aϕregular)
∣
∣
particle.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 4 / 18
Effective source (puncture field) regularization
Assume a δ-function particle with scalar charge q.
The particle’s physical (retarded) scalar field ϕ satisfies �ϕ = δ(
x − xparticle(t))
.ϕ is singular at the particle.
If we knew the Detwiler-Whiting decomposition ϕ = ϕsingular + ϕregular explicitly,we could compute the self-force via Fa = q(∇aϕregular)
∣
∣
particle. But it’s very hard
to explicly compute the decomposition.
Instead we choose ϕpuncture ≈ ϕsingular so that ϕresidual := ϕ− ϕpuncture
is finite and “differentiable enough” at the particle.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 4 / 18
Effective source (puncture field) regularization
Assume a δ-function particle with scalar charge q.
The particle’s physical (retarded) scalar field ϕ satisfies �ϕ = δ(
x − xparticle(t))
.ϕ is singular at the particle.
If we knew the Detwiler-Whiting decomposition ϕ = ϕsingular + ϕregular explicitly,we could compute the self-force via Fa = q(∇aϕregular)
∣
∣
particle. But it’s very hard
to explicly compute the decomposition.
Instead we choose ϕpuncture ≈ ϕsingular so that ϕresidual := ϕ− ϕpuncture
is finite and “differentiable enough” at the particle. It’s then easy to see that
�ϕresidual =
{
0 at the particle−�ϕpuncture elsewhere
}
:= Seffective
If we can solve this equation for ϕresidual, then we can compute the self-force(exactly!) via Fa = q(∇aϕresidual)
∣
∣
particle.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 4 / 18
Puncture field and effective source
We choose ϕpuncture so that |ϕpuncture − ϕsingular| = O(
‖x − xparticle‖n)
where the puncture order n ≥ 2 is a parameter.
ϕresidual is then C n−2 at the particle, and Seffective is C0.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 5 / 18
Puncture field and effective source
We choose ϕpuncture so that |ϕpuncture − ϕsingular| = O(
‖x − xparticle‖n)
where the puncture order n ≥ 2 is a parameter.
ϕresidual is then C n−2 at the particle, and Seffective is C0.
The choice of the puncture order n is a tradeoff:Higher n ⇒ ϕresidual is smoother at the particle (good),but ϕpuncture and Seffective are more complicated (expensive) to compute.
We choose n = 4 ⇒ ϕresidual is C2 at the particle.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 5 / 18
Puncture field and effective source
We choose ϕpuncture so that |ϕpuncture − ϕsingular| = O(
‖x − xparticle‖n)
where the puncture order n ≥ 2 is a parameter.
ϕresidual is then C n−2 at the particle, and Seffective is C0.
The choice of the puncture order n is a tradeoff:Higher n ⇒ ϕresidual is smoother at the particle (good),but ϕpuncture and Seffective are more complicated (expensive) to compute.
We choose n = 4 ⇒ ϕresidual is C2 at the particle.
The actual computation of ϕpuncture and Seffective involves lengthly seriesexpansions in Mathematica, then machine-generated C code. SeeWardell, Vega, Thornburg, and Diener, PRD 85, 104044 (2012) = arXiv:1112.6355
for details.
Computing Seffective at a single event requires ∼ 12 × 106 arithmetic operations.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 5 / 18
The worldtube
Problems:
• ϕpuncture and Seffective are only defined in a neighbourhood of the particle
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 6 / 18
The worldtube
Problems:
• ϕpuncture and Seffective are only defined in a neighbourhood of the particle
• far-field outgoing-radiation BCs apply to ϕ, not ϕresidual
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 6 / 18
The worldtube
Problems:
• ϕpuncture and Seffective are only defined in a neighbourhood of the particle
• far-field outgoing-radiation BCs apply to ϕ, not ϕresidual
Solution:
introduce finite worldtube containing the particle worldline
• define “numerical field” ϕnumerical =
{
ϕresidual inside the worldtubeϕ outside the worldtube
• compute ϕnumerical by numerically solving
�ϕnumerical =
{
Seffective inside the worldtube0 outside the worldtube
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 6 / 18
The worldtube
Problems:
• ϕpuncture and Seffective are only defined in a neighbourhood of the particle
• far-field outgoing-radiation BCs apply to ϕ, not ϕresidual
Solution:
introduce finite worldtube containing the particle worldline
• define “numerical field” ϕnumerical =
{
ϕresidual inside the worldtubeϕ outside the worldtube
• compute ϕnumerical by numerically solving
�ϕnumerical =
{
Seffective inside the worldtube0 outside the worldtube
• now Seffective is only needed inside the worldtube
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 6 / 18
The worldtube
Problems:
• ϕpuncture and Seffective are only defined in a neighbourhood of the particle
• far-field outgoing-radiation BCs apply to ϕ, not ϕresidual
Solution:
introduce finite worldtube containing the particle worldline
• define “numerical field” ϕnumerical =
{
ϕresidual inside the worldtubeϕ outside the worldtube
• compute ϕnumerical by numerically solving
�ϕnumerical =
{
Seffective inside the worldtube0 outside the worldtube
• now Seffective is only needed inside the worldtube
• ϕnumerical has a ±ϕpuncture jump discontinuity across worldtube boundary⇒ finite difference operators that cross the worldtube boundary
must compensate for the jump discontinuity
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 6 / 18
m-mode decomposition
Instead of numerically solving �ϕnumerical =
{
Seffective inside the worldtube0 outside the worldtube
in 3+1 dimensions, we Fourier-decompose into e imφ modes and solve for eachFourier mode in 2+1 dimensions via
�m ϕnumerical,m =
{
Seffective,m inside the worldtube0 outside the worldtube
numerically
solve this
for each m
in 2+1D
The self-force is given (exactly!) by Fa = q∞∑
m=0
(∇aϕnumerical,m)∣
∣
particle
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 7 / 18
m-mode decomposition
Instead of numerically solving �ϕnumerical =
{
Seffective inside the worldtube0 outside the worldtube
in 3+1 dimensions, we Fourier-decompose into e imφ modes and solve for eachFourier mode in 2+1 dimensions via
�m ϕnumerical,m =
{
Seffective,m inside the worldtube0 outside the worldtube
numerically
solve this
for each m
in 2+1D
The self-force is given (exactly!) by Fa = q∞∑
m=0
(∇aϕnumerical,m)∣
∣
particle
Advantages (vs. direct solution in 3 + 1 dimensions):
• can use different numerical parameters for different m
⋆ (this is crucial for our hoped-for solution to theLorenz-gauge instabilities in the gravitational case)
• each individual m’s evolution is smaller ⇒ test/debug code on laptop
• get moderate parallelism “for free” (run different m’s evolutions in parallel)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 7 / 18
Moving the worldtube
We actually do m-mode decomposition before introducing worldtube⇒ worldtube “lives” in (t, r , θ) space, not full spacetime
The worldtube must contain the particle in (r , θ).But for a non-circular orbit, the particle moves in (r , θ) during the orbit.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 8 / 18
Moving the worldtube
We actually do m-mode decomposition before introducing worldtube⇒ worldtube “lives” in (t, r , θ) space, not full spacetime
The worldtube must contain the particle in (r , θ).But for a non-circular orbit, the particle moves in (r , θ) during the orbit.
Small eccentricity: can use a worldtube big enough to contain the entire orbit
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 8 / 18
Moving the worldtube
We actually do m-mode decomposition before introducing worldtube⇒ worldtube “lives” in (t, r , θ) space, not full spacetime
The worldtube must contain the particle in (r , θ).But for a non-circular orbit, the particle moves in (r , θ) during the orbit.
Small eccentricity: can use a worldtube big enough to contain the entire orbit
Large eccentricity:
• must move the worldtube in (r , θ) to follow the particle around the orbit
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 8 / 18
Moving the worldtube
We actually do m-mode decomposition before introducing worldtube⇒ worldtube “lives” in (t, r , θ) space, not full spacetime
The worldtube must contain the particle in (r , θ).But for a non-circular orbit, the particle moves in (r , θ) during the orbit.
Small eccentricity: can use a worldtube big enough to contain the entire orbit
Large eccentricity:
• must move the worldtube in (r , θ) to follow the particle around the orbit
• recall that our numerically-evolved field is
ϕnumerical :=
{
ϕ− ϕpuncture inside the worldtubeϕ outside the worldtube
this means then if we move the worldtube, we must adjust the evolvedϕnumerical: add ±ϕpuncture at spatial points which change from being insidethe worldtube to being outside, or vice versa
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 8 / 18
Code Validation
Comparison withfrequency-domainresults kindly providedby Niels Warburton
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 9 / 18
Code Validation
Comparison withfrequency-domainresults kindly providedby Niels Warburton
Typical example:(a, p, e) = (0.9, 10M , 0.5)⇒ results agree to
∼ 10−5 relative error
We have also compareda variety of otherconfigurations, withfairly similar results
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
6 8 10 12 14 16 18 20
(r/M)3 ||Fa||+(r
/M)3 ||
Fa|
| + (
q2 /M)
r (M)
(r/M)3 ||Fa dissipative part||+(r/M)3 ||Fa conservative part||+(r/M)3 ||difference in Fa dissipative part||+(r/M)3 ||difference in Fa conservative part||+
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 9 / 18
e = 0.8 orbit
(a, p, e) = (0.6, 8M , 0.8)
t
0
100
200
300
4 6 8 10 15 20 30 40
r3 Ft (
10-3
q2 /M
)
r (M)
spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop
outwardsinwards
r
-120
-100
-80
-60
-40
-20
0
20
40
4 6 8 10 15 20 30 40
r3 Fr (
10-3
q2 /M
)
r (M)
spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop
outwardsinwards
φ
-2500
-2000
-1500
-1000
-500
0
4 6 8 10 15 20 30 40
r3 Fφ
(10-3
q2 /M
)
r (M)
spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop
outwardsinwards
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 10 / 18
e = 0.8 orbit
(a, p, e) = (0.6, 8M , 0.8)
t
0
100
200
300
4 6 8 10 15 20 30 40
r3 Ft (
10-3
q2 /M
)
r (M)
spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop
outwardsinwards
r
-120
-100
-80
-60
-40
-20
0
20
40
4 6 8 10 15 20 30 40
r3 Fr (
10-3
q2 /M
)
r (M)
spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop
outwardsinwards
φ
-2500
-2000
-1500
-1000
-500
0
4 6 8 10 15 20 30 40
r3 Fφ
(10-3
q2 /M
)
r (M)
spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop
outwardsinwards
Notice the “bump” in Fr
near r = 15M moving outwards
Maybe a caustic crossing?
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 10 / 18
Wiggles!
Higher-eccentricity orbit:(a, p, e) = (0.99, 7M , 0.9)
t
-200
0
200
400
4 6 8 10 15 20 30 40 50 60 70
r3 Ft (
10-3
q2 /M
)
r (M)
spin=0.99 p=7M e=0.9 ∆R*=M/64 Self-force Loop
outgoingingoing
r
-200
-100
0
100
4 6 8 10 15 20 30 40 50 60 70
r3 Fr (
10-3
q2 /M
)
r (M)
spin=0.99 p=7M e=0.9 ∆R*=M/64 Self-force Loop
outgoingingoing
φ
-2000
-1000
0
1000
4 6 8 10 15 20 30 40 50 60 70
r3 Fφ
(10-3
q2 /M
)
r (M)
spin=0.99 p=7M e=0.9 ∆R*=M/64 Self-force Loop
outgoingingoing
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 11 / 18
Wiggles!
Higher-eccentricity orbit:(a, p, e) = (0.99, 7M , 0.9)
t
-200
0
200
400
4 6 8 10 15 20 30 40 50 60 70
r3 Ft (
10-3
q2 /M
)
r (M)
spin=0.99 p=7M e=0.9 ∆R*=M/64 Self-force Loop
outgoingingoing
r
-200
-100
0
100
4 6 8 10 15 20 30 40 50 60 70
r3 Fr (
10-3
q2 /M
)
r (M)
spin=0.99 p=7M e=0.9 ∆R*=M/64 Self-force Loop
outgoingingoing
φ
-2000
-1000
0
1000
4 6 8 10 15 20 30 40 50 60 70
r3 Fφ
(10-3
q2 /M
)
r (M)
spin=0.99 p=7M e=0.9 ∆R*=M/64 Self-force Loop
outgoingingoing
Notice:
• wiggles on outgoing leg of orbit
• wiggles not seen on ingoing leg
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 11 / 18
Wiggles as Kerr Quasinormal Modes: Mode Fit
Leor Barack suggested that wiggles might be quasinormal modes of the(background) Kerr spacetime, excited by the particle’s close flyby. Test this byfitting decay of wiggles to a damped sinusoid with corrections for motion of theobserver (particle):
rF [m=1]r
(u) = background(u) + A exp(
−u − u0
τ
)
sin(
φ0 + 2πu − u0
T
)
where u := t − r∗
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 50 100 150 200
wiggle period = 12.892
wiggle 1/e decay time = 25.142
104 F
r
u = t-r*
spin=0.99 p=7M e=0.9 rFr(u = t-r*) wiggles fit
data (orbit 5)fn(u)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 12 / 18
Wiggles as Kerr Quasinormal Modes: Mode Frequencies
Now compare wiggle-fit complex frequency ω := 2π/T − i/τvs. known Kerr quasinormal mode frequencies computed by Emanuele Berti.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 13 / 18
Wiggles as Kerr Quasinormal Modes: Mode Frequencies
Now compare wiggle-fit complex frequency ω := 2π/T − i/τvs. known Kerr quasinormal mode frequencies computed by Emanuele Berti.
⇒ Nice agreement with least-damped corotating QNM!
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Im[ω
]
Re[ω]
spin=0.99 p=7M e=0.9 rFr(u = t-r*) wiggles decay vs Kerr QNMs
wiggles decay fitKerr QNMs (m=+1)Kerr QNMs (m=-1)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 13 / 18
Wiggles as Kerr Quasinormal Modes: Varying BH Spin
Repeat wiggle-fit procedure for other Kerr spins (0.99, 0.95, 0.9, and 0.8)⇒ Nice agreement with least-damped corotating QNM for all BH spins!
a = 0.99
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Im[ω
]
Re[ω]
spin=0.99 p=7M e=0.9 rFr(u = t-r*) wiggles decay vs Kerr QNMs
wiggles decay fitKerr QNMs (m=+1)Kerr QNMs (m=-1)
a = 0.95
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Im[ω
]
Re[ω]
spin=0.95 p=7M e=0.9 rFr(u = t-r*) wiggles decay vs Kerr QNMs
wiggles decay fitKerr QNMs (m=+1)Kerr QNMs (m=-1)
a = 0.9
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Im[ω
]
Re[ω]
spin=0.9 p=7M e=0.9 rFr(u = t-r*) wiggles decay vs Kerr QNMs
wiggles decay fitKerr QNMs (m=+1)Kerr QNMs (m=-1)
a = 0.8
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Im[ω
]
Re[ω]
spin=0.8 p=7M e=0.9 rFr(u = t-r*) wiggles decay vs Kerr QNMs
wiggles decay fitKerr QNMs (m=+1)Kerr QNMs (m=-1)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 14 / 18
More wiggles
-300
-200
-100
0
100
200
300
-2000
-1000
0
1000
2000e9 configuration(a,p,e) = (0.99,7M,0.9)
(r/M
)3 {F
t,Fr}
(10
-3 q
2 /M)
(r/M
)3 Fφ
(10-3
q2 /M
)
(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)
-600
-400
-200
0
200
400
600
-2000
-1000
0
1000
2000
e95 configuration(a,p,e) = (0.99,7M,0.95)
(r/M
)3 {F
t,Fr}
(10
-3 q
2 /M)
(r/M
)3 Fφ
(10-3
q2 /M
)
(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
-100 0 100 200 300 400-15000
-10000
-5000
0
5000
10000
15000ze98 configuration(a,p,e) = (0.99,2.4M,0.98)
(r/M
)3 {F
t,Fr}
(10
-3 q
2 /M)
(r/M
)3 Fφ
(10-3
q2 /M
)
t (M)
(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)
We now see wiggles formany configurations with
• high Kerr spin
• highly-eccentricprograde orbitwith close-inperiastron
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 15 / 18
More wiggles
-300
-200
-100
0
100
200
300
-2000
-1000
0
1000
2000e9 configuration(a,p,e) = (0.99,7M,0.9)
(r/M
)3 {F
t,Fr}
(10
-3 q
2 /M)
(r/M
)3 Fφ
(10-3
q2 /M
)
(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)
-600
-400
-200
0
200
400
600
-2000
-1000
0
1000
2000
e95 configuration(a,p,e) = (0.99,7M,0.95)
(r/M
)3 {F
t,Fr}
(10
-3 q
2 /M)
(r/M
)3 Fφ
(10-3
q2 /M
)
(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
-100 0 100 200 300 400-15000
-10000
-5000
0
5000
10000
15000ze98 configuration(a,p,e) = (0.99,2.4M,0.98)
(r/M
)3 {F
t,Fr}
(10
-3 q
2 /M)
(r/M
)3 Fφ
(10-3
q2 /M
)
t (M)
(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)
We now see wiggles formany configurations with
• high Kerr spin
• highly-eccentricprograde orbitwith close-inperiastron
Note non-sinsoidalwiggle shapes⇒ multiple QNMs?
maybe caustic crossingsare also important?(Ottewill & Wardell)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 15 / 18
Gravitation: Unstable Lorenz Gauge Modes
How to extend this work to the full gravitational field?
• Work in Lorenz gauge ⇒ metric perturbation is isotropic at the particle.
• The effective-source (puncture-field) regularization works fine for thegravitational case.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 16 / 18
Gravitation: Unstable Lorenz Gauge Modes
How to extend this work to the full gravitational field?
• Work in Lorenz gauge ⇒ metric perturbation is isotropic at the particle.
• The effective-source (puncture-field) regularization works fine for thegravitational case.
• Dolan and Barack [PRD 87, 084066 (2013) = arXiv:1211.4586] found that them = 0 and m = 1 modes have Lorenz gauge instabilities. These are modeswhich are consistent with the Lorenz gauge condition at any finite time, butblow up as t → ∞. They were ableto stabilize the m = 0 mode using adynamically-driven generalized Lorenzgauge (analogous to the Z
4 evolutionsystem in full nonlinear numericalrelativity).
• They were not able to stabilize them = 1 mode. The instability islinear in time. 0
10
20
30
40
50
60
70
0 50 100 150 200
||sta
te v
ecto
r||
t
standard
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 16 / 18
(Maybe) Stabilizing the m = 1 Lorenz Gauge Mode
Basic idea (schematic): [Joint work with Sam Dolan]
• First define (choose) an inner product on state vectors u.
• Then compute the growing mode G(t, x i ) := tugrowing(xi ). Notice that this
is a homogeneous solution of the evolution equation. ugrowing depends onlyon the spacetime, not on the initial data or particle orbit.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 17 / 18
(Maybe) Stabilizing the m = 1 Lorenz Gauge Mode
Basic idea (schematic): [Joint work with Sam Dolan]
• First define (choose) an inner product on state vectors u.
• Then compute the growing mode G(t, x i ) := tugrowing(xi ). Notice that this
is a homogeneous solution of the evolution equation. ugrowing depends onlyon the spacetime, not on the initial data or particle orbit.
• Write m = 1 gravitational perturbation evolution as dudt
= R(u)
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 17 / 18
(Maybe) Stabilizing the m = 1 Lorenz Gauge Mode
Basic idea (schematic): [Joint work with Sam Dolan]
• First define (choose) an inner product on state vectors u.
• Then compute the growing mode G(t, x i ) := tugrowing(xi ). Notice that this
is a homogeneous solution of the evolution equation. ugrowing depends onlyon the spacetime, not on the initial data or particle orbit.
• Write m = 1 gravitational perturbation evolution as dudt
= R(u)
• Replace evolution eqn with dudt
= R(u)
where R(u) := R(u) + λugrowing
• Choose (update) λ “every so often”such that R(u) ⊥ ugrowing
(⊥ w.r.t. our chosen inner product).
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 17 / 18
(Maybe) Stabilizing the m = 1 Lorenz Gauge Mode
Basic idea (schematic): [Joint work with Sam Dolan]
• First define (choose) an inner product on state vectors u.
• Then compute the growing mode G(t, x i ) := tugrowing(xi ). Notice that this
is a homogeneous solution of the evolution equation. ugrowing depends onlyon the spacetime, not on the initial data or particle orbit.
• Write m = 1 gravitational perturbation evolution as dudt
= R(u)
• Replace evolution eqn with dudt
= R(u)
where R(u) := R(u) + λugrowing
• Choose (update) λ “every so often”such that R(u) ⊥ ugrowing
(⊥ w.r.t. our chosen inner product).
• This is equivalent to Gram-Schmidtorthogonalizing R w.r.t. ugrowing.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 17 / 18
(Maybe) Stabilizing the m = 1 Lorenz Gauge Mode
Basic idea (schematic): [Joint work with Sam Dolan]
• First define (choose) an inner product on state vectors u.
• Then compute the growing mode G(t, x i ) := tugrowing(xi ). Notice that this
is a homogeneous solution of the evolution equation. ugrowing depends onlyon the spacetime, not on the initial data or particle orbit.
• Write m = 1 gravitational perturbation evolution as dudt
= R(u)
• Replace evolution eqn with dudt
= R(u)
where R(u) := R(u) + λugrowing
• Choose (update) λ “every so often”such that R(u) ⊥ ugrowing
(⊥ w.r.t. our chosen inner product).
• This is equivalent to Gram-Schmidtorthogonalizing R w.r.t. ugrowing.
0
10
20
30
40
50
60
70
0 50 100 150 200-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
||sta
te v
ecto
r||
λ
t
standardorthogonalizedλ
This seems to work: evolutions are stable!
• Now trying it for sourced evolution. . .
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 17 / 18
Conclusions
Methods
• effective source/puncture field regularization works very well
• Zenginoglu’s hyperboloidal slices work very well
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 18 / 18
Conclusions
Methods
• effective source/puncture field regularization works very well
• Zenginoglu’s hyperboloidal slices work very well
• moving worldtube allows highly-eccentric orbits• we have done up to e = 0.98• higher is possible but expensive with current code
• details → Thornburg & Wardell, arXiv:1607.?????
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 18 / 18
Conclusions
Methods
• effective source/puncture field regularization works very well
• Zenginoglu’s hyperboloidal slices work very well
• moving worldtube allows highly-eccentric orbits• we have done up to e = 0.98• higher is possible but expensive with current code
• details → Thornburg & Wardell, arXiv:1607.?????
Wiggles
• if high Kerr spin, a close particle passage excites Kerr quasinormal modes;these show up as “wiggles” in the local self-force
• are caustic crossings also important?
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 18 / 18
Conclusions
Methods
• effective source/puncture field regularization works very well
• Zenginoglu’s hyperboloidal slices work very well
• moving worldtube allows highly-eccentric orbits• we have done up to e = 0.98• higher is possible but expensive with current code
• details → Thornburg & Wardell, arXiv:1607.?????
Wiggles
• if high Kerr spin, a close particle passage excites Kerr quasinormal modes;these show up as “wiggles” in the local self-force
• are caustic crossings also important?
• Maarten van de Meent has now found wiggles in Ψ4 flux at J +
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 18 / 18
Conclusions
Methods
• effective source/puncture field regularization works very well
• Zenginoglu’s hyperboloidal slices work very well
• moving worldtube allows highly-eccentric orbits• we have done up to e = 0.98• higher is possible but expensive with current code
• details → Thornburg & Wardell, arXiv:1607.?????
Wiggles
• if high Kerr spin, a close particle passage excites Kerr quasinormal modes;these show up as “wiggles” in the local self-force
• are caustic crossings also important?
• Maarten van de Meent has now found wiggles in Ψ4 flux at J +
Gravitation [Joint work with Sam Dolan]
⋆ May have found a way to stabilize Lorenz gauge modes⇒ can do gravitational self-force etc.
Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 18 / 18