On the Schwarzschild-Levi-Civita Metric

This study investigates gravitational lensing and null geodesics in the Schwarzschild-Levi-Civita Metric (SLCM), an axially symmetric solution to the vacuum Einstein equations. The SLCM resembles the Ernst black hole but differs fundamentally in being a vacuum solution rather than one sourced by electromagnetic fields. Utilizing the Gauss–Bonnet theorem and the optical metric approach developed by Gibbons, Werner, and Ishihara et al., the deflection angle of light in the SLCM is derived, including finite-distance corrections for source and observer positions. An explicit analytic expression for the deflection angle is obtained, featuring both Levi-Civita-like terms and corrections from the SLCM’s cosmological horizon. Numerical analysis of null geodesics confirms that all light rays are eventually drawn toward the horizon, regardless of initial velocity direction. These findings offer insights into the lensing properties of axially symmetric, non-asymptotically flat spacetimes in general relativity.