- #1

chaksome

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- TL;DR Summary
- A discussion about special relativity.

I want to know why an else solution can not get the right answer. And want to know the way to correct this solution.Supposed that a frame S'' is moving in the lab frame at ##\beta_x## in the x-direction, ##\beta_y## in the y-direction, now I want to find out the Lorentz transformation between these frames.

Applying the vector transformation in special relative, I've already worked out the solution like this(and I think it is the correct answer):$$\begin{cases}x''=\frac{\gamma \beta_x^2+\beta_y^2}{\beta_x^2+\beta_y^2}x+\frac{(\gamma-1)\beta_x\beta_y}{\beta_x^2+\beta_y^2}y-\gamma\beta_x ct\\y''=\frac{\gamma \beta_y^2+\beta_x^2}{\beta_x^2+\beta_y^2}y+\frac{(\gamma-1)\beta_x\beta_y}{\beta_x^2+\beta_y^2}x-\gamma\beta_y ct，(1)\\ct''=\gamma ct-\gamma \beta_x x-\gamma\beta_y y\end{cases}$$(and## \gamma=\frac{1}{\sqrt{1-(\beta_x^2+\beta_y^2)}}##)(if there is anything wrong, please point it out)Now I want to solve this problem in a different way (which is valid if the 3 frames are moving in one direction, but fail to work out the right answer in this scene)First, I make a Lorentz transformation between S and S'(which is moving in the frame S at##\beta_x##in the x-direction),and I get the equation$$\begin{cases}x'=\gamma_x(x-\beta_x ct)\\y'=y\\ct'=\gamma_x(ct-\beta_xx)\end{cases}$$Then,I make a Lorentz transformation between S' and S''(according to the velocity-addition formula,S'' is moving in S' at ##\beta_y'=\gamma_x\beta_y##in the y-direction),and I get the equation$$\begin{cases}x''=x'\\y''=\gamma_y'(y'-\beta_y' ct')\\t''=\gamma_y'(ct'-\beta_y'y')\end{cases}$$OK,now we solve these equation, and find that$$\begin{cases}x''=\gamma_xx+0y-\gamma_x\beta_xct\\y''=\frac{\gamma}{\gamma_x}y+\gamma\beta_x\beta_yx-\gamma\gamma_x\beta_yct, (2)\\ct''=\gamma ct-\gamma\beta_x x-\gamma\beta_y y\end{cases}$$We can see that (1) is obviously different from (2)I hope to find out the essential cause of this phenomenon(maybe it will refer to the general relativity):What causes the discrepancy between (1) and (2)?Why the ct'' stay consistence?Thanks a million!

Applying the vector transformation in special relative, I've already worked out the solution like this(and I think it is the correct answer):$$\begin{cases}x''=\frac{\gamma \beta_x^2+\beta_y^2}{\beta_x^2+\beta_y^2}x+\frac{(\gamma-1)\beta_x\beta_y}{\beta_x^2+\beta_y^2}y-\gamma\beta_x ct\\y''=\frac{\gamma \beta_y^2+\beta_x^2}{\beta_x^2+\beta_y^2}y+\frac{(\gamma-1)\beta_x\beta_y}{\beta_x^2+\beta_y^2}x-\gamma\beta_y ct，(1)\\ct''=\gamma ct-\gamma \beta_x x-\gamma\beta_y y\end{cases}$$(and## \gamma=\frac{1}{\sqrt{1-(\beta_x^2+\beta_y^2)}}##)(if there is anything wrong, please point it out)Now I want to solve this problem in a different way (which is valid if the 3 frames are moving in one direction, but fail to work out the right answer in this scene)First, I make a Lorentz transformation between S and S'(which is moving in the frame S at##\beta_x##in the x-direction),and I get the equation$$\begin{cases}x'=\gamma_x(x-\beta_x ct)\\y'=y\\ct'=\gamma_x(ct-\beta_xx)\end{cases}$$Then,I make a Lorentz transformation between S' and S''(according to the velocity-addition formula,S'' is moving in S' at ##\beta_y'=\gamma_x\beta_y##in the y-direction),and I get the equation$$\begin{cases}x''=x'\\y''=\gamma_y'(y'-\beta_y' ct')\\t''=\gamma_y'(ct'-\beta_y'y')\end{cases}$$OK,now we solve these equation, and find that$$\begin{cases}x''=\gamma_xx+0y-\gamma_x\beta_xct\\y''=\frac{\gamma}{\gamma_x}y+\gamma\beta_x\beta_yx-\gamma\gamma_x\beta_yct, (2)\\ct''=\gamma ct-\gamma\beta_x x-\gamma\beta_y y\end{cases}$$We can see that (1) is obviously different from (2)I hope to find out the essential cause of this phenomenon(maybe it will refer to the general relativity):What causes the discrepancy between (1) and (2)?Why the ct'' stay consistence?Thanks a million!

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