- The principle of relativity
- Galilean relativity
- Frames of reference
- Galiliean coordinate transformations
- Invariance

### The principle of relativity

The principle of relativity states that the laws of physics are independent of the inertial frame of reference in which they are studied.

For our purposes, an **inertial frame of reference** means a set of spatial coordinates and clocks. We can imagine space being filled with an imaginary lattice of measuring rods, each having the same (arbitrary) length. Points $(x, y, z)$, in space are at the ends of these rods, and at each point we imagine there is a clock measuring time, $t$.

We could also imagine a second frame of reference, constructed in exactly the same way, with coordinates $(x', y', z')$ at time $t'$, but which is moving at a uniform velocity relative to the first frame.

Suppose an identical experiment is being done in each of the two frames. Then, an observer at rest in the first frame will obtain a certain set of results from the experiment. The principle of relativity states that an observer at rest in the second frame will obtain an *identical* set of results.

### Galilean relativity

Galileo and Newton would have said the clocks in both frames would tick off time at exactly the same rate, $t' = t$. That is, time is *universal* and once the clocks are *synchronised*, everybody would agree what the time is at a particular instant.

The concept of universal time had to be given up after Einstein introduced special relativity, but the idea of relativity itself went back way before Einstein.

Imagine being inside a train carriage - a perfect carriage, with no rattling or other disturbances - moving at a perfectly uniform speed along a track. We can compare the laws of physics as studied by someone on the train with those studied by someone standing beside the track.

In particular, Professor Susskind considers the motion of a ball launched directly upwards. It's clear that for the observer beside the track, a ball launched directly upwards (in this frame of reference) will move only vertically and would land back on the same point from which it was launched (neglecting wind and other disturbances).

In fact, a person launching a ball upwards on the moving train will observe exactly the same motion - the ball will move only vertically and land on the same point from which it was launched.

This may well have been a surprise to people, pre-Galileo. They may have thought that the ball on the train would land somewhere towards the back of the carriage, depending on the speed that the train was moving at, relative to the earth. Galileo realised this was wrong and Newton spelled it out in detail.

### Frames of reference

In order to compare the laws of physics in two different reference frames, we need to consider transformations of properties between the two frames, and in particular, transformations of coordinates.

For now, we shall restrict our attention to motion only along the $x$-axis and assume time is universal, meaning that the clocks in both frames tick off time at the same - universal - rate.

We first construct a stationary frame of reference.

We imagine our meter sticks and clocks laid out on the horizontal $x$-axis. The dashed vertical lines represent *world-lines* of the ends of the meter sticks (stationary objects in space have trajectories in space-time).

The $t$-axis is defined to be the line $$x = 0$$ so it must be in the vertical direction. Horizontal dashed lines are lines of constant time.

Nb. The Galilean space-time lattice of world-lines and lines of constant time can be made as finely-grained as necessary. We could be using metres, kilometres, seconds or years.

We can overlay a second frame of reference defined by the coordinates, $(t', x')$, moving at speed $v$ relative to the first frame, onto this diagram.

We imagine that an observer at rest in the moving frame has laid out a set of measuring sticks and synchronised clocks in the same way as for the stationary frame.

In Figure 1.2, the frame in blue is moving at speed $v$ in the (common) $x, x'$ axes, hence the line joining the two origins is given by $x = vt$. This is the time-axis for the moving frame, since it also satisfies $x' = 0$.

The dashed, blue lines are the world-lines of the clocks of the moving frame, as measured in the stationary frame.

We do not need to include new lines of constant time since time is universal for Galileo and Newton, and so they are the same horizontal lines as for the stationary frame.

### Galiliean coordinate transformations

Universal time means that $t' = t$, but it's clear from figure 1.2 that $x' \neq x$.

In figure 1.3, the event marked $E$ can be written in terms of coordinates in both frames, $$\begin{align*}E &\rightarrow (t, x)\\E &\rightarrow (t', x')\end{align*}$$

The difference between the $x'$-coordinate in the moving frame and the $x$-coordinate in the stationary frame is exactly the distance travelled by the frame in time $t$.

Thus, the coordinate transformation relating the two sets of coordinates is given by $$\begin{align*}t' &= t\\x' &= x - vt\end{align*}$$

We could of course look at things from the point of view of an observer at rest in the moving frame, as in figure 1.4.

From this viewpoint, the other frame would be moving at velocity $-v$ and we would write $$\begin{align*}t &= t'\\x &= x' + vt'\end{align*}$$

### Invariance

The notion of invariance is important in physics, both invariance of physical quantities or properties, and the invariance of equations relating such quantities.

#### Events

A fundamental example of an invariant quantity, in all forms of relativity, is an event in space-time. To be explicit, suppose two events, $E_1, E_2$, have the same space-time coordinates in a particular inertial frame of reference, $$\begin{align*} E_1 &\rightarrow \{t_1, x_1, y_1, z_1\} \\ E_2 &\rightarrow \{t_2, x_2, y_2, z_2\} \end{align*}$$ such that $$\begin{align*} t_1 &= t_2 \\ x_1 &= x_2 \\ y_1 &= y_2 \\ z_1 &= z_2 \end{align*}$$ then they will have the same space-time coordinates in **every** inertial frame of reference.

That is, if $$\begin{align*} E_1 &\rightarrow \{t'_1, x'_1, y'_1, z'_1\} \\ E_2 &\rightarrow \{t'_2, x'_2, y'_2, z'_2\} \end{align*}$$ then $$\begin{align*} t'_1 &= t'_2 \\ x'_1 &= x'_2 \\ y'_1 &= y'_2 \\ z'_1 &= z'_2 \end{align*}$$

#### Simultaneity

In Galilean relativity, time is invariant, and so we can consider simultaneous events, which are events that occur at the same time, but not necessarily at the same location. Then, if two events are simulltaneous in one frame, they will be simultaneous in any other frame.

This, along with time-invariance itself, gets dropped in special relativity.

#### Spatial separation

Following directly from the invariance of simultaneity, consider the spatial separation between two simultaneous events - the two ends of a meter stick moving at speed $v$ say.

In figure 1.5, at time $t = t'$, the ends of the stick are at $(t, x_1), (t, x_2)$ in the stationary frame and at $(t', x'_1), (t', x'_2)$ in the moving frame (which is at rest relative to the stick).

The spatial separation as measured in the moving frame is then $$\begin{align*}x'_2 - x'_1 &= (x_2 - vt) - (x_1 - vt)\\&= x_2 - vt - x_1 + vt\\&= x_2 - x_1\end{align*}$$ which is the spatial separation as measured in the stationary frame.

Hence, in Galilean relativity, spatial separation is invariant.

#### Acceleration

Suppose a particle has a trajectory in space (we can stay in one dimension for simplicity), and the coordinates of the trajectory are $x(t)$ in one frame and $x'(t)$ in another (where $t' = t$, of course), moving at uniform speed $v$ relative to the first.

Now, the *position* of the particle is *not* an invariant quantity, since $$x'(t) = x(t) - vt$$

And neither is the *velocity* of the particle, since $$\frac{\mathrm{d} x'}{\mathrm{d} t} = \frac{\mathrm{d} x}{\mathrm{d} t} - v$$

But **acceleration** is invariant, since $$\frac{\mathrm{d}^2 x'}{\mathrm{d} t^2} = \frac{\mathrm{d}^2 x}{\mathrm{d} t^2}$$

#### Mass

Given the fact that position and velocity are not invariant, and that acceleration is invariant, it would make sense to formulate the laws of physics in terms of acceleration.

Newton expressed this explicitly in his force law as $F = ma$. Suppose we have two particles (on the $x$-axis for simplicity) exerting forces on each other. We always assume that fundamental forces are conservative forces, meaning that they depend only on the distance between the particles. Since we know already that distance - spatial separation - is invariant in Galilean relativity, then the forces acting on the particles are also invariant.

Writing Newton's law as $$m = \frac{F}{a}$$ we see that mass is also invariant since it is a quotient of invariants.

#### Newton's Law

Finally, we note that the law itself is invariant. That is, $$F = ma$$ holds in all reference frames.