Special relativity was introduced by Einstein in 1905 to reconcile inconsistencies between Newtonian mechanics and Maxwell’s electromagnetism. In turned out that Newton’s theory had to be brought into line with Maxwell’s, with the reformulated mechanics then able to correctly describe motion approaching or at light speed. Just as significantly, the theory demanded a radical reappraisal of the relationship between space and time. The notion that our local geometry is purely spatial with time somehow distinct, absolute and universal, had to be abandoned. Space and time though different in character had to be regarded as combining to play equally important roles in a richer spacetime geometry.

The theory rests upon the simple yet profound principle of relativity:

The laws of physics are identical in all inertial frames of reference.

The giants of physics, Galileo, Newton, Maxwell and Einstein, have each had a hand in shaping our understanding of this principle. Here I’ll try to place our current understanding of its meaning in the context of its historical development.

Galileo’s ship

In his 1632 “Dialogue Concerning the Two Chief World Systems”, Galileo Galilei (1564-1642) described a picturesque scene below deck of a ship featuring “small winged creatures”, fish, dripping bottles as well as a game of catch and some jumping about to illustrate a phenomenon well known to us all. Stuff behaves the same whether we and our immediate environment are stationary or moving uniformly. Sitting in an aeroplane, the windows shuttered and headphones on are you parked on the runway or cruising at 30,000 feet? You have no way of distinguishing between these two possibilities. It’s only if the plane hits turbulence, its velocity suddenly changing, that you appreciate the importance of keeping your seat belt fastened and realise you’re perhaps further from the ground than you’d like! Galileo had identified that the way things move, mechanics, is identical whether our frame of reference, a laboratory in which we can measure distances and times, is stationary or moving with constant velocity. The laws of physics, as far as they were then understood, do not and cannot distinguish between frames of reference in uniform relative motion. This was Galileo’s principle of relativity.

Galilean invariance of Newton’s laws

Isaac Newton (1642-1727) formalised the laws of mechanics in terms of his three laws of motion and law of universal gravity. These were presented, along with a great deal else, in his monumental “Principia” published in 1687. The first law of motion states that every body continues in its state of rest or uniform motion in a straight line unless compelled by some external force to change that state. Explicitly, this says that a free particle (one not acted on by any force) has constant velocity. But also, implicitly, that there exists a frame of reference in which this is the case. Such a reference frame is precisely what is meant by an inertial frame of reference. In other words, the validity of Newton’s first law tests whether or not we are in an inertial frame. Furthermore, given one inertial frame, Galileo’s principle of relativity tells us that any other frame of reference moving uniformly with respect to it is also an inertial frame. One obvious question is where (on earth!) are these free particles — nothing escapes gravity! There are of course special situations, a puck on an ice rink, particles with no mass, in which gravity can clearly be ignored, but Newton posits that quite generally were it not for gravity the natural state of all things is rest or uniform motion. Some justification for setting aside gravity in this way is provided by the following observation. Recall that Newton’s law of gravity says that the gravitational force exerted on a point pass \(m\) by another point mass \(M\) is given by
\begin{equation}
F=G\frac{Mm}{r^2}
\end{equation}where \(G\) is the gravitational constant and \(r\) is the distance separating the point masses. Now normally, in accordance with Newton’s second law, \(F=ma\), the acceleration due to an applied force is inversely proportional to the mass. In the case of gravity though, since the force itself is proportional to the mass on which it acts, it accelerates all bodies equally regardless of their mass and so can be regarded as a kind of overlay upon the existing physics.

Mathematically, a frame of reference may be regarded as a coordinate system. To specify the location of a particle we need four coordinates, \(x,y,z,t\). Three, \(x,y,z\), to specify its where and one, \(t\), specifies its when. Let’s call this coordinate system \(S\) and assume it corresponds to an inertial frame of reference. Of course any other coordinate system which is simply spatially translated and/or rotated with respect to \(S\) also corresponds to an inertial reference frame. More interesting though would be one which was also in relative motion with respect to \(S\). Let’s call the corresponding coordinate system \(S’\). To keep things simple let’s focus on the relative motion and assume that we’ve arranged that at \(t=0\) the coordinate systems are aligned with \(S’\) moving with a velocity \(\mathbf{v}=(v,0,0)\), that is, with speed \(v\) in the positive \(x\)-direction relative to \(S\).

Then at some time \(t\) the spatial coordinates of a point with respect to \(S’\) are related to its coordinates in \(S\) according to the simple equations,
\begin{align}
x’&=x-vt\nonumber\\
y’&=y\label{eq:Galilean_space}\\
z’&=z.\nonumber
\end{align}Notice that we’ve implicitly assumed that there is a single, absolute time. That is, time in \(S’\) is assumed to be the same as time in \(S\),
\begin{equation}
t’=t.\label{eq:Galilean_time}\\
\end{equation}Together, the equations \eqref{eq:Galilean_space} and \eqref{eq:Galilean_time} are called the Galilean transformations relating the inertial coordinate systems \(S\) and \(S’\).

An immediate consequence of the Galilean transformations is that velocities add. That is, if \(u_x\) and \(u’_x\) are the \(x\)-components of the velocities \(\mathbf{u}\) and \(\mathbf{u}’\) of a particle as measured in \(S\) and \(S’\) respectively then,
\begin{equation*}
u_x=\frac{dx}{dt}=\frac{dx’}{dt’}+v=u’_x+v,\\
\end{equation*}so that, together with the obvious relations for the \(y\)- and \(z\)-components, \(\mathbf{u}=\mathbf{u}’+\mathbf{v}\). In particular there is no notion of absolute rest. As Galileo had observed, one person’s state of rest is another’s uniform motion, it is a matter of perspective.

Coordinate systems related by Galilean transformations are the mathematical abstraction of inertial frames of reference and Galileo’s principle of relativity set in this context is the statement that the mathematical expression of the laws of physics should be invariant under Galilean transformations. Take Newton’s second law as an example, \(\mathbf{F}=m\mathbf{a}\), now expressed in terms of 3-dimensional vectors, \(\mathbf{F}=(F_x,F_y,F_z)\) and \(\mathbf{a}=(a_x,a_y,a_z)\). If \(\mathbf{a}’\) is the acceleration as measured in \(S’\) then we have, say for its \(x\)-component, \(a’_x\),
\begin{equation*}
a’_x=\frac{d^2x’}{dt’^2}=\frac{d^2x}{dt^2}=a_x,\\
\end{equation*}and similarly for the \(y\)- and \(z\)-components, so \(\mathbf{a}’=\mathbf{a}\), that is, acceleration is invariant under Galilean transformation. To confirm that Newton’s second law is true in all inertial frames of reference then becomes the mathematical problem of checking, on a case by case basis, that all forces of interest are also invariant under Galilean transformations. In fact, most forces encountered in Newtonian dynamics depend only on relative position, relative velocity and time. So, since each of these are invariant under Galilean transformations so are the forces. In particular, this is the case for the force of gravity between two objects since it is inversely proportional to the square of their separation.

In most cases a frame of reference fixed to earth, such as the room in which you’re sitting, is a good approximation to an inertial frame. Technically though it isn’t, consider for example earth’s rotation about its axis which constitutes a radially directed acceleration. When working with such noninertial frames the acceleration of the frame manifests itself in the form of ‘fictitious’ forces — in the case of rotating frames, the Coriolis and centrifugal forces. Incidentally, it is a feature of such fictitious forces that, like gravity they are always proportional to the mass of the object whose motion is being studied. Could it be that gravity is also somehow a fictitious force? This idea turns out to have considerable legs!

Galilean relativity in action: What happens when you drop a soccer ball with a table-tennis ball sitting on top?

If you’ve never tried it you should — seeing the table-tennis ball ping high in the air is pretty dramatic. Understanding this behaviour provides a nice example of the power of translating between inertial frames of reference using the Galilean transformations. It will be intuitively obvious that a table-tennis ball hitting a soccer ball will simply bounce back with essentially the same speed but in the opposite direction leaving the football unmoved. Now, when we drop the football with the table-tennis ball on top, for a split second after the football hits the ground, we have the two balls colliding with each other with equal and opposite velocities. Schematically, and imagined horizontally, the situation we wish to understand is this:

Now let us consider the situation from the perspective of a frame of reference moving to the right with velocity \(v\). In this frame the football is at rest and, thanks to the way velocities add in Galilean transformations, we know that the table-tennis ball is on a collision course travelling at a speed of \(2v\). As already mentioned we know what happens is this situation – the table-tennis ball simply bounces back travelling in the opposite direction with the same speed \(2v\) and the football remains at rest. To understand the original problem we simply translate this outcome back to the original frame of reference to find the football still travelling a \(v\) but the table tennis ball travelling at \(3v\). In other words, when we drop a football with a table tennis ball on top the table tennis ball bounces back up at three times the speed at which the pair hit the ground!