Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Linearized Gravitational Waves Near Space-Like and Null Infinity Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Relativistic Positioning Systems in Flat Space-Time: The Location Problem

Authors : Bartolomé Coll, Joan Josep Ferrando, Juan Antonio Morales-Lladosa

Published in: Progress in Mathematical Relativity, Gravitation and Cosmology

Publisher: Springer Berlin Heidelberg

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The location problem in relativistic positioning is considered in flat space-time. When two formal solutions are possible for a user (receiver) of the system, its true location may be obtained from a standard set of emission data extended with an observational rule. The covariant expression giving the location of the user in inertial coordinates is decomposed with respect to an inertial observer.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Radiation from a D-Dimensional Collision of Shock Waves: Numerics and a Charged Case
next chapter BSSN Equations in Spherical Coordinates Without Regularization
Footnotes
1
We use the following notation: (−, +, +, +) is the signature of the Minkowski metric g; i() denotes the interior product (if x is a vector and T a covariant 2-tensor, \([i(x)T]_{\nu } = {x}^{\mu }T_{\mu \nu }\)); ∧ stands for the exterior product; the asterisk ∗ denotes the Hodge dual operator associated to the metric volume element η, \(\eta _{\alpha \beta \gamma \delta } = -\sqrt{-\mathit{det } \,g}\,\epsilon _{\alpha \beta \gamma \delta }\), where \(\epsilon _{\alpha \beta \gamma \delta }\) is Levi–Civita permutation symbol, ε 0123 = 1. For a given inertial observer of unit velocity u, u 2 = −1, any vector x splits as \(x = {x}^{0}u +\vec{ x}\) where x 0 = −x ⋅ u and \(\vec{x} \in E_{\perp }\) with E  ⊥  the three-space of u. For vectors \(\vec{x}\), \(\vec{y} \in E_{\perp }\), the vector product is given by \(\vec{x} \times \vec{ y} = {\ast}(u \wedge \vec{ x} \wedge \vec{ y})\) and \((\vec{x},\vec{y},\vec{z}) \equiv (\vec{x} \times \vec{ y}) \cdot \vec{ z}\) (with \(\vec{z} \in E_{\perp }\)) is the scalar triple product.
 
2
It seems that Abel and Chaffee [1, 2] were the first authors in considering the location problem in connection with Global Positioning System (GPS) by using Lorentzian algebra.
 
3
The region \({\mathcal{C}}^{C} \equiv \{ x \in \mathcal{C}\,{\vert \,\chi }^{2} \leq 0\}\) is called the central region of the RPS. The orientation \(\hat{\epsilon }\) is constant on \({\mathcal{C}}^{C}\), and may be evaluated from the sole standard data set E. The bifurcation problem always appears in the time-like configuration region \(\mathcal{C}_{t} \equiv \{ x \in \mathcal{C}\,{\vert \,\chi }^{2} > 0\} = \mathcal{C}-{\mathcal{C}}^{C}\).
 
Literature
1.
Abel, J. S., Chaffee, J. W.: IEEE Trans. Aerosp. Electron. Syst. 27, 952 (1991)CrossRef
2.
Chaffee, J. W., Abel, J. S.: IEEE Trans. Aerosp. Electron. Syst. 30, 1021 (1994)CrossRef
3.
Coll, B., Pozo, J. M.: Constuction in 4D: Algebraic properties and special observers. Lecture delivered at the School on Relativistic Coordinates, Reference and Positioning Systems, (Salamanca 2005). See also arXiv: gr-qc/0601125
4.
5.
Coll, B., Ferrando, J. J., Morales-Lladosa, J. A.: Phys. Rev. D 86, 084036 (2012)CrossRef
6.
Puchades, N., Sáez, D.: Astrophys. Space. Sci. 341, 631 (2012). See also arXiv:11126054
7.
Sáez, D., Puchades, N.: Locating objects away from Earth surface: positioning accuracy. In these proceedings.
Metadata
Title
Relativistic Positioning Systems in Flat Space-Time: The Location Problem
Authors
Bartolomé Coll
Joan Josep Ferrando
Juan Antonio Morales-Lladosa
Copyright Year
2014
Publisher
Springer Berlin Heidelberg
Book
Progress in Mathematical Relativity, Gravitation and Cosmology

Print ISBN: 978-3-642-40156-5

Electronic ISBN: 978-3-642-40157-2

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-642-40157-2_24

Premium Partner

    Image Credits