VCE Physics/Unit 3/AoS 3/Special relativity

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Introduction

Relativity is a concept that appears in Newtonian physics in terms of "inertial reference frames" and "relative motion". These relativistic ideas go back at least as far as Galileo and are often referred to as "Galielean Relativity". So we will first discuss Galilean Relativity before moving on to Special Relativity. This will help you understand what is "special" about "Special Relativity".

Inertial Reference Frames

One Inertial Reference Frame

A reference frame is a set of objects we can use as a reference for the location and motion of other objects. So imagine we are sitting in a train carriage. We will take the train carriage as our reference frame. We will take the length of the train carriage to be our x-axis. We will call the left hand end x = 0 (the origin) and the positive x-direction will be from left to right.

diagram

Throughout this topic we will keep things simple by only using one dimension (we will not bother with a y and z axes).

We are sitting facing in the positive x-direction. There is a table in front of where we are sitting. You place a block of ice on the table (using ice so we can ignore friction between the table and the object). If the ice stays where we place it then our carriage is an "inertial frame of reference". The carriage is not accelerating - it is either at rest relative to the train tracks, or moving at a constant velocity. So, an inertial frame of reference is a frame of reference that is not undergoing acceleration.

If the carriage accelerated in the positive x-direction we would feel like we were being pushed back into our chair, the ice would move towards us across the table. The carriage would no longer be an inertial reference frame - we experience "pseudo-forces" due to the acceleration of the carriage.

We are cheating slightly. If we hold the ice in the air and let it go, the ice falls to the ground. There is an "acceleration" towards the ground. To experience an inertial reference frame we should really be in a spaceship in outer space away from any gravitational influences. However, this is an unfamiliar environment and so, to be able to at least sometimes use a common environment for our examples, we will take the surface of the earth as an approximate "inertial reference frame".

Other Inertial Reference Frames

We now pass through a train station. There are people standing on a platform. The platform, and the people on the platform, are in a different "inertial reference frame". Why is this important? Well, if they make measurements of things. They are different to our measurements. BUT the physics is the same.

Speed of light

You should be familiar with modelling light as an electromagnetic wave. You should be familiar with various types of waves such as water waves, sounds waves and waves on a string (such as the waves on a guitar string when you pluck a guitar). You should understand that

• the speed of a wave is relative to the medium through which it travels.
• light "does not have a medium".
• the speed of light is the same in every inertial frame of reference.

Time dilation

How do we measure the time of a relatively moving clock? two events that happen in the same location in one frame of reference = one clock vs two clocks

Length contraction

How do we measure the length of a relatively moving object? Measuring the distance between the simultaneous location of the ends of a relatively moving object.

Space-time

It might not seem like it, but the behaviour of space (length) and time in special relativity (in terms of length contraction and time dilation) sensibly fit together.

The following scenario is an example of how this happens.

You are at the front of a train with a watch. The train is passing a platform at a station. The proper length of the platform is $L_{0}$. How does the measurement of the speed of the train (and you) by the people on the platform, compare to the speed you measure the platform to be moving past you?

In your IFoR:

You are stationary and the platform is an object moving relative to you. So in your IFoR, the platform is length contracted. So in your IFoR, you measure the length of the platform as $L_{0}/\gamma$.

The start of the platform passes you, and then the end of the platform passes you. The two events happen in the same location i.e. where you are at the front of the train. Or, similarly, you only need to use one clock to determine the time between the two events. So you are measuring the proper time $t_{0}$.

So simply using speed = distance/time, you determine the speed of the platform passing you as $v={\frac {L_{0}/\gamma }{t_{0}}}={\frac {L_{0}}{\gamma t_{0}}}$

In the IFoR of the platform:

The platform is stationary, the train (and you) are moving past the platform. The platform is stationary in the IFoR of the platform, so in the IFoR of the platform the length you have travelled in this time is just the proper length of the platform, $L_{0}$.

In this IFoR, the two events occur at different locations (to the left and right of the platform). Or, similarly, people on the platform need two clocks, one either end of the platform. So the people on the platform measure dilated time between the two events $t=\gamma t_{0}$.

So simply using speed = distance/time, people on the platform measure the speed of the train (your speed)as $v={\frac {L_{0}}{\gamma t_{0}}}$. Which is the same speed as you measured for the platform moving past you.

For the world "to work" the measured speeds need to be the same. The way in which length contraction and time dilation work leads to this being true. This is just one example of how time dilation and length contraction "work together" to provide a coherent theory.

Energy

Some stuff about energy.

Relevant dot points from the Study Design

• describe Einstein’s two postulates for his theory of special relativity that:
  • the laws of physics are the same in all inertial(non-accelerated)frames of reference
  • the speed of light has a constant value for all observers regardless of their motion or the motion of the source
• compare Einstein’s theory of special relativity with the principles of classical physics
• describe proper time ($t_{0}$) as the time interval between two events in a reference frame where the two events occur at the same point in space
• describe proper length ($L_{0}$) as the length that is measured in the frame of reference in which objects are at rest
• model mathematically time dilation and length contraction at speeds approaching c using the equations:

$t=t_{0}$ and $L={\frac {L_{0}}{\gamma }}$ where $\gamma =\left(1-{\frac {v^{2}}{c^{2}}}\right)^{-{\frac {1}{2}}}$

• explain why muons can reach Earth even though their half-lives would suggest that they should decay in the outer atmosphere.

• interpret Einstein’s prediction by showing that the total ‘mass-energy’ of an object is given by: $E_{tot}=E_{k}+E_{0}=\gamma mc^{2}$ where $E_{0}=mc^{2}$, and where kinetic energy can be calculated by: $E_{k}=\left(\gamma -1\right)mc^{2}$
• describe how matter is converted to energy by nuclear fusion in the Sun, which leads to its mass decreasing and the emission of electromagnetic radiation.