Review Derivation: Physics Derivation Graph

Return to navigation page or list derivations

Review Schwarzschild radius for non-rotating black hole

abstract:
reference: https://en.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solution

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.	5404822208 $v_{\rm escape} = \sqrt{2 G \frac{m}{r}}$	3663007361 $2$	8946383937 $v_{\rm escape}^2 = 2 G \frac{m}{r}$	valid
2	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	8946383937 $v_{\rm escape}^2 = 2 G \frac{m}{r}$	9933742680 $r_{\rm Schwarzschild}$ 2660368546 $r$ 1238593037 $c$ 8362338572 $v_{\rm escape}$	4275004561 $c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}$	LHS diff is -pdg00045672 + pdg00086562 RHS diff is 2pdg0005156pdg0006277(-pdg0002530 + pdg0004518)/(pdg0002530pdg0004518)
3	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	4275004561 $c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}$	7194432406 $r_{\rm Schwarzschild}$	2883079365 $r_{\rm Schwarzschild} c^2 = 2 G m$	valid
4	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	2883079365 $r_{\rm Schwarzschild} c^2 = 2 G m$	7263534144 $c^2$	6800170830 $r_{\rm Schwarzschild} = \frac{2 G m}{c^2}$	valid

Symbols used in Schwarzschild radius for non-rotating black hole

0000004567: $ c $
0000006277: $ G $
0000005156: $ m $
0000002530: $ r $
0000004518: $ r_{\rm Schwarzschild} $
0000008656: $ v_{\rm escape} $

d3js visualization of steps and expressions in Schwarzschild radius for non-rotating black hole

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.

Actions: Edit Derivation

Generate Tex file or PDF file

Delete Derivation and all associated steps

This does not remove expressions, symbols, and operations.

timing of Neo4j queries:

0.007 seconds for pdg_app/to_review_derivation 910f1eb1-c3ff-485a-960d-eabdfe532535
0.006 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationc2189742-2d8c-4def-be47-37407b678838
0.013 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDc2189742-2d8c-4def-be47-37407b678838
0.017 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTc2189742-2d8c-4def-be47-37407b678838
0.017 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDc2189742-2d8c-4def-be47-37407b678838
0.018 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTc2189742-2d8c-4def-be47-37407b678838
0.023 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepc2189742-2d8c-4def-be47-37407b678838
0.013 seconds for pdg_app/ 910f1eb1-c3ff-485a-960d-eabdfe532535

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.	5404822208 \(v_{\rm escape} = \sqrt{2 G \frac{m}{r}}\)	3663007361 \(2\)	8946383937 \(v_{\rm escape}^2 = 2 G \frac{m}{r}\)	valid
2	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	8946383937 \(v_{\rm escape}^2 = 2 G \frac{m}{r}\)	9933742680 \(r_{\rm Schwarzschild}\) 2660368546 \(r\) 1238593037 \(c\) 8362338572 \(v_{\rm escape}\)	4275004561 \(c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}\)	LHS diff is -pdg00045672 + pdg00086562 RHS diff is 2pdg0005156pdg0006277(-pdg0002530 + pdg0004518)/(pdg0002530pdg0004518)
3	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	4275004561 \(c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}\)	7194432406 \(r_{\rm Schwarzschild}\)	2883079365 \(r_{\rm Schwarzschild} c^2 = 2 G m\)	valid
4	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	2883079365 \(r_{\rm Schwarzschild} c^2 = 2 G m\)	7263534144 \(c^2\)	6800170830 \(r_{\rm Schwarzschild} = \frac{2 G m}{c^2}\)	valid