Navigation in curved space–time

Bahder, Thomas B.

doi:10.1119/1.1326078

Cited by 42 publications

(55 citation statements)

References 17 publications

(18 reference statements)

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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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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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“…An application of the world function to problems of navigation and time transfer can be found in Ref. [26]. Compared to the enormous attention given to tensors, the world function has been used very little by physicists.…”

Section: A the World Functionmentioning

confidence: 99%

“…Calculations of the world function for spaces other than Euclidean spaces, namely four-dimensional space-time, can be found in Refs. [15,26,27,28,29]. In what follows, I restrict myself to a three-dimensional space.…”

Section: A the World Functionmentioning

confidence: 99%

Transformation Properties of the Lagrangian and Eulerian Strain Tensors

Bahder¹

2002

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A coordinate independent derivation of the Eulerian and Lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a three-point tensor.The derivation explicitly shows that the Eulerian and Lagrangian strain tensors are two-point tensors, each a function of both the spatial and material coordinates. The Eulerian strain is a two-point tensor that transforms as a second rank tensor under transformation of spatial coordinates and transforms as a scalar under transformation of the material coordinates. The Lagrangian strain is a two-point tensor that transforms as scalar under transformation of spatial coordinates and transforms as a second rank tensor under transformation of the material coordinates. These transformation properties are needed when transforming the strain tensors from one frame of reference to another moving frame.

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“…This means that, for its development, it needs no other physical concepts than the ones contained in its specific foundations, or those that can be coherently deduced from them. But in practice, despite efforts made in this sense [3][4][5][6][7][8][9][10][11][12], to develop physical applications must, for the moment, resort to Newtonian concepts and post-Newtonian methods. This situation reduces Relativity theory, with few exceptions, to a role of corrective algorithm for Newtonian theory, relegating its best specific concepts to a simple historically admirable, but otherwise ineffective, method of setting the main equations of the theory, the Einstein equations.…”

mentioning

confidence: 97%

Four Causal Classes of Newtonian Frames

2009

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The causal characters (spacelike, lightlike, timelike) of the coordinate lines, coordinate surfaces and coordinate hypersurfaces of a coordinate system in Relativity define what is called its causal class. It is known that, in any relativistic space-time, there exist one hundred and ninety nine such causal classes. But in Newtonian physics (where only spacelike and timelike characters exist) the corresponding causal classes have not been discussed until recently. Here it is shown that, in sharp contrast with the relativistic case, in Newtonian space-time the different causal classes of coordinate systems are drastically reduced to four. These causal classes admit simple geometric descriptions and physical interpretations. For example, it is shown that one can generate coordinate systems of the four causal classes by means of the sole linear synchronization group, i.e. by coordinate transformations that only change the origin of time linearly. The relativistic analogs of these examples are also considered.

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Navigation in curved space–time

Cited by 42 publications

References 17 publications

Approaches to relativistic positioning around Earth and error estimations

Approaches to relativistic positioning around Earth and error estimations

Transformation Properties of the Lagrangian and Eulerian Strain Tensors

Four Causal Classes of Newtonian Frames

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