Review Derivation: Physics Derivation Graph

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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	111483: 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 $v_{\rm escape}^2=2 G \frac{m}{r}$	recognized infrule but not yet supported
2	111984: 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 $v_{\rm escape}^2=2 G \frac{m}{r}$	1238593037 $c$ 2660368546 $r$ 8362338572 $v_{\rm escape}$ 9933742680 $r_{\rm Schwarzschild}$	4275004561 $c^2=2 G \frac{m}{r_{\rm Schwarzschild}}$	list index out of range
3	111182: 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	111975: 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

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timing of Neo4j queries:

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step	inference rule	input	feed	output	step validity (as per SymPy)
1	111483: 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 \(v_{\rm escape}^2=2 G \frac{m}{r}\)	recognized infrule but not yet supported
2	111984: 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 \(v_{\rm escape}^2=2 G \frac{m}{r}\)	1238593037 \(c\) 2660368546 \(r\) 8362338572 \(v_{\rm escape}\) 9933742680 \(r_{\rm Schwarzschild}\)	4275004561 \(c^2=2 G \frac{m}{r_{\rm Schwarzschild}}\)	list index out of range
3	111182: 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	111975: 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