Optics in the Schwarzschild spacetime

Realistic modeling of radiation transfer in and from variable accretion disks around black holes requires the solution of the problem: find the constants of motion and equation of motion of a lightlike geodesic connecting two arbitrary points in space. Here we give the complete solution of this problem in the Schwarzschild spacetime.

Figure 1

Left panel: The orbital plane in equatorial coordinates: $\widehat{n}$ unit normal, $\epsilon$ inclination, $\Omega$ longitude of the ascending node, $\omega$ longitude of the periastron and $\lambda$ the true anomaly. Right panel: The initial ${\mathcal{P}}_i=({\theta }_i,{\phi }_i)$ and the final ${\mathcal{P}}_f=({\theta }_f,{\phi }_f)$ points.

Figure 2

Left: Some orbits of type A with parameters $r_i=20M$ and $a_{\mathrm{crit}}-a\in \{0.2,0.15,0.1,0.05,0.005\}$. Middle: Orbits of type B with parameters $r_i=20M$ and $a-a_{\mathrm{crit}}\in \{2,0.7,0.2,0.05,0.005\}$. Right: Orbits of type C with parameters $r_i=2.00001M$ and $a_{\mathrm{crit}}-a\in \{0.01,0.005,0.001\}$. The radius of the black circle is the Schwarzschild radius.

Figure 3

For a known initial point ${\mathcal{P}}_i$, the two critical orbits through this point divide the orbital plane into three regions. Left panel: initial point at $r>3M$. If the final point ${\mathcal{P}}_f$ is in region 3, then all orbits from ${\mathcal{P}}_i$ are of type A. If ${\mathcal{P}}_f$ is in region 1, then all orbits from ${\mathcal{P}}_i$ are of type B. If ${\mathcal{P}}_f$ is in region 2, then orbits leading to ${\mathcal{P}}_f$ are of type A if the trajectory winds around the black hole, and of type B if it goes directly from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$. Right panel: initial point at $r<3M$. If the final point ${\mathcal{P}}_f$ is in region 3, then all orbits from ${\mathcal{P}}_i$ are of type B. If ${\mathcal{P}}_f$ is in region 1, then orbits from ${\mathcal{P}}_i$ are of type C. If ${\mathcal{P}}_f$ is in region 2, then orbits leading to ${\mathcal{P}}_f$ are of type C if the trajectory winds around the black hole, and of type B if it goes directly from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$.

Figure 4

Functions $\mathrm{cn}(x\mid m)$ and $\mathrm{sn}(x\mid m)$ along orbits: left panel, type A; middle panel, type B; right panel, type C. $x_{\min }$ and $x_{\max }$ are at the end points of the orbit and are designated by ${\mathcal{P}}_{-}$ and ${\mathcal{P}}_{+}$ if they are at spatial infinity, and ${\mathcal{P}}_{h-}$ and ${\mathcal{P}}_{h+}$ if they are at the horizon of the black hole. Interval definitions for the end points follow after Eqs. (10, 13, 15). $x_i=z$ and $x_f=v$ are at the initial and the final points of the orbit.

Figure 5

Determining the angular momentum parameter for a type A orbit. Left panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ does not pass periastron. Right panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f^{\prime }$ passes periastron. The designations ${\mathrm{right}}_1$ and ${\mathrm{right}}_2$ refer to the branches (20, 21) respectively.

Figure 6

Connecting ${\mathcal{P}}_i$ and ${\mathcal{P}}_f$ with a type B orbit and a type A winding orbit (winding number $k=1$); no $k=0$ type A orbit exists in this case. Left panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ of type A. Right panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ of type B.

Figure 7

Determining the angular momentum parameter for a type B orbit. Left panel: a direct orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$. Right panel: a winding orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ (winding number $k=1$).

Figure 8

Determining the angular momentum parameter for a type C orbit. Left panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ does not pass apastron. Right panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f^{\prime }$ passes apastron. The designations ${\mathrm{right}}_1$ and ${\mathrm{right}}_3$ refer to the branches (20, 22) respectively.

Figure 9

Connecting ${\mathcal{P}}_i$ and ${\mathcal{P}}_f$ with a type B orbit and a type C winding orbit (winding number $k=1$); no $k=0$ type C orbit exists in this case. Left panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ of type C . Right panel: the orbit from ${\mathcal{P}}_i$ to ${\mathcal{P}}_f$ of type B.

Figure 10

Graph $r=r(t)$ in units of $M$ for all three types of orbits. The type A orbit starts at infinity at $t\to -\infty$, passes the periastron at $t=0$ and continues to infinity as $t\to \infty$. The type B orbit starts at the event horizon at $t\to -\infty$, crosses the critical $r=3M$ at $t=0$ and continues to infinity as $t\to \infty$. The type C orbit starts at the event horizon at $t\to -\infty$, passes the apastron at $t=M$ and again approaches the event horizon as $t\to \infty$. In the above example $a=(2/3\sqrt{3})-{10}^{-3}$ for types A and C orbits and $a=(2/3\sqrt{3})+5\times {10}^{-4}$ for type B orbit.