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review derivation: Schwarzschild radius for non-rotating black hole

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Notes for this derivation:
https://en.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solution

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
2 change two variables in expr
  1. 8946383937; locally 2478510:
    \(v_{\rm escape}^2 = 2 G \frac{m}{r}\)
    \(pdg_{8656}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{2530}}\)
  1. 8362338572:
    \(v_{\rm escape}\)
    \(pdg_{8656}\)
  2. 1238593037:
    \(c\)
    \(pdg_{4567}\)
  3. 2660368546:
    \(r\)
    \(pdg_{2530}\)
  4. 9933742680:
    \(r_{\rm Schwarzschild}\)
    \(pdg_{4518}\)
  1. 4275004561; locally 5459812:
    \(c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}\)
    \(pdg_{4567}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{4518}}\)
valid 8946383937:
4275004561: dimensions are consistent
 8946383937:
4275004561: N/A
4 divide both sides by
  1. 2883079365; locally 9932666:
    \(r_{\rm Schwarzschild} c^2 = 2 G m\)
    \(pdg_{4518} pdg_{4567}^{2} = 2 pdg_{5156} pdg_{6277}\)
  1. 7263534144:
    \(c^2\)
    \(pdg_{4567}^{2}\)
  1. 6800170830; locally 9994959:
    \(r_{\rm Schwarzschild} = \frac{2 G m}{c^2}\)
    \(pdg_{4518} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{4567}^{2}}\)
valid 2883079365: dimensions are consistent
6800170830:
2883079365: N/A
6800170830:
1 raise both sides to power
  1. 5404822208; locally 1044984:
    \(v_{\rm escape} = \sqrt{2 G \frac{m}{r}}\)
    \(pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5156} pdg_{6277}}{pdg_{2530}}}\)
  1. 3663007361:
    \(2\)
    \(2\)
  1. 8946383937; locally 2478510:
    \(v_{\rm escape}^2 = 2 G \frac{m}{r}\)
    \(pdg_{8656}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{2530}}\)
no check is performed 5404822208:
8946383937:
5404822208:
8946383937:
3 multiply both sides by
  1. 4275004561; locally 5459812:
    \(c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}\)
    \(pdg_{4567}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{4518}}\)
  1. 7194432406:
    \(r_{\rm Schwarzschild}\)
    \(pdg_{4518}\)
  1. 2883079365; locally 9932666:
    \(r_{\rm Schwarzschild} c^2 = 2 G m\)
    \(pdg_{4518} pdg_{4567}^{2} = 2 pdg_{5156} pdg_{6277}\)
valid 4275004561: dimensions are consistent
2883079365: dimensions are consistent
 4275004561: N/A
2883079365: N/A
Physics Derivation Graph: Steps for Schwarzschild radius for non-rotating black hole

Symbols for this derivation

See also all 215 symbols
symbol ID category latex scope dimension name value Used in derivations references
4567 constant c
\(c\)		 ['real']
  • length: 1
  • time: -1
 speed of light in vacuum 299792458   meters/second
 32
2530 variable r
\(r\)		 ['real']
  • length: 1
 radius 34
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 46
4518 variable r_{\rm Schwarzschild}
\(r_{\rm Schwarzschild}\)		 real
  • length: 1
 Schwarzschild radius; event horizon 5
8656 variable v_{\rm escape}
\(v_{\rm escape}\)		 real
  • length: 1
  • time: -1
 escape velocity 12
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 57
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