Gravitational frequency shift of light in equatorial plane of a radially moving Schwarzschild black hole

TL;DR

This paper investigates the impact of radial motion of a Schwarzschild black hole on the frequency shift of light, revealing it to be comparable to transversal effects near the lens, especially at small impact parameters.

Contribution

It demonstrates that radial velocity effects on light frequency shifts are significant and comparable to transversal effects, contrary to previous assumptions that they are negligible.

Findings

Radial velocity effects are of the same order as transversal effects near the lens.

The velocity effect is transient due to relative motion.

Significant effects occur at impact parameter scales close to the lens.

Abstract

The kinematical effect induced by the transversal motion of a gravitational lens on the frequency shift of light has been investigated in detail, while the effect of the radial motion is thought to be much smaller than the transversal one and thus has usually been neglected. In this work, we find that the radial velocity effect on the frequency shift has the same order of magnitude as that of the transversal velocity effect, when the light emitter (or the receiver) is close to the gravitational lens with the distance between them being an impact parameter scale. The significant velocity effect is usually transient due to the motion of the gravitational lens relative to the light emitter or the receiver.