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## Galilean Relativity

Relativity has been in the news quite a bit recently with the detection of neutrinos apparently travelling faster than the speed of light. Although most people don’t know the details of Einstein’s theory of Relativity, many are aware of the “cosmic speed limit” predicted in it, and also the most famous equation in physics which came from his theory – $E=mc^{2}$.

Many people, however, are unaware that the idea of relativity had been around long before Einstein. In fact, we can trace the idea of relativity back to Galileo. Galileo was one of the first scientists to do experiments on the motions of bodies (what we would now call mechanics), and was also one of the first scientists to use “thought experiments” to make scientific arguments.

Galileo

Galileo started thinking about whether mechanical experiments would behave differently if one were in motion or at rest. For example, if a ship is anchored in the port and one were to drop a stone from the top of the mast, we all know that it would strike the deck at the bottom of the mast, i.e. vertically below the place from where it was dropped (as long as we were careful not to give it any sideways motion). This is, of course, a pretty obvious statement.

But, what would happen if the ship were in motion? Let us suppose the ship is sailing at 5 metres per second $(5m/s)$ in some direction on a perfectly smooth lake. If someone were now to drop a stone from the mast, surely it would fall behind the mast because the ship has moved forwards whilst the stone was dropping. If the stone were to take 1 second to drop to the deck, surely the stone would land 5 metres behind the bottom of the mast, rather than at the bottom, because the ship has moved 5 metres forwards in that 1 second.

NO, Galileo argued, this would not be the case. He argued that it would hit the deck at the bottom of the mast, just as in the case when the ship is not moving. If you think about it carefully you can see why.

When the person drops the stone from the mast, they are moving forwards with the ship. So the stone is actually given a forwards motion as it is dropped, and it is this forwards motion which leads it to land at the bottom of the mast, not behind it. As the ship moves forwards at $5 m/s$, so does the stone. By performing this simple mechanical experiment one would not be able to tell whether the ship were anchored in the port, or moving on a smooth lake.

If the person at the port were able to see the motion of the stone against some sort of background, he would see the stone move in a parabola, which is exactly the motion a falling object which is also given some sideways velocity has. But, at every point of its travel down towards the deck, it will be next to the mast, as this is moving forwards as the stone falls.

Galileo then went on to generalise this specific thought experiment to say that there was no mechanical experiment that one could perform which would be able to tell the difference between being at rest or moving with a constant velocity (that is, with no acceleration). This principle is know as Galilean relativity, and we define a set of equations known as the Galilean transforms which allow us to switch between what we would see in two frames of reference, for example what someone standing on the shore would measure and what someone on a moving ship would measure.

If the ship is moving with a constant velocity $v$ then in time $t$ it will move a distance $v t$ (distance = velocity x time). To make it easier for ourselves we will set up the $x,y,z$ axes so that the ship is moving only along our $x-axis$. If we refer to the position and time of any event in the person on the shore’s frame of reference as $(x,y,z,t)$ and those in the frame of reference of someone on the ship as $(x^{\prime},y^{\prime},z^{\prime},t^{\prime})$ then the equations which relate the two (known as the Galilean transforms) are:

$\begin{array}{lcl} x^{\prime} & = & x + vt \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & t \end{array}$

What these equations mean is that the only variable which is different in the two frames of reference is the x-displacement. The y and z-displacements are unaltered (as the ship is only moving in the x-direction), and time is the same for the two frames of reference. Let us look at how the x-displacement is transformed in going from one frame of reference to the other.

Suppose the ship is moving in the positive x-direction at $5 m/s$. We want to measure the position of an object which is on the deck of the ship, let’s say the mast, as time goes by. For the person on the ship, it’s position is say, $15m$ in front of the stern of the ship. This is clearly not going to change with time, the mast does not move relative to the ship! So, we shall call this $x$.

For the person on the shore, the position of the mast is going to change as the ship sails away from him. So if the ship is sailing away at $5 m/s$ and the mast is initially $15m$ away from the person on the shore, then after $1 second$ it will be $15+(5 \times 1)=15+5=20m$ away. This is the x-position $x^{\prime}$, the x-position for the person in the other frame of reference, as given by the Galilean transformation equations above.

You can get this straight from using the equation $x^{\prime} = x + v t = 15 + (5)(1) = 20m$

The Galilean transforms are mathematically very simple, and conceptually simple too. As I will discuss in a future blog, the idea of performing experiments to determine between a state of rest or uniform motion, which Galileo argued could not be done, haunted scientists for centuries. In the 19th Century, with the development of electrodynamics (the study of the electricity and magnetism of moving bodies), physicists thought they could devise experiments to distinguish one’s state of uniform motion. They were wrong in thinking this, and it led to Einstein overthrowing the whole ideas of absolute time and absolute space in his Special Theory of Relativity.