Binary Black Holes

A GENERIC Approach

Ref Bari

Advisor: Prof. Brendan Keith

Approach

❌ Rotation

❌ Inclined

❌ 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]

Approach

❌ Rotation

❌ Inclined

❌ Dissipation

❌ Rotation

❌ Inclined

✅ Dissipation

M\nabla S=?

Approach

❌ Rotation

❌ Inclined

❌ Dissipation

❌ Rotation

❌ Inclined

✅ Dissipation

✅ Rotation

❌ Inclined

✅ Dissipation

Approach

❌ Rotation

❌ Inclined

❌ Dissipation

❌ Rotation

❌ Inclined

✅ Dissipation

✅ Rotation

❌ Inclined

✅ Dissipation

✅ Unaligned Spins

❌ Inclined

✅ Dissipation

Approach

❌ Rotation

❌ Inclined

❌ Dissipation

❌ Rotation

❌ Inclined

✅ Dissipation

✅ Rotation

❌ Inclined

✅ Dissipation

✅ Unaligned Spins

❌ Inclined

✅ Dissipation

✅ 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 = ?