Binary Black Holes
A GENERIC Approach
Ref Bari
Advisor: Prof. Brendan Keith
Approach
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
Approach
M\nabla S=?
Approach
Approach
Approach
✅ Unaligned Spins
✅ Inclined
✅ Dissipation
Keplerian Approximation
Simple!
Circular Orbits
Weak Field Limit
The basic physics of the binary black hole merger GW150914
Keplerian Approximation
No Precession
Only Circular Orbits
Weak Field Limit
The basic physics of the binary black hole merger GW150914 (LIGO & VIRGO Collaborations, 2016)
\dot{\phi}=\sqrt{\frac{G M}{r^3}}, \quad \dot{r}=-\frac{2}{3} \frac{r \dot{\omega}}{\omega}
Limiting Case
Keplerian Approximation
Simple!
Circular Orbits
Weak Field Limit
Peters (1964)
Quadrupole Model
More Complex
Eccentric Orbits
Weak Field Limit
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses
Any Mass Ratio!
Keplerian Approximation
Simple!
Circular Orbits
Weak Field Limit
Peters (1964)
Quadrupole Model
More Complex
Eccentric Orbits
Weak Field Limit
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses (Peters, 1964)
\left\langle\frac{d E}{d t}\right\rangle=-\frac{32}{5} \frac{G^4 m_1^2 m_2^2\left(m_1+m_2\right)}{c^5 a^5\left(1-e^2\right)^{7 / 2}}f(e)
\left\langle\frac{d L}{d t}\right\rangle=-\frac{32}{5} \frac{G^{7 / 2} m_1^2 m_2^2\left(m_1+m_2\right)^{1 / 2}}{c^5 a^{7 / 2}\left(1-e^2\right)^2}f_2(e)
Any Mass Ratio!
Keplerian Approximation
Simple!
Circular Orbits
Weak Field Limit
Peters (1964)
Quadrupole Model
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses (Peters, 1964)
\left\langle\frac{d E}{d t}\right\rangle=-\frac{32}{5} \frac{G^4 m_1^2 m_2^2\left(m_1+m_2\right)}{c^5 a^5\left(1-e^2\right)^{7 / 2}}f(e)
\left\langle\frac{d L}{d t}\right\rangle=-\frac{32}{5} \frac{G^{7 / 2} m_1^2 m_2^2\left(m_1+m_2\right)^{1 / 2}}{c^5 a^{7 / 2}\left(1-e^2\right)^2}f_2(e)
\left\langle\frac{d a}{d t}\right\rangle=-\frac{64}{5} \frac{G^3 m_1 m_2\left(m_1+m_2\right)}{c^5 a^3\left(1-e^2\right)^{7 / 2}}\left(1+\frac{73}{24} e^2+\frac{37}{96} e^4\right)
\left\langle\frac{d e}{d t}\right\rangle=-\frac{304}{15} e \frac{G^3 m_1 m_2\left(m_1+m_2\right)}{c^5 a^4\left(1-e^2\right)^{5 / 2}}\left(1+\frac{121}{304} e^2\right)
Keplerian Approximation
Peters (1964)
Quadrupole Model
Cutler & Poisson (1994)
Dissipation Model
Simple!
More Complex
Eccentric Orbits
Weak Field Limit
Circular Orbits
Weak Field Limit
Complicated!
Any eccentricity!
Any distance!
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses
Gravitational radiation reaction for bound motion around a Schwarzschild black hole
Any Mass Ratio!
Only EMRIs
Keplerian Approximation
Peters (1964)
Quadrupole Model
Cutler & Poisson (1994)
Dissipation Model
Simple!
More Complex
Eccentric Orbits
Weak Field Limit
Circular Orbits
Weak Field Limit
Complicated!
Any eccentricity!
Any distance!
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses
Gravitational radiation reaction for bound motion around a Schwarzschild black hole (Cutler et. al., 1994)
Simple!
Circular Orbits
Weak Field Limit
Only EMRIs
Keplerian Approximation
Peters (1964)
Quadrupole Model
Cutler & Poisson (1994)
Dissipation Model
Simple!
More Complex
Eccentric Orbits
Weak Field Limit
Circular Orbits
Weak Field Limit
Complicated!
Any eccentricity!
Any distance!
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses
Gravitational radiation reaction for bound motion around a Schwarzschild black hole (Cutler et. al., 1994)
Simple!
Circular Orbits
Weak Field Limit
\mu
M
\mu/M <<1
Only EMRIs
Keplerian Approximation
Peters (1964)
Quadrupole Model
Cutler & Poisson (1994)
Dissipation Model
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses
Gravitational radiation reaction for bound motion around a Schwarzschild black hole (Cutler et. al., 1994)
M
Keplerian Approximation
Peters (1964)
Quadrupole Model
Cutler & Poisson (1994)
Dissipation Model
The basic physics of the binary black hole merger GW150914
Gravitational Radiation and the Motion of Two Point Masses
M
Simple!
More Complex
Eccentric Orbits
Circular Orbits
Weak Field Limit
Simple!
Circular Orbits
Weak Field Limit
\mu
M
M
More Complex
Eccentric Orbits
Weak Field Limit
p
M
More Complex
Eccentric Orbits
Weak Field Limit
p
e=0
e=0.5
e=0.7
p
0\leq e<1, p\geq 6+2e
Condition for Bound Orbits
0\leq e<1, p\geq 6+2e
Condition for Bound Orbits
\text{Bound Orbits}
\text{Plunge Orbits}
p=6
p=7
p=8
p=9
e=0
e=0.2
e=0.4
e=0.6
Stable Circular Orbits
p=6+2e
Unstable Circular Orbits
Eccentric Orbits
p=6
p=7
p=8
p=9
e=0
e=0.2
e=0.4
e=0.6
Stable Circular Orbits
p=6+2e
Unstable Circular Orbits
Eccentric Orbits
(p,e)
p=6
p=7
p=8
p=9
e=0
e=0.2
e=0.4
e=0.6
Stable Circular Orbits
p=6+2e
Unstable Circular Orbits
Eccentric Orbits
\vec{v}=\braket{1,de/dp}
\text{Case I: } p>>6
\text{Case II: } p-6-2e<<4e
\text{Case I: } e<
\{\dot{L}, \dot{E}\}
\{\dot{p}, \dot{e}\}
\{\dot{t}, \dot{r}, \dot{\theta}, \dot{\phi}\}
\{\dot{r}, \dot{\phi}\}
M\nabla S
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Mission Statement
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
H(\vec{x})
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{Neural ODE}
x(t)
\textbf{Neural ODE}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{ODE Solver}
x(t)
\textbf{Neural Net}
+
\dot{x}=f(x)
x(t)
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{ODE Solver}
x(t)
\textbf{Neural Net}
+
\dot{x}=f_{NN}(\vec{x}(t,\eta))
x(t)
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x)
\textbf{ODE Solver}
x(t)
\textbf{Neural Net}
+
\dot{x}=f_{NN}(\vec{x}(t,\eta))
x(t)
\text{min}_{\eta} \mathcal{L} = \sum_{i=1}^N |x(t)-\hat{x}(t)|^2
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x,\mathcal{F}(x))
x(t)
\textbf{Neural ODE}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\dot{x}=f(x,\mathcal{F}(x))
x(t)
\textbf{Neural ODE}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\textbf{Discover Hamiltonian (in GENERIC Form) of EMRIs}
\textbf{from Gravitational Waves using Neural ODEs}
H
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\textbf{Discover Hamiltonian (in GENERIC Form) of EMRIs}
\textbf{from Gravitational Waves using Neural ODEs}
H=H_{\text{conservative}}+H_{\text{dissipative}}
Mission Statement
\textbf{Learn Physics of Binary Black Holes!}
\textbf{Learn Physics of Eccentric Mass Ratio Inspirals (EMRIs)}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{Discover Hamiltonian of EMRIs from Gravitational Wave}
\textbf{using Neural ODEs}
\textbf{Discover Hamiltonian (in GENERIC Form) of EMRIs}
\textbf{from Gravitational Waves using Neural ODEs}
H=H_{\text{conservative}}+H_{\text{dissipative}}
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Introduction to GENERIC
G
E
N
E
R
I
C
Introduction to GENERIC
G
E
N
E
R
I
C
eneral
quation for
on
quilibrium
eversible
rreversible
oupling
Introduction to GENERIC
dU = dW + dQ
Introduction to GENERIC
dU = \underbrace{dW} +\underbrace{dQ}
"reversible"
"irreversible"
dU = \underbrace{dW} +\underbrace{dQ}
Introduction to GENERIC
dU = \underbrace{dW} +\underbrace{dQ}
"reversible"
"irreversible"
dU = -pdV+TdS
Introduction to GENERIC
dU = \underbrace{dW} +\underbrace{dQ}
"reversible"
"irreversible"
\dot{x} = \dot{x}_{\text{reversible}} + \dot{x}_{\text{irreversible}}
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\underbrace{\begin{pmatrix}
\dot{p}\\
\dot{q}
\end{pmatrix}}_{\dot{x}} =
\underbrace{\begin{pmatrix}
0 & -1\\
+1 & 0
\end{pmatrix}}_{L}
\underbrace{\begin{pmatrix}
\frac{\partial H}{\partial p}\\
\frac{\partial H}{\partial q}
\end{pmatrix}}_{\nabla E}
Conservative Dynamics
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\underbrace{\begin{pmatrix}
\dot{p}\\
\dot{q}
\end{pmatrix}}_{\dot{x}} =
\underbrace{\begin{pmatrix}
? & ?\\
? & ?
\end{pmatrix}}_{M}
\underbrace{\begin{pmatrix}
\frac{\partial S}{\partial p}\\
\frac{\partial S}{\partial q}
\end{pmatrix}}_{\nabla S}
Dissipative Dynamics
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\frac{dU}{dt} = 0
M \nabla E =0
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\frac{dU}{dt} = 0
M \nabla E =0
L \nabla S =0
\frac{dS}{dt} \geq 0
Introduction to GENERIC
dU = {dW} +{dQ}
\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
Equilibrium
Inequilibrium
dU = {-p dV} +{T dS}
\dot{x} = L \nabla E + M \nabla S
\frac{dU}{dt} = 0
M \nabla E =0
L \nabla S =0
\frac{dS}{dt} \geq 0
L^T=-L
M^T=M, M\geq0
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
- Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Examples of GENERIC
Examples of GENERIC
x=\{q,p,S\}
State Variables
x=\{q,p,S\}
State Variables
Examples of GENERIC
x=\{q,p,S\}
State Variables
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
Examples of GENERIC
x=\{q,p,S\}
State Variables
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
S = S
Energy
Entropy
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
S = S
Energy
Entropy
Examples of GENERIC
x=\{q,p,S\}
State Variables
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
Energy
S = S
Energy
Entropy
\nabla E =(kq,p/m,T)
\nabla S = (0,0,1)
Entropy
\nabla E =(kq,p/m,T)
\nabla S = (0,0,1)
Examples of GENERIC
x=\{q,p,S\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
\nabla E =(kq,p/m,T)
\nabla S = (0,0,1)
L={\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
M=\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
\dot{q}=p
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
\dot{q}=p
\dot{p}=-q-\gamma p
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
S = S
{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}+
{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
\dot{S}=\gamma p^2
\dot{q}=p
\dot{p}=-q-\gamma p
Examples of GENERIC
{\begin{pmatrix}
\dot{q} \\
\dot{p}\\
\dot{S}
\end{pmatrix}}=
S = S
\underbrace{\begin{pmatrix}
kq \\
p/m\\
T
\end{pmatrix}}_{\nabla E}+
\underbrace{\begin{pmatrix}
0 \\
0\\
1
\end{pmatrix}}_{\nabla S}
\underbrace{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}_{L}
\underbrace{\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}}_{M}
\dot{S}=\gamma p^2
\dot{q}=p
\dot{p}=-q-\gamma p
\dot{x} = L \nabla E + M \nabla S
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
Examples of GENERIC
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
L=\begin{pmatrix}\mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0\end{pmatrix}
L=
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
L=\frac{1}{m}\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & p_1 A_{\mathrm{c}} & -p_2 A_{\mathrm{c}} \\ 0 & -p_1 A_{\mathrm{c}} & 0 & 0 \\ 0 & p_2 A_{\mathrm{c}} & 0 & 0\end{array}\right)
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
L=\begin{pmatrix}\mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0\end{pmatrix}
L=
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
L=\frac{1}{m}\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & p_1 A_{\mathrm{c}} & -p_2 A_{\mathrm{c}} \\ 0 & -p_1 A_{\mathrm{c}} & 0 & 0 \\ 0 & p_2 A_{\mathrm{c}} & 0 & 0\end{array}\right)
Examples of GENERIC
M=\begin{pmatrix}\begin{array}{cccccc}\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \kappa \frac{\theta_b}{\theta_a} & -\kappa \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\kappa & \kappa \frac{\theta_a}{\theta_b}\end{array}\end{pmatrix}
M=\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
M=\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha & -\alpha \\ 0 & 0 & -\alpha & \alpha\end{array}\right)
x=\{q,p,S\}
x=\{q_1,p_1,s_a, q_2, p_2, s_b\}
x=\{q,v,E_1,E_2\}
E =\frac{p^2}{2m}+\frac{1}{2}kq^2+TS
E = {\frac{|\vec{p}_1|^2}{2m}+ \frac{|\vec{p}_2|^2}{2m}}+{e_a(\lambda_a, s_a)}
+e_b(\lambda_b, s_b)
E =\frac{1}{2}mv^2+E_1+E_2
S=s_a+s_b
S=S(E_1,qA_c,N)
+S(E_2,(2L_g-q)A_c,N)
S=S
L=\begin{pmatrix}\mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0\end{pmatrix}
L=
{\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}}
L=\frac{1}{m}\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & p_1 A_{\mathrm{c}} & -p_2 A_{\mathrm{c}} \\ 0 & -p_1 A_{\mathrm{c}} & 0 & 0 \\ 0 & p_2 A_{\mathrm{c}} & 0 & 0\end{array}\right)
M=\begin{pmatrix}\begin{array}{cccccc}\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \kappa \frac{\theta_b}{\theta_a} & -\kappa \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\kappa & \kappa \frac{\theta_a}{\theta_b}\end{array}\end{pmatrix}
M=\frac{\gamma}{mT}{\begin{pmatrix}
0 & 0 & 0 \\
0 & (mT)^2 & -pmT \\
0 & -pmT & p^2
\end{pmatrix}}
M=\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha & -\alpha \\ 0 & 0 & -\alpha & \alpha\end{array}\right)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
- Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
GENERIC for Black Holes (No Dissipation)
H = \frac{p^2}{2}
GENERIC for Black Holes (No Dissipation)
H = \frac{p^2}{2}=\text{Don't be lazy, Ref! Write it on the board.}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
\begin{pmatrix}
\dot{t} \\
\dot{r} \\
\dot{\theta} \\
\dot{\phi} \\
\dot{p_t} \\
\dot{p_r} \\
\dot{p_\theta} \\
\dot{p_\phi}
\end{pmatrix} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\
\end{pmatrix}
\begin{pmatrix}
{\partial H} / {\partial {t}} \\
\partial H / \partial {r} \\
\partial H / \partial {\theta} \\
\partial H / \partial {\phi} \\
\partial H / \partial {p_t} \\
\partial H / \partial {p_r} \\
\partial H / \partial {p_\theta} \\
\partial H / \partial {p_\phi}
\end{pmatrix}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
\begin{pmatrix}
\dot{t} \\
\dot{r} \\
\dot{\theta} \\
\dot{\phi} \\
\dot{p_t} \\
\dot{p_r} \\
\dot{p_\theta} \\
\dot{p_\phi}
\end{pmatrix} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\
\end{pmatrix}
\begin{pmatrix}
{\partial H} / {\partial {t}} \\
\partial H / \partial {r} \\
\partial H / \partial {\theta} \\
\partial H / \partial {\phi} \\
\partial H / \partial {p_t} \\
\partial H / \partial {p_r} \\
\partial H / \partial {p_\theta} \\
\partial H / \partial {p_\phi}
\end{pmatrix}
L
\nabla E
\dot{x}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
\dot{t} = \left(1-\frac{2M}{r}\right)^{-1}E
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}
\dot{\theta} = r^{-2}{p_\theta}
\dot{\phi} = r^{-2}\sin^{-2}\theta{L}
\dot{p_t} =\dot{E}=0
\dot{p_r} = -\frac{1}{2}\left[\left(1-\frac{2M}{r}\right)^{-2}\left( \frac{2M}{r^2}\right) (p_t)^2 + \frac{2M}{r^2}(p_r)^2-2r^{-3}(p_{\theta})^2-2r^{-3}\sin^{-2}\theta (p_{\phi})^2\right]
\dot{p_\theta} =-\frac{1}{2}\left[ -2r^{-2}\sin^{-3}\theta\cos\theta\right](p_\phi)^2
\dot{p_\phi} =\dot{L}=0
\dot{x}
GENERIC for Black Holes (No Dissipation)
H = \frac{1}{2}\left[-\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\left(1-\frac{2M}{r}\right)(p_r)^2+\\
r^{-2}(p_\theta)^2+
r^{-2}\sin^{-2}\theta(p_\phi)^2\right]
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
GENERIC for Black Holes (No Dissipation)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
Binary Black Holes
- Neural ODE DynAMO Proposal Goal
- Introduction to GENERIC Formalism
-
Examples of GENERIC
- Damped Harmonic Oscillator
- Thermolastic Double Pendulum
- Two Boxes Exchanging Energy
- GENERIC for Black Holes (No Dissipation)
- GENERIC for Black Holes (Gravitational Waves)
GENERIC for Black Holes (Gravitational Waves)
M\nabla S = ?
Binary Black Holes | 06/12 Update
By Ref Bari
