Special Relativity Concepts Cheat Sheet

Key Facts

• The speed of light in vacuum is constant for all inertial observers and has value $c = 3.00 \times 10^8\ \text{m/s}$.
• The Lorentz factor is $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, and it is always greater than or equal to $1$ for $0 \le v < c$.
• Time dilation is $\Delta t = \gamma \Delta t_0$, where $\Delta t_0$ is the proper time measured by a clock at rest with the events.
• 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 for motion along one line is $u' = \frac{u - v}{1 - \frac{uv}{c^2}}$, which prevents any massive object from being measured faster than $c$.
• The spacetime interval is $s^2 = c^2\Delta t^2 - \Delta x^2$ for one spatial dimension, and its value is invariant between inertial frames.
• Relativistic momentum is $p = \gamma mv$, so momentum increases more rapidly than the classical value $mv$ near light speed.
• Total relativistic energy is $E = \gamma mc^2$, rest energy is $E_0 = mc^2$, and kinetic energy is $K = (\gamma - 1)mc^2$.

Vocabulary

Inertial reference frame

A frame of reference moving at constant velocity where an object with no net force moves in a straight line at constant speed.

Proper time

The time interval $\Delta t_0$ measured by an observer who is at rest relative to the clock or both events.

Proper length

The length $L_0$ of an object measured in the reference frame where the object is at rest.

Lorentz factor

The factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ that determines how strongly relativistic effects appear.

Spacetime interval

A quantity such as $s^2 = c^2\Delta t^2 - \Delta x^2$ that stays the same for all inertial observers.

Rest energy

The energy $E_0 = mc^2$ an object has because of its mass, even when it is not moving.