The Doppler shift in a Schwarzschild spacetime

Abstract

We demonstrate that in the case of Schwarzschild spacetime the Doppler shift is partially factorized into terms representing relativistic, kinematical and the gravitational contributions. The condition for the complete factorization is derived. Application of these results to the simplest cases and possible implementation in the framework of GPS is briefly discussed.Comment: Accepted for publishing in Physics Letters

On the nature of cosmological redshift and spectral shift in Schwarzschild-like and other spacetimes

“…We shall consider two co-radial observers in Schwarzschild spacetime, in this section denoted as A and B. The relation between frequency ω A and A's speed v A as measured by a local resting observer R, is [3]:…”

“…In this way kinematical and gravitational components factorize [3]. The relative speed may be determined via frequency shift, by means of the scalar product (7).…”

“…Relative speed B versus A does not vanish and is given in terms of the strength of the gravitational field In this case v B = 0, v A = 0; simple calculation shows (see [3]) that the frequency shift turns out to be purely kinematical (the classical Doppler effect), see also [11] …”

“…This is a combined effect involving, in the general case, entangled contributions of kinematical and gravitational origins. Recently, various aspects of this problem have been discussed, displaying decoupling and in consequence, factorization or partial factorization into gravitational and kinematical (time-dilation-like and classical Doppler shift) contributions [1][2][3] in the case of Schwarzschild spacetime. Nevertheless, as shown long ago by Synge [4], the ratio of the frequencies, the generalized Doppler shift, measured by observer O and sent from source S may be expressed as…”

On the Doppler effect for light from orbiting sources in Kerr-type metrics

“…One such example is treated in Radosz et al (2009), analyzing situations in which the Doppler shift factorizes into the kinematic contribution and the general relativistic contribution (in general, these contributions are intricately entangled). Another example where the formula derived in this paper is useful is in the analysis presented by Schönenbach et al (2014), contrasting the predictions of emission from the accretion disks of central galactic nuclei in General Relativity versus pseudo-complex General Relativity.…”