Physics Derivation Graph navigation Sign in

review derivation: Schwarzschild radius for non-rotating black hole

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

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

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

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 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
4518 variable r_{\rm Schwarzschild}
\(r_{\rm Schwarzschild}\)		 real
  • length: 1
 Schwarzschild radius; event horizon 5
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 60
2530 variable r
\(r\)		 ['real']
  • length: 1
 radius 60
4567 constant c
\(c\)		 ['real']
  • length: 1
  • time: -1
 speed of light in vacuum 299792458   meters/second
 32
8656 variable v_{\rm escape}
\(v_{\rm escape}\)		 real
  • length: 1
  • time: -1
 escape velocity 12
MESSAGE: