Binary Black Holes

A GENERIC Approach

Ref Bari

Advisor: Prof. Brendan Keith

Learning orbital dynamics of binary black hole systems from gravitational wave measurements (B. Keith et. al., PRD, 2021)

m_1
m_2
m_1>>m_2
(t, r, \theta, \phi)
(t, r, \theta, \phi)
(p, e)
(t, r, \theta, \phi)
(p, e)
(t(p, e, \chi), r(p,e,\chi), \theta(p,e,\chi), \phi(p,e,\chi))
(p, e)
(t(p, e, \chi), r(p,e,\chi), \theta(p,e,\chi), \phi(p,e,\chi))
\theta=\pi/2, \dot{\theta}=0
(p, e)
(t(p, e, \chi), r(p,e,\chi), \theta(p,e,\chi), \phi(p,e,\chi))
(t, r, \theta, \phi)
?
R_s

Anatomy of a Black Hole

R_s

Anatomy of a Black Hole

R_{PR}
R_s
R_{ISCO}
R_{PR}
R_s=\frac{2GM}{c^2}
R_{ISCO}=\frac{6GM}{c^2}
R_{PR}=\frac{3GM}{c^2}

Schwarzschild Radius

Photon Ring

Innermost Stable Circular Orbit

R_s=2M
R_{ISCO}=6M
R_{PR}=3M

Schwarzschild Radius

Photon Ring

Innermost Stable Circular Orbit

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]
(t, r, \theta, \phi)
(t, r, \theta, \phi)
\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
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]
(t, r, \theta, \phi)
(t, r, \theta, \phi)
\dot{t} = \left(1-\frac{2M}{r}\right)^{-1}E
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}
\dot{\theta} = 0
\dot{\phi} = r^{-2}{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_{\phi})^2\right]
\dot{p_\theta} =0
\dot{p_\phi} =\dot{L}=0
\theta=\pi/2, \dot{\theta}=0
r(\chi)=\frac{pM}{1+e\cos\chi}
r(\chi)=\frac{pM}{1+e\cos\chi}
r(\chi)=\frac{pM}{1+e\cos\chi}
r(\chi)=\frac{pM}{1+e\cos\chi}
M
r(\chi)=\frac{pM}{1+e\cos\chi}
e=0
e=0.5
r(\chi)=\frac{pM}{1+e\cos\chi}
\chi
r_{max}(\chi)=\frac{pM}{1+e\cos\chi}
r_{max}
r_{max}(\pi)=\frac{pM}{1-e}
r_{max}:\chi = \pi
r_{min}(0)=\frac{pM}{1+e}
r_{min}:\chi =0
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
r_{min}+r_{max}=pM\left(\frac{1}{1+e}+\frac{1}{1-e}\right)
r_{min}-r_{max}=pM\left(\frac{1}{1+e}-\frac{1}{1-e}\right)
r_{min}-r_{max}=pM\left(\frac{1}{1+e}-\frac{1}{1-e}\right)
r_{min}+r_{max}=pM\left(\frac{1}{1+e}+\frac{1}{1-e}\right)
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
pM\left(\frac{1}{1+e}+\frac{1}{1-e}\right)
pM\left(\frac{1}{1+e}-\frac{1}{1-e}\right)
r_{min}-r_{max}
r_{min}+r_{max}
=
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
\frac{1-e+(1+e)}{1-e^2}
\frac{1-e-(1+e)}{1-e^2}
r_{min}-r_{max}
r_{min}+r_{max}
=
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
2
-2e
r_{min}-r_{max}
r_{min}+r_{max}
=
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
r_{min}
r_{max}=\frac{p^2M^2}{1-e^2}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
r_{min}
r_{max}=\frac{p^2M^2}{1-e^2}
r_{min}+
r_{max}=\frac{2pM}{1-e^2}
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
r_{min}
r_{max}
r_{min}+
r_{max}
\frac{pM}{2}
=
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
L = mvr
m
v
r
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
L = mvr\to \mathcal{L}=vr
v
F_c
F_g
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
L = mvr\to \mathcal{L}=vr
F_c
v
F_g
=
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
L = mvr\to \mathcal{L}=vr
F_c
v
F_g\to \frac{mv^2}{r}=\frac{GMm}{r^2}
=
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
L = mvr\to \mathcal{L}=vr
v
v=\sqrt{\frac{GM}{r}}=\sqrt{\frac{M}{r}}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}=\sqrt{\frac{M}{r}}r
v
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}^2=Mr
v
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}^2=Mr
c
R_{PR}=3M
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}^2=\frac{Mr}{3M}
c
R_{PR}=3M
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}^2=\frac{Mr}{r/3M}
c
R_{PR}=3M
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}^2=\frac{Mr}{1-r/3M}
c
R_{PR}=3M
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\mathcal{L}^2=\frac{Mr}{1-3M/r}
c
R_{PR}=3M
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
c
R_{PR}=3M
r\to \infty:\mathcal{L}^2=Mr
\mathcal{L}^2=\frac{Mr}{1-3M/r}
r\to 3M:\mathcal{L}^2=\infty

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{Mr}{1-3M/r}

But we want this in terms of p and e ...

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{Mr}{1-3M/r}

For a circular orbit ...

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r^2}{2Mr}=p
c
R_{PR}=3M
\mathcal{L}^2=\frac{Mr}{1-3M/r}

For a circular orbit:

r_{min}=r_{max}=r
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{r}{M}=p\to r = pM
c
R_{PR}=3M
\mathcal{L}^2=\frac{Mr}{1-3M/r}

For a circular orbit:

r_{min}=r_{max}=r
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{M(pM)}{1-3M/(pM)}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{pM^2}{1-3/p}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{p^2M^2}{p-3}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
c
R_{PR}=3M
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
  • Eccentric orbits require greater angular momentum!
  • No particle with mass can orbit at the photon ring!
  • Must reduce to Newtonian angular momentum for large r
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E=?
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\left(\frac{dr}{d\tau} \right)^2 = E^2-V^2
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\left(\frac{dr}{d\tau} \right)^2 = E^2-V^2

Kinetic Energy

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\left(\frac{dr}{d\tau} \right)^2 = E^2-V^2

Fixed Constant Energy of Particle

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
\left(\frac{dr}{d\tau} \right)^2 = E^2-V^2

Gravitational Potential created by Black Hole

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
0= E^2-V^2

radius is constant for a circular orbit!

r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=V^2
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{\mathcal{L}^2}{r^2}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{\mathcal{L}^2}{r^2}\right)
\dot{t} = \left(1-\frac{2M}{r}\right)^{-1}E
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}
\dot{\theta} = 0
\dot{\phi} = \frac{\mathcal{L}}{r^2}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{\mathcal{L}^2}{r^2}\right)
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}
\dot{\phi} = \frac{\mathcal{L}}{r^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{\mathcal{L}^2}{r^2}\right)
\mathcal{L}^2=\frac{Mr}{1-3M/r}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{\frac{Mr}{1-3M/r}}{r^2}\right)
\mathcal{L}^2=\frac{Mr}{1-3M/r}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{Mr}{r^2(1-\frac{3M}{r})}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(1+\frac{Mr}{r^2-3Mr}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(\frac{r^2-3Mr}{r^2-3Mr}+\frac{Mr}{r^2-3Mr}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(\frac{r^2-2Mr}{r^2-3Mr}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)\left(\frac{1-2M/r}{1-3M/r}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\left(1-\frac{2M}{r}\right)^2/\left(1-\frac{3M}{r}\right)
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{\left(1-\frac{2M}{r}\right)^2}{\left(1-\frac{3M}{r}\right)}
\frac{r}{M}=p\to r = pM

For a circular orbit:

r_{min}=r_{max}=r
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{\left(1-\frac{2M}{pM}\right)^2}{\left(1-\frac{3M}{pM}\right)}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{\left(1-\frac{2}{p}\right)^2}{\left(1-\frac{3}{p}\right)}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{1-\frac{4}{p}+\frac{4}{p^2}}{1-\frac{3}{p}}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{p^2-4p+4}{p^2-3p}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{(p-2)(p+2)}{p(p-3)}
r_{min}=\frac{pM}{1+e}
r_{max}=\frac{pM}{1-e}
c
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e
r_{max}-r_{min}
r_{max}+r_{min}
=
\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
p
=
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t_{\mathrm{p}}}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \\ & \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \varphi_{\mathrm{p}}}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=?
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{dt/d\tau}{dr/d\tau}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{dt/d\tau}{dr/d\tau}
\dot{t} = \left(1-\frac{2M}{r}\right)^{-1}E
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
r(\chi)=\frac{pM}{1+e\cos\chi}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\left(\frac{dr}{d\tau} \right)^2 = E^2-V^2
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\left(\frac{dr}{d\tau} \right)^2 = E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\left(\frac{dr}{d\tau} \right)^2 = E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)
\left(\frac{dr}{d\tau} \right)= \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\left(1-\frac{2M}{r}\right){p_r}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\left(\frac{dr}{d\tau} \right)^2 = E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)
\left(\frac{dr}{d\tau} \right)= \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}=
\frac{d t}{d r}=\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}\cdot
\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\frac{d t}{d \chi}=\frac{dt}{dr}\frac{dr}{d\chi}=
\frac{\left(1-\frac{2M}{r}\right)^{-1}E}{\sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}\cdot
\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\frac{d t}{d \chi}=\frac{\left(1-\frac{2M}{r}\right)^{-1}\sqrt{\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}}}{\sqrt{\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}-\left( 1-\frac{2M}{r}\right)\left(1+\frac{\frac{p^2M^2}{p-3-e}}{r^2}\right)}}\cdot\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=?
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}
\frac{d \phi}{d r}=\frac{d\phi/d\tau}{dr/d\tau}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}
\frac{d \phi}{d r}=\frac{d\phi/d\tau}{dr/d\tau}
\dot{r} = \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}
\dot{\phi} = \frac{\mathcal{L}}{r^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}
\frac{d \phi}{d r}=\frac{d\phi/d\tau}{dr/d\tau}=\frac{\mathcal{L}/r^2}{ \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\dot{r} = \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}
\dot{\phi} = \frac{\mathcal{L}}{r^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}
\frac{d \phi}{d r}=\frac{\mathcal{L}/r^2}{ \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}
\frac{d \phi}{d r}=\frac{\mathcal{L}/r^2}{ \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\frac{d\phi}{dr}\frac{dr}{d\chi}=\frac{}{}
\frac{\mathcal{L}/r^2}{ \sqrt{E^2-\left( 1-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)}}
\cdot \frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

Why does this term keep showing up?

\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

Ah ... it's a sign of the event horizon of the black hole!

ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2 d\Omega^2
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2 d\Omega^2
r=2M
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

\left(1-\frac{2M}{r}\right)
r=2M
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

\left(1-\frac{2M}{r}\right)
r=2M
r=\frac{pM}{1+e\cos\chi}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

\left(1-\frac{2M}{r}\right)
r=2M
\frac{2M}{r}=\frac{2M(1+e\cos\chi)}{pM}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

\left(1-\frac{2M}{r}\right)
r=2M
1-\frac{2M}{r}=1-\frac{2M(1+e\cos\chi)}{pM}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

\left(1-\frac{2M}{r}\right)
r=2M
1-\frac{2M}{r}=1-\frac{2(1+e\cos\chi)}{p}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}

The metric becomes singular at the event horizon of the black hole!

\left(1-\frac{2M}{r}\right)
r=2M
1-\frac{2M}{r}=\frac{p-2-2e\cos\chi}{p}
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}
r=2M

What if we had a circular orbit?

\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2)} \times \sqrt{\frac{(p-2)^2}{p-6}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6}}
r=2M

What if we had a circular orbit?

\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2)} \times \sqrt{\frac{(p-2)^2}{p-6}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6}}
r=2M

Time slows down dramatically near p=6! Why?

\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2)} \times \sqrt{\frac{(p-2)^2}{p-6}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6}}
r=2M

Time slows down dramatically near p=6! Why?

R_s=2M
R_{ISCO}=6M
R_{PR}=3M
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2)} \times \sqrt{\frac{(p-2)^2}{p-6}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6}}
r=2M

Time stops at the event horizon!

R_s=2M
R_{ISCO}=6M
R_{PR}=3M
\mathcal{L}^2=\frac{p^2M^2}{p-3-e}
e=\frac{r_{max}-r_{min}}{r_{max}+r_{min}}
p=\frac{2r_{min}r_{max}}{M(r_{min}+r_{max})}
E^2=\frac{(p-2-2e)(p+2+2e)}{p(p-3-e^2)}
r=2M
\begin{aligned} \frac{d t}{d \chi}= & \frac{M p^2}{(p-2-2 e \cos \chi)(1+e \cos \chi)^2} \times \sqrt{\frac{(p-2-2 e)(p-2+2 e)}{p-6-2 e \cos \chi}}\end{aligned}
\frac{d \phi}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}
r_{min}=\frac{pM}{1+e}
r_{min}:\chi =0
r_{max}=\frac{pM}{1-e}
r_{min}+r_{max}=pM\left(\frac{1}{1+e}+\frac{1}{1-e}\right)
r_{min}-r_{max}=pM\left(\frac{1}{1+e}-\frac{1}{1-e}\right)

Binary Black Holes

\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)
\textbf{>} \text{Preliminary Dissipation Model}
\textbf{>} \text{GENERIC Form of } M\nabla S
\textbf{>} \text{What orbits are possible around black holes?}
\textbf{>} \text{What are the constraints on } (p,e) \text{ and }(r,\phi) \text{ for each orbit?}
\textbf{>} \text{What is the mapping between } (p,e) \text{ and }(r,\phi) \text{?}
\textbf{>} \text{How can we verify that both descriptions are equivalent?}
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)

\text{Transformation between } (p,e) \to (r,\phi)

Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)

Stable Circular Orbit

\text{Transformation between } (p,e) \to (r,\phi)

Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)

Elliptical Orbit

\text{Transformation between } (p,e) \to (r,\phi)

Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)

Hyperbolic Orbit

\text{Transformation between } (p,e) \to (r,\phi)

Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)

Unstable Circular Orbit

\text{Transformation between } (p,e) \to (r,\phi)

Analytical time-like geodesics in Schwarzschild space-time (Kostic, Gen. Rel. Grav., 2012)

Plunge Orbit

\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Circular

Elliptical

Hyperbolic

Plunge

\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Elliptical

Hyperbolic

Plunge

Circle

\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Elliptical

Hyperbolic

Plunge

Circle

\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Elliptical

Hyperbolic

Plunge

Circle

\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Hyperbolic

Plunge

Circle

\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)

Ellipse

0.95<\tilde{E}^2<1
\tilde{J}^2>12M^2
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, \left(1-\frac{2M}{r}\right){p_r}, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,\dot{p_r},0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, p_{r,0}, 0, L)
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Circle

\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)

Ellipse

0.95<\tilde{E}^2<1
\tilde{J}^2>12M^2
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, \left(1-\frac{2M}{r}\right){p_r}, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,\dot{p_r},0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, p_{r,0}, 0, L)

Ellipse

Unbound

:(
\textbf{>} \text{Transformation between } (p,e) \to (r,\phi)

Circle

\tilde{E}^2 = \left(1-\frac{2M}{r} \right)^2/\left(1-\frac{3M}{r} \right)
\tilde{J}^2 = \frac{Mr}{1-3M/r}
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, 0, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,0,0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, 0, 0, L)

Ellipse

0.95<\tilde{E}^2<1 \to 0\leq e < 1
\tilde{J}^2>12M^2 \to p>6+2e
(\dot{t},\dot{r},\dot{\theta},\dot{\phi})=\left(\left(1-\frac{2M}{r}\right)^{-1}E, \left(1-\frac{2M}{r}\right){p_r}, 0, L/r^2\right)
(\dot{p_t},\dot{p_r},\dot{p_\theta},\dot{p_\phi})=(\dot{E}, \dot{p_r}, 0, \dot{L})=(0,\dot{p_r},0,0)
(t_0, r_0, \theta_0, \phi_0) = (0, r_0, \pi/2,0)
(p_{t,0},p_{r,0}, p_{\theta,0}, p_{\phi,0})=(-E, p_{r,0}, 0, L)

Ellipse

Constraints:

0.95<\tilde{E}^2<1
\tilde{J}^2>12M^2

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 = ?