Lorentz Transformations and Relativistic Kinematics

Manual: General · Subject: Physics

This lesson derives and uses the transformation rules connecting moving observers.

From Galilean to Lorentzian transformation

Why Galilean transformations fail

In Newtonian physics, coordinates transform as $x' = x - vt$ and $t' = t$ . This keeps time absolute, but it does not preserve the invariance of the speed of light.

Lorentz transformations

For motion along the $x$ -axis, the Lorentz transformations are $x' = \gamma (x - vt), \quad t' = \gamma \left(t - \frac{vx}{c^2}\right),$ where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$

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Interpretation

The factor $\gamma$ grows rapidly as $v$ approaches $c$ , which is why relativistic effects become dramatic at high speed.

Velocity addition

If an object moves at speed $u$ in one frame, another frame moving at speed $v$ measures $u' = \frac{u - v}{1 - uv/c^2}.$ This keeps all measured speeds below $c$ when $u$ and $v$ are below $c$ .

What is the Lorentz factor $\gamma$ ?

\sqrt{1-v^2/c^2}

1/\sqrt{1-v^2/c^2}

v/c^2

c/v

Write the relativistic velocity addition formula for collinear motion.

Problem-solving strategy

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Step 1: Identify the two inertial frames and the relative speed $v$ .
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Step 2: Choose the motion axis and write the Lorentz transformation.
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Step 3: Substitute known coordinates or velocities.
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Step 4: Check the low-speed limit $v \ll c$ to confirm the result reduces to classical physics.

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