Generalization of the k coefficient method in relativity to an arbitrary angle between the velocity of an observer (source) and the direction of the light ray from (to) a faraway source (observer) at rest

Abstract

The k coefficient method proposed by Bondi is extended to the general case where the angle α between the velocity of a signal from a distant source at rest and the velocity of the observer does not coincide with 0 or π, as considered by Bondi, but takes an arbitrary value in the interval 0 ⩽ α ⩽ π, and to the opposite case where the source is moving and the observer is at rest, while the angle α between the source velocity and the direction of the signal to the observer takes any value between 0 and π. Functions k*(β, α) and k+(β, α) of the angle and relative velocity are introduced for the ratio ω / ω′ of proper frequencies of the source and observer. Their explicit expressions are obtained without using Lorentz transformations, from the condition that the coherence of a bunch of rays is preserved in passing from the source frame to the observer frame. Owing to the analyticity of these functions in α, the ratio of frequencies in the cases mentioned is given by the formulas ω / ω′ = k*(β, α) and ω / ω′ = k+(β, π – α) ≡ 1 / k*(β, α), which coincide with those for the Doppler effect, in which the angle α, the velocity β, and one of the frequencies are measured in the rest frame. A ray emitted by the source at an angle α to the observer’s velocity in the source frame is directed at an angle α′ to the same velocity in the observer frame. Owing to light aberration, the angles α and α′ are functionally related through k*(β, α) = k+(β, β′). The functions α′(α, β) and α (α′,β) are expressed as antiderivatives of k*(β, α) and k*(β, π – α′). The analyticity of the functions k*(β, z) and k+(β, z) in z ≡ α in the interval 0 ⩽ z ⩽ π is extended to the entire plane of complex z, where k* has poles at i ln cos α1 (see ()), and k+ has zeros at the same points shifted by π. The spatiotemporal asymmetry of the Doppler and light aberration effects is explained by the closeness of these singularities to the real axis.