## 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

symbol ID category latex scope dimension name value Used in derivations references
2530 variable r
$$r$$
['real']
• length: 1
60
8656 variable v_{\rm escape}
$$v_{\rm escape}$$
real
• length: 1
• time: -1
escape velocity
12
5156 variable m
$$m$$
['real']
• mass: 1
mass
69
4567 constant c
$$c$$
['real']
• length: 1
• time: -1
speed of light in vacuum 299792458   meters/second
32
4518 variable r_{\rm Schwarzschild}
$$r_{\rm Schwarzschild}$$
real
• length: 1
$$G$$