MESSAGES:

Return to navigation page or list derivations

Review Lorentz transformation

step inference rule input feed output step validity (as per SymPy)
1
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 4662369843
    \(x'=\gamma (x - v t)\)
 no validation is available for declarations
2
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 2983053062
    \(x=\gamma (x' + v t')\)
 no validation is available for declarations
3
  • 111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 4662369843
    \(x'=\gamma (x - v t)\)
  1. 2983053062
    \(x=\gamma (x' + v t')\)
  1. 3426941928
    \(x=\gamma ( \gamma (x - v t) + v t' )\)
 LHS diff is 0 RHS diff is pdg0001790**2*(pdg0001464 - pdg0004037)
4
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 3426941928
    \(x=\gamma ( \gamma (x - v t) + v t' )\)
  1. 2096918413
    \(x=\gamma ( \gamma x - \gamma v t + v t' )\)
 LHS diff is 0 RHS diff is pdg0001790*(pdg0001357*pdg0004989 - pdg0001790*(pdg0001357*pdg0001467 - pdg0004037)) - pdg0001790(-pdg0001357*pdg0001467*pdg0001790 + pdg0001357*pdg0004989 + pdg0001790*pdg0004037)
5
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 2096918413
    \(x=\gamma ( \gamma x - \gamma v t + v t' )\)
  1. 7741202861
    \(x=\gamma^2 x - \gamma^2 v t + \gamma v t'\)
 LHS diff is 0 RHS diff is pdg0001357*pdg0001467*pdg0001790**2 - pdg0001357*pdg0001790*pdg0004989 - pdg0001790**2*pdg0004037 + pdg0001790(-pdg0001357*pdg0001467*pdg0001790 + pdg0001357*pdg0004989 + pdg0001790*pdg0004037)
6
  • 111282: subtract X from both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 7741202861
    \(x=\gamma^2 x - \gamma^2 v t + \gamma v t'\)
  1. 7337056406
    \(\gamma^2 x\)
  1. 4139999399
    \(x - \gamma^2 x=- \gamma^2 v t + \gamma v t'\)
 valid
7
  • 111613: factor out X from LHS
  • number of inputs: 1; feeds: 1; outputs: 1
  • Factor $#1$ from the LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 4139999399
    \(x - \gamma^2 x=- \gamma^2 v t + \gamma v t'\)
  1. 3495403335
    \(x\)
  1. 9031609275
    \(x (1 - \gamma^2 )=- \gamma^2 v t + \gamma v t'\)
 invalid syntax (<string>, line 0)
8
  • 111530: add X to both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 9031609275
    \(x (1 - \gamma^2 )=- \gamma^2 v t + \gamma v t'\)
  1. 8014566709
    \(\gamma^2 v t\)
  1. 9409776983
    \(x (1 - \gamma^2 ) + \gamma^2 v t=\gamma v t'\)
 invalid syntax (<string>, line 0)
9
  • 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}.
  1. 9409776983
    \(x (1 - \gamma^2 ) + \gamma^2 v t=\gamma v t'\)
  1. 2226340358
    \(\gamma v\)
  1. 1974334644
    \(\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v}=t'\)
 invalid syntax (<string>, line 0)
10
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 1974334644
    \(\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v}=t'\)
  1. 8730201316
    \(\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t=t'\)
 invalid syntax (<string>, line 0)
11
  • 111268: swap LHS with RHS
  • number of inputs: 1; feeds: 0; outputs: 1
  • Swap LHS of Eq.~\\ref{eq:#1} with RHS; yields Eq.~\\ref{eq:#2}.
  1. 8730201316
    \(\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t=t'\)
  1. 5148266645
    \(t'=\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t\)
 invalid syntax (<string>, line 0)
12
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 4287102261
    \(x^2 + y^2 + z^2=c^2 t^2\)
 no validation is available for declarations
13
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 1201689765
    \(x'^2 + y'^2 + z'^2=c^2 t'^2\)
 no validation is available for declarations
14
  • 111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an assumption.
  1. 7057864873
    \(y'=y\)
 no validation is available for declarations
15
  • 111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an assumption.
  1. 8515803375
    \(z'=z\)
 no validation is available for declarations
16
  • 111797: substitute LHS of four expressions into expression
  • number of inputs: 5; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} and LHS of Eq.~\\ref{eq:#4} into Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}.
  1. 5148266645
    \(t'=\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t\)
  1. 8515803375
    \(z'=z\)
  1. 7057864873
    \(y'=y\)
  1. 1201689765
    \(x'^2 + y'^2 + z'^2=c^2 t'^2\)
  1. 4662369843
    \(x'=\gamma (x - v t)\)
  1. 9805063945
    \(\gamma^2 (x - v t)^2 + y^2 + z^2=c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2\)
 recognized infrule but not yet supported
17
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 9805063945
    \(\gamma^2 (x - v t)^2 + y^2 + z^2=c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2\)
  1. 1935543849
    \(\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2=c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2\)
 LHS diff is 0 RHS diff is pdg0004567**2*(-pdg0001357**2*pdg0001467**2*pdg0001790**4 + 2*pdg0001357**2*pdg0001467*pdg0001790*pdg0004037*(pdg0001790**2 - 1) + pdg0001357**2*pdg0004037**2*(pdg0001790**2 - 1) + (pdg0001357*pdg0001467*pdg0001790**2 - pdg0004037*(pdg0001790**2 - 1))**2)/(pdg0001357**2*pdg0001790**2)
18
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 1935543849
    \(\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2=c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2\)
  1. 1586866563
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right)=t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
 LHS diff is pdg0001357**2*pdg0001467**2*pdg0001790**2 - 2*pdg0001467*pdg0001790**2*pdg0004037*pdg0004567**2/pdg0001357 + 2*pdg0001467*pdg0004037*pdg0004567**2/pdg0001357 + pdg0004037**2*pdg0004567**2*(pdg0001790**2 - 1)**2/(pdg0001357**2*pdg0001790**2) RHS diff is (pdg0001357**2*pdg0001467**2*pdg0001790**4 - 2*pdg0001467*pdg0001790*pdg0004037*pdg0004567**2*(pdg0001790**2 - 1) - pdg0004037**2*pdg0004567**2*(pdg0001790**2 - 1))/pdg0001790**2
19
  • 111768: expr 1 is equivalent to expr 2 under the condition
  • number of inputs: 2; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is equivalent to Eq.~\\ref{eq:#2} under the condition in Eq.~\\ref{eq:#3}.
  1. 1586866563
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right)=t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
  1. 4287102261
    \(x^2 + y^2 + z^2=c^2 t^2\)
  1. 3182633789
    \(\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}=1\)
 recognized infrule but not yet supported
20
  • 111768: expr 1 is equivalent to expr 2 under the condition
  • number of inputs: 2; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is equivalent to Eq.~\\ref{eq:#2} under the condition in Eq.~\\ref{eq:#3}.
  1. 1586866563
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right)=t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
  1. 4287102261
    \(x^2 + y^2 + z^2=c^2 t^2\)
  1. 1916173354
    \(-\gamma^2 v^2 + c^2 \gamma^2=c^2\)
 recognized infrule but not yet supported
21
  • 111768: expr 1 is equivalent to expr 2 under the condition
  • number of inputs: 2; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is equivalent to Eq.~\\ref{eq:#2} under the condition in Eq.~\\ref{eq:#3}.
  1. 4287102261
    \(x^2 + y^2 + z^2=c^2 t^2\)
  1. 1586866563
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right)=t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
  1. 2076171250
    \(-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v}=0\)
 recognized infrule but not yet supported
22
  • 111282: subtract X from both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 3182633789
    \(\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}=1\)
  1. 5284610349
    \(\gamma^2\)
  1. 2417941373
    \(- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}=1 - \gamma^2\)
 valid
23
  • 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}.
  1. 2417941373
    \(- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}=1 - \gamma^2\)
  1. 5787469164
    \(1 - \gamma^2\)
  1. 1639827492
    \(- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2}=1\)
 valid
24
  • 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}.
  1. 1639827492
    \(- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2}=1\)
  1. 5669500954
    \(v^2 \gamma^2\)
  1. 5763749235
    \(-c^2 + c^2 \gamma^2=v^2 \gamma^2\)
 valid
25
  • 111530: add X to both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 5763749235
    \(-c^2 + c^2 \gamma^2=v^2 \gamma^2\)
  1. 6408214498
    \(c^2\)
  1. 2999795755
    \(c^2 \gamma^2=v^2 \gamma^2 + c^2\)
 valid
26
  • 111282: subtract X from both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 2999795755
    \(c^2 \gamma^2=v^2 \gamma^2 + c^2\)
  1. 3412946408
    \(v^2 \gamma^2\)
  1. 2542420160
    \(c^2 \gamma^2 - v^2 \gamma^2=c^2\)
 valid
27
  • 111613: factor out X from LHS
  • number of inputs: 1; feeds: 1; outputs: 1
  • Factor $#1$ from the LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 2542420160
    \(c^2 \gamma^2 - v^2 \gamma^2=c^2\)
  1. 7743841045
    \(\gamma^2\)
  1. 7513513483
    \(\gamma^2 (c^2 - v^2)=c^2\)
 valid
28
  • 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}.
  1. 7513513483
    \(\gamma^2 (c^2 - v^2)=c^2\)
  1. 8571466509
    \(c^2 - \gamma^2\)
  1. 7906112355
    \(\gamma^2=\frac{c^2}{c^2 - \gamma^2}\)
 Algebraic error: LHS diff is pdg0001790**2*(pdg0001357**2 - pdg0001790**2)/(pdg0001790**2 - pdg0004567**2), RHS diff is 0
29
  • 111524: square root both sides
  • number of inputs: 1; feeds: 0; outputs: 2
  • Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}.
  1. 7906112355
    \(\gamma^2=\frac{c^2}{c^2 - \gamma^2}\)
  1. 1528310784
    \(\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)
  1. 8360117126
    \(\gamma=\frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}\)
 recognized infrule but not yet supported
30
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 1528310784
    \(\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)
 no validation is available for declarations


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
   xor   

Delete Derivation and all associated steps

This does not remove expressions, symbols, and operations.

timing of Neo4j queries: