Binary Black Holes
A GENERIC Approach
Ref Bari
Advisor: Prof. Brendan Keith
Binary Black Holes
A GENERIC Approach
- Proved equivalence between two parameterizations of Schwarzschild Metric
- Parameter Estimation for Schwarzschild Metric
- AI for Schwarzschild Metric (in Hamiltonian form)!
(r(\tau),\phi(\tau)) \text{ vs. } (r(t, p, e), \phi(t, p, e))
Binary Black Holes
A GENERIC Approach
- Proved equivalence between two parameterizations of Schwarzschild Metric
- Parameter Estimation for Schwarzschild Metric
- AI for Schwarzschild Metric (in Hamiltonian form)!
(r(\tau),\phi(\tau)) \text{ vs. } (r(t, p, e), \phi(t, p, e))
\{
r=r(\tau)
\phi = \phi(\tau)
\{
t=t(\chi)
\phi = \phi(\chi)
r=r(\chi)
t=t(\tau)
r=r(\chi)
\frac{d\phi}{dt}=\frac{d\phi/d\chi}{dt/d\chi}
\{
\phi = \phi(t, p, e)
\chi = \chi(t, p, e)
\frac{d\chi}{dt}=\left(\frac{dt}{d\chi}\right)^{-1}
\{
r=r(\tau)
\phi = \phi(\tau)
t=t(\chi)
\phi = \phi(\chi)
r=r(\chi)
t=t(\tau)
\{
\phi = \phi(t, p, e)
\chi = \chi(t, p, e)
\{
\frac{dr}{d\tau} = \left(1-\frac{2M}{r}\right){p_r}
\frac{d\phi}{d\tau} = r^{-2}{L}
\{
\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 \varphi_{\mathrm{p}}}{d \chi}=\sqrt{\frac{p}{p-6-2 e \cos \chi}}
\frac{dr}{d\chi}=\frac{pM\cdot e\sin \chi}{(1+e\cos\chi)^2}
\frac{dt}{d\tau} = \left(1-\frac{2M}{r}\right)^{-1}E
r(\chi)=\frac{pM}{1+e\cos\chi}
\frac{d\phi}{dt}=\frac{d\phi/d\chi}{dt/d\chi}
\{
\dot{\phi}=\frac{(p-2-2 e \cos \chi)(1+e \cos \chi)^2}{M p^{3 / 2}\left[(p-2)^2-4 e^2\right]^{1 / 2}}
\dot{\chi}=\frac{(p-2-2 e \cos \chi)(1+e \cos \chi)^2[p-6-2 e \cos \chi]^{1 / 2}}{M p^2\left[(p-2)^2-4 e^2\right]^{1 / 2}}
\frac{d\chi}{dt}=\left(\frac{dt}{d\chi}\right)^{-1}
Binary Black Holes
A GENERIC Approach
- Proved equivalence between two parameterizations of Schwarzschild Metric
- Parameter Estimation for Schwarzschild Metric
- AI for Schwarzschild Metric (in Hamiltonian form)!
(r(\tau),\phi(\tau)) \text{ vs. } (r(t, p, e), \phi(t, p, e))
Binary Black Holes
A GENERIC Approach
- Proved equivalence between two parameterizations of Schwarzschild Metric
- Parameter Estimation for Schwarzschild Metric
- AI for Schwarzschild Metric (in Hamiltonian form)!
(r(\tau),\phi(\tau)) \text{ vs. } (r(t, p, e), \phi(t, p, e))
Parameter Estimation Results
Elliptical Orbits
Learning Rate
LR=1e-4
LR=1e-3
LR=1e-2
LR=1e-1
(E=0.97, L = 3.78)
Circular Orbits
(E=0.95, L = 3.78)
,\textit{Epochs = 50, Training Fraction= 5\%}
^*\textit{p}_{guess}=(0.95, 3.6)
Parameter Estimation Results
Elliptical Orbits
Learning Rate
LR=1e-4
LR=1e-3
LR=1e-2
LR=1e-1
(E=0.97, L = 3.78)
Circular Orbits
(E=0.95, L = 3.78)
,\textit{Epochs = 50, Training Fraction= 5\%}
^*\textit{p}_{guess}=(0.95, 3.6)
Parameter Estimation Results
Elliptical Orbits
Learning Rate
LR=1e-2
(E=0.97, L = 3.78)
Circular Orbits
(E=0.95, L = 3.78)
,\textit{Epochs = 50, Training Fraction= 5\%}
^*\textit{p}_{guess}=(0.95, 3.6)
Parameter Estimation Results
Circular Orbits
Epochs
E=25
E=50
E=100
E=200
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Exact Schwarzschild
Hamiltonian
Geodesic Equations of Motion
Integrate Equations to get Orbit
Convert Orbit to Waveform
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Exact Schwarzschild Hamiltonian
Geodesic Equations of Motion
Integrate Equations to get Orbit
Convert Orbit to Waveform
\begin{align*}
H = -&\frac{1}{2}\left(1-\frac{2M}{r}\right)^{-1}(p_t)^2+
\frac{1}{2}\left(1-\frac{2M}{r}\right)(p_r)^2\\
&+\frac{1}{2}r^{-2}(p_\theta)^2+
\frac{1}{2}r^{-2}\sin^{-2}\theta(p_\phi)^2]\end{align*}
\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}
t(\tau), r(\tau), \phi(\tau)
h^{22}=\frac{\ddot{Q}}{R}, Q = \int \rho (r^2 \delta_{ij}-r_{i}r_{j})
\tau(t), r(t), \phi(t)
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Define Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
endDefine Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
endDefine Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
endDefine Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
endDefine Loss Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
function loss(pn)
newprob = remake(prob, p = pn)
sol = solve(newprob, Tsit5(), saveat=0.1)
predicted_waveform_plus = compute_waveform(0.1, sol, 1.0)[1]
predicted_waveform_cross = compute_waveform(0.1, sol, 1.0)[2]
# Compare only the overlapping portion
n_pred = length(predicted_waveform_plus)
n_train = length(h_plus_training)
n_compare = min(n_pred, n_train)
loss = sum(abs2, predicted_waveform_plus[1:n_compare] .- h_plus_training[1:n_compare])
loss += sum(abs2, predicted_waveform_cross[1:n_compare] .- h_cross_training[1:n_compare])
return loss
endL = \sum |h_{+,pred}-h_+|^2+\sum |h_{\times,pred}-h_\times|^2
Generate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Define Callback Function
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
losses = []
function callback(pn, loss; dotrain = true)
if dotrain
push!(losses, loss);
@printf("Epoch: %d, Loss: %15.12f \n",length(losses),loss);
p = plot(losses, label = "Loss")
display(p)
else
prinln(l)
end
return false
endGenerate Training Data
Define
Loss Function
Define
Callback
Optimize!
Compare Predicted & Truth
Declare Success
Parameter Estimation Pipeline
Binary Black Holes
A GENERIC Approach
- Proved equivalence between two parameterizations of Schwarzschild Metric
- Parameter Estimation for Schwarzschild Metric
- AI for Schwarzschild Metric (in Hamiltonian form)!
(r(\tau),\phi(\tau)) \text{ vs. } (r(t, p, e), \phi(t, p, e))
Binary Black Holes
A GENERIC Approach
- Proved equivalence between two parameterizations of Schwarzschild Metric
- Parameter Estimation for Schwarzschild Metric
- AI for Schwarzschild Metric (in Hamiltonian form)!
(r(\tau),\phi(\tau)) \text{ vs. } (r(t, p, e), \phi(t, p, e))
Predicted Hamiltonian
Equations of Motion
Predicted Waveform
Compare to True Waveform
Adjust Hamiltonian
Keep Training Network
Neural Network Pipeline
Predicted Hamiltonian
Equations of Motion
Predicted Waveform
Compare to True Waveform
Adjust Hamiltonian
Keep Training Network
H_{guess}
\{t,r,\phi\}
h^{22}(t)
\sum |h^{22}(t)-h_{true}|^2
H_{guess}+\Delta H
H_{guess}+\Delta H
Neural Network Pipeline
Learning orbital dynamics of binary black hole systems from gravitational wave measurements (B. Keith et. al., 2021)
\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) (E)^2 + \frac{2M}{r^2}(p_r)^2-2r^{-3}(L)^2\right]
\dot{p_\theta} =0
\dot{p_\phi} =\dot{L}=0
\dot{t} = \left(1-\frac{2M}{r}\right)^{-1}E \to \dot{t}=1\cdot(1+\mathcal{F}_{NN}[1])
\dot{r} = \left(1-\frac{2M}{r}\right){p_r}\to \dot{r}=p_r\cdot(1+\mathcal{F}_{NN}[2])
\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] \to \left(-\frac{M}{r^2}+\frac{L^2}{r^3}\right)\cdot(1+\mathcal{F}_{NN}[3])
\dot{p_\theta} =0
\dot{p_\phi} =\dot{L}=0
\begin{aligned} \dot{\phi} & =\frac{(1+e \cos (\chi))^2}{M p^{3 / 2}}\left(1+\mathcal{F}_1(\cos (\chi), p, e)\right) \\ \dot{\chi} & =\frac{(1+e \cos (\chi))^2}{M p^{3 / 2}}\left(1+\mathcal{F}_2(\cos (\chi), p, e)\right) \\ \dot{p} & =\mathcal{F}_3(p, e) \\ \dot{e} & =\mathcal{F}_4(p, e)\end{aligned}
\begin{aligned} \dot{t}&=1\cdot(1+\mathcal{F}_{NN}[1]) \\ \dot{r}&=p_r\cdot(1+\mathcal{F}_{NN}[2]) \\ \dot{\theta} &= 0\\ \dot{\phi} &= r^{-2}{L}\\
\dot{p_t}&=\dot{E}=0 \\
\dot{p_r}&=\left(-\frac{M}{r^2}+\frac{L^2}{r^3}\right)\cdot(1+\mathcal{F}_{NN}[3])\\
\dot{p_\theta} &=0\\
\dot{p_\phi} &=\dot{L}=0
\end{aligned}
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 = ?