5 – Relativity
and the Speed of Light
and the Speed of Light
Relativity was used initially to help physicists in proving the idea of luminiferous ether, a mysterious substance in which light waves had to travel. The thoughts in those days was that all waves had to travel in a material. This was at a time when it had been conclusively demonstrated that light was a wave. The idea of photons had yet to catch on.
We will look at relativity and its use in trying to prove the existence of ether, and the way that the speed of light was measured. As well as his experiment to show ether, Albert Michelson demonstrated the speed of light, the value of which is accepted today.
When we measure movement, we do so against a fixed reference point. A car travelling at 30 m s-1 is moving at 30 m s-1 relative to the road. Suppose we have two cars, A travelling at 30 m s-1 and B travelling at 20 m s-1.
Relative to B, car A is travelling 10 m s-1 faster, i.e. +10 m s-1;
Relative to A, car B is travelling 10 m s-1 slower, i.e. –10 m s-1.
We can use any of these frames of reference:
Another example is an aeroplane flying at 90o to the wind:
plane is heading due North at 75
m s-1 and the wind is blowing from West to East
m s-1. We can easily work out
the resultant velocity to be 76.5
There can be three frames of reference on the ground:
The speed is 75 m s-1 heading due North;
Or 15 m s-1 due East
Or a resultant velocity of 76.5 m s-1 at 11.5o east of north.
The question that bothered physicists was whether there was an absolute fixed point relative to which all speeds could be measured.
Galileo Galilei (1564 - 1642) was one of the first people to carry out experiments on the motion of objects (which we now call mechanics). He also carried out the first thought experiments and one of these forms the basis of understanding frames of reference. Would a mechanics experiment behave in a different way depending on whether it was stationary or moving?
Consider a ship which is stationary. A stone is resting on a platform on the mast of the ship (it's a ship of Galileo's era) like this:
We would be right in expecting that the stone would fall vertically to the deck.
Now suppose the ship is moving at 5 m s-1 along a perfectly flat sea. It does not rock from side to side (roll) or pitch forwards or backwards. What happens if we drop the stone from the platform now? We will assume that the time taken for the stone to fall is 1.0 s, and this gives a mast height of 4.9 m (how?).
Does the stone hit the deck where it did before, or does it hit the deck 5.0 m behind? The answer is of course that the stone hits the deck in exactly the same place as before. This is because the stone is moving at 5.0 m s-1 along with the rest of the ship. The idea is shown below:
If you are on board the ship, you would not be able to tell the difference in the outcome. However if you are watching the stone falling while you are standing on the quayside, you will see this:
The stone would move on a parabolic path. If you want to review the behaviour of the stone as it falls, go to Mechanics 5 to revise projectile motion.
The key point is that there is no difference in outcome between a mechanics experiment being carried out in a stationary environment and one carried out in an environment that is moving at a constant velocity with zero acceleration. (As soon as there is acceleration, the outcome will be different.) We say that the ship forms one frame of reference, while the quayside is a second frame of reference. In each frame of reference, there is an observer to see what happens. The observer on the ship will see the stone falling in a vertical path, while the observer on the quayside will see the stone falling in a parabolic path.
We call this principle Galilean Relativity.
When something moves, we can describe its movement in terms of the x-axis (horizontal axis across the page), the y-axis (horizontal axis into the page) and the z-axis (upwards). Let's say that the ship is travelling at v m s-1. In a time t seconds, it will travel vt metres. For the observer on the quayside, the position of the ship can be described as a set of coordinates:
(x, y, z, t )
For the observer on the ship, the coordinates will be:
(xʹ, yʹ, zʹ, tʹ )
(xʹ is pronounced "ex-prime" or "ex-dashed" or "ex-stroke". It depends on your tutor.)
However in this case, we will keep things simple and consider the movement solely in terms of the x-axis, i.e., 1 dimension. The equations that relate the two are called Galilean transforms and are listed below:
xʹ = x + vt
yʹ = y
zʹ = z
tʹ = t
In this case, y remains the same, as is z, as is t. Time t does not vary at all in Galilean Relativity. It is called a physical invariant.
Let's suppose that the mast is at a point 15 m from the stern, as shown:
(The ship here is a modern replica.)
For an observer at the stern of the ship, the mast will not move. It will remain 15 m in front of the observer. For the observer on the quay, let's say that at time 0, the stern passes the observer. The mast is 15 m in front. If we then plot the position every second:
xʹ0 = 15 m + 5.0 m s-1 × 0 s = 15 m
xʹ1 = 15 m + 5.0 m s-1 × 1 s = 20 m
xʹ2 = 15 m + 5.0 m s-1 × 2 s = 25 m
and so on...
We can plot a distance time graph like this:
A more modern context would be a juggler playing on an aeroplane travelling at 200 m s-1 at a steady altitude of 10 000 m.
Among the many contributions that Isaac Newton has made to our understanding of physics are his two postulates. (A postulate is a thing suggested or assumed as true as the basis for reasoning, discussion, or belief.)
The first one is that there is absolute time. This is independent of any observer. We cannot observe absolute time, only the passage of time, which we do using clocks, days, months, seasons, etc.
Perhaps a way of thinking about this is to imagine getting on a train at an intermediate station. We know that the train started somewhere, but we don't know the exact time it left (yes, we can use the timetable, but we cannot be sure that it left on the dot). When we get off the train at another intermediate station, we know that the train will go on to its terminal station, but we don't give a damn whether it arrives on time there. All we are interested in is the time we are on the train.
Absolute time, according to Newton, was something that could only be understood mathematically.
Newton's second postulate was that space was absolute. It formed a backdrop to events. Every object has an absolute state of motion. For example the Earth is orbiting the Sun in a constant motion at a speed of 30 000 m s-1. The Sun is moving as well. As I sit here typing this, listening to Classic FM, I am stationary, as is my house and my garden. My surroundings are stationary in a relative sense. However they are moving in an absolute sense.
Absolute motion is the movement of objects in absolute space from one absolute position to another. Relative motion is the motion of objects from one relative position to another relative position.
A Boat Race
Both boats, X and Y have a speed of 5 m s-1. Boat X has to cross the river from A to B and back to A, while Boat Y has to travel from A to C and back again.
(a) What is the velocity of X relative to the river bed (Use a vector diagram)?
(b) What is the time for X to travel from A to B to A?
(c) What is the velocity of Y relative to the river bed going from A to C and the time?
(d) What is the velocity of the boat Y going from C to A and its time?
(e) Which boat wins and by how much?
Write down Newton’s First Law of Motion.
An inertial frame of reference is one in which Newton I is valid. If you are in a train travelling at constant speed, all objects behave as if they were stationary in the stationary train. The train is travelling at 60 m s-1, the passengers and their luggage are all travelling at 60 m s-1.
Suppose now that you are in an aeroplane. Against all airline regulations, there is a drinks trolley free (not secured) in the central aisle. The aeroplane accelerates down the runway. From within the plane the trolley appears to accelerate towards the back of the plane.
From the ground, the trolley obeys Newton I since there is zero force acting on it, hence zero movement.
From within the plane, an accelerating frame of reference, the trolley appears to accelerate, which is not consistent with Newton I.
Why is it not consistent with Newton I?
Now consider this situation. A person is standing at the centre of a roundabout. He has a gun. A target is placed outside the roundabout as shown
When the roundabout is stationary, it is easy to see that the path of the bullet is straight. What about when the roundabout is turning?
For an observer on the ground the path of the bullet will be a straight line. For the person on the roundabout, the path will appear curved.
Why is this not consistent with Newton I?
What kind of frame of reference is the roundabout? Explain your answer.
In order to explain wave phenomena such as light waves, the late nineteenth century physicists depended on a medium called ether. (It has nothing to do with diethyl ether, an explosively flammable compound used in organic chemistry.) Ether was a mass-less and non-viscous material that was needed to carry waves. Ether is used nowadays as a poetical word to describe radio-broadcasting.
If ether permeated the whole of space, then it would provide a perfect frame of reference to determine absolute motion. The experiment was carried out in 1887 by Albert Abraham Michelson (1852 – 1931) and Edward Williams Morley (1838 – 1923). Their idea was to measure the speed of light parallel to the Earth’s motion with the speed of light perpendicular to it. It would be rather like the boat race example we saw above.
They used the physics of optical interference in a set up like this:
Light is split into two perpendicular beams.
They travel to the mirrors and superpose as they return to give interference fringes.
If the distance between the half-silvered mirror and m1 is the same as the distance between the half silvered mirror and m2, the time taken would be different.
This would indicate a shift in the expected interference pattern.
The experiment was repeated with the equipment set at 90o to the orientation of the first experiment, so that the motion in the ether would be observed in two different directions.
It was repeated at different times of the year in case the sun at one point or another was moving in the same frame of reference.
Although the Earth orbits about the Sun, and is technically an accelerating frame of reference, we can generally treat it as an inertial frame of reference.
The results were the most important null (nothing) result of the time:
There was no difference in the speed of light whichever way the experiment was done, or whatever the time of year;
There was no absolute reference point;
There was no such thing as ether.
Michelson never gave up on the idea of ether. He tried again and again, but never found it.
Einstein's Theory of Special Relativity suggested
that ether could not exist, and that the speed of light is a universal constant.
Experiments to Measure the Speed of Light
The measurement of the speed of light foxed the early physicists. Galileo tried it, but his results were inconclusive. It is not easy. Many thought that the speed of light was infinite. The first scientist to give an estimate of the speed of light was Ole Römer (1644 - 1710), a Danish astronomer. From astronomical observations of one of Jupiter's many moons, Io, he came up with a figure of about 3.1 × 108 m s-1.
Why was it so hard to measure the speed of light?
The way that the speed of light can be measured is to chop it up into very small pulses. This can be achieved with rotating mirrors or with a toothed wheel. The apparatus below was devised by Hippolyte Fizeau (1819 - 1896).
Light from a bright monochromatic light source is focused onto a focal point. There is a half-silvered mirror to bend the light through 90 degrees onto the focal point. The light is picked up by Lens 1, which makes parallel rays to travel a distance d to Lens 2. The rays reflected by a parabolic mirror back to Lens 2 and Lens 1. The light passes through the focal point, and passes through the half-silvered mirror to the eyepiece.
The toothed wheel is spinning. The idea is that the ray going out is allowed to pass through the gap. The ray goes to the mirror and returns. In the meantime, the wheel has turned a small amount, but enough to block off the returning ray:
Therefore no light is observed at the eyepiece. The first time this is observed would be when the next tooth replaces the gap and blocks off the light that has returned.
The light travels from the transmitter to the reflector, a distance d (which is quite large, several kilometres). It then has to travel back again, covering a total distance of 2d. To work out the speed, we also need to know the time. That is not so easy, but can be worked out if we know how fast the toothed wheel is spinning.
Suppose the wheel, which has N teeth, is turning at n revolutions per second. Remember that for N teeth, there are N gaps. In the picture above above, there are 8 teeth, and 8 gaps. The time taken for the light to get out and back is given by:
Since speed = distance ÷ time, we can write:
Using these data:
d = 8.63 km;
N = 720 teeth;
n = 12.6 revolutions per second;
we can now work out the speed of light:
c = 4 × 8.63 × 103 m × 720 × 12.6 s-1 = 3.13 × 108 m s-1
There were some difficulties. The toothed wheel was made by a clock maker, with 720 teeth and 720 gaps. It was not easy to make. Measuring long distances could be done by triangulation with good precision. The main uncertainty would be determining the precise rate of turning of the toothed wheel. Since the experiment was done in 1849, the technology was not as sophisticated as today. So a reliable measure of the rate of rotation was less likely. Also the toothed wheel was spun, not with an electric motor, but with a clockwork mechanism. Also the light source would not be that bright - lasers were not invented. There would also have been problems with the scattering of light as it passed through the air. The experiment would need to be done on a clear day (or night).
The experiment can be reproduced nowadays with a toothed wheel driven by an electric motor. An electronic tachometer can give an accurate read out of the speed.
A similar experiment was carried out by Leon Foucault (1819 - 1868), but with a spinning mirror.
In 1926 Michelson gave the first accurate measurement of the speed of light using a system of concave mirrors and a rotating octagonal mirror. The parabolic mirror on the left was set up on a mountain that was about 36 km from the base station. The experiment had to be set up very carefully. (Light experiments in the lab are difficult to set up without an optical bench. Imaging how much more difficult it would be to set up on a hilltop.)
The experiment is done like this:
The light is reflected off the bottom face of the rotating octagonal mirror.
It passes to the parabolic mirror on the right, and is reflected to the mirror on the left which is many kilometres away.
It is reflected back to the parabolic mirror on the right, and reflected to the telescope.
When the mirror is parallel with the rays and stationary, the slit through which the light comes is visible to the observer through the telescope.
If the mirror is slightly rotated, the beam from the source gets reflected away and does not travel to the far mirror. This is shown below:
Therefore the image of the slit disappears from the observer.
If the octagonal mirror is spinning at a certain fast speed of rotation, the image appears again when the next surface appears parallel to the rays of light:
The appearance is quite abrupt, and the rate of turning has to be exact.
The time taken for the light to travel out and back is the same time for the mirror to rotate 1/8th of a turn. If the mirror is turning at n revolutions per second, the time taken for the light to travel is:
The distance travelled by the light out and back 2d. So using speed = distance ÷ time, we combine these to:
The equipment that Michelson had available was more sophisticated than Fizeau's. The octagonal mirror was spun at 512 revolutions per second using an air-turbine. The rate of turning was determined by a stroboscope. The distance was 36.3 km. So simple substitution gives us:
c = 16 × 36.3 × 103 m × 512 s-1 = 2.97 × 108 m s-1
Michelson's final answer was that the speed of light is 299 792 458 m s-1. The quoted figure of 3.00 × 108 m s-1 is quite good enough for most purposes.
There were some compromises in that during the series of experiments, there was an earthquake. Some estimates suggest that the mountain may have moved by about a metre. It was not always possible to keep the mirror turning at 512 s-1. Also at that rate of turning (31 000 rpm) the mirror could fly apart, which could be highly dangerous.
In a school physics lab it is possible to get a reasonable estimate of the speed of light in an optical fibre by sending a pulse up a length of fibre optic cable to a receiver. The transmitted and received pulses are displayed on a CRO. The time period between the pulses is measured, and the length of the fibre optic cable is measured, so it’s possible to measure the speed. The diagram shows the idea:
The CRO shows the transmitted pulse, and a fraction of a microsecond later, the received pulse. The period T can be measured, and the time worked out, knowing the time-base setting.
Would this give the speed of light quoted above?
While we may scoff at the notion of ether, we should remember that physicists had picked up their knowledge and understanding from their own teachers and colleagues. They genuinely believed it, and many found it hard to conceive that ether did not exist. We now know and take for granted the idea of light as photons that can travel in a vacuum. However the quantum models that photons obey is difficult to understand.
We have also seen how experiments, carried out in trying conditions, using quite simple, if not primitive, equipment can give us the speed of light to a value that we now take for granted.
Maxwell and the constancy of the Speed of Light
In Turning Points 3, we saw that James Clerk-Maxwell worked out the speed of electromagnetic waves using the equation that he derived:
A simple calculation using this gives us the speed of electromagnetic waves as 2.99 × 108 m s-1. You are not expected to use the Maxwell Equations; they are difficult, and will be covered at university level.
Light travels at 3.0 × 108 m s-1, regardless of whether its source is moving towards an observer, or away from the observer. Nothing can catch up with a propagating electromagnetic wave, so he argued. At those days, light was regarded as a wave. Nowadays we know that light consists of photons, which are trains of waves. Therefore nothing can travel at a speed greater than the speed of light, and this formed the basis of Einstein's Theory of Special Relativity, which we will study in the next tutorial, Turning Points 6.