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An example is the apparent (longitudinal) singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude ) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (i.e., jumping from longitude 0 to longitude 180 degrees).
The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates defined for r > 2GM and < < can be extended, as an analytic function, at least to the first singularity which occurs at =. Thus the above metric is a solution of Einstein's equations throughout this region.
Kruskal–Szekeres coordinates, a chart covering the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity, Eddington–Finkelstein coordinates, an alternative chart for static spherically symmetric spacetimes,
Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric—a spherically symmetric solution to the Einstein field equations in vacuum—introduced by Georges Lemaître in 1932. [1] Changing from Schwarzschild to Lemaître coordinates removes the coordinate singularity at the Schwarzschild radius.
A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the ...
The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at r = r s is an illusion however; it is an instance of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates or coordinate ...
The Rindler coordinate chart has a coordinate singularity at x = 0, where the metric tensor (expressed in the Rindler coordinates) has vanishing determinant. This happens because as x → 0 the acceleration of the Rindler observers diverges.
When a coordinate system is chosen in the Euclidean space, this defines coordinates on : the coordinates of a point of are defined as the coordinates of (). The pair formed by a chart and such a coordinate system is called a local coordinate system , coordinate chart , coordinate patch , coordinate map , or local frame .