Abstract

The Theory of Special Relativity, developed by Albert Einstein in 1905, was a discovery that redefined physics, and one which contributed to the paradigm shift from Classical to Modern Physics. Drawing from the discoveries of Newton, Galileo, Maxwell and Lorentz, Einstein found meaning within certain equations. The theory destroyed one of the most fundamental assumptions that Classical Physics had made: that time and space were absolute quantities that were common to all observers. Einstein showed that it is in fact the speed of light that is absolute, while time dilates and space contracts in order to account for this. The findings of Einstein have allowed us to discover key facts about the universe, such as the speed of light being the limit of speed in the universe.

Introduction

At the end of the 19th Century, contemporary scientists believed they had unravelled all the mysteries of the universe, that they knew all there was about the world around them. Newtonian physics could explain almost all phenomena scientists met, applying to new fields conceived long after Newton’s death, such as Electricity and Thermodynamics. It seemed as though the giant shoulders of Newton would carry scientists through every new problem they encountered. At the turn of the 20th Century, however, cracks started to appear in Classical Mechanics that raised doubts about the very foundations of science. Inconsistencies were found in areas such as blackbody radiation, or the Photoelectric Effect, which would later be explained by Einstein. One of the most apparent problems was exposed by the results of Maxwell’s equations concerning light. Maxwell concluded that light was a wave consisting of oscillating electric and magnetic fields that self-propagated. The speed of light, $c$, was found to be $c=\frac{1}{\sqrt{\mu_o \epsilon_o}}$, where $\mu_o$ is the permeability of free space and $\epsilon_0$ is the permittivity of free space. If it were any faster, it would induce larger and larger magnetic fields, defying energy laws; any slower and it would slow down and eventually stop. This accurately predicted the speed of light, however, it did not establish a reference frame to which this speed was relative, suggesting that the speed of light was absolute regardless of reference frame. This directly contradicted classical mechanics in which the speed of an object was always relative to some reference point. Despite large amounts of scepticism from the wider scientific community, Einstein developed Special Relativity in 1905, relying on the work of Lorentz and Maxwell, which provided an answer to the problem of the speed of light. He established two postulates upon which it relied. The first was Galilean Relativity which stated that the laws of physics were the same for all inertial reference frames. The second postulate was the conclusion of Maxwell’s equations, that the speed of light, c, is the same for all observers, regardless of the speed of the source or the speed of the observer. Through his equations, Einstein showed that, since the speed of light is absolute, it is, in fact, time and space that are relative and dynamic, undermining any sense of human intuition about the world and previous ideas about the nature of reality. Fundamentally, this breakthrough about the nature of time and space defined the paradigm shift that occurred, developing Classical Physics into the Modern Physics of the past century.

Method

In order to demonstrate how Special Relativity works, Einstein used the following thought experiment:

Imagine there is a person called person A on a train travelling at velocity $v$. Inside this train is a light clock: an extremely accurate clock that reflects a photon off of a mirror into a detector. If the mirror is distance $L$ away, then due to the relationship of speed, distance and time, the time it takes for one tick, or unit of time, for Person A is: $\Delta t_0 = \frac{2L}{c}$

We use $\Delta t_0$ to establish that the observer is in the same inertial reference frame as the phenomenon being observed, while $c$ is the speed of light.

From person B’s perspective, however, the photon does not travel exclusively vertically, but also horizontally with the train. The overall horizontal distance it travels is $v\Delta t$ from the speed, distance, time relationship. We can therefore use Pythagoras to show that the relationship between the distance observed from outside the train, $D$, to the distance observed inside the train, $L$:

$D^2 = L^2 + (0.5v \Delta t)^2$

In classical mechanics, the horizontal speed $v$ and the vertical speed $c$ would be added, giving the photon a diagonal speed of $\sqrt{v^2 + c^2}$. However, Einstein did not do this. We recall his second postulate: the speed of light is $c$ regardless of frame of reference, leading the time for one tick or unit of time on the moving train from person B’s perspective to be:

$\Delta t = \frac{2D}{c}$

If we square we get:

$\Delta t^2 = \frac{4D^2}{c^2}$

We can then substitute in:

$D^2 = L^2 + (0.5v \Delta t)^2$

To get:

$\Delta t^2 = \frac{4(L^2 + (0.5v \Delta t)^2)}{c^2}$

Simplification:

$\Delta t^2 = \frac{4L^2 + v^2 \Delta t^2}{c^2}$

We substitute in the original equation of time from person A’s perspective:

$\Delta t_0^2 = \frac{4L^2}{c^2}$

To get:

$\Delta t^2 = \Delta t_0^2 + \frac{v^2 \Delta t^2}{c^2}$

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

Factorise:

$\Delta t_0^2 = \Delta t^2(1 - \frac{v^2}{c^2})$

Take the square root:

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

Therefore:

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

We express the Lorentz factor, $\sqrt{1-\frac{v^2}{c^2}}$, as $\gamma$, to be left with:

$\Delta t = \gamma \Delta t_0$

A similar derivation can be done about length contraction. However, from the result of time dilation, given that speed is constant, it is intuitive that:

$\gamma L= L_0$

as the $\gamma$ terms must cancel when $L$ is divided by $\Delta t$.

Results

Time Dilation goes against all human intuition and tells us that there is no such thing as absolute time, showing how the relativistic times relate at different speeds. Despite being an innovation in the world of science, the discovery doesn’t affect our everyday lives to any noticeable degree. This is due to the relatively low speeds we travel compared to the Earth, which is the default reference frame for humans. The fastest humans have ever travelled relative to Earth was the crew of the Apollo 10 Mission, reaching a speed of 24,791 mph, which is 11,083 m/s and results in a $\gamma$ of around $10^{-9}$, which is negligible. The equations do, however, provide us with important information about the universe. It suggests that the maximum speed possible to travel is the speed of light, as otherwise $\gamma$ would be an imaginary number.

Many people were sceptical of such a counterintuitive idea that they tried to invent ‘paradoxes’ to disprove special relativity. The most famous of these is the twin paradox, in which a twin sets out from earth at a very high speed and returns some time later to the other twin back on earth. It was thought that according to special relativity both twins would view the other twin as ‘younger’ (having aged less). However, this paradox cannot be a logical paradox as it does not include exclusively inertial frames of reference as the twin who leaves Earth must change direction. Another key discovery of time dilation was that, if the technology were available, one could travel previously impossibly large distances within a lifetime, opening avenues for space exploration in the future.