gen-define-lorentz-transformations-and-give-one-short-worked-exa-662e

Define Lorentz transformations and give one short worked example.

# Lorentz Transformations **Definition:** Lorentz transformations are mathematical equations that relate space and time coordinates between two inertial reference frames moving at constant velocity relative to each other. They preserve the speed of light in all frames and form the foundation of special relativity. For frames moving with relative velocity *v* along the x-axis: $$x' = \gamma(x - vt)$$ $$y' = y$$ $$z' = z$$ $$t' = \gamma\left(t - \frac{vx}{c^2}\right)$$ where $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor. ## Worked Example **Problem:** A spacecraft moves at v = 0.6c relative to Earth. An event occurs at x = 900 m, t = 5 μs in Earth's frame. Find the coordinates in the spacecraft's frame. **Solution:** First, calculate γ: $$\gamma = \frac{1}{\sqrt{1-(0.6)^2}} = \frac{1}{\sqrt{0.64}} = \frac{1}{0.8} = 1.25$$ Apply transformations: $$x' = 1.25(900 - 0.6 \times 3 \times 10^8 \times 5 \times 10^{-6})$$ $$x' = 1.25(900 - 900) = 0 \text{ m}$$ $$t' = 1.25\left(5 \times 10^{-6} - \frac{0.6 \times 3 \times 10^8 \times 900}{(3 \times 10^8)^2}\right)$$ $$t' = 1.25(5 - 1.8) \times 10^{-6} = 4.0 \text{ μs}$$ **Answer:** x' = 0 m, t' = 4.0 μs