Einstein’s Theory of Special Relativity (HSC SSCE Physics): Revision Notes

Einstein's Theory of Special Relativity

Introduction to Einstein's revolutionary theory

In 1905, Albert Einstein published a groundbreaking paper that fundamentally changed our understanding of space and time. This paper introduced the theory of special relativity, which challenged the most basic assumptions that scientists had taken for granted for centuries.

Einstein's approach was bold but effective - he questioned the fundamental nature of space and time itself. By doing this, he was able to resolve puzzling contradictions about how light behaves. His theory eliminated the need for the "aether" (a hypothetical medium through which light was thought to travel) and demonstrated that there is no absolute, privileged frame of reference in the universe.

The two postulates of special relativity

Einstein's entire theory rests on two fundamental propositions. These postulates form the bedrock of special relativity, and all the strange effects we'll discuss flow logically from them.

To understand why this is so strange, consider this example: imagine two rockets, one moving toward a distant pulsar (a type of star that emits pulses of light) and one moving away from it, each travelling at $0.25c$ (one-quarter the speed of light).

This constancy of the speed of light was consistent with the results of the famous Michelson-Morley experiment. Einstein may or may not have known about this experiment when he developed his theory - he was primarily focused on Maxwell's equations of electromagnetism, which showed that light speed should be constant. Regardless, the Michelson-Morley results provided crucial experimental support for special relativity.

Time dilation

The theory of special relativity requires us to abandon our everyday intuitions about space and time. One of the most striking consequences is that time itself is not absolute - observers in different inertial frames will disagree about time measurements.

Understanding time dilation through a thought experiment

Let's imagine a train carriage moving at a high velocity $v$ relative to the ground. On the carriage are two mirrors ($m_1$ and $m_2$) arranged so that light pulses can bounce back and forth between them. Fred stands on the train, while Pierre stands on the ground watching the train pass by.

Fred measures the time ($t_0$) it takes for a pulse of light to travel from mirror $m_1$ to mirror $m_2$ and back again. Both Fred and Pierre agree that the perpendicular distance between the mirrors is $w$. This perpendicular distance is the same in both reference frames because length contraction only occurs in the direction of motion.

From Fred's perspective on the train, the situation is straightforward. The light simply travels straight up and down between the two stationary mirrors.

Fred observes the light travelling a total distance of $2w$, so the time he measures is:

$t_0 = \frac{2w}{c}$

However, Pierre sees something quite different. From his viewpoint on the ground, the mirrors are moving horizontally with velocity $v$ while the light pulse travels between them. This means the light doesn't travel straight up and down - instead, it follows a diagonal path, forming triangles.

For half the journey (from $m_1$ to $m_2$), the light pulse travels along the hypotenuse of a right-angled triangle. The three sides of this triangle have lengths:

• Vertical side: $w$ (the distance between mirrors)
• Horizontal side: $\frac{vt}{2}$ (the distance the mirrors move during the transit)
• Hypotenuse: $\frac{ct}{2}$ (the distance the light travels)

Using Pythagoras' theorem:

$\left(\frac{ct}{2}\right)^2 = \left(\frac{vt}{2}\right)^2 + w^2$

We can rearrange this equation:

$\frac{t^2}{4}\left(1 - \frac{v^2}{c^2}\right) = \frac{w^2}{c^2}$

$t\sqrt{1 - \frac{v^2}{c^2}} = \frac{2w}{c} = t_0$

Therefore:

$t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}$

This is the time dilation equation. The time measured by Pierre (who is moving relative to the mirrors) is longer than the time measured by Fred (who is at rest relative to the mirrors). In other words, Pierre sees Fred's clock running slow!

Proper time

Proper time is the time interval between two events occurring at the same place in an inertial frame, as measured by an observer in that inertial frame.

The Lorentz factor

The expression $\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ appears repeatedly in relativity calculations. It's given the special symbol $\gamma$ (gamma) and is called the Lorentz factor:

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

We can also define $\beta = \frac{v}{c}$ (the velocity as a fraction of light speed), so:

$\gamma = \frac{1}{\sqrt{1-\beta^2}}$

The Lorentz factor tells us how significant relativistic effects will be. When $v$ is small compared to $c$ (as it is in everyday life), $\gamma$ is very close to 1, and relativistic effects are negligible. But as $v$ approaches $c$, $\gamma$ increases dramatically.

Here's a table showing how $\gamma$ varies with velocity:

$\beta = \frac{v}{c}$	$\gamma$
0	1
0.0010	1.000 000 5
0.010	1.000 05
0.10	1.005
0.20	1.021
0.50	1.155
0.80	1.667
0.90	2.294
0.94	2.931
0.99	7.089
0.999	22.37

Notice how $\gamma$ stays close to 1 until velocities reach a significant fraction of light speed, then increases rapidly. At $v = 0.999c$, time is dilated by more than a factor of 22!

The reciprocal nature of time dilation

Here's a fascinating question: what would happen if we repeated the experiment with the mirrors on the ground and Fred on a moving train? From Fred's perspective, his train could be considered stationary and Pierre moving at velocity $-v$ relative to him. In this case, Fred would find that Pierre's clock runs slow compared to his!

Length contraction

If time measurements differ between inertial frames, it makes sense that measurements of length might also be affected. Indeed, special relativity predicts that objects in motion appear contracted (shortened) in the direction of motion.

Understanding length contraction through a thought experiment

Imagine a train travelling at velocity $v_0$. On the train, there's a light source that emits a short pulse towards a mirror, and the light bounces back to the source.

An observer on the train measures the distance from the source to the mirror as $w_0$. This is the proper length because it's measured in the rest frame of the mirror. The time for the light to travel to the mirror and back is:

$t_0 = \frac{2w_0}{c}$

Now consider an observer on the ground, watching the train pass. This observer is stationary relative to the moving frame of the train.

From the ground observer's perspective, during the time the light travels to the mirror, the mirror moves forward by a distance $v_0t_1$. The light must travel an extra distance to catch up with the moving mirror.

For the outward journey (source to mirror), the ground observer sees:

• The initial distance to the mirror: $w$
• The mirror moves forward: $v_0t_1$
• The light travels: $L_1 = w + v_0t_1$

Since the speed of light is constant in all frames:

$ct_1 = w + v_0t_1$

Rearranging:

$t_1 = \frac{w}{c - v_0}$

For the return journey (mirror back to source), the source is moving toward the returning light pulse:

$L_2 = w - v_0t_2$

Again, since the light speed is constant:

$ct_2 = w - v_0t_2$

$t_2 = \frac{w}{c + v_0}$

The total time for the round trip, as measured by the ground observer, is:

$t_1 + t_2 = \frac{w}{c-v_0} + \frac{w}{c+v_0}$

$= \frac{w(c+v_0) + w(c-v_0)}{c^2-v_0^2}$

$= \frac{2wc}{c^2-v_0^2}$

Now we can use the time dilation relationship. We know that:

$t_1 + t_2 = \frac{t_0}{\sqrt{1-\frac{v_0^2}{c^2}}}$

Therefore:

$\frac{2wc}{c^2-v_0^2} = \frac{t_0}{\sqrt{1-\frac{v_0^2}{c^2}}}$

Substituting $t_0 = \frac{2w_0}{c}$:

$\frac{2wc}{c^2-v_0^2} = \frac{2w_0}{c\sqrt{1-\frac{v_0^2}{c^2}}}$

Simplifying:

$\frac{wc}{c^2-v_0^2} = \frac{w_0}{\sqrt{1-\frac{v_0^2}{c^2}}}$

$\frac{w}{\sqrt{1-\frac{v_0^2}{c^2}}} = \frac{w_0}{\sqrt{1-\frac{v_0^2}{c^2}}}$

$w = w_0\sqrt{1-\frac{v_0^2}{c^2}}$

This is the length contraction equation! The distance $w$ measured by the ground observer is shorter than the proper length $w_0$ measured in the train's reference frame.

The length contraction formula

For any object moving with velocity $v_0$ relative to an observer:

$l = l_0\sqrt{1-\frac{v_0^2}{c^2}}$

where:

• $l_0$ is the proper length (length measured at rest in the object's frame)
• $l$ is the contracted length measured by an observer moving relative to the object
• $v_0$ is the relative velocity between the frames

The factor $\sqrt{1-\frac{v_0^2}{c^2}}$ is always less than 1 (for any velocity less than $c$), which means $l < l_0$. Objects moving relative to an observer always appear shortened in the direction of motion.

Important characteristics of length contraction

The Lorentz factor: historical context

Before Einstein developed special relativity, two scientists independently proposed an explanation for the null result of the Michelson-Morley experiment. George Fitzgerald from Ireland and Hendrik Lorentz from the Netherlands suggested that all objects physically contract in the direction of motion through the aether by a factor of $\sqrt{1-\frac{v^2}{c^2}}$.

Their motivation was to preserve Maxwell's equations when applied to moving charges. They saw this contraction as a real, physical change in the size of objects as they moved through the stationary aether.

Remember!