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

The $k$ coefficient method proposed by Bondi is extended to the general case where the angle $\alpha$ between the velocity of a signal from a distant source at rest and the velocity of the observer does not coincide with 0 or $\pi$, as considered by Bondi, but takes an arbitrary value in the interval $0\le \alpha \le \pi$, and to the opposite case where the source is moving and the observer is at rest, while the angle $\alpha$ between the source velocity and the direction of the signal to the observer takes any value between 0 and $\pi $. Functions $k_*(\beta,\alpha)$ and $k_+(\beta,\alpha)$ of the angle and relative velocity are introduced for the ratio $\omega /\omega^{'}$ 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 $\alpha$, the ratio of frequencies in the cases mentioned is given by the formulas $\omega /\omega ^{'}=k_*(\beta,\alpha)$ and $\omega /\omega^{'}=k_+(\beta,\pi -\alpha )\equiv 1/k_*(\beta,\alpha)$, which coincide with those for the Doppler effect, in which the angle $\alpha$, the velocity $\beta$, and one of the frequencies are measured in the rest frame. A ray emitted by the source at an angle $\alpha$ to the observer's velocity in the source frame is directed at an angle $\alpha^{'}$ to the same velocity in the observer frame. Owing to light aberration, the angles $\alpha$ and $\alpha^{'}$ are functionally related through $k_*(\beta,\alpha)=k_+(\beta,\alpha^{'})$. The functions $\alpha^{'}(\alpha,\beta)$ and $\alpha (\alpha ^{'},\beta)$ are expressed as antiderivatives of $k_*(\beta,\alpha)$ and $k_*(\beta,\pi -\alpha ^{'})$. The analyticity of the functions $k_*(\beta, z)$ and $k_+(\beta, z)$ in $z\equiv \alpha$ in the interval $0\le z\le \pi $ is extended to the entire plane of complex $z$, where $k_*$ has poles at $z^\pm _n=2\pi n\mp \rm i \ln \cos \alpha _1$ (see (17)), and $k_+$ has zeros at the same points shifted by $\pi$. The spatiotemporal asymmetry of the Doppler and light aberration effects is explained by the closeness of these singularities to the real axis.