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review derivation: Lorentz transformation

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Notes for this derivation:
source: \cite{1999_Tipler_Llewellyn}, page 21; see also https://en.wikipedia.org/wiki/Lorentz_transformation and https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
20 expr 1 is equivalent to expr 2 under the condition
  1. 4287102261; locally 6319661:
    \(x^2 + y^2 + z^2 = c^2 t^2\)
    \(pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}\)
  2. 1586866563; locally 4202425:
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
    \(- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)\)
  1. 1916173354; locally 3640931:
    \(-\gamma^2 v^2 + c^2 \gamma^2 = c^2\)
    \(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{4567}^{2}\)
no check performed 4287102261:
1586866563:
1916173354:
4287102261:
1586866563:
1916173354:
based on the comparison of the t^2 terms
1 declare initial expr
  1. 4662369843; locally 5427510:
    \(x' = \gamma (x - v t)\)
    \(pdg_{5456} = pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{1464}\right)\)
no validation is available for declarations 4662369843:
4662369843:
equation 1-13 on page 21 in \cite{1999_Tipler_Llewellyn}
15 declare assumption
  1. 8515803375; locally 7666907:
    \(z' = z\)
    \(pdg_{4306} = pdg_{6728}\)
no validation is available for declarations 8515803375:
8515803375:
11 swap LHS with RHS
  1. 8730201316; locally 7546640:
    \(\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'\)
    \(pdg_{1790}\)
  1. 5148266645; locally 1693888:
    \(t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t\)
    \(pdg_{1790}\)
Nothing to split 8730201316:
5148266645:
8730201316:
5148266645:
12 declare initial expr
  1. 4287102261; locally 6319661:
    \(x^2 + y^2 + z^2 = c^2 t^2\)
    \(pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}\)
no validation is available for declarations 4287102261:
4287102261:
25 add X to both sides
  1. 5763749235; locally 8195408:
    \(-c^2 + c^2 \gamma^2 = v^2 \gamma^2\)
    \(pdg_{1790}^{2} pdg_{4567}^{2} - pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2}\)
  1. 6408214498:
    \(c^2\)
    \(pdg_{4567}^{2}\)
  1. 2999795755; locally 6913493:
    \(c^2 \gamma^2 = v^2 \gamma^2 + c^2\)
    \(pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{4567}^{2}\)
valid 5763749235:
2999795755:
5763749235:
2999795755:
6 subtract X from both sides
  1. 7741202861; locally 6463955:
    \(x = \gamma^2 x - \gamma^2 v t + \gamma v t'\)
    \(pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989} + pdg_{1790}^{2} pdg_{4037}\)
  1. 7337056406:
    \(\gamma^2 x\)
    \(pdg_{1790}^{2} pdg_{4037}\)
  1. 4139999399; locally 8494407:
    \(x - \gamma^2 x = - \gamma^2 v t + \gamma v t'\)
    \(- pdg_{1790}^{2} pdg_{4037} + pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989}\)
valid 7741202861:
4139999399:
7741202861:
4139999399:
23 divide both sides by
  1. 2417941373; locally 7403799:
    \(- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2\)
    \(- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1 - pdg_{1790}^{2}\)
  1. 5787469164:
    \(1 - \gamma^2\)
    \(1 - pdg_{1790}^{2}\)
  1. 1639827492; locally 4052253:
    \(- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1\)
    \(- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1\)
valid 2417941373:
1639827492:
2417941373:
1639827492:
24 multiply both sides by
  1. 1639827492; locally 4052253:
    \(- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1\)
    \(- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1\)
  1. 5669500954:
    \(v^2 \gamma^2\)
    \(pdg_{1357}^{2} pdg_{1790}^{2}\)
  1. 5763749235; locally 8195408:
    \(-c^2 + c^2 \gamma^2 = v^2 \gamma^2\)
    \(pdg_{1790}^{2} pdg_{4567}^{2} - pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2}\)
valid 1639827492:
5763749235:
1639827492:
5763749235:
10 simplify
  1. 1974334644; locally 5995189:
    \(\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'\)
    \(pdg_{1467} pdg_{1790} + \frac{\operatorname{pdg}_{4037}{\left(1 - pdg_{1790}^{2} \right)}}{pdg_{1357} pdg_{1790}} = pdg_{4989}\)
  1. 8730201316; locally 7546640:
    \(\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'\)
    \(pdg_{1790}\)
Nothing to split 1974334644:
8730201316:
1974334644:
8730201316:
4 simplify
  1. 3426941928; locally 4471422:
    \(x = \gamma ( \gamma (x - v t) + v t' )\)
    \(pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)\right)\)
  1. 2096918413; locally 7169020:
    \(x = \gamma ( \gamma x - \gamma v t + v t' )\)
    \(pdg_{4037} = \operatorname{pdg}_{1790}{\left(- pdg_{1357} pdg_{1467} pdg_{1790} + pdg_{1357} pdg_{4989} + pdg_{1790} pdg_{4037} \right)}\)
LHS diff is 0 RHS diff is pdg1790*(pdg1357*pdg4989 - pdg1790*(pdg1357*pdg1467 - pdg4037)) - pdg1790(-pdg1357*pdg1467*pdg1790 + pdg1357*pdg4989 + pdg1790*pdg4037) 3426941928:
2096918413:
3426941928:
2096918413:
19 expr 1 is equivalent to expr 2 under the condition
  1. 4287102261; locally 6319661:
    \(x^2 + y^2 + z^2 = c^2 t^2\)
    \(pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}\)
  2. 1586866563; locally 4202425:
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
    \(- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)\)
  1. 3182633789; locally 2562123:
    \(\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1\)
    \(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1\)
no check performed 4287102261:
1586866563:
3182633789:
4287102261:
1586866563:
3182633789:
based on the comparison of the x^2 terms
22 subtract X from both sides
  1. 3182633789; locally 2562123:
    \(\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1\)
    \(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1\)
  1. 5284610349:
    \(\gamma^2\)
    \(pdg_{1790}^{2}\)
  1. 2417941373; locally 7403799:
    \(- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2\)
    \(- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1 - pdg_{1790}^{2}\)
valid 3182633789:
2417941373:
3182633789:
2417941373:
solve for \gamma
29 square root both sides
  1. 7906112355; locally 7595841:
    \(\gamma^2 = \frac{c^2}{c^2 - \gamma^2}\)
    \(pdg_{1790}^{2} = \frac{pdg_{4567}^{2}}{- pdg_{1790}^{2} + pdg_{4567}^{2}}\)
  1. 1528310784; locally 3040283:
    \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)
    \(pdg_{1790} = \frac{1}{\sqrt{- \frac{pdg_{1357}^{2}}{pdg_{4567}^{2}} + 1}}\)
  2. 8360117126; locally 6010461:
    \(\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}\)
    \(pdg_{1790} = - \frac{1}{\sqrt{- \frac{pdg_{1357}^{2}}{pdg_{4567}^{2}} + 1}}\)
no check performed 7906112355:
1528310784:
8360117126:
7906112355:
1528310784:
8360117126:
17 simplify
  1. 9805063945; locally 4326342:
    \(\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2\)
    \(pdg_{1790}^{2} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1790}^{2} pdg_{4567}^{2} \left(pdg_{1467} + \frac{pdg_{4037} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357} pdg_{1790}^{2}}\right)^{2}\)
  1. 1935543849; locally 6066191:
    \(\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2\)
    \(pdg_{1357}^{2} pdg_{1467}^{2} pdg_{1790}^{2} - 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{1790}^{2} pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{1790}^{2} pdg_{4567}^{2} + \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}} + \frac{pdg_{4037}^{2} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}^{2}}\)
LHS diff is 0 RHS diff is pdg4567**2*(-pdg1357**2*pdg1467**2*pdg1790**4 + 2*pdg1357**2*pdg1467*pdg1790*pdg4037*(pdg1790**2 - 1) + pdg1357**2*pdg4037**2*(pdg1790**2 - 1) + (pdg1357*pdg1467*pdg1790**2 - pdg4037*(pdg1790**2 - 1))**2)/(pdg1357**2*pdg1790**2) 9805063945:
1935543849:
9805063945:
1935543849:
expanded the squared terms
30 declare final expr
  1. 1528310784; locally 3040283:
    \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)
    \(pdg_{1790} = \frac{1}{\sqrt{- \frac{pdg_{1357}^{2}}{pdg_{4567}^{2}} + 1}}\)
no validation is available for declarations 1528310784:
1528310784:
Lorentz factor definition
13 declare initial expr
  1. 1201689765; locally 5649086:
    \(x'^2 + y'^2 + z'^2 = c^2 t'^2\)
    \(pdg_{1888}^{2} + pdg_{4306}^{2} + pdg_{5456}^{2} = pdg_{4567}^{2} pdg_{4989}^{2}\)
no validation is available for declarations 1201689765:
1201689765:
5 simplify
  1. 2096918413; locally 7169020:
    \(x = \gamma ( \gamma x - \gamma v t + v t' )\)
    \(pdg_{4037} = \operatorname{pdg}_{1790}{\left(- pdg_{1357} pdg_{1467} pdg_{1790} + pdg_{1357} pdg_{4989} + pdg_{1790} pdg_{4037} \right)}\)
  1. 7741202861; locally 6463955:
    \(x = \gamma^2 x - \gamma^2 v t + \gamma v t'\)
    \(pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989} + pdg_{1790}^{2} pdg_{4037}\)
LHS diff is 0 RHS diff is pdg1357*pdg1467*pdg1790**2 - pdg1357*pdg1790*pdg4989 - pdg1790**2*pdg4037 + pdg1790(-pdg1357*pdg1467*pdg1790 + pdg1357*pdg4989 + pdg1790*pdg4037) 2096918413:
7741202861:
2096918413:
7741202861:
14 declare assumption
  1. 7057864873; locally 6316097:
    \(y' = y\)
    \(pdg_{1888} = pdg_{5647}\)
no validation is available for declarations 7057864873:
7057864873:
3 substitute LHS of expr 1 into expr 2
  1. 4662369843; locally 5427510:
    \(x' = \gamma (x - v t)\)
    \(pdg_{5456} = pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{1464}\right)\)
  2. 2983053062; locally 2283140:
    \(x = \gamma (x' + v t')\)
    \(pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{5456}\right)\)
  1. 3426941928; locally 4471422:
    \(x = \gamma ( \gamma (x - v t) + v t' )\)
    \(pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)\right)\)
LHS diff is 0 RHS diff is pdg1790**2*(pdg1464 - pdg4037) 4662369843:
2983053062: dimensions are consistent
3426941928:
4662369843:
2983053062: N/A
3426941928:
solve output expr for t'
9 divide both sides by
  1. 9409776983; locally 6047713:
    \(x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'\)
    \(pdg_{1790}\)
  1. 2226340358:
    \(\gamma v\)
    \(pdg_{1357} pdg_{1790}\)
  1. 1974334644; locally 5995189:
    \(\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'\)
    \(pdg_{1467} pdg_{1790} + \frac{\operatorname{pdg}_{4037}{\left(1 - pdg_{1790}^{2} \right)}}{pdg_{1357} pdg_{1790}} = pdg_{4989}\)
Nothing to split 9409776983:
1974334644:
9409776983:
1974334644:
18 simplify
  1. 1935543849; locally 6066191:
    \(\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2\)
    \(pdg_{1357}^{2} pdg_{1467}^{2} pdg_{1790}^{2} - 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{1790}^{2} pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{1790}^{2} pdg_{4567}^{2} + \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}} + \frac{pdg_{4037}^{2} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}^{2}}\)
  1. 1586866563; locally 4202425:
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
    \(- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)\)
LHS diff is pdg1357**2*pdg1467**2*pdg1790**2 - 2*pdg1467*pdg1790**2*pdg4037*pdg4567**2/pdg1357 + 2*pdg1467*pdg4037*pdg4567**2/pdg1357 + pdg4037**2*pdg4567**2*(pdg1790**2 - 1)**2/(pdg1357**2*pdg1790**2) RHS diff is (pdg1357**2*pdg1467**2*pdg1790**4 - 2*pdg1467*pdg1790*pdg4037*pdg4567**2*(pdg1790**2 - 1) - pdg4037**2*pdg4567**2*(pdg1790**2 - 1))/pdg1790**2 1935543849:
1586866563:
1935543849:
1586866563:
grouped by terms for x^2, xt, and t^2
27 factor out X from LHS
  1. 2542420160; locally 5207615:
    \(c^2 \gamma^2 - v^2 \gamma^2 = c^2\)
    \(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{4567}^{2}\)
  1. 7743841045:
    \(\gamma^2\)
    \(pdg_{1790}^{2}\)
  1. 7513513483; locally 8842089:
    \(\gamma^2 (c^2 - v^2) = c^2\)
    \(pdg_{1790}^{2} \left(- pdg_{1357}^{2} + pdg_{4567}^{2}\right) = pdg_{4567}^{2}\)
valid 2542420160:
7513513483:
2542420160:
7513513483:
21 expr 1 is equivalent to expr 2 under the condition
  1. 4287102261; locally 6319661:
    \(x^2 + y^2 + z^2 = c^2 t^2\)
    \(pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}\)
  2. 1586866563; locally 4202425:
    \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
    \(- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)\)
  1. 2076171250; locally 6685577:
    \(-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0\)
    \(pdg_{1790}\)
Nothing to split 4287102261:
1586866563:
2076171250:
4287102261:
1586866563:
2076171250:
based on the comparison of the (x t) terms
16 substitute LHS of four expressions into expr
  1. 8515803375; locally 7666907:
    \(z' = z\)
    \(pdg_{4306} = pdg_{6728}\)
  2. 7057864873; locally 6316097:
    \(y' = y\)
    \(pdg_{1888} = pdg_{5647}\)
  3. 5148266645; locally 1693888:
    \(t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t\)
    \(pdg_{1790}\)
  4. 4662369843; locally 5427510:
    \(x' = \gamma (x - v t)\)
    \(pdg_{5456} = pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{1464}\right)\)
  5. 1201689765; locally 5649086:
    \(x'^2 + y'^2 + z'^2 = c^2 t'^2\)
    \(pdg_{1888}^{2} + pdg_{4306}^{2} + pdg_{5456}^{2} = pdg_{4567}^{2} pdg_{4989}^{2}\)
  1. 9805063945; locally 4326342:
    \(\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2\)
    \(pdg_{1790}^{2} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1790}^{2} pdg_{4567}^{2} \left(pdg_{1467} + \frac{pdg_{4037} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357} pdg_{1790}^{2}}\right)^{2}\)
Nothing to split 8515803375:
7057864873:
5148266645:
4662369843:
1201689765:
9805063945:
8515803375:
7057864873:
5148266645:
4662369843:
1201689765:
9805063945:
8 add X to both sides
  1. 9031609275; locally 3992172:
    \(x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'\)
    \(pdg_{1790}\)
  1. 8014566709:
    \(\gamma^2 v t\)
    \(pdg_{1357} pdg_{1467} pdg_{1790}^{2}\)
  1. 9409776983; locally 6047713:
    \(x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'\)
    \(pdg_{1790}\)
Nothing to split 9031609275:
9409776983:
9031609275:
9409776983:
26 subtract X from both sides
  1. 2999795755; locally 6913493:
    \(c^2 \gamma^2 = v^2 \gamma^2 + c^2\)
    \(pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{4567}^{2}\)
  1. 3412946408:
    \(v^2 \gamma^2\)
    \(pdg_{1357}^{2} pdg_{1790}^{2}\)
  1. 2542420160; locally 5207615:
    \(c^2 \gamma^2 - v^2 \gamma^2 = c^2\)
    \(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{4567}^{2}\)
valid 2999795755:
2542420160:
2999795755:
2542420160:
28 divide both sides by
  1. 7513513483; locally 8842089:
    \(\gamma^2 (c^2 - v^2) = c^2\)
    \(pdg_{1790}^{2} \left(- pdg_{1357}^{2} + pdg_{4567}^{2}\right) = pdg_{4567}^{2}\)
  1. 8571466509:
    \(c^2 - \gamma^2\)
    \(- pdg_{1790}^{2} + pdg_{4567}^{2}\)
  1. 7906112355; locally 7595841:
    \(\gamma^2 = \frac{c^2}{c^2 - \gamma^2}\)
    \(pdg_{1790}^{2} = \frac{pdg_{4567}^{2}}{- pdg_{1790}^{2} + pdg_{4567}^{2}}\)
LHS diff is pdg1790**2*(pdg1357**2 - pdg1790**2)/(pdg1790**2 - pdg4567**2) RHS diff is 0 7513513483:
7906112355:
7513513483:
7906112355:
7 factor out X from LHS
  1. 4139999399; locally 8494407:
    \(x - \gamma^2 x = - \gamma^2 v t + \gamma v t'\)
    \(- pdg_{1790}^{2} pdg_{4037} + pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989}\)
  1. 3495403335:
    \(x\)
    \(pdg_{1464}\)
  1. 9031609275; locally 3992172:
    \(x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'\)
    \(pdg_{1790}\)
Nothing to split 4139999399:
9031609275:
4139999399:
9031609275:
2 declare initial expr
  1. 2983053062; locally 2283140:
    \(x = \gamma (x' + v t')\)
    \(pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{5456}\right)\)
no validation is available for declarations 2983053062: dimensions are consistent
 2983053062: N/A
 equation 1-14 on page 21 in \cite{1999_Tipler_Llewellyn}
Physics Derivation Graph: Steps for Lorentz transformation

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
4306 variable z'
\(z'\)		 ['real']
  • length: 1
 position in moving reference frame 2
6728 variable z
\(z\)		 ['real']
  • length: 1
 position 5
1467 variable t
\(t\)		 ['real']
  • time: 1
 time 121
4989 variable t'
\(t'\)		 real
  • time: 1
 time in moving reference frame
  • str_note
7
1464 variable x
\(x\)		 ['real'] dimensionless 140
1790 variable \gamma
\(\gamma\)		 ['real'] dimensionless Lorentz factor 35
4567 constant c
\(c\)		 ['real']
  • length: 1
  • time: -1
 speed of light in vacuum 299792458   meters/second
 32
5647 variable y
\(y\)		 ['real']
  • length: 1
 position 14
1888 variable y'
\(y'\)		 ['real']
  • length: 1
 position in moving reference frame 2
5456 variable x'
\(x'\)		 ['real']
  • length: 1
 position in moving reference frame 3
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
MESSAGE: