Trigonometry in Lorentzian Geometry

1984 ◽  
Vol 91 (9) ◽  
pp. 543 ◽  
Author(s):  
Graciela S. Birman ◽  
Katsumi Nomizu
Keyword(s):  
Lorentzian Geometry

scholarly journals A MAXWELL-LIKE FORMULATION OF GRAVITATIONAL THEORY IN MINKOWSKI SPACE–TIME

2007 ◽  
Vol 16 (06) ◽  
pp. 1027-1041 ◽  
Author(s):  
EDUARDO A. NOTTE-CUELLO ◽  
WALDYR A. RODRIGUES
Keyword(s):  
Gravitational Field ◽  
Minkowski Space ◽  
Gravitational Theory ◽  
Space Time ◽  
Lorentzian Geometry ◽  
Field Equations ◽  
Yang Mills ◽  
Minkowski Space Time ◽  
Clifford Bundle ◽  
Gauge Fixing

Using the Clifford bundle formalism, a Lagrangian theory of the Yang–Mills type (with a gauge fixing term and an auto interacting term) for the gravitational field in Minkowski space–time is presented. It is shown how two simple hypotheses permit the interpretation of the formalism in terms of effective Lorentzian or teleparallel geometries. In the case of a Lorentzian geometry interpretation of the theory, the field equations are shown to be equivalent to Einstein's equations.

The Splitting Problem in Global Lorentzian Geometry

2017 ◽  
pp. 501-566
Author(s):  
John K. Beem ◽  
Paul E. Ehrlich ◽  
Kevin L. Easley
Keyword(s):  
Lorentzian Geometry ◽  
Splitting Problem

Lorentzian Geometry

2006 ◽  
pp. 343-349
Author(s):  
P.E. Ehrlich ◽  
S.B. Kim
Keyword(s):  
Lorentzian Geometry

Global dynamical time for binary point-like particle systems

2020 ◽  
Vol 17 (09) ◽  
pp. 2050131
Author(s):  
Osvaldo M. Moreschi
Keyword(s):  
Equations Of Motion ◽  
Minkowski Spacetime ◽  
Particle Systems ◽  
Lorentzian Geometry ◽  
Field Equations ◽  
Retarded Time ◽  
Dynamical Time ◽  
The Difference ◽  
Multiple Particle ◽  
Equations Of Motions

A geometrical construction for a global dynamical time for binary point-like particle systems, modeled by relativistic equations of motions, is presented. Thus, we provide a convenient tool for the calculation of the dynamics of recent models for the dynamics of black holes that use individual proper times. The construction is naturally based on the local Lorentzian geometry of the spacetime considered. Although in this presentation we are dealing with the Minkowskian spacetime, the construction is useful for gravitational models that have as a seed Minkowski spacetime. We present the discussion for a binary system, but the construction is obviously generalizable to multiple particle systems. The calculations are organized in terms of the order of the corresponding relativistic forces. In particular, we improve on the Darwin and Landau–Lifshitz approaches, for the case of electromagnetic systems. We discuss the possibility of a Lagrangian treatment of the retarded effects, depending on the nature of the relativistic forces. The higher-order contractions are based on a Runge–Kutta type procedure, which is used to calculate the quantities at the required retarded time, by increasing evaluations of the forces at intermediate times. We emphasize the difference between approximation orders in field equations and approximation orders in retarded effects in the equations of motion. We show this difference by applying our construction to the binary electromagnetic case.

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