celestial mechanics in kerr spacetime
TRANSCRIPT
Celestial mechanics in Kerr spacetime
Alexandre Le Tiec
Laboratoire Univers et TheoriesObservatoire de Paris / CNRS
Collaborators: R. Fujita, S. Isoyama,H. Nakano, N. Sago, and T. Tanaka
CQG 31 (2014) 097001, arXiv:1311.3836 [gr-qc]
PRL 113 (2014) 161101, arXiv:1404.6133 [gr-qc]
CQG 34 (2017) 134001, arXiv:1612.02504 [gr-qc]
Outline
1 Gravitational waves
2 EMRIs and the gravitational self-force
3 Geodesic motion in Kerr spacetime
4 Beyond the geodesic approximation
5 Innermost stable circular orbits
UFF – November 26, 2018 Alexandre Le Tiec
Outline
1 Gravitational waves
2 EMRIs and the gravitational self-force
3 Geodesic motion in Kerr spacetime
4 Beyond the geodesic approximation
5 Innermost stable circular orbits
UFF – November 26, 2018 Alexandre Le Tiec
What is a gravitational wave ?
A gravitational wave is a tiny ripple in the curvature ofspacetime that propagates at the vacuum speed of light
gravitational wave background curvature
hab + 2Rabcdhcd = −16πTab
Key prediction of Einstein’s general theory of relativity
UFF – November 26, 2018 Alexandre Le Tiec
The gravitational-wave spectrum
UFF – November 26, 2018 Alexandre Le Tiec
Gravitational-wave science
Fundamental physics• Strong-field tests of GR• Black hole no-hair theorem• Cosmic censorship conjecture• Dark energy equation of state• Alternatives to general relativity
Astrophysics• Formation and evolution of compact binaries• Origin and mechanisms of γ-ray bursts• Internal structure of neutron stars
Cosmology• Cosmography and measure of Hubble’s constant• Origin and growth of supermassive black holes• Phase transitions during primordial Universe
UFF – November 26, 2018 Alexandre Le Tiec
Gravitational-wave science
Fundamental physics4 Strong-field tests of GR• Black hole no-hair theorem• Cosmic censorship conjecture• Dark energy equation of state
4 Alternatives to general relativity
Astrophysics4 Formation and evolution of compact binaries
4 Origin and mechanisms of γ-ray bursts
4 Internal structure of neutron stars
Cosmology4 Cosmography and measure of Hubble’s constant• Origin and growth of supermassive black holes• Phase transitions during primordial Universe
UFF – November 26, 2018 Alexandre Le Tiec
Ground-based interferometric detectors
LIGO Hanford LIGO Livingston
VirgoGEO600
UFF – November 26, 2018 Alexandre Le Tiec
UFF – November 26, 2018 Alexandre Le Tiec
Need for accurate template waveforms
0 0.1 0.2 0.3 0.4 0.5
-2×10-19
-1×10-19
0
1×10-19
2×10-19
t (sec.)
n(t)
h(t)
10 100 1000 1000010-24
10-23
10-22
10-21
f (Hz)
h(f)
S1/2(f)
If the expected signal is known in advance then n(t) can be filteredand h(t) recovered by matched filtering −→ template waveforms
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: L. Lindblom)
Need for accurate template waveforms
0 0.1 0.2 0.3 0.4 0.5
-2×10-19
-1×10-19
0
1×10-19
2×10-19
t (sec.)
n(t)
h(t)
100 h(t)
10 100 1000 1000010-24
10-23
10-22
10-21
f (Hz)
h(f)
S1/2(f)
If the expected signal is known in advance then n(t) can be filteredand h(t) recovered by matched filtering −→ template waveforms
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: L. Lindblom)
A recent example: the event GW151226
UFF – November 26, 2018 Alexandre Le Tiec[PRL 116 (2016) 241103]
LISA: a gravitational antenna in space
The LISA mission proposal was accepted by ESA in 2017 for L3slot, with a launch planned for 2034 [http://www.lisamission.org]
UFF – November 26, 2018 Alexandre Le Tiec
LISA sources of gravitational waves
UFF – November 26, 2018 Alexandre Le Tiec
Outline
1 Gravitational waves
2 EMRIs and the gravitational self-force
3 Geodesic motion in Kerr spacetime
4 Beyond the geodesic approximation
5 Innermost stable circular orbits
UFF – November 26, 2018 Alexandre Le Tiec
Extreme mass ratio inspirals
• LISA sensitive to MBH ∼ 105 − 107M → q ∼ 10−7 − 10−4
• Torb ∝ MBH ∼ hr and Tinsp ∝ MBH/q ∼ yrs
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: L. Barack)
Geodesy
Botriomeladesy
Test of the black hole no-hair theorem
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: S. Drasco)
Geodesy
Botriomeladesy
Test of the black hole no-hair theorem
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: S. Drasco)
Geodesy Botriomeladesy
Test of the black hole no-hair theorem
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: S. Drasco)
Geodesy Botriomeladesy
Test of the black hole no-hair theorem
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: S. Drasco)
Geodesy
M` arbitrary
Botriomeladesy
M` + iS` = M(ia)`
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: S. Drasco)
Testing the black hole no-hair theorem
UFF – November 26, 2018 Alexandre Le Tiec
Q = −S2/M
[PRD 95 (2017) 103012]
Gravitational self-force
• Dissipative component ←→ gravitational waves
• Conservative component ←→ some secular effects
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Gravitational self-force
Initial Configuration
Later Configuration
Conservative
self- force
Dissipative
self- force
Direction of
apsidal advance
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: Osburn et al. 2016)
Gravitational self-force
Spacetime metric
gαβ = gαβ
+ hαβ
Small parameter
q ≡ m
M 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m → 0
M
u
UFF – November 26, 2018 Alexandre Le Tiec
Gravitational self-force
Spacetime metric
gαβ = gαβ
+ hαβ
Small parameter
q ≡ m
M 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m
M
u
UFF – November 26, 2018 Alexandre Le Tiec
Gravitational self-force
Spacetime metric
gαβ = gαβ + hαβ
Small parameter
q ≡ m
M 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m
M
u
h
UFF – November 26, 2018 Alexandre Le Tiec
Gravitational self-force
Spacetime metric
gαβ = gαβ + hαβ
Small parameter
q ≡ m
M 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m
M
f
u
UFF – November 26, 2018 Alexandre Le Tiec
Gravitational self-force
Spacetime metric
gαβ = gαβ + hαβ
Small parameter
q ≡ m
M 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m
M
f
u u
f
UFF – November 26, 2018 Alexandre Le Tiec
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βh
tailλσ −∇λhtail
βσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βh
tailλσ −∇λhtail
βσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βh
tailλσ −∇λhtail
βσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βh
tailλσ −∇λhtail
βσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
self-acc. motion in gαβ ⇐⇒ geodesic motion in gαβ + hRαβ
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
self-acc. motion in gαβ ⇐⇒ geodesic motion in gαβ + hRαβ
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
self-acc. motion in gαβ ⇐⇒ geodesic motion in gαβ + hRαβ
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
self-acc. motion in gαβ ⇐⇒ geodesic motion in gαβ + hRαβ
UFF – November 26, 2018 Alexandre Le Tiec
(Credit: A. Pound)
Outline
1 Gravitational waves
2 EMRIs and the gravitational self-force
3 Geodesic motion in Kerr spacetime
4 Beyond the geodesic approximation
5 Innermost stable circular orbits
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut
• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
Canonical Hamiltonian
H(x , u) =1
2gαβ(x) uαuβ
Equations of motion
xα =∂H
∂uα, uα = − ∂H
∂xα
Constants of motion
• On-shell value H = − 12
• Energy E = −tαuα = −ut• Ang. mom. L = φαuα = uφ
• Carter constant Q = Kαβuαuβ
UFF – November 26, 2018 Alexandre Le Tiec
Generalized action-angle variables[Carter, PRD 1968; Schmidt, CQG 2002]
• The Hamilton-Jacobi equation is completely separable
• Canonical transfo. to coordinate-invariant action-anglevariables (wα, Jα), with w i + 2π ≡ w i and
Jt =1
2π
∫ 2π
0ut dt = −E , Jr =
1
2π
∮ur (r) dr
Jφ =1
2π
∮uφ dφ = L , Jθ =
1
2π
∮uθ(θ)dθ
• Hamilton’s canonical equations of motion for H(J):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
= 0
UFF – November 26, 2018 Alexandre Le Tiec
Generalized action-angle variables[Carter, PRD 1968; Schmidt, CQG 2002]
• The Hamilton-Jacobi equation is completely separable
• Canonical transfo. to coordinate-invariant action-anglevariables (wα, Jα), with w i + 2π ≡ w i and
Jt =1
2π
∫ 2π
0ut dt = −E , Jr =
1
2π
∮ur (r) dr
Jφ =1
2π
∮uφ dφ = L , Jθ =
1
2π
∮uθ(θ) dθ
• Hamilton’s canonical equations of motion for H(J):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
= 0
UFF – November 26, 2018 Alexandre Le Tiec
Generalized action-angle variables[Carter, PRD 1968; Schmidt, CQG 2002]
• The Hamilton-Jacobi equation is completely separable
• Canonical transfo. to coordinate-invariant action-anglevariables (wα, Jα), with w i + 2π ≡ w i and
Jt =1
2π
∫ 2π
0ut dt = −E , Jr =
1
2π
∮ur (r) dr
Jφ =1
2π
∮uφ dφ = L , Jθ =
1
2π
∮uθ(θ) dθ
• Hamilton’s canonical equations of motion for H(J):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
= 0
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian first law of mechanics[Le Tiec, CQG 2014]
• Varying H(J) and using Hamilton’s equations, as well as theon-shell constraint H = −1
2 , yields the variational formula
δH = ωαδJα = 0
• Since H(λJ) = λ2H(J), we also have the algebraic relation
ωαJα =∂H
∂JαJα = 2H = −1
• Using non-specific variables Jα ≡ mJα and the fundamentalt-frequencies Ωα ≡ ωα/ωt ≡ z ωα, this gives
δE = Ωϕ δL+ Ωr δJr + Ωθ δJθ + z δm
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian first law of mechanics[Le Tiec, CQG 2014]
• Varying H(J) and using Hamilton’s equations, as well as theon-shell constraint H = −1
2 , yields the variational formula
δH = ωαδJα = 0
• Since H(λJ) = λ2H(J), we also have the algebraic relation
ωαJα =∂H
∂JαJα = 2H = −1
• Using non-specific variables Jα ≡ mJα and the fundamentalt-frequencies Ωα ≡ ωα/ωt ≡ z ωα, this gives
δE = Ωϕ δL+ Ωr δJr + Ωθ δJθ + z δm
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian first law of mechanics[Le Tiec, CQG 2014]
• Varying H(J) and using Hamilton’s equations, as well as theon-shell constraint H = −1
2 , yields the variational formula
δH = ωαδJα = 0
• Since H(λJ) = λ2H(J), we also have the algebraic relation
ωαJα =∂H
∂JαJα = 2H = −1
• Using non-specific variables Jα ≡ mJα and the fundamentalt-frequencies Ωα ≡ ωα/ωt ≡ z ωα, this gives
δE = Ωϕ δL+ Ωr δJr + Ωθ δJθ + z δm
UFF – November 26, 2018 Alexandre Le Tiec
Outline
1 Gravitational waves
2 EMRIs and the gravitational self-force
3 Geodesic motion in Kerr spacetime
4 Beyond the geodesic approximation
5 Innermost stable circular orbits
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
• We consider only the conservative piece of the self-force:
hRαβ = 12
(hretαβ + hadv
αβ
)− hSαβ
• The geodesic motion of a self-gravitating mass m in themetric gαβ(x) + hRαβ(x ; γ) derives from the Hamiltonian
H(x , u; γ) = H(0)(x , u)︸ ︷︷ ︸12g
αβ(x)uαuβ
+ H(1)(x , u; γ)︸ ︷︷ ︸−1
2hαβR (x ;γ)uαuβ
• Canonical transform. to action-angle variables (wα, Jα)
• Hamilton’s canonical equations of motion for H(w , J; γ):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
6= 0
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
• We consider only the conservative piece of the self-force:
hRαβ = 12
(hretαβ + hadv
αβ
)− hSαβ
• The geodesic motion of a self-gravitating mass m in themetric gαβ(x) + hRαβ(x ; γ) derives from the Hamiltonian
H(x , u; γ) = H(0)(x , u)︸ ︷︷ ︸12g
αβ(x)uαuβ
+ H(1)(x , u; γ)︸ ︷︷ ︸−1
2hαβR (x ;γ)uαuβ
• Canonical transform. to action-angle variables (wα, Jα)
• Hamilton’s canonical equations of motion for H(w , J; γ):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
6= 0
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
• We consider only the conservative piece of the self-force:
hRαβ = 12
(hretαβ + hadv
αβ
)− hSαβ
• The geodesic motion of a self-gravitating mass m in themetric gαβ(x) + hRαβ(x ; γ) derives from the Hamiltonian
H(x , u; γ) = H(0)(x , u)︸ ︷︷ ︸12g
αβ(x)uαuβ
+ H(1)(x , u; γ)︸ ︷︷ ︸−1
2hαβR (x ;γ)uαuβ
• Canonical transform. to action-angle variables (wα, Jα)
• Hamilton’s canonical equations of motion for H(w , J; γ):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
6= 0
UFF – November 26, 2018 Alexandre Le Tiec
Hamiltonian formulation
• We consider only the conservative piece of the self-force:
hRαβ = 12
(hretαβ + hadv
αβ
)− hSαβ
• The geodesic motion of a self-gravitating mass m in themetric gαβ(x) + hRαβ(x ; γ) derives from the Hamiltonian
H(x , u; γ) = H(0)(x , u)︸ ︷︷ ︸12g
αβ(x)uαuβ
+ H(1)(x , u; γ)︸ ︷︷ ︸−1
2hαβR (x ;γ)uαuβ
• Canonical transform. to action-angle variables (wα, Jα)
• Hamilton’s canonical equations of motion for H(w , J; γ):
wα =
(∂H
∂Jα
)w
≡ ωα , Jα = −(∂H
∂wα
)J
6= 0
UFF – November 26, 2018 Alexandre Le Tiec
No secular change in the actions
• For any function f of the canonical variables (x , u) we definethe long (proper) time average
〈f 〉 ≡ limT→∞
1
2T
∫ T
−Tf (x(τ), u(τ))dτ
• The rate of change Jα of the actions can be split into anaverage piece and an oscillatory component:
Jα =⟨Jα⟩
+ δJα with⟨δJα⟩
= 0
• The average rate of change of the actions vanishes:
⟨Jα⟩
= −⟨(
∂H(1)
∂wα
)J
⟩= 0
UFF – November 26, 2018 Alexandre Le Tiec
No secular change in the actions
• For any function f of the canonical variables (x , u) we definethe long (proper) time average
〈f 〉 ≡ limT→∞
1
2T
∫ T
−Tf (x(τ), u(τ))dτ
• The rate of change Jα of the actions can be split into anaverage piece and an oscillatory component:
Jα =⟨Jα⟩
+ δJα with⟨δJα⟩
= 0
• The average rate of change of the actions vanishes:
⟨Jα⟩
= −⟨(
∂H(1)
∂wα
)J
⟩= 0
UFF – November 26, 2018 Alexandre Le Tiec
No secular change in the actions
• For any function f of the canonical variables (x , u) we definethe long (proper) time average
〈f 〉 ≡ limT→∞
1
2T
∫ T
−Tf (x(τ), u(τ))dτ
• The rate of change Jα of the actions can be split into anaverage piece and an oscillatory component:
Jα =⟨Jα⟩
+ δJα with⟨δJα⟩
= 0
• The average rate of change of the actions vanishes:
⟨Jα⟩
= −⟨(
∂H(1)
∂wα
)J
⟩= 0
UFF – November 26, 2018 Alexandre Le Tiec
Gauge transformations
• A gauge transformation xα → xα + ξα induces a canonicaltransformation with generator Ξ(x , u) = uαξ
α(x) such that
δξxα =
(∂Ξ
∂uα
)x
, δξuα = −(∂Ξ
∂xα
)u
• The action-angle variables (wα, Jα) are affected by thisgauge transformation in a similar manner:
δξwα =
(∂Ξ
∂Jα
)w
, δξJα = −(∂Ξ
∂wα
)J
• Despite that, we have the gauge-invariant identity:
wαJα = ωαJα = −1
UFF – November 26, 2018 Alexandre Le Tiec
Gauge transformations
• A gauge transformation xα → xα + ξα induces a canonicaltransformation with generator Ξ(x , u) = uαξ
α(x) such that
δξxα =
(∂Ξ
∂uα
)x
, δξuα = −(∂Ξ
∂xα
)u
• The action-angle variables (wα, Jα) are affected by thisgauge transformation in a similar manner:
δξwα =
(∂Ξ
∂Jα
)w
, δξJα = −(∂Ξ
∂wα
)J
• Despite that, we have the gauge-invariant identity:
wαJα = ωαJα = −1
UFF – November 26, 2018 Alexandre Le Tiec
Gauge transformations
• A gauge transformation xα → xα + ξα induces a canonicaltransformation with generator Ξ(x , u) = uαξ
α(x) such that
δξxα =
(∂Ξ
∂uα
)x
, δξuα = −(∂Ξ
∂xα
)u
• The action-angle variables (wα, Jα) are affected by thisgauge transformation in a similar manner:
δξwα =
(∂Ξ
∂Jα
)w
, δξJα = −(∂Ξ
∂wα
)J
• Despite that, we have the gauge-invariant identity:
wαJα = ωαJα = −1
UFF – November 26, 2018 Alexandre Le Tiec
Gauge-invariant information
• Neither the action variables nor the fundamental frequenciesare gauge invariant:
δξJα 6= 0 , δξ ωα 6= 0
• But the averaged frequencies have gauge-invariant meaning:
δξ〈ωα〉 = 0
• The averaged perturbed Hamiltonian is also gauge invariant:
δξ⟨H(1)
⟩= 0
• The relationship between the average redshift z ≡ 1/ 〈ωt〉 andthe average t-frequencies (Ωr ,Ωθ,Ωφ) is gauge-invariant
UFF – November 26, 2018 Alexandre Le Tiec
Gauge-invariant information
• Neither the action variables nor the fundamental frequenciesare gauge invariant:
δξJα 6= 0 , δξ ωα 6= 0
• But the averaged frequencies have gauge-invariant meaning:
δξ〈ωα〉 = 0
• The averaged perturbed Hamiltonian is also gauge invariant:
δξ⟨H(1)
⟩= 0
• The relationship between the average redshift z ≡ 1/ 〈ωt〉 andthe average t-frequencies (Ωr ,Ωθ,Ωφ) is gauge-invariant
UFF – November 26, 2018 Alexandre Le Tiec
Gauge-invariant information
• Neither the action variables nor the fundamental frequenciesare gauge invariant:
δξJα 6= 0 , δξ ωα 6= 0
• But the averaged frequencies have gauge-invariant meaning:
δξ〈ωα〉 = 0
• The averaged perturbed Hamiltonian is also gauge invariant:
δξ⟨H(1)
⟩= 0
• The relationship between the average redshift z ≡ 1/ 〈ωt〉 andthe average t-frequencies (Ωr ,Ωθ,Ωφ) is gauge-invariant
UFF – November 26, 2018 Alexandre Le Tiec
Gauge-invariant information
• Neither the action variables nor the fundamental frequenciesare gauge invariant:
δξJα 6= 0 , δξ ωα 6= 0
• But the averaged frequencies have gauge-invariant meaning:
δξ〈ωα〉 = 0
• The averaged perturbed Hamiltonian is also gauge invariant:
δξ⟨H(1)
⟩= 0
• The relationship between the average redshift z ≡ 1/ 〈ωt〉 andthe average t-frequencies (Ωr ,Ωθ,Ωφ) is gauge-invariant
UFF – November 26, 2018 Alexandre Le Tiec
A special gauge choice
• The gauge freedom allows us to choose a gauge such that
Jα = 0 , ωα = 0 =⇒ ωα = 〈ωα〉
• Additionally, the gauge-invariant interaction HamiltonianHint(J) ∝ 〈H(1)〉 is required to satisfy⟨(
∂H(1)
∂Jα
)w
⟩=
1
2
∂Hint
∂Jα
• These gauge conditions can indeed be imposed consistently
UFF – November 26, 2018 Alexandre Le Tiec
A special gauge choice
• The gauge freedom allows us to choose a gauge such that
Jα = 0 , ωα = 0 =⇒ ωα = 〈ωα〉
• Additionally, the gauge-invariant interaction HamiltonianHint(J) ∝ 〈H(1)〉 is required to satisfy⟨(
∂H(1)
∂Jα
)w
⟩=
1
2
∂Hint
∂Jα
• These gauge conditions can indeed be imposed consistently
UFF – November 26, 2018 Alexandre Le Tiec
A special gauge choice
• The gauge freedom allows us to choose a gauge such that
Jα = 0 , ωα = 0 =⇒ ωα = 〈ωα〉
• Additionally, the gauge-invariant interaction HamiltonianHint(J) ∝ 〈H(1)〉 is required to satisfy⟨(
∂H(1)
∂Jα
)w
⟩=
1
2
∂Hint
∂Jα
• These gauge conditions can indeed be imposed consistently
UFF – November 26, 2018 Alexandre Le Tiec
An effective Hamiltonian
• In this particular gauge, Hamilton’s equations of motion takethe remarkably simple form
wα = ωα = ωα(0)(J) +1
2
∂Hint
∂Jα, Jα = 0
• Hence the dynamics is completely reproduced as a flow by theeffective Hamiltonian
H(J) = H(0)(J) +1
2Hint(J)
• Despite the freedom to change the constant actions Jα by agauge transformation, this effective Hamiltonian is unique
UFF – November 26, 2018 Alexandre Le Tiec
An effective Hamiltonian
• In this particular gauge, Hamilton’s equations of motion takethe remarkably simple form
wα = ωα = ωα(0)(J) +1
2
∂Hint
∂Jα, Jα = 0
• Hence the dynamics is completely reproduced as a flow by theeffective Hamiltonian
H(J) = H(0)(J) +1
2Hint(J)
• Despite the freedom to change the constant actions Jα by agauge transformation, this effective Hamiltonian is unique
UFF – November 26, 2018 Alexandre Le Tiec
An effective Hamiltonian
• In this particular gauge, Hamilton’s equations of motion takethe remarkably simple form
wα = ωα = ωα(0)(J) +1
2
∂Hint
∂Jα, Jα = 0
• Hence the dynamics is completely reproduced as a flow by theeffective Hamiltonian
H(J) = H(0)(J) +1
2Hint(J)
• Despite the freedom to change the constant actions Jα by agauge transformation, this effective Hamiltonian is unique
UFF – November 26, 2018 Alexandre Le Tiec
First law of mechanics
• Using the effective Hamiltonian H(J), the test-mass first lawof mechanics can be extended to relative O(q):
δE = Ωϕ δL+ Ωr δJr + Ωθ δJθ + z δm
• It involves the renormalized actions and the average redshift
Jα ≡ mJα ≡ mJα(
1− 12Hint
)z = z(0) + z(1) = z(0)
(1 + Hint
)• The actions Jα and the average redshift z , as functions of
(Ωr ,Ωθ,Ωφ), include conservative self-force corrections fromthe gauge-invariant interaction Hamiltonian Hint
UFF – November 26, 2018 Alexandre Le Tiec
First law of mechanics
• Using the effective Hamiltonian H(J), the test-mass first lawof mechanics can be extended to relative O(q):
δE = Ωϕ δL+ Ωr δJr + Ωθ δJθ + z δm
• It involves the renormalized actions and the average redshift
Jα ≡ mJα ≡ mJα(
1− 12Hint
)z = z(0) + z(1) = z(0)
(1 + Hint
)
• The actions Jα and the average redshift z , as functions of(Ωr ,Ωθ,Ωφ), include conservative self-force corrections fromthe gauge-invariant interaction Hamiltonian Hint
UFF – November 26, 2018 Alexandre Le Tiec
First law of mechanics
• Using the effective Hamiltonian H(J), the test-mass first lawof mechanics can be extended to relative O(q):
δE = Ωϕ δL+ Ωr δJr + Ωθ δJθ + z δm
• It involves the renormalized actions and the average redshift
Jα ≡ mJα ≡ mJα(
1− 12Hint
)z = z(0) + z(1) = z(0)
(1 + Hint
)• The actions Jα and the average redshift z , as functions of
(Ωr ,Ωθ,Ωφ), include conservative self-force corrections fromthe gauge-invariant interaction Hamiltonian Hint
UFF – November 26, 2018 Alexandre Le Tiec
Outline
1 Gravitational waves
2 EMRIs and the gravitational self-force
3 Geodesic motion in Kerr spacetime
4 Beyond the geodesic approximation
5 Innermost stable circular orbits
UFF – November 26, 2018 Alexandre Le Tiec
Innermost stable circular orbit (ISCO)
0.87
0.9
0.93
0.96
2 4 6 8 10 12 14 16 18
V
r / M
L2 = 15
UFF – November 26, 2018 Alexandre Le Tiec
r2 = E − V (r ; L)
Innermost stable circular orbit (ISCO)
0.87
0.9
0.93
0.96
2 4 6 8 10 12 14 16 18
V
r / M
L2 = 15
L2 = 14
UFF – November 26, 2018 Alexandre Le Tiec
r2 = E − V (r ; L)
Innermost stable circular orbit (ISCO)
0.87
0.9
0.93
0.96
2 4 6 8 10 12 14 16 18
V
r / M
L2 = 15
L2 = 14
L2 = 13
UFF – November 26, 2018 Alexandre Le Tiec
r2 = E − V (r ; L)
Innermost stable circular orbit (ISCO)
0.87
0.9
0.93
0.96
2 4 6 8 10 12 14 16 18
V
r / M
L2 = 15
L2 = 14
L2 = 13
L2 = 12
UFF – November 26, 2018 Alexandre Le Tiec
r2 = E − V (r ; L)
Innermost stable circular orbit (ISCO)
• The innermost stable circular orbit is identified by a vanishingrestoring radial force under small-e perturbations:
∂2H
∂r2= 0 −→ Ωisco
• The minimum energy circular orbit is the most bound orbitalong a sequence of circular orbits:
∂E
∂Ω= 0 −→ Ωmeco
• For Hamiltonian systems,it can be shown that
Ωisco = Ωmeco
M,S
m
ISCO
no stable circular orbit
UFF – November 26, 2018 Alexandre Le Tiec
Kerr ISCO frequency vs black hole spin[Bardeen et al., ApJ 1972]
0
0.1
0.2
0.3
0.4
0.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
MΩ
kerr
isco
χ = S / M2
UFF – November 26, 2018 Alexandre Le Tiec
Kerr ISCO frequency vs black hole spin[Bardeen et al., ApJ 1972]
0
0.1
0.2
0.3
0.4
0.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
MΩ
kerr
isco
χ = S / M2
UFF – November 26, 2018 Alexandre Le Tiec
Astrophysical relevance:measure of a black hole spin fromthe inner edge of its accretion disk
UFF – November 26, 2018 Alexandre Le Tiec
Spins of supermassive black holes[Reynolds, CQG 2013]
χ
M (106 M)
UFF – November 26, 2018 Alexandre Le Tiec
Frequency shift of the Kerr ISCO[Isoyama et al., PRL 2014]
• The orbital frequency of the Kerr ISCO is shifted under theeffect of the conservative self-force:
(M + m)Ωisco = MΩ(0)isco(χ)︸ ︷︷ ︸
test massresult
1 + q CΩ(χ)︸ ︷︷ ︸
self-forcecorrection
+ O(q2)
• The frequency shift can be computed from a stability analysisof slightly eccentric orbits near the Kerr ISCO
• Combining the Hamiltonian first law with the MECO conditio∂E/∂Ω = 0 yields the same result:
CΩ =1
2
z ′′(1)(Ω(0)isco)
E ′′(0)(Ω(0)isco)
UFF – November 26, 2018 Alexandre Le Tiec
Frequency shift of the Kerr ISCO[Isoyama et al., PRL 2014]
• The orbital frequency of the Kerr ISCO is shifted under theeffect of the conservative self-force:
(M + m)Ωisco = MΩ(0)isco(χ)︸ ︷︷ ︸
test massresult
1 + q CΩ(χ)︸ ︷︷ ︸
self-forcecorrection
+ O(q2)
• The frequency shift can be computed from a stability analysisof slightly eccentric orbits near the Kerr ISCO
• Combining the Hamiltonian first law with the MECO conditio∂E/∂Ω = 0 yields the same result:
CΩ =1
2
z ′′(1)(Ω(0)isco)
E ′′(0)(Ω(0)isco)
UFF – November 26, 2018 Alexandre Le Tiec
Frequency shift of the Kerr ISCO[Isoyama et al., PRL 2014]
• The orbital frequency of the Kerr ISCO is shifted under theeffect of the conservative self-force:
(M + m)Ωisco = MΩ(0)isco(χ)︸ ︷︷ ︸
test massresult
1 + q CΩ(χ)︸ ︷︷ ︸
self-forcecorrection
+ O(q2)
• The frequency shift can be computed from a stability analysisof slightly eccentric orbits near the Kerr ISCO
• Combining the Hamiltonian first law with the MECO conditio∂E/∂Ω = 0 yields the same result:
CΩ =1
2
z ′′(1)(Ω(0)isco)
E ′′(0)(Ω(0)isco)
UFF – November 26, 2018 Alexandre Le Tiec
ISCO frequency shift vs black hole spin[Isoyama et al., PRL 2014]
1.2
1.24
1.28
1.32
1.36
1.4
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
CΩ
χ = S / M2
method (i)
method (ii)
UFF – November 26, 2018 Alexandre Le Tiec
ISCO frequency shift vs black hole spin[Isoyama et al., PRL 2014]
UFF – November 26, 2018 Alexandre Le Tiec
[Barack & Sago, PRL 2009]
[Le Tiec et al., PRL 2012]
ISCO frequency shift vs black hole spin[Isoyama et al., PRL 2014]
0.01
0.1
1
10
100
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
δC
Ω /C
Ω
χ = S / M2
3PN (ISCO)
PN (MECO)
EOB (ISCO)
UFF – November 26, 2018 Alexandre Le Tiec
ISCO frequency shift vs black hole spin[Isoyama et al., PRL 2014]
0.01
0.1
1
10
100
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
δC
Ω /C
Ω
χ = S / M2
3PN (ISCO)
PN (MECO)
EOB (ISCO)
UFF – November 26, 2018 Alexandre Le Tiec
Strong-field “benchmark”
ISCO frequency shift vs black hole spin[van de Meent, PRL 2017]
CΩ from:redshift
GSF
-1 -0.5 0 0.5 0.9 0.99 0.999 1
1.24
1.26
1.28
1.30
1.32
1.34
1.36
1.38
a
10-13 10-16 10-19-9 ·10-5-6 ·10-5-3 ·10-50 ·10-5
δCΩ vs. δa
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF
Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form
Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec
Summary
• EMRIs are prime targets for the planned LISA observatory
• Highly accurate template waveforms are a prerequisite fordoing science with gravitational-wave observations
• We introduced a Hamiltonian formulation of the conservativeself-force (SF) dynamics in the Kerr geometry, allowing us to:
Extract the gauge-invariant information contained in the SF Describe the binary dynamics including SF in a concise form Derive a “first law” of binary mechanics including SF effects
• We computed the shift in the Kerr ISCO frequency inducedby the conservative piece of the SF
• This result provides an accurate strong-field “benchmark”for comparison with other analytical methods (PN, EOB)
UFF – November 26, 2018 Alexandre Le Tiec