On visible effects in the double Schwarzschild solution

TL;DR

This work analyzes how a Weyl strut between two static Schwarzschild black holes modifies null geodesics and observable images via ray-tracing in the double Schwarzschild spacetime. By solving the exact metric in Weyl coordinates and integrating the null geodesic equations with conserved quantities $E$ and $L$, the authors quantify defocusing by the strut and its competition with the BHs' attraction, showing that large separations resemble a single BH while small separations produce rich caustics, shadow distortions, and higher-order image copies. The study yields concrete results on horizon accessibility, photon surfaces, caustics, and trajectory classifications (including a free-group description of path topologies), and demonstrates pronounced imaging differences relative to a single BH that depend on $R_0$ and mass symmetry. The findings have implications for interpreting binary BH lensing in static configurations and set the stage for extending to rotating binaries and non-equal masses, where frame-dragging and weaker struts are anticipated to alter the observed phenomena. The work advances visualization-based analysis of exact multi-BBH spacetimes and highlights observable signatures of the Weyl strut in gravitational lensing and shadows.

Abstract

Physical aspects of a static solution to the Einstein equations with two black holes are studied via ray tracing. The exact solution for this double Schwarzschild solution is known in explicit form. The black holes are separated by a singularity called \emph{Weyl strut}. The effect of this strut on null geodesics is shown to be defocusing in contrast to the focusing effect of the black holes. It is shown that black holes with a large separation essentially lead to similar behavior of the null geodesics as a single black hole, whereas nearby holes display a widely changed behavior due to the Weyl strut.

Figures (21)

• Figure 1: The metric functions (\ref{['fk']}) for the Schwarzschild solution for $m=1$ , on the left $f$, on the right $e^{2k}$.
• Figure 2: The metric functions (\ref{['fk']}) for the double Schwarzschild solution for $m_{1}=m_{2}=1$ and $R_{0}=4$, on the left $f$, on the right $e^{2k}$.
• Figure 3: Diagram with the horizon (line on the axis) and photon sphere of each black hole.
• Figure 4: Left hand side: initially parallel light rays on the equatorial plane in a single Schwarzschild space-time. Right hand side: light rays that orbit the BH once and then fall inside the horizon or escape to infinity. The black dot is the 2D projection of the horizon.
• Figure 5: Initially parallel light rays with fixed $m_1=m_2=1$ and decreasing values of $R_0$. The dot is the 2D projection of the horizons.