Intuitions for Special Relativity

Special Relativity is a theory of space and time that describes how space and time are affected by motion. It follows the natural consequences of the constancy of the speed of light in all reference frames. The special in special relativity refers to the fact that it is a special case of general relativity: situations when objects are moving at constant velocities in a vacuum.

Special Relativity Intuitions

The speed of light is constant in all inertial reference frames, regardless of the motion of the observer or the source of the light. This is a fundamental postulate of special relativity.

Inertial reference frames

The first key insight is to approach the problem with the understanding that motion is relative to the inertial reference frame of the observer. In any situation, we can always choose an inertial reference frame in which one object is at rest and everything else is moving relative to it.

From Brock’s perspective:

• The train is moving towards the station at speed $v$.
• The station is at rest.

Ash and Brock experience the world from their respective inertial reference frames. Each of their observations are true, and they are all valid.

The second key insight is to understand that the speed of light is constant for every observer regardless of their motion or the source of the light.

The speed of light is constant

The first postulate of special relativity is that the speed of light is constant in all inertial reference frames.

Another way to think about this is to consider a perfect “light clock” that each observer watches. The light clock consists of a photon that bounces between two mirrors once per second. Since the speed of light is constant in all inertial reference frames, an observer watching their clock will always observe the photon moving at the speed of light, $c$.

Let’s give a light clock to Ash and Brock and see what happens.

From Ash’s perspective:

• The station is moving towards the train at speed $v$.
• Light clock A is at rest.
• The photon in the light clock A is moving at speed $c$.
• Light clock B is moving towards the train at speed $v$.
• The photon in the light clock B is moving at speed $c$.

From Brock’s perspective:

• The train is moving towards the station at speed $v$.
• Light clock A is moving towards the station at speed $v$.
• The photon in the light clock A is moving at speed $c$.
• Light clock B is at rest.
• The photon in the light clock B is moving at speed $c$.

Light clock A is moving at speed $v$ relative to Brock, but he observes the photon inside to be moving at speed $c$, not $c+v$. How can this be?

The emergence of time dilation

Let’s follow the math and see what happens when we assume all of the observations are true.

$$\text{speed}=\frac{\text{distance}}{\text{elapsed time}}$$

                distance traveled in one second
Brock  * 0 m
Ash    *--* 100 m
Light  *~~~~~~~~~~~~~~~~~~~~~~~~~~~~~* 300_000_000 m

$$\begin{array}{rl}c& =\frac{300,000,000\text{ m}}{t_B\text{ s}}\\ c& =\frac{299,999,900\text{ m}}{t_A\text{ s}}\end{array}$$

In order for the speed of light to be constant in both cases, the elapsed time must be different for each observer!

$$\begin{array}{rl}\text{if}3.00\times 10^8\text{ m/s}& =\frac{300,000,000\text{ m}}{t_B}⟹t_B=1.00\text{ s}\\ \text{then}3.00\times 10^8\text{ m/s}& =\frac{299,999,900\text{ m}}{t_A}⟹t_A=0.999999667\text{ s}\end{array}$$

The time experienced by an observer at rest is called its proper time. Remember that it an observer only ever sees themself at rest. The proper time is always “normal” for the observer. (This is where the twin paradox comes from.)