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Fermi coordinates
About: Fermi coordinates is a research topic. Over the lifetime, 115 publications have been published within this topic receiving 1870 citations.
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TL;DR: In this article, a discussion of the use of the modern, coordinate-free concept of a vector and of computations which are simplified by introducing a vector instead of its components is presented.
Abstract: Fermi coordinates, where the metric is rectangular and has vanishing first derivatives at each point of a curve, are constructed in a particular way about a geodesic. This determines an expansion of the metric in powers of proper distance normal to the geodesic, of which the second‐order terms are explicitly computed here in terms of the curvature tensor at the corresponding point on the base geodesic. These terms determine the lowest‐order effects of a gravitational field which can be measured locally by a freely falling observer. An example is provided in the Schwarzschild metric. This discussion of Fermi Normal Coordinate provides numerous examples of the use of the modern, coordinate‐free concept of a vector and of computations which are simplified by introducing a vector instead of its components. The ideas of contravariant vector and Lie Bracket, as well as the equation of geodesic deviation, are reviewed before being applied.
330 citations
TL;DR: In this article, the existence of conformally related metrics of zero scalar curvature and constant mean curvature on the boundary, under suitable hypotheses on the Weyl tensor, was proved.
Abstract: Given a compact Riemannian manifold with umbilic boundary, we prove the existence of conformally related metrics of zero scalar curvature and constant mean curvature on the boundary, under suitable hypotheses on the Weyl tensor. In order to carry out the estimates on the Sobolev quotient, we also prove the existence of conformal Fermi coordinates.
153 citations
TL;DR: The Fermi coordinate system of an observer in arbitrary motion in an arbitrary weak gravitational field valid to all orders in the geodesic distance from the world line of the observer is derived to make the correspondence between general relativity and Newtonian gravity more precise.
Abstract: We derive the Fermi coordinate system of an observer in arbitrary motion in an arbitrary weak gravitational field valid to all orders in the geodesic distance from the world line of the observer. In flat space-time this leads to a generalization of Rindler space for arbitrary acceleration and rotation. The general approach is applied to the special case of an observer resting with respect to the weak gravitational field of a static mass distribution. This allows us to make the correspondence between general relativity and Newtonian gravity more precise.
99 citations
TL;DR: In this paper, quasi-inertial Fermi coordinates are employed to discuss the dynamics of a free test particle in these spacetimes and the results are compared to the corresponding generalized Jacobi equations that contain only the lowest-order tidal terms.
Abstract: Fermi coordinates are directly constructed in de Sitter and G\"odel spacetimes and the corresponding exact coordinate transformations are given explicitly. The quasi-inertial Fermi coordinates are then employed to discuss the dynamics of a free test particle in these spacetimes and the results are compared to the corresponding generalized Jacobi equations that contain only the lowest-order tidal terms. The domain of validity of the generalized Jacobi equation is thus examined in these cases. Furthermore, the difficulty of constructing explicit Fermi coordinates in black hole spacetimes is demonstrated.
90 citations
TL;DR: In this paper, fractional Sobolev and Besov spaces on non-compact Riemannian manifolds with bounded geometry are studied and sufficient conditions on the corresponding coordinates resulting in equivalent norms are given.
Abstract: We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel-Lizorkin spaces on manifolds, improving [11].
79 citations