Coordinate Charts in General Relativity: Comoving Distance, Born Coordinates, Rindler Coordinates, Schwarzschild Coordinates

9781155523484: Coordinate Charts in General Relativity: Comoving Distance, Born Coordinates, Rindler Coordinates, Schwarzschild Coordinates

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 29. Chapters: Comoving distance, Born coordinates, Rindler coordinates, Schwarzschild coordinates, Friedmann-Lemaître-Robertson-Walker metric, Gullstrand-Painlevé coordinates, Kruskal-Szekeres coordinates, Coordinate conditions, Isotropic coordinates, Harmonic coordinate condition, Lemaître coordinates, Eddington-Finkelstein coordinates, Ehlers-Geren-Sachs theorem, Gaussian polar coordinates, Synchronous coordinates, Boyer-Lindquist coordinates, Brinkmann coordinates. Excerpt: In relativistic physics, the Born coordinate chart is a coordinate chart for (part of) Minkowski spacetime, the flat spacetime of special relativity. It is often used to analyze the physical experience of observers who ride on a ring or disk rigidly rotating at relativistic speeds. This chart is often attributed to Max Born, due to his 1909 work on the relativistic physics of a rotating body - see Born rigidity. To motivate the Born chart, we first consider the family of Langevin observers represented in an ordinary cylindrical coordinate chart for Minkowski spacetime. The world lines of these observers form a timelike congruence which is rigid in the sense of having a vanishing expansion tensor. They represent observers who rotate rigidly around an axis of cylindrical symmetry. From the line element we can immediately read off a frame field representing the local Lorentz frames of stationary (inertial) observers Here, is a timelike unit vector field while the others are spacelike unit vector fields; at each event, all four are mutually orthogonal and determine the infinitesimal Lorentz frame of the static observer whose world line passes through that event. Simultaneously boosting these frame fields in the direction, we obtain the desired frame field describing the physical experience of the Langevin observers, namely This frame was apparently first introd...

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