# How does this canonical transformation on a Schwarzschild black hole work?

In this paper "Holography of the Photon Ring" the authors use a canonical transformation in section 2.4 in eqs. (2.52)-(2.55).

It is basically a transformation from spherical coordinates for a Schwarzschild black hole on a reduced phase space $$(r, \phi, p_r, p_\phi)$$ to coordinates in which the symmetries are very visible $$(T, \Phi, H, L)$$ where $$H$$ and $$L$$ are the conserved quantities related to the stationarity and axial symmetry, while $$T$$ and $$\Phi$$ are in essence Hamilton's equations.

From Hamilton's equations the authors derive $$$$\label{eq:prSchw} p_r(r, H, L) = \pm_r \frac{\sqrt{\mathcal{R}(r)}}{r^2 - 2Mr}, \qquad \qquad \mathcal{R}(r) = r^4H^2 - \left(r^2-2Mr\right)L^2.$$$$ from which it is computed that \begin{align} d T &= \frac{Hr^4}{\left(r^2-2Mr\right)\sqrt{\mathcal{R}(r)}} d r \\ d \Phi &= d \phi - \frac{L}{\sqrt{\mathcal{R}(r)}} d r. \end{align} The authors claim that this defines a canonical transformation.

However I am quite confused on how this makes sense as these expressions are not closed in the sense that $$d^2T \neq 0$$ and $$d^2 \Phi \neq 0$$, because of the $$p_r$$ and $$p_\phi$$ dependence in the expressions for $$H$$ and $$L$$.

I understand that $$H$$ and $$L$$ are conserved along geodesic motion, however I don't understand why this can be used in doing the canonical transformation.

1. We restrict phase space to the equatorial plane $$\theta=\frac{\pi}{2}$$ and $$p_{\theta}=0$$.

2. We consider the Schwarzschild solution \begin{align} ds^2~=~&-f(r)dt^2+\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta d\phi^2)\cr~=~&-f(r)dt^2+\frac{dr^2}{f(r)}+r^2 d\phi^2.\tag{2.1}\end{align}

3. The Lagrangian for the null geodesics is standard.

4. We impose that the 4-momentum is null \begin{align} 0~=~&-m^2~=~p_{\mu}g^{\mu\nu}p_{\nu}\cr ~=~&-\frac{p_t^2}{f(r)}+f(r)p_r^2 +\frac{p_{\theta}^2}{r^2}+\frac{p_{\phi}^2}{r^2\sin^2\theta}\cr ~=~&-\frac{p_t^2}{f(r)}+f(r)p_r^2 +\frac{p_{\phi}^2}{r^2},\end{align} which should be viewed as a definition of the radial momentum $$p_r(r,p_t,p_{\phi})~:=~\sqrt{\frac{\cal W}{f(r)}}, \qquad {\cal W}~:=~\frac{p_t^2}{f(r)}-\frac{p_{\phi}^2}{r^2}.$$

5. The variables $$t$$ and $$\phi$$ are cyclic, and hence $$p_t$$ and $$p_{\phi}$$ are COM. We will consider a 4-dimensional mini phase space with fundamental variables $$(r,\phi,p_t,p_{\phi})$$.

6. Define radial antiderivative $$I(\cdot)~:=~\left.\int_{r_{\ast}}^r\! \mathrm{d}r(\cdot)\right|_{p_t,p_{\phi},\phi},$$ where $$r_{\ast}$$ is a radial reference point. Define restricted exterior derivative $$\mathrm{d}_|~:=~\mathrm{d} - \mathrm{d}r\frac{\partial}{\partial r}~=~\mathrm{d}p_t\frac{\partial}{\partial p_t}+\mathrm{d}p_{\phi}\frac{\partial}{\partial p_{\phi}}+\mathrm{d}\phi\frac{\partial}{\partial \phi}.$$ Note that $$I$$ and $$\mathrm{d}_|$$ commute.

7. Define functions \begin{align} g(r,p_t,p_{\phi})~:=~&\frac{\partial p_r}{\partial p_t}~=~\frac{p_t}{f^2p_r}, \cr e(r,p_t,p_{\phi})~:=~&-\frac{\partial p_r}{\partial p_{\phi}}~=~\frac{p_{\phi}}{r^2fp_r}.\end{align} Note that since $$p_r$$ is a homogeneous function of the momenta $$(p_t,p_{\phi})$$ with weight 1, we have $$p_r~=~\left(p_t\frac{\partial }{\partial p_t}+p_{\phi}\frac{\partial }{\partial p_{\phi}}\right)p_r~=~p_tg-p_{\phi}e .$$

8. Define 1-forms \begin{align} \tau~:=~&g\mathrm{d}r, \cr \eta~:=~&e\mathrm{d}r. \end{align} Then $$p_r\mathrm{d}r~=~p_t\tau-p_{\phi}\eta.$$

9. Define new coordinates \begin{align} T(r,p_t,p_{\phi})~:=~&I(g)~=~\int_{r_{\ast}}^r\! \tau, \cr \phi-\Phi(r,\phi,p_t,p_{\phi}) ~:=~&I(e)~=~\int_{r_{\ast}}^r\! \eta. \end{align} Then \begin{align} \mathrm{d}T~=~&\tau+I(\mathrm{d}_|g) , \cr \mathrm{d}\phi-\mathrm{d}\Phi ~=~&\eta+I(\mathrm{d}_|e) . \end{align} Therefore we may identify \begin{align} \mathrm{d}T~\approx~&\tau , \cr \mathrm{d}\phi-\mathrm{d}\Phi ~\approx~&\eta , \end{align} along a geodesic where $$\mathrm{d}p_t\approx 0\approx\mathrm{d}p_{\phi}$$.

10. Let us consider the tautological symplectic potential 1-form \begin{align} \vartheta~:=~&p_r\mathrm{d}r +p_{\phi}\mathrm{d}\phi\cr ~=~&p_t\tau-p_{\phi}\eta+p_{\phi}\mathrm{d}\phi\cr ~=~& p_t\mathrm{d}T+p_{\phi}\mathrm{d}\Phi-I\left(p_t\mathrm{d}_|g-p_{\phi}\mathrm{d}_|e\right)\cr ~=~& p_t\mathrm{d}T+p_{\phi}\mathrm{d}\Phi-I\left(\mathrm{d}_| (p_tg-p_{\phi}e)-g\mathrm{d}_|p_t+e\mathrm{d}_|p_{\phi}\right)\cr ~=~& p_t\mathrm{d}T+p_{\phi}\mathrm{d}\Phi-I\left(\mathrm{d}_| p_r-\frac{\partial p_r}{\partial p_t}\mathrm{d}p_t-\frac{\partial p_r}{\partial p_{\phi}}\mathrm{d}p_{\phi}\right)\cr ~=~& p_t\mathrm{d}T+p_{\phi}\mathrm{d}\Phi, \end{align} \tag{2.55} which shows that $$(r,\phi;p_r,p_{\phi}) \quad\longrightarrow\quad(T,\Phi;p_t,p_{\phi})$$ is a symplectomorphism in the mini phase space.

References:

1. S. Hadar, D. Kapec, A. Lupsasca & A. Strominger, Holography of the Photon Ring, arXiv:2205.05064; section 2.4.
• Wonderful answer! I just have a few questions: (1) the step in 9. where you say we can identify $dT$ and $d\Phi$ as $\tau$ and $\eta$ along a geodesic seems strange to me. Why do we lose the second term? (2) Does the differential operator $d$ also commute with both $I$ and $d_\vert$? Commented May 25 at 17:13
• (1) I updated the answer. (2) No, essentially because integration and differentiation do not commute due to a possible integration constant. Commented May 25 at 18:17
• (1) still confuses me because $dp_t \approx 0$ would imply that the symplectomorphism does not work right? Or is the statement more like $dp_t \approx 0$ is an informal way to see this? (2) So if i understand correctly, the first expressions in 9. for $dT$ and $d\Phi$ obey $d^2 T = d^2\Phi = 0$? Commented May 25 at 18:56
• (1) That is observed but not used anywhere in the answer, in particular not in section 10. (2) Yes. Commented May 25 at 23:56
• (1) In 9. it is. This is precisely where my confusion stemmed from! But I guess the formal expression is precisely the first ones you give for $dT$ and $d\Phi$. Commented May 26 at 8:38