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 $t′=tgamma$

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

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

1 Asked on February 3, 2021 by rixte-de-wolff

1 Asked on February 3, 2021

1pi effective action effective field theory quantum field theory renormalization terminology

1 Asked on February 3, 2021 by stephen-blake

adiabatic condensed matter perturbation theory quantum mechanics statistical mechanics

5 Asked on February 3, 2021 by qyo

5 Asked on February 3, 2021 by ghost-repeater

centrifugal force centripetal force forces homework and exercises rotation

0 Asked on February 3, 2021

double slit experiment interferometry measurement problem quantum mechanics

0 Asked on February 3, 2021

differential geometry integration kinetic theory plasma physics statistical mechanics

2 Asked on February 3, 2021

2 Asked on February 3, 2021 by ethant

field theory measurement problem observables quantum field theory

3 Asked on February 3, 2021 by trytryagain

0 Asked on February 3, 2021

3 Asked on February 3, 2021

2 Asked on February 3, 2021 by yash-sinhal

1 Asked on February 3, 2021 by incrediblesimon

1 Asked on February 3, 2021 by thetada

2 Asked on February 3, 2021

0 Asked on February 3, 2021

fourier transform hilbert space operators quantum field theory quantum mechanics

1 Asked on February 3, 2021 by zegkljan

1 Asked on February 3, 2021

4 Asked on February 3, 2021 by charlotte-gleeson

Get help from others!

Recent Questions

- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?
- Does Google Analytics track 404 page responses as valid page views?

Recent Answers

- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?

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