Lorentz Transformations Reference Cheat Sheet

Lorentz transformations describe how measurements of space and time change between observers moving at constant velocity relative to each other. This cheat sheet helps students connect algebraic formulas to physical ideas in special relativity. It is most useful when comparing events measured in two inertial reference frames.

The focus is on clear use of variables, signs, and units.

Key Facts

The Lorentz factor is $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ , where $v$ is the relative speed and $c$ is the speed of light.
For a boost along the $x$ -axis, the position transformation is $x' = \gamma(x - vt)$ .
For a boost along the $x$ -axis, the time transformation is $t' = \gamma\left(t - \frac{vx}{c^2}\right)$ .
The inverse Lorentz transformations are $x = \gamma(x' + vt')$ and $t = \gamma\left(t' + \frac{vx'}{c^2}\right)$ .
Time dilation is $\Delta t = \gamma \Delta \tau$ , where $\Delta \tau$ is the proper time measured in the frame where the events occur at the same place.
Length contraction is $L = \frac{L_0}{\gamma}$ , where $L_0$ is the proper length measured in the object's rest frame.
Relativistic velocity addition along one line is $u' = \frac{u - v}{1 - \frac{uv}{c^2}}$ and $u = \frac{u' + v}{1 + \frac{u'v}{c^2}}$ .
The spacetime interval $s^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2$ is invariant for all inertial observers.

Vocabulary

Inertial reference frame: An inertial reference frame is a frame of observation moving at constant velocity where Newton's first law holds.
Lorentz boost: A Lorentz boost is a transformation between two inertial frames moving at a constant relative velocity.
Lorentz factor: The Lorentz factor $\gamma$ measures how strongly relativistic effects appear at speed $v$ compared with the speed of light $c$ .
Proper time: Proper time $\Delta \tau$ is the time interval measured by a clock present at both events.
Proper length: Proper length $L_0$ is the length of an object measured in the object's own rest frame.
Spacetime interval: The spacetime interval is a quantity combining space and time separations that has the same value in every inertial frame.

Common Mistakes to Avoid

Using $\gamma = \sqrt{1 - \frac{v^2}{c^2}}$ instead of $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is wrong because relativistic effects require $\gamma \ge 1$ .
Mixing up proper time and dilated time is wrong because $\Delta \tau$ is measured in the frame where the two events happen at the same location, while $\Delta t = \gamma \Delta \tau$ is measured from another frame.
Applying length contraction to the wrong direction is wrong because only lengths parallel to the relative motion contract, while perpendicular dimensions do not contract.
Using ordinary velocity addition, such as $u' = u - v$ , is wrong at high speeds because it can produce speeds greater than $c$ and ignores the relativistic denominator.
Forgetting the sign convention in $x' = \gamma(x - vt)$ is wrong because changing which frame moves in the positive direction changes whether the formula uses $-vt$ or $+vt$ .

Practice Questions

1 A spaceship moves at $v = 0.80c$ relative to Earth. Calculate $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ .
2 A clock on a moving spacecraft measures a proper time of $\Delta \tau = 5.0\ \text{s}$ while moving at $v = 0.60c$ . What time interval $\Delta t$ is measured on Earth?
3 A rod has proper length $L_0 = 12\ \text{m}$ and moves parallel to its length at $v = 0.75c$ . Find its contracted length $L = \frac{L_0}{\gamma}$ .
4 Explain why the spacetime interval $s^2 = c^2\Delta t^2 - \Delta x^2$ can stay the same even when two observers disagree about $\Delta t$ and $\Delta x$ .

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Lorentz Transformations Reference Cheat Sheet

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