Binary Black Holes

A GENERIC Approach

Ref Bari

Advisor: Prof. Brendan Keith

\dot E_{\text{particle}} = 0
\dot E_{\text{system}} = 0
\text{Conservative}
\text{Dissipation}
p=13.37
p=12.37
\text{Dissipation}

Energy of Particle:

E_{\text{particle}}

Energy taken away by Gravitational Wave:

E_{\text{GW}}
t h
1.155987123132360184e+00 -5.609346551357122235e-02
2.285489553960637910e+00 -6.036403746984250751e-02
3.402211606323863879e+00 -6.318515807196223300e-02
4.523268886770305031e+00 -6.485891351192739351e-02
5.651430592468121183e+00 -6.641091336216202456e-02

Raw SXS Data

2021 SXS Waveform

t h
1.155987123132368399e+00 -5.717062632140130357e-02
2.285489553960653897e+00 -6.130264068534922728e-02
3.402211606323888304e+00 -6.379464706226181669e-02
4.523268886770337005e+00 -6.530329070047019568e-02
5.651430592468161151e+00 -6.716188456065923240e-02
\text{Dissipation}

Energy of Particle:

E_{\text{particle}}+

Energy taken away by Gravitational Wave:

E_{\text{GW}}

Total Energy of System:

E_{\text{system}}=E_{\text{particle}}+E_{GW}
\dot E_{\text{system}} = 0
\text{Dissipation}

Energy of Particle:

E_{\text{particle}}+

Energy taken away by Gravitational Wave:

E_{\text{GW}}

Total Energy of System:

E_{\text{system}}=E_{\text{particle}}+E_{GW}
\dot E_{\text{system}} = 0
\text{Dissipation}

Energy of Particle:

E_{\text{particle}}+

Energy taken away by Gravitational Wave:

E_{\text{GW}}

Total Energy of System:

E_{\text{system}}=E_{\text{particle}}+E_{GW}
\dot E_{\text{system}} = 0 \to \dot E_{\text{particle}}=- \dot E_{GW}

Energy of Particle:

Energy taken away by Gravitational Wave:

GENERIC

What should our energy functional E(x) be?

What should our entropy functional S(x) be?

What should our friction matrix M be?

Should reduce to conservative dynamics if "T=0"

Should conserve the total energy of the closed system

\dot{x} = L \nabla E + M \nabla S

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

Poisson Matrix

Energy Functional

Friction Matrix

Entropy Functional

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

Energy Functional

Friction Matrix

Entropy Functional

L

L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}
\begin{aligned} \frac{d r}{d t} & =\left(\frac{A}{B}\right)^{1 / 2} \frac{\partial \hat{H}_{\mathrm{EOB}}}{\partial p_{r_*}} \\ \frac{d \varphi}{d t} & =\frac{\partial \hat{H}_{\mathrm{EOB}}}{\partial p_{\varphi}} \equiv \Omega \end{aligned}
\begin{aligned}\frac{d p_{r_*}}{d t} & =-\left(\frac{A}{B}\right)^{1 / 2} \frac{\partial \hat{H}_{\mathrm{EOB}}}{\partial r} \\ \frac{d p_{\varphi}}{d t} & =\hat{\mathcal{F}}_{\varphi}\end{aligned}

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

Friction Matrix

Entropy Functional

L

L

E

E
E(x)=H-TS
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, T\right)
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

Entropy Functional

L

L

E

E
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}

M

M
M=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
E(x)=H-TS
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, T\right)

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}

M

M
M=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1/T\right)
E(x)=H-TS
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, T\right)

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}

M

M
M=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1\right)
\nabla S=\left(0, 0, 0, 0,1\right)
E(x)=H-TS
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, T\right)

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
L\nabla E = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}

M

M
M\nabla S=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1\right)
\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\1\end{pmatrix}
=\begin{pmatrix}M_{15} \\ M_{25} \\ M_{35} \\ M_{45} \\ M_{55}\end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}
=\begin{pmatrix} \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\ - \frac{\partial H}{\partial r} \\ 0 \\ 0\end{pmatrix}
E(x)=H-TS
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, 1\right)

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
E(x)=H+T(E_{\text{particle}}+E_{\text{GW}})
\dot E(x)=\dot H+T(\dot E_{\text{particle}}+\dot E_{\text{GW}})=\dot H + T \dot E_{\text{system}} = 0
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, 1\right)
L\nabla E = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot E_{GW}\end{pmatrix}=

M

M
M\nabla S=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1\right)
\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\1\end{pmatrix}
\begin{pmatrix}M_{15} \\ M_{25} \\ M_{35} \\ M_{45} \\ M_{55}\end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}
\begin{pmatrix} \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\ - \frac{\partial H}{\partial r} \\ 0 \\ 0\end{pmatrix}+
M\nabla E=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_{\phi}}+M_{55}=0
\dot r \propto -r^{-3}\to r\left(t\right)=\left(r_{0}^{4}-4t\right)^{\frac{1}{4}}
\dot{p}_{\phi}=-f_{NN}(r)\cdot\dot\phi/r^2

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
E(x)=H+T(E_{\text{particle}}+E_{\text{GW}})
\dot E(x)=\dot H+T(\dot E_{\text{particle}}+\dot E_{\text{GW}})=\dot H + T \dot E_{\text{system}} = 0
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, 1\right)
L\nabla E = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}

M

M
M\nabla S=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1\right)
\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\1\end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}
\dot r \propto -r^{-3}\to r\left(t\right)\sim\left(r_{0}^{4}-4t\right)^{\frac{1}{4}}
L\sim \int |\dot h|^2d\Omega
\frac{dE}{dt}\sim \frac{\mu^2M^3}{r^5}

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
E(x)=H+T(E_{\text{particle}}+E_{\text{GW}})
\dot E(x)=\dot H+T(\dot E_{\text{particle}}+\dot E_{\text{GW}})=\dot H + T \dot E_{\text{system}} = 0
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, 1\right)
L\nabla E = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}

M

M
M\nabla S=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1\right)
\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\1\end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}
\dot r \propto -r^{-3}\to r\left(t\right)\sim\left(r_{0}^{4}-4t\right)^{\frac{1}{4}}
\frac{dE}{dr}\frac{dr}{dt}\sim \frac{\mu^2M^3}{r^5}
E=-\frac{GM\mu}{2r}
\frac{\mu M}{\left(2 r^2\right)}\frac{dr}{dt}\sim \frac{\mu^2 M^3}{r^5}
\frac{dr}{dt}\propto \frac{\mu M^2}{r^3}

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

L

L

E

E
E(x)=H+T(E_{\text{particle}}+E_{\text{GW}})
\dot E(x)=\dot H+T(\dot E_{\text{particle}}+\dot E_{\text{GW}})=\dot H + T \dot E_{\text{system}} = 0
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, 1\right)
L\nabla E = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot E_{GW}\end{pmatrix}=

M

M
M\nabla S=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

S

S
S(x)=\frac{E_{GW}}{T} \to E_{GW} = TS
\nabla S=\left(0, 0, 0, 0,1\right)
\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\1\end{pmatrix}
\begin{pmatrix}-Cr^{-3} \\ M_{25} \\ M_{35} \\ M_{45} \\ M_{55}\end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}
\begin{pmatrix} \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\ - \frac{\partial H}{\partial r} \\ 0 \\ 0\end{pmatrix}+
M\nabla E=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
\begin{pmatrix}\frac{\partial H}{\partial r}\\ 0 \\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi} \\1\end{pmatrix}=0
M_{15}\frac{\partial H}{\partial r}+M_{55}=0\to M_{55}=-M_{15}\frac{\partial H}{\partial r}
-Cr^{-3}

Simplification: 

M_{35}=M_{45}=0

No Neural Network

Neural Network

\texttt{datasize = 100}

Neural Network

\texttt{datasize = 100}

Neural Network

\texttt{datasize = 1000}

Neural Network

\texttt{datasize = 1000}
p_r=0
NN = Chain(
    Dense(2, 4, tanh), 
    Dense(4, 4, tanh),
    Dense(4, 1),
)
r, p_\phi
M15 = -1 * (250) * (log(1+exp(nn_schwarzschild[1]))) * (1/x[2]^3)
Dissipation = [
                    0, # t
                    M15, # r
                    0, # θ
                    M25, # Ï•
                    0, # p_t
                    M35, # p_r
                    0,  # p_θ
                    M45] # p_Ï•
du_dτ = Conservative + Dissipation

Energy of Particle:

Energy taken away by Gravitational Wave:

\dot{x} = L \nabla E + M \nabla S

Entropy Functional

L

L

E

E
E(x)=H+T(E_{\text{particle}}+E_{\text{GW}})
\dot E(x)=\dot H+T(\dot E_{\text{particle}}+\dot E_{\text{GW}})=\dot H + T \dot E_{\text{system}} = 0
\nabla E=\left(\frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, T\right)
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
x=\begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ E_{GW}\end{pmatrix}

M

M
M=\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}

Introduction to GENERIC

\dot{x} = \dot{x}_{\text{R}} + \dot{x}_{\text{I}}
\dot{x} = L \nabla E + M \nabla S
\frac{dE}{dt} = 0
M \nabla E =0
L \nabla S =0
\frac{dS}{dt} \geq 0
L^T=-L
M^T=M, M\geq0
\dot{x} = L \nabla E + M \nabla S
\dot{x} = L \nabla E + M \nabla S
E(x) = H - TS
S(x) = S
\dot{x} = L \nabla E + M \nabla S
E(x) = H - TS \to \dot E=0
S(x) = S\to \dot S\geq 0
S(x) = E_{GW}\to \dot E_{GW}\geq 0
\dot{x} = L \nabla E + M \nabla S
E(x) = H - TS
S(x) = S
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
M = \begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
\dot{x} = L \nabla E + M \nabla S
E(x) = H - TS
S(x) = S
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
M = \begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
x = \begin{pmatrix} r \\ \phi \\ p_r \\ p_\phi \\ S \end{pmatrix}
\dot{x} = L \nabla E + M \nabla S
E(x) = H - TS \to \nabla E = \begin{pmatrix} \frac{\partial H}{\partial r}, 0, \frac{\partial H}{\partial p_r}, \frac{\partial H}{\partial p_\phi}, -1 \end{pmatrix}
S(x) = S \to \nabla S = (0, 0, 0, 0, 1)
L = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
M = \begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
x = \begin{pmatrix} r, \phi, p_r, p_\phi, S \end{pmatrix}
\dot{x} = L \nabla E + M \nabla S
\begin{pmatrix} \frac{\partial H}{\partial r}\\ 0\\ \frac{\partial H}{\partial p_r}\\ \frac{\partial H}{\partial p_\phi}\\ -1 \end{pmatrix} +
\begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}
\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix}
\dot x = \begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r\\ \dot p_\phi\\ \dot S \end{pmatrix}=
\begin{pmatrix}0\\ 0\\ 0\\ 0\\ 1\end{pmatrix}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{\partial H}{\partial r} \\ 0 \\ \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ 1 \end{pmatrix} + \begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} M_{15} \\ M_{25} \\ M_{35} \\ M_{45} \\ M_{55} \end{pmatrix}
M_{i5} \text{ contributes to } \dot x
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} M_{15} \\ M_{25} \\ M_{35} \\ M_{45} \\ M_{55} \end{pmatrix}
S(x)=-S
E= H-TS
\begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} \\ M_{12} & M_{22} & M_{23} & M_{24} & M_{25} \\ M_{13} & M_{23} & M_{33} & M_{34} & M_{35} \\ M_{14} & M_{24} & M_{34} & M_{44} & M_{45} \\ M_{15} & M_{25} & M_{35} & M_{45} & M_{55} \end{pmatrix} \begin{pmatrix} \frac{\partial H}{\partial r} \\ 0 \\ \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -1 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}
M_{11}\frac{\partial H}{\partial r}+M_{13}\frac{\partial H}{\partial p_r}+M_{14}\frac{\partial H}{\partial p_\phi}+M_{15}=0
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_\phi}-M_{55}=0
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ 0\\ 0 \\ M_{45} \\ M_{55} \end{pmatrix}
M_{55}>0
M_{15}=M_{35}=0
M_{45}\frac{\partial H}{\partial p_\phi}-M_{55}=0\to \boxed{M_{45}=\left(\frac{\partial H}{\partial p_\phi} \right)^{-1}M_{55}}
M_{45}>0
M_{55}=1\mathrm{e}{-5}
M_{55}=0 \text{ (reference)}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0\\ M_{35} \\ 0 \\ M_{55} \end{pmatrix}
M_{55}>0
M_{15}=M_{45}=0
M_{35}\neq0
M_{55}=1\mathrm{e}{-7}
M_{55}=0 \text{ (reference)}
M_{35}\frac{\partial H}{\partial p_r}-M_{55}=0 \to \boxed{M_{35} = \left(\frac{\partial H}{\partial p_r} \right)^{-1} M_{55}}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} M_{15} \\ 0\\ 0 \\ 0 \\ M_{55} \end{pmatrix}
M_{55}>0
M_{35}=M_{45}=0
M_{15}\neq0
M_{55}=1\mathrm{e}{-6}
M_{55}=0 \text{ (reference)}
M_{15}\frac{\partial H}{\partial r}-M_{55}=0 \to \boxed{M_{15}=\left(\frac{\partial H}{\partial r} \right)^{-1} M_{55}}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ 0\\ 0 \\ M_{45}(p_\phi, r) \\ M_{55} \end{pmatrix}
M_{55}>0
M_{15}=M_{35}=0
M_{45}\frac{\partial H}{\partial p_\phi}-M_{55}=0\to \boxed{M_{55}=\left(\frac{\partial H}{\partial p_\phi} \right)M_{45}}
M_{45}>0
M_{55}=1\mathrm{e}{-5}
M_{55}=0 \text{ (reference)}
M\nabla E = 0, L \nabla S = 0
L\nabla E \cdot M \nabla S = 0
\dot \phi = \frac{\partial H}{\partial p_\phi}-TM_{25}, M_{25}>0
\dot \phi \text{ for different }(T,M_{25})
(1\mathrm{e}{-5},1\mathrm{e}{-5})
(1\mathrm{e}{-4},1\mathrm{e}{-4})
(1\mathrm{e}{-2},1\mathrm{e}{-2})
(1\mathrm{e}{-1},1\mathrm{e}{-1})
(1,1)
(1\mathrm{e}{-3},1\mathrm{e}{-3})
M_{11}\frac{\partial H}{\partial r}+M_{13}\frac{\partial H}{\partial p_r}+M_{14}\frac{\partial H}{\partial p_\phi}+M_{15}=0
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_\phi}-M_{55}=0
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} - \begin{pmatrix} M_{15} \\ M_{25} \\ M_{35} \\ M_{45} \\ M_{55} \end{pmatrix}
M_{11}\frac{\partial H}{\partial r}+M_{13}\frac{\partial H}{\partial p_r}+M_{14}\frac{\partial H}{\partial p_\phi}+M_{15}=0
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_\phi}+M_{55}=0
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} M_{15} \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}
M_{15}>0
M_{11}=M_{13}=0
M_{14}\frac{\partial H}{\partial p_\phi}+M_{15}=0\to \boxed{M_{14}=-\left(\frac{\partial H}{\partial p_\phi} \right)^{-1}M_{15}}
\boxed{M_{15}>0}
M_{15}=0
M_{15}=5\mathrm{e}{-2}
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_\phi}+M_{55}=0
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ M_{25} \\ 0 \\ 0 \\ 0 \end{pmatrix}
M_{25}>0
M_{25}=0
M_{25}=5\mathrm{e}{-4}
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{12}=M_{23}=0
M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0\to \boxed{M_{24}=-\left(\frac{\partial H}{\partial p_\phi} \right)^{-1}M_{25}}
\boxed{M_{25}>0}
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_\phi}+M_{55}=0
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0\\ M_{35} \\ 0 \\ 0 \end{pmatrix}
M_{35}>0
M_{35}=0
M_{35}=1.44\mathrm{e}{-5}
M_{13}=M_{33}=0
M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0\to \boxed{M_{34}=-\left(\frac{\partial H}{\partial p_\phi} \right)^{-1}M_{35}}
\boxed{M_{35}>0}
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{12}\frac{\partial H}{\partial r}+M_{23}\frac{\partial H}{\partial p_r}+M_{24}\frac{\partial H}{\partial p_\phi}+M_{25}=0
M_{13}\frac{\partial H}{\partial r}+M_{33}\frac{\partial H}{\partial p_r}+M_{34}\frac{\partial H}{\partial p_\phi}+M_{35}=0
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{15}\frac{\partial H}{\partial r}+M_{35}\frac{\partial H}{\partial p_r}+M_{45}\frac{\partial H}{\partial p_\phi}+M_{55}=0
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0\\ 0 \\ M_{45} \\ 0 \end{pmatrix}
M_{45}>0
M_{45}=0
M_{45}=2\mathrm{e}{-4}
M_{14}\frac{\partial H}{\partial r}+M_{34}\frac{\partial H}{\partial p_r}+M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0
M_{14}=M_{34}=0
M_{44}\frac{\partial H}{\partial p_\phi}+M_{45}=0\to \boxed{M_{44}=-\left(\frac{\partial H}{\partial p_\phi} \right)^{-1}M_{45}}
\boxed{M_{45}>0}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0\\ M_{35} \\ 0 \\ M_{55} \end{pmatrix}
M_{55}>0
M_{15}=M_{45}=0
M_{35}=-\left(\frac{\partial H}{\partial p_r} \right)^{-1}M_{55}
M_{35}\frac{\partial H}{\partial p_r}+M_{55}=0 \to \boxed{M_{35} = -\left(\frac{\partial H}{\partial p_r} \right)^{-1} M_{55}}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} M_{15} \\ 0\\ 0 \\ 0 \\ M_{55} \end{pmatrix}
M_{55}>0
M_{35}=M_{45}=0
M_{15}=-\left(\frac{\partial H}{\partial r} \right)^{-1}M_{55}
M_{15}\frac{\partial H}{\partial r}+M_{55}=0 \to \boxed{M_{15}=\left(\frac{\partial H}{\partial r} \right)^{-1} M_{55}}
\begin{pmatrix} \dot r \\ \dot \phi \\ \dot p_r \\ \dot p_\phi \\ \dot S \end{pmatrix}= \begin{pmatrix} \frac{\partial H}{\partial p_r} \\ \frac{\partial H}{\partial p_\phi} \\ -\frac{\partial H}{\partial r} \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0\\ 0 \\ - p_\phi \\ p_\phi \end{pmatrix}
\dot p_\phi \propto -p_\phi

Binary Black Holes | 08/25 Update

By Ref Bari

Binary Black Holes | 08/25 Update

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