## review derivation: Lorentz transformation

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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

symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
$$x$$
['real'] dimensionless 140
4306 variable z'
$$z'$$
['real']
• length: 1
position in moving reference frame
2
1790 variable \gamma
$$\gamma$$
['real'] dimensionless Lorentz factor
35
4037 variable x
$$x$$
['real']
• length: 1
position
53
1357 variable v
$$v$$
['real']
• length: 1
• time: -1
velocity
83
1467 variable t
$$t$$
['real']
• time: 1
time
121
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
6728 variable z
$$z$$
['real']
• length: 1
position
5
4989 variable t'
$$t'$$
real
• time: 1
time in moving reference frame
• str_note
7
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
MESSAGE:
• local variable 'all_df' referenced before assignment