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  2. Kruskal–Szekeres coordinates - Wikipedia

    en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

    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.

  3. Coordinate singularity - Wikipedia

    en.wikipedia.org/wiki/Coordinate_singularity

    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).

  4. Singularity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Singularity_(mathematics)

    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 ...

  5. Lemaître coordinates - Wikipedia

    en.wikipedia.org/wiki/Lemaître_coordinates

    The time coordinate used in the Lemaître coordinates is identical to the "raindrop" time coordinate used in the Gullstrand–Painlevé coordinates. The other three: the radial and angular coordinates r , θ , ϕ {\displaystyle r,\theta ,\phi } of the Gullstrand–Painlevé coordinates are identical to those of the Schwarzschild chart.

  6. Gullstrand–Painlevé coordinates - Wikipedia

    en.wikipedia.org/wiki/Gullstrand–Painlevé...

    Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial ...

  7. Rindler coordinates - Wikipedia

    en.wikipedia.org/wiki/Rindler_coordinates

    These hyperbolic coordinates can be separated into two main variants depending on the accelerated observer's position: If the observer is located at time T = 0 at position X = 1/α (with α as the constant proper acceleration measured by a comoving accelerometer), then the hyperbolic coordinates are often called Rindler coordinates with the ...

  8. Pinch point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Pinch_point_(mathematics)

    Section of the Whitney umbrella, an example of pinch point singularity.. In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface.. The equation for the surface near a pinch point may be put in the form

  9. Eddington–Finkelstein coordinates - Wikipedia

    en.wikipedia.org/wiki/Eddington–Finkelstein...

    This is despite the fact that the probe itself can nonetheless travel past the horizon. It is also why the space-time metric of the black hole, when expressed in Schwarzschild coordinates, becomes singular at the horizon – and thereby fails to be able to fully chart the trajectory of an infalling probe.