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Relativity, Cosmology, and Time


Schwarzschild's Solution and Black Hole (2018 Edition)

Madel One half of the 2020 Nobel Prize in Physics is awarded to Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity", the other half jointly to Reinhard Genzel and Andrea Ghez for the discovery of a "Supermassive Compact Object at the Centre of our Galaxy." While Roger Penrose have done some works on black hole (see for example "Gravitational Collapse and Space-time Singularities", and "Penrose Diagram"), the original discovery should go to Karl Schwarzschild, who provided the first exact solution to the field equations of general relativity in 1915 (for the case of a single spherical non-rotating mass). Then, there is an even more dubious process of the Nobel Committees to accept the nomination of Donald Trump to the Peace Prize twice; while he has done irreparable damages to the global environment (see "How Trump damaged science and why it could take decades to recover" by Nature, 05 October 2020; and "What is the Trump administration's track record on the environment?" by Brookings Institute, 04 August 2020).

Anyway back to the science of the cosmos, the equations of metric tensors can be solved exactly for the case of a centrally symmetric field in vacuum with mass M at the center. In terms of spherical coordinates and ct, the "world line" has the curvilinear form :

ds2 = (1 - 2GM/c2r) c2dt2 - dr2 / (1 - 2GM/c2r) - r2 (sin2 d2 + d2) ---------- (13)

This is the celebrated Schwarzschild solution. The integration constant (often referred to as the Schwarzschild's radius or event horizon) rs = 2GM/c2 appears in the formula in order to relate to Newton's inverse square law, otherwise it is rather arbitrary. Some special properties about this metric are listed in the followings :

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