Chapter 4, Problem 4.1P

To determine

The derivation of Lorentz transformation Equations.

Answer to Problem 4.1P

The derived equations are.

$\begin{array}{l}{x}^{\prime }=\frac{x-ut}{\sqrt{1-\frac{u^2}{c^2}}}\\ y^{\prime }=y\\ z^{\prime }=z\\ t^{\prime }=\frac{t-\frac{ux}{c^2}}{\sqrt{1{-}^{u^2{c}^2}}}\end{array}$

Explanation of Solution

Assume an inertial frame with space and time coordinate as $S=\left(z,y,z,t\right)$ and another inertial frame as $S^{\prime }=\left(x^{\prime },y^{\prime },z^{\prime },t^{\prime }\right)$.

The coordinates of second inertial frame can be expressed as

    $\begin{array}{l}{x}^{\prime }=a_{11}\left(x-ut\right)\\ y^{\prime }=y\\ z^{\prime }=z\\ t^{\prime }=a_{41}x+a_{44}t\end{array}$

Write the expression for spherical wave front.

    $x^2+y^2+z^2={\left(ct\right)}^2$        (1)

Here, $ct$ is the radius and coordinates are $x$, $y$ and $z$.

Write the expression of wave front for second inertial frame.

    ${x^{\prime }}^2+{y^{\prime }}^2+{z^{\prime }}^2={\left(c{t}^{\prime }\right)}^2$        (2)

Conclusion:

Substitute $a_{11}\left(x-ut\right)$ for $x^{\prime }$, $y$ for $y^{\prime }$, $z$ for $z^{\prime }$ and $a_{41}x+a_{44}t$ for $t^{\prime }$ in Equation (2).

    $\begin{array}{l}{\left(a_{11}\left(x-ut\right)\right)}^2+y^2+z^2={\left(c\left(a_{41}x+a_{44}t\right)\right)}^2\\ x^2\left(a_{11}^2-a_{41}^2{c}^2\right)-2\left(a_{11}^{2}u+2a_{41}{a}_{44}{c}^2\right)xt+y^2+z^2=\left(-a_{11}^{2}u+a_{44}^2{c}^2\right)t^2\end{array}$        (3)

Compare Equation (3) with (1).

$\begin{array}{l}{a}_{11}^2-a_{41}^2{c}^2=1\\ a_{11}^{2}u+2a_{41}{a}_{44}{c}^2=0\\ -a_{11}^{2}u+a_{44}^2{c}^2=c^2\end{array}$

Solve the above equations.

$\begin{array}{l}{x}^{\prime }=\frac{x-ut}{\sqrt{1-\frac{u^2}{c^2}}}\\ y^{\prime }=y\\ z^{\prime }=z\\ t^{\prime }=\frac{t-\frac{ux}{c^2}}{\sqrt{1{-}^{u^2{c}^2}}}\end{array}$

Thus, derived Equations are

$\begin{array}{l}{x}^{\prime }=\frac{x-ut}{\sqrt{1-\frac{u^2}{c^2}}}\\ y^{\prime }=y\\ z^{\prime }=z\\ t^{\prime }=\frac{t-\frac{ux}{c^2}}{\sqrt{1{-}^{u^2{c}^2}}}\end{array}$