A5-1. Special Relativity: Reference Frames and Galilean RelativityHL ONLY

1. Reference Frames

In physics, motion is entirely relative. To measure velocity, displacement, or time, we must first establish a point of view. A reference frame (or frame of reference) is a set of coordinates used to record the position and time of events.

Inertial vs. Non-Inertial Frames:

  • Inertial Reference Frame: A frame of reference that is either entirely completely at rest or moving at a constant velocity. In an inertial frame, Newton's First Law holds true (an object at rest stays at rest unless acted upon by a force). There is no "fictitious" acceleration.
  • Non-Inertial Reference Frame: A frame of reference that is accelerating or rotating. In these frames, objects appear to accelerate without a physical force acting on them (e.g., being pushed back into your seat when a car accelerates).

Special relativity deals exclusively with inertial reference frames.

2. Galilean Relativity & Transformations

Before Einstein, classical mechanics relied on Galilean relativity. Galileo proposed that the fundamental laws of mechanics are identical in all inertial reference frames. If you are in a train moving at a perfectly constant velocity, tossing a ball straight up looks exactly the same as if you were standing on the ground.

Galilean Transformation Equations:

Suppose Frame $S'$ is moving at a constant velocity $v$ relative to a stationary Frame $S$ along the x-axis. For an event occurring at position $x$ and time $t$, Galilean transformations allow us to translate coordinates between the frames:

$$ \begin{aligned} x' &= x - vt \\[0.1cm] u' &= u - v \\[0.1cm] t' &= t \end{aligned} $$
  • $x, x'$: Position of the event in Frame S and Frame S'.
  • $u, u'$: Velocity of an object in Frame S and Frame S'.
  • $v$: Relative velocity between the two frames.
  • $t, t'$: Time. Crucially, in Galilean relativity, time is absolute. It flows at the exact same rate for all observers everywhere ($t' = t$).

Example 1: Galilean Velocity Addition

Problem: A train (Frame S') is moving right at $v = 20\text{ m/s}$ relative to the ground (Frame S). A passenger on the train throws a ball forward at $u' = 5\text{ m/s}$ relative to the train. What is the velocity of the ball ($u$) relative to an observer standing on the ground?

Solution:

  • Using the Galilean velocity transformation: $u' = u - v$
  • Rearranging for $u$: $u = u' + v$
  • $u = 5\text{ m/s} + 20\text{ m/s} = \mathbf{25\text{ m/s}}$
  • Insight: This simple addition perfectly describes classical mechanics, but it critically fails when speeds approach the speed of light.