Numerical determination of time transfer in general relativity

Miguel, A. San

doi:10.1007/s10714-007-0499-y

Cited by 13 publications

(13 citation statements)

References 13 publications

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“…To illustrate once more the pervasive presence of the linear differential equation (4) let us mention reference [168] in which the aim is to determine the time elapsed between two events when the space-time is treated as in General Relativity. Then it turns out to be necessary to solve a two-point boundary value problem for null geodesics.…”

Section: General Relativitymentioning

confidence: 99%

See 1 more Smart Citation

The Magnus expansion and some of its applications

Blanes¹,

Casas²,

Oteo³

et al. 2009

Physics Reports

1,103

1,300

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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution.The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.

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Section: General Relativitymentioning

confidence: 99%

“…In so doing one needs to know a Jacobian whose expression involves a 8 × 8 matrix function obeying the basic equation (4). In [168] an eighth order numerical method from [22] is used, which is proved to be an efficient scheme.…”

Section: General Relativitymentioning

confidence: 99%

The Magnus expansion and some of its applications

Blanes¹,

Casas²,

Oteo³

et al. 2009

Physics Reports

1,103

1,300

View full text Add to dashboard Cite

show abstract

“…The time-of-flight (ToF) of the signal has several contributions: geometrical, ionospheric, tropospheric and Shapiro (see equations (12)- (14)). The geometrical ToF is calculated numerically by solving a two-point boundary value problem by means of the shooting method [28]. The ionospheric delay depends on the electron density profile in the atmosphere (see equations (28)- (29)) and on the magnetic field.…”

Section: Data Analysis and Simulation 41 Implementation Of The Simulmentioning

confidence: 99%

Atomic clock ensemble in space (ACES) data analysis

Meynadier¹,

Delva²,

Poncin-Lafitte³

et al. 2018

Class. Quantum Grav.

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The Atomic Clocks Ensemble in Space (ACES/PHARAO mission, ESA & CNES) will be installed on board the International Space Station (ISS) next year. A crucial part of this experiment is its two-way MicroWave Link (MWL), which will compare the timescale generated on board with those provided by several ground stations disseminated on the Earth. A dedicated Data Analysis Center (DAC) is being implemented at SYRTE -Observatoire de Paris, where our team currently develops theoretical modelling, numerical simulations and the data analysis software itself.In this paper, we present some key aspects of the MWL measurement method and the associated algorithms for simulations and data analysis. We show the results of tests using simulated data with fully realistic effects such as fundamental measurement noise, Doppler, atmospheric delays, or cycle ambiguities. We demonstrate satisfactory performance of the software with respect to the specifications of the ACES mission. The main scientific product of our analysis is the clock desynchronisation between ground and space clocks, i.e. the difference of proper times between the space clocks and ground clocks at participating institutes. While in flight, this measurement will allow for tests of General Relativity and Lorentz invariance at unprecedented levels, e.g. measurement of the gravitational redshift at the 3 × 10 −6 level.As a specific example, we use real ISS orbit data with estimated errors at the 10 m level to study the effect of such errors on the clock desynchronisation obtained from MWL data. We demonstrate that the resulting effects are totally negligible.

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“…Various concepts and techniques being useful to develop the 1-order RPS have been found in previous papers, among them, we may point out the definition and uses of the world function (Synge, 1931;Bahder, 2001;Bini et al, 2008;San Miguel, 2007) and the time transfer function, the form of this last function in the S-ST (Teyssandier and Le Poncin-Lafitte, 2008), and a method to find the user position coordinates by using the time transfer function (Čadež and Kostić, 2005;Čadež et al, 2010;Delva et al, 2011). Here, this last method is modified by using the analytical formula derived by Coll et al (2010) -instead of numerical iterations-to work with photons moving in M-ST The Earth's center is at rest in the asymptotic M-ST; hence, the S-ST may be considered as a perturbation of the asymptotic M-ST with a static metric g αβ = η αβ +s αβ , where η αβ is the Minkowski metric, and s αβ are perturbation terms depending on GM ⊕ /R, where R is the Schwarzschild radial coordinate.…”

Section: Relativistic Positioning In S-st: the 1-order Rpsmentioning

confidence: 99%

Approaches to relativistic positioning around Earth and error estimations

Puchades

Sáez

2016

Advances in Space Research

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In the context of relativistic positioning, the coordinates of a given user may be calculated by using suitable information broadcast by a 4-tuple of satellites. Our 4-tuples belong to the Galileo constellation. Recently, we estimated the positioning errors due to uncertainties in the satellite world lines (U-errors). A distribution of U-errors was obtained, at various times, in a set of points covering a large region surrounding Earth. Here, the positioning errors associated to the simplifying assumption that photons move in Minkowski space-time (S-errors) are estimated and compared with the U-errors. Both errors have been calculated for the same points and times to make comparisons possible. For a certain realistic modeling of the world line uncertainties, the estimated S-errors have proved to be smaller than the U-errors, which shows that the approach based on the assumption that the Earth's gravitational field produces negligible effects on photons may be used in a large region surrounding Earth. The applicability of this approach -which simplifies numerical calculations-to positioning problems, and the usefulness of our S-error maps, are pointed out. A better approach, based on the assumption that photons move in the Schwarzschild space-time governed by an idealized Earth, is also analyzed. More accurate descriptions of photon propagation involving non symmetric space-time structures are not necessary for ordinary positioning and spacecraft navigation around Earth.

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Numerical determination of time transfer in general relativity

Cited by 13 publications

References 13 publications

The Magnus expansion and some of its applications

The Magnus expansion and some of its applications

Atomic clock ensemble in space (ACES) data analysis

Approaches to relativistic positioning around Earth and error estimations

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