A Universal Tool for Determining the Time Delay and the Frequency Shift of Light: Synge's World function

Abstract. Modeling most of the tests of general relativity requires to know the function relating light travel time to the coordinate time of reception and to the spatial coordinates of the emitter and the receiver. We call such a function the reception time transfer function. Of course, an emission time transfer function may as well be considered. We present here a recursive procedure enabling to expand each time transfer function into a perturbative series of ascending powers of the Newtonian gravitational constant G (general post-Minkowskian expansion). Our method is self-sufficient, in the sense that neither the integration of null geodesic equations nor the determination of Synge's world function are necessary. To illustrate the method, the time transfer function of a threeparameter family of static, spherically symmetric metrics is derived within the post-linear approximation.

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“…[14] and [15]. § Equations (4) and (5) can also be derived from the general-relativistic version of Fermat's principle established in Ref.…”

Section: Time Transfer Functionsmentioning

confidence: 99%

“…Ref. [15]). It is the purpose of this paper to present a stand-alone method which totally avoids performing the calculation of the world function.…”

Section: Introductionmentioning

confidence: 99%

General post-Minkowskian expansion of time transfer functions

Teyssandier¹,

Poncin-Lafitte²

2008

Class. Quantum Grav.

Self Cite

100

175

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“…The integration is performed along the light path, from the transmission point (labeled 1) to the receiving point (labeled 2). The instantaneous relativistic Doppler shift derives from the fact that the ratio between the received and the transmitted frequencies can be expressed as [20,21]…”

Section: B Observables Given By the Geometrymentioning

confidence: 99%

“…where ν 1 and ν 2 are the transmitted and the received frequencies respectively, u α denotes the 4-velocity vector, k α is the covariant form of the 4-wave vector which is tangent to light ray, andû i andk i are respectively given byû i = u i /u 0 , andk i = k i /k 0 (see [21,22] for definitions and notations). In this expression, the subscripts 1 and 2 specify that the terms between parenthesis must be evaluated at the position of the transmitter and the receiver respectively.…”

Section: B Observables Given By the Geometrymentioning

confidence: 99%

Analytical ray-tracing in planetary atmospheres

Bourgoin

Zannoni

Tortora

2019

A&A

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Context. Ground-based astro-geodetic observations and atmospheric occultations, are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes in observables (range, Doppler shift, or signal attenuation) are geometrically related to changes in the photon path and the light time of the received electromagnetic signal. In the context of geometrical optics, the change in the physical properties of the signal are related to the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path and the light time evolution is of prime importance to clearly understand observational features. Analytical studies usually focused on spherically symmetric atmospheres and only few aimed at exploring the effect of the non-spherical symmetry on the observables.Aims. In this paper, we analytically perform the integration of the photon path and the light time of rays traveling across a planetary atmosphere. We do not restrict our attention to spherically symmetric atmospheres and introduce a comprehensive mathematical framework which allows to handle any kind of analytical studies in the context of geometrical optics.Methods. Assuming that the index of refraction of the medium is a linear function of the Newtonian potential, we derive an exact solution to the equations of geometrical optics. The solution's arbitrary constants of integration are parametrized by the refractive response of the medium to the gravitational potential, and by the first integrals of the problem. Varying the constants, we are able to reformulate the equation of geometrical optics into a new set of osculating equations describing the constants' evolution for any arbitrary changes in the index of refraction profile.Results. The osculating equations are identical to the equation of geometrical optics, however, they offer a comprehensive framework to handle analytical studies which aim at exploring non-radial dependencies in the refractive profile. To highlight the capabilities of this new formalism, we carry out five realistic applications for which we derive analytical solutions. The accuracy of the method of integration is assessed by comparing our results to a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment J 2 . This shows that the analytical solution leads to the determination of the light time and the refractive bending with relative errors at the level of one part in 10 8 and one part in 10 5 , for typical values of the refractivity and the J 2 parameter at levels of 10 −4 and 10 −2 , respectively. * adrien.bourgoin@unibo.it arXiv:1901.08461v1 [physics.class-ph]

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“…[93] for an implementation of this idea). Quantities like r 2 ij (τ ) have to be replaced with the Synge world function [82,87,89] 39 . This problem is another reason why extended objects tend to be replaced with point-like multipoles, which, however, do not span a canonical basis of phase space (see Refs.…”

Section: The Center-of-mass Problem In General Relativitymentioning

confidence: 99%

From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

Lusanna¹

2014

Springer Proceedings in Physics

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A Universal Tool for Determining the Time Delay and the Frequency Shift of Light: Synge's World function

Cited by 18 publications

References 24 publications

General post-Minkowskian expansion of time transfer functions

General post-Minkowskian expansion of time transfer functions

Analytical ray-tracing in planetary atmospheres

From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

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