Arthur Eddington gave alternative forms in isotropic coordinates.
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Alternatively, we can plug in specific functions ( possibly depending on some parameters ) to obtain an isotropic coordinate chart on a specific Lorentzian spacetime.
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Schwarzschild coordinates are fairly popular, but there are some disadvantages to those coordinates, and various advantages to other coordinate choices, such as isotropic coordinates, Eddington coordinates, Gaussian polar coordinates, Kruskal Szekeres coordinates, or Eddington Finkelstein coordinates.
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