The laws of physics are the same in all inertial reference frames.

• The speed of light c is the same in all inertial reference frames.

• No particle or causal influence can travel at a speed greater than c.

• Base the analysis on events.

• Time intervals can often be found using time dilation.

• Distances can often be found using length contraction.

• Use the Lorentz transformations for general problems.

Spatial measurements depend on the motion of the experimenter relative to the events. An object’s length is the difference between simultaneous measurements of the positions of both ends.

Proper length $\ell$ is the length of an object measured in a reference frame in which the object is at rest. The object’s length in a frame in which the object moves with velocity v is

This is called length contraction.

The law of conservation of momentum is valid in all inertial reference frames if the momentum of a particle with velocity u is $p={\gamma }_{\text{p}}mu$, where

The momentum approaches $\text{∞}$ as $u\to c$.

Invariants are quantities that have the same value in all inertial reference frames.

Spacetime interval: $s^2={\left(c\text{Δ}t\right)}^2-{\left(\text{Δ}x\right)}^2$

Particle rest energy: $E_0^2={\left(m{c}^2\right)}^2=E^2-{\left(pc\right)}^2$

Time measurements depend on the motion of the experimenter relative to the events. Events that are simultaneous in reference frame S are not simultaneous in frame $\text{S}\prime$ moving relative to S.

Proper time $\mathbf{\text{Δ}}\mathit{\tau }$ is the time interval between two events measured in a reference frame in which the events occur at the same position. The time interval between the events in a frame moving with relative velocity v is

This is called time dilation.

The law of conservation of energy is valid in all inertial reference frames if the energy of a particle with velocity u is $E={\gamma }_{\text{p}}m{c}^2=E_0+K$.

Rest energy $E_0=m{c}^2$

Kinetic energy $K=({\gamma }_{\text{p}}-1)m{c}^2$

Mass m can be transformed into energy $\text{Δ}E=m{c}^2$.

Energy can be transformed into mass $m=\text{Δ}E/c^2$.