Lorentz transformations Questions
1. Prove $x^2 + y^2 + z^2 = c^2t^2$ is invariant under Lorentz transformation.
Suppose we transform from $S$ frame to $S'$ frame using Lorentz transformation. So we would replace $x,y,z,t$ with $x',y',z',t'$ using the following relation:-
$$
x = γ(x'+vt') \\
y = y' \\
z = z' \\
t = γ(t' + \frac{vx'}{c^2})
$$
where $v$ is the velocity at which $S'$ frame is moving with respect to the $S$ frame along the $X-X'$ frame. $γ = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is relativistic factor. $\\$
so,
$$
= [γ(x'+vt')]^2 + (y')^2 + (z')^2 - c^2[γ^2(t' + \frac{vx'}{c^2})^2] \\
= γ^2((x')^2+(vt')^2+2x'vt') + (y')^2 + (z')^2 - c^2γ^2[t'+\frac{vx'}{c^2}]^2 \\
= x'^2+y'^2+z'^2-c^2t'^2
$$
Hence we saw that upon transformation from $S$ to $S'$, one could see there is no change in the nature of the equation $x^2 + y^2 + z^2 - c^2t^2$. Hence the given quantity is invariant under the Lorentz transformation.
2. A muon is travelling through the laboratory at three-fifths the speed
of light. How long does it last?
The lorentz factor is given by \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \)
$$\\$$
here \(v = \frac{3}{5}c \), so \( \gamma = \frac{1}{\sqrt{1-(\frac{3}{5})^2}} = \frac{5}{4}, \\ \)
so it lives longer (than at rest) by a factor of \(\frac{5}{4} :\\ \)
$$\\$$
\( \frac{5}{4} \times (2 \times 10^{-6})s = 2.5 \times 10^{-6}s \)