Special Relativity

A frame of reference is the viewpoint you use to measure position, time and motion. It is essentially a coordinate system tied to an observer (for example: “the platform frame” or “the train frame”). In everyday life we often say “the train is moving”, but equally you could say “the platform is moving backwards” – motion depends on the chosen frame.

Thought experiment (train with blinds down):

Imagine a person inside a train who falls asleep with the blinds down. When they wake up, they some do experiments inside the carriage (drop a ball, swing a pendulum, bounce a ball off the wall). If the train is moving at a steady speed in a straight line, all the results look normal and there is no experiment they can do to prove they are “really moving”. This is called an inertial frame of reference.

An inertial frame of reference is a frame moving at constant velocity (including ). In an inertial frame, Newton’s first law appears to work normally: if the resultant force on an object is zero, it stays at rest or moves in a straight line at constant velocity.

If the train speeds up, slows down, or turns a corner, the passenger can detect that they are not in an inertial frame because ‘fictitious forces’ seem to appear (for example, you feel pushed backwards when the train accelerates and the ball on the ground will start rolling without any apparent force on it).

We will only work with inertial frames of reference in this topic.

Classical Mechanics- Galilean transformation

In classical mechanics, speeds add (or subtract) between inertial frames. If object A moves at speed u towards object B which is moving at v then the relative speed will be , (i.e. Object A will see object B coming towards it at , which is faster). If object A and object B move at the same speed and direction the speeds subtract to give a relative speed of .

If Train 1 moves right at and Train 2 moves left at , each train measures the other approaching at . That seems obvious because you ‘add’ the speeds.

Same direction example: If Train 1 moves right at and Train 2 moves right at , then Train 1 measures Train 2 moving backwards at ().

By the mid-1800s, interference and diffraction showed that light behaves as a wave. Most physicists therefore expected light waves to need a medium, just like sound needs air. The proposed medium was the (luminiferous) ether: a fixed ‘background’ filling all space. The Earth was thought to move through this either.

If the ether existed, light should travel at speed c relative to the ether. So an observer moving through the ether might measure different light speeds depending on direction. In other words, physicists thought if a train was going towards a beam of light, a person on the train would see the light approach at a speed greater than .

The Michelson–Morley (MM) experiment: searching for absolute motion

Michelson and Morley designed an interferometer to compare the travel time of two light beams at right angles. A beam splitter (semi-silvered glass block) sends half the light along one arm and half along the perpendicular arm. Each beam reflects from a mirror and returns to recombine at the beam splitter, producing an interference pattern at a viewing telescope.

A plane glass block is placed in one path to ensure equal optical path lengths through glass. Without it, the beam going upward would pass through more glass than the other, creating a built-in phase difference.

Prediction: If the apparatus is moving through the ether, the beam travelling parallel to motion of the apparatus through the ether should take a longer compared with the beam travelling perpendicular to it. That would produce a phase difference and shift the interference fringes.

Key test: Rotate the apparatus by 90° at different times in the year as the earth moves around the sun. If the ether exists, the roles of the two arms swap, so the fringe pattern should shift as the apparatus is rotated.

What they found: no fringe shift (a null result). This strongly suggested there is no detectable ether and therefore no ‘absolute rest frame’ for light. The speed of light is the same for all observers irrespective of their velocity.

Einstein (1905) resolved the problem by starting from two postulates:

• Postulate 1: The laws of physics have the same form in all inertial frames of reference.

• Postulate 2: The speed of light in free space (vacuum), , is invariant for all observers.

The first postulate basically says that there is no experiment we can do to determine if we’re in a inertial frame of reference that is moving with respect to some “truly stationary” frame of reference.

The second postulate says that even if you run towards a beam of light (or run away from it), it will still approach you at c. This is consistent with Maxwell’s equations which predict electromagnetic waves that travel in free space at . This value depends only on fundamental constants, not on the motion of the source or observer. Einstein took this seriously: if the equations are true in one inertial frame, they must be true in all inertial frames.

This is strange at first. In everyday life, if you run towards something, you expect its speed relative to you to be higher. But for light, every inertial observer still measures the same value . The solution to this problem has two parts 1) time dilation and 2) length contraction (next sections of the topic).