Chapter 12: Q12P (page 523)

Solve Eqs. 12.18 for $x, y, z, t$ in terms of $x, y, z, t$ and check that you recover Eqs. 12.19.

Short Answer

Expert verified

The coordinates of the S frame in terms of coordinates in $S$ are found as:

$x = γ (\begin{matrix} x + y t \end{matrix}) y = y z = z t = γ (\begin{matrix} t + \frac{v}{c^{2}} x \end{matrix})$

Step by step solution

Expression for the Lorentz transformation equations:

Write the expression for the Lorentz transformation equations of coordinates $x, y, z$ and tin $S$ frame.

First equation:

$x = Y (x - vt)$ …… (1)

Second equation:

$y = y$

Third equation:

$z = z$

Fourth equation:

localid="1654594577273" $t = Y (1 - \frac{VX}{c^{2}})$ …… (2)

Determine the coordinate of x in terms of x:

Rearrange the equation (2) in terms of t.

$t = \frac{t}{γ} + \frac{vx}{c^{2}}$ ……. (3)

Substitute the value of equation (3) in equation (1).

$x = γ (\begin{matrix} x - v (\begin{matrix} \frac{t}{γ} + \frac{v x}{c^{2}} \end{matrix}) \end{matrix}) x = γ \times (\begin{matrix} 1 - \frac{v^{2}}{c^{2}} \end{matrix}) - v t$

Rearrange the above expression in terms of x.

$x = γ (\begin{matrix} x + v t \end{matrix})$ …… (4)

Determine the coordinate of t in terms of t :

Substitute the value of equation (4) in equation (3).

$t = γ (\begin{matrix} 1 - \frac{v (\begin{matrix} γ (\begin{matrix} x + v t \end{matrix}) \end{matrix})}{c^{2}} \end{matrix}) t = γ [t - \frac{v}{c^{2}} (\begin{matrix} \frac{x}{γ} + v t \end{matrix})]$

Rearrange the above expression in terms of t.

$t = γ (\begin{matrix} t - \frac{v}{c^{2}} x \end{matrix})$

Therefore, the coordinates of the S frame in terms of coordinates in are found as:

$x = γ (\begin{matrix} x + v t \end{matrix}) y = y z = z t = γ (\begin{matrix} t + \frac{v}{c^{2}} x \end{matrix})$

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Speed of $\to$ Relative to $↓$	Ground	Police	Outlaws	Bullet	Do they escape?
Ground	0	role="math" localid="1654061605668" $\frac{1}{2} c$	$\frac{3}{4} c$
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Solve Eqs. 12.18 for $x, y, z, t$ in terms of $x, y, z, t$ and check that you recover Eqs. 12.19.

Short Answer

Step by step solution

Expression for the Lorentz transformation equations:

Determine the coordinate of x in terms of x:

Determine the coordinate of t in terms of t :

One App. One Place for Learning.

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Solve Eqs. 12.18 forx,y,z,tin terms ofx,y,z,t and check that you recover Eqs. 12.19.

Short Answer

Step by step solution

Expression for the Lorentz transformation equations:

Determine the coordinate of x in terms of x:

Determine the coordinate of t in terms of t :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Famous Physicists

Electricity and Magnetism

Electromagnetism

Quantum Physics

Oscillations

Study anywhere. Anytime. Across all devices.

Solve Eqs. 12.18 for $x, y, z, t$ in terms of $x, y, z, t$ and check that you recover Eqs. 12.19.