A.5 Galilean and Special Relativity (HL only)

Galilean and Lorentz Transformations

An inertial reference frame is a reference point for measuring position and time that is not accelerating. Usually denoted S and S'. The relative speed between two frames is v.

There are two postulates (assumptions) in special relativity.

1. Physics is the same in all inertial reference frames.

2. The speed of light (c = 3e8 m/s) is the same for all observers.

Light, radio waves, and other electromagnetic radiation travels at c; everything else must travel slower than this.

A useful quantity to calculate is the gamma factor (γ). Make sure you can calculate the gamma factor with your calculator (e.g., if v = 0.6c then γ = 1.25). The gamma factor has a range 1 ≤ γ < ∞, so if you get γ < 1, you made a calculation mistake.

At low speeds (say, v < 0.1c), γ ≈ 1 and so we can use the Galilean Transformations (see the data booklet).

At high speeds (say, v > 0.1c), we need to use the Lorentz Transformations (data booklet) in order to meet the postulate that light travels at the same speed for all observers (and thus v > c is impossible). Like the Galilean Transformations, the Lorentz Transformations tell you how to determine positions, times, and speeds in frame S' if you know the equivalent quantity in frame S. Other than using the transformations to switch frames, you may also need to do some constant-velocity kinematics in order to solve problems, so don't forget that speed = distance / time!

Time Dilation and Length Contraction

Consider two "events", each of which has coordinates in space-time. These could be starting and stopping a timer, measuring the two ends of a rod at a specific time, two fireworks detonating, etc. If the two events happen at the same position (so ∆x = 0), in a given frame S, then the time interval between those events is the proper time. Likewise, if the two events happen at the same time (so ∆t = 0), in a given frame S, then the distance between those events is a proper length. Now, if you look at the Lorentz Transformations, you will see that they simplify, resulting in the simpler equations for time dilation (∆t = γ ∆t₀) and length contraction (L = L₀ / γ).

Be careful when applying the time dilation and length contraction formulas, because they only apply if you are working with a proper time and proper length. In problems involving a radio signal being transmitted from one location to another, for example, this does not apply.

Space-Time Diagrams