AnswerBun.com

Lorentz transformation matrix for all 3 spatial axes

Physics Asked on January 4, 2022

The lorentz transformation matrix (for all 3 spatial axes, not just a single dimension boost) appears to be commonly defined as the following:
$$
begin{bmatrix}
gamma &-gamma v_x/c &-gamma v_y/c &-gamma v_z/c \
-gamma v_x/c&1+(gamma-1)dfrac{v_x^2} {v^2}& (gamma-1)dfrac{v_x v_y}{v^2}& (gamma-1)dfrac{v_x v_z}{v^2} \
-gamma v_y/c& (gamma-1)dfrac{v_y v_x}{v^2}&1+(gamma-1)dfrac{v_y^2} {v^2}& (gamma-1)dfrac{v_y v_z}{v^2} \
-gamma v_z/c& (gamma-1)dfrac{v_z v_x}{v^2}& (gamma-1)dfrac{v_z v_y}{v^2}&1+(gamma-1)dfrac{v_z^2} {v^2}
end{bmatrix}
$$

I tried to derive it myself by combining the matrices for the individual boost directions and making $v=|vec{v}|$and ended up at
$$
begin{bmatrix} ct’ \ x’ \ y’ \ z’ \ end{bmatrix} = begin{bmatrix} gamma & -beta_xgamma& -beta_ygamma & -beta_zgamma \ -frac{beta_y} {gamma_{v_x}} & frac{1}{gamma_{v_x}} & 0 & 0 \ -frac{beta_y}{gamma_{v_y}} & 0 & frac{1}{gamma_{v_y}} & 0\-frac{beta_z}{gamma_{v_z}} & 0 & 0 & frac{1}{gamma_{v_z}} \ end{bmatrix} begin{bmatrix} ct \ x \ y \ z \ end{bmatrix}
$$

Where $gamma = displaystylefrac{1}{sqrt{1-displaystylefrac{|vec{v}|^2}{c^2}}} $ and
$gamma_{v_x} = displaystylefrac{1}{sqrt{1-displaystylefrac{v_x^2}{c^2}}}$

2 questions. Where do the bottom right 9 terms come from in the common definition and why is the top $gamma$ and not $frac{1}{gamma}$ given that $l′=frac{l}{gamma}$ but $t′=tgamma$

One Answer

$$Delta x =x_f-x_i=gamma(Delta x'+vDelta t')=gamma(x'_f-x'_i+v(t'_f-t'_i))$$ $$ Delta t'=t'_f-t'_i=0$$ $$Delta t'=t'_f-t'_i=gamma(Delta t-frac{v(x_f-x_i)}{c^2})$$ $$Delta x=x_f-x_i=0$$

We deduce these facts: $Delta x=gamma Delta x'=l=gamma Delta l'$ and $Delta t'= gamma Delta t$ not $t'=gamma t$.The reason why is that we don't talk about a time of an event in spacetime. What we're interested in is time difference between two events.

Answered by Fatma Keskin on January 4, 2022

Add your own answers!

Related Questions

Interpreting recession velocity (red / blue shift)

1  Asked on February 3, 2021 by thetada

   

Ask a Question

Get help from others!

© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP