1.5.1 einstein and relativity

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Topic 9.2

1.5.1 Einstein and Relativity

Galilean Relativity

In the Galilean view point, the laws of physics should hold true in all inertial frames of reference.That is, so long as there is no acceleration, all experiments should yield the same result.An experiment to measure g by using a pendulum should give the same result when it is stationary, or when it is moving at a steady speed on a train.

Frames of Reference

A frame of reference is simply a background against which measurements can be measured.Imagine a fully tiled swimming pool.The tiles form a grid on the walls and floor.The absolute position of anything can be measured relative to this grid.The absolute displacement of anything can be defined using the corner of the pool as the origin.

Galliean Relativity

Differences in observation between different frames of reference can be explained by considering the relative motion of the two frames.Consider an observer in the red frame of reference looking at the clock.They would see the clock ticking but not movingConsider an observer in the black frame of reference looking at the red clockThey would see the clock ticking and moving with speed vx. Its position would be s(x,y,z)=(vt,y,z)If the black frame were also moving (say at speed ux then the absolute velocity of the red clock would be ux - vx

Newtonian Frames of Reference

A frame of reference is said to be inertial if Newtons first law is valid for it.That is if it is not accelerating in any dimensionA frame of reference is said to be non-inertial if Newtons second law is valid for it.That is it is accelerating in at least one dimension.Newton realised that no frame of reference would be more correct than any other. Therefore the concept of absolute position is meaningless.This is especially true in space where you have no background grid to work with.Different observers can have different frames of reference, but in Newtons view, they should agree about when an event happened even if they both say it was in a different position.

A First Thought Experiment

You are sat on a (very good) train in a tunnel.You cannot hear anything.You cannot feel any movement.You are looking out of the window and see another train move from left to right past your window.What is your train doing?


Are you sat stationary in the red train and the blue train moves from left to right at speed v?

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Are you sat in the red train moving from right to left at speed v and the blue train is stationary?

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Are you sat in the red train moving from right to left at speed v and the blue train is moving at v in the other direction?

v

v


Or is there something else going on?Does it matter to the Physics?

3/2 v

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Imagine that your friend on the red train now hold 2 charges in her hands. What do you see if you are on the red train? On the blue train?

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On the red train you only see the force of electrostatic repulsion.On the blue train you see this same electrostatic force but because the charges are moving, there appears to be a current, therefore there is a magnetic force in the opposite direction.The blue train always measures more force than the red trainThe laws of physics are different to different observers!!!

FQ

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Waves and Media

All waves, such as sound waves and water waves, had been observed to require a medium in order to propagate.

Light waves should also require a medium.

This medium was called the Aether (also ether)

The Aether was thought to be a transparent, massless, colourless medium that was present everywhere including in the vacuum.

This was the theory in the 1600s through until the end of the 1800s!

Michelson and Morley

Michelson and Morley set out to detect the Aether in 1887.

They assumed that the Aether was moving with constant velocity and that the Earth moved relative to it.

They likened their experiment to two boats moving on a fast flowing river.

The two boats are capable of exactly the same speed.

One moves parallel to the bank,

The other perpendicular to it.

They both travel the same distance before turning round and coming back to the start.

Michelson and Morley

The speed of the boat is u and the speed of the water is v.The boat moving parallel to the stream has speed u+v and then u-v which averages to u.The boat moving across the stream has speed in both directions As the speeds are different, but the distance travelled the same, the time taken to travel will be different!

v

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u-v

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The Interferometer

Michelson devised a device known as an interferometer to test this theory.Because light is a wave, it can interfere with other light waves and form an interference pattern.Differences in times of travel result in a phase difference which causes a change in the pattern.As the Earth rotated around the Sun, it was expected that changes in the interference pattern would be observed.

The Interferometer

The Results

Michelson and Morley detected no systematic change in the interference pattern.Any changes that were observed were random errors.More sensitive equipment has since been built and to this day no detectable change has been reliably observed.

A Failed Experiment?

The Michelson-Morley experiment is possibly the most famous failed experiment.However, whilst the experiment failed to detect an Aether, it did suggest that the postulate that the Aether exists was incorrect.This caused other Hypotheses to be put forward, including Einsteins special relativity.So the experiment did not truly fail.

An Unexpected Result

As shown using vectors, the speed of light should be different in the two directions.However, the time of travel is observed to be the same, as is the distance of travel. Therefore the speed of light is the same in all directions, regardless of motion.This is compatible with Maxwells equations which showed that the speed of light is given by:

The implications of Einsteins Postulates

1) Two events that are simultaneous to one observer may not necessarily appear simultaneous to another observer in a different frame of reference.

Simultaneity of Events

Consider a train moving at high speed, v, through a station.As the train passes through 2 small explosions are set off at either end of the train.The two explosions are equidistant from a stationary observer on the platform.By simple calculation, the time taken for light to travel from the head of the train (t) is given by and from the tail of the train (t) is

To the observer on the platform the events were simultaneous.

d'

d

Simultaneity of Events

Consider an observer now on the train moving with high speed v.The two explosions are still set off as before.However, in the time taken for the light to travel, the observer has moved forward by vt metres.By simple calculation, the time taken for light to travel from the head of the train (t) is now given by and from the tail of the train (t) is

To the observer in the train the events were not simultaneous.

d'

d

d'

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vt

Time Dilation

Zoe is travelling in her car at speed v past Jasper on his verandah.She has a type of clock that measures the time by reflecting a light ray between two mirrors.Jasper can also see the light ray bouncing and use this to measure his own time.

Time Dilation

Before the experiment begins, Zoe parks her car next to Jaspers verandah and they both agree on the timing and both get the same result.Zoe then drives from left to right at 0.8c according to Jaspers frame of reference.Of course in Zoes frame of reference it is Jaspers verandah that moves from right to left at 0.8c!

Time Dilation

The red counters count each time Jasper or Zoe sees a reflection.

Time Dilation

Say the car is w wide and the time Zoe measures between reflections is t.Then the distance is given by

However, if t is the time to tick on Jaspers clock, he records the distance travelled (using Pythagoras) as

Time Dilation

Combining the previous equations gives:

This rearranges to:

This is also written as:Where tv is the time on the
moving clock as measured
by a stationary observer. t0 is the time on the moving clock as
measured by an observer in the
same moving frame of reference.

Time Dilation

In summary, a moving clock appears to run slower according to

Length Contraction

If the speed of light is constant, and time gets shorter (dilates) for a moving object, then the length of the object must get smaller as measured in the direction on motion.

lv is the length of the object when it is movingl0 is the rest length of the object.

Mass Dilation

Experiments in the early 1900s carried out by Kaufmann showed that the charge/mass ratio of high speed electrons decreased with increasing speed.However, the charge carried by the electrons was a constant.Therefore the mass must have been increasing.

Mass Dilation

Einsteins theory of special relativity showed how this could happen.

Practice.

Consider a pen of mass 100g and length 15cm. What would be its length and mass, as measured by a stationary observer, when it is moving at 340 ms-1 and 3x107 ms-1 relative to the observer?

Practice

NASA have invented a rocket that can travel at 95% of the speed of light. How long will it take to get to the nearest star, 4.5ly away as measured by the scientists at Houston and by the astronauts on board?

Practice

A type of subatomic particle called a muon has a half-life of 2s. What will be the half-life of these particles if they are travelling at 0.99c in the laboratory?

What is a Thought Experiment

Einstein's understanding of relativity came about due to his use of thought experiments.

These are experiments where the logic of the situation, and hence the results, is flawless BUT the situation is usually impossible (or at least highly improbable) to replicate in reality.

However, just because the experiment cannot really be done does not mean that the result cannot be significant!

Common limitations of thought experiments.

Trains and spacecraft cannot travel at relativistic velocities (due to the consequences of special relativity!!)

An observer outside of the train (travelling at nearly c) would find it impossible to physically make any valid or accurate observations of events inside the train.


It is impossible to see the light beam travelling from the source to the mirror and back.

Either the whole train carriage lights up (the light spreads out)

Or you need a laser beam and to have dust particles in the path to scatter it.

Either way the light becomes spread out.


It is impossible engineer an infinitely long train and perfectly identical fireworks.

The relative simultaneity experiment (fireworks at either end of a long train) requires both of these in order to register any noticeable effect.

The implications of Special Relativity.

As an object increases its speed towards c, its mass increases (or dilates)

This means that it becomes even more difficult to accelerate the object ( )

Even at the maximum theoretical speed (~0.99c), with the biggest engines possible, it would take over 4 years to reach just out to our nearest star. The rest of the galaxy is just too far away!

The implications of Special Relativity

Consider two identical twins. One remains on Earth whilst the other takes a long trip to a distant star on a spacecraft that can travel at 0.99c.

From the Earth twin's point of view, the space twin is moving whilst they stand still. Therefore the Space twin's clock is running more slowly. Therefore the Space twin will be younger when they return.

However, from the space twin's point of view, the Earth twin may just as well be moving whilst they sit still. Therefore the Earth twin's clock runs slower and therefore the Earth twin will be younger.

They can't both be younger can they?

This is the so called twins paradox.


The twins paradox is really not a major problem once you consider the limitation of Special Relativity.

Special relativity is ONLY valid in an inertial frame of reference!

The space twin must have accelerated to 0.99c, turned around the distant star, and decelerated again.

Therefore the space twin has not always been in an inertial frame of reference, therefore their conclusion is invalid.

The space twin would indeed be the younger of the two.


When the spacecraft is moving forward through space at relativistic speeds it appears that the space is moving towards the spacecraft.

Therefore the distance will appear shorter to the pilots than the distance measured by an external observer.

This is a consequence of a slower ticking clock (time dilation) and a constant speed of light.

Measuring Time

The passage of time in an inertial frame of reference can be very accurately measured using a pendulum of a specific length.

However, this requires knowledge of the gravitational field strength to be useful.

It is also mostly useless in non-inertial frames of references (i.e. the deck of a boat that is rocking up and down)

Measuring Time

Watchmakers developed more and more precise devices to measure time but the definition of the second was still as being 1/8600 of the time for the Earth to rotate on its axis.

In 1967, following observations that the Earth's rotation was slowing (very) slightly a new definition of the second was formulated.

1 second is the time taken for an atom of Cs-133 to oscillate 9129631770 times. In theory this definition will remain accurate everywhere in the Universe.

Measuring Distance

The metre was originally defined as being one ten-millionth of the distance from the North pole to the Equator passing through Paris.

This distance was then scored onto 3 platinum bars kept in Paris.

This allowed a common scale of distance to be used by everyone and eventually has replaced a much more complex English system of Feet, Yards, Fathoms, Chains, Furlongs and Miles in most of the world.

Measuring Distance

As the surveyors had actually made a slight mistake in their calculations for the metre, the bars became the standard rather than the original definition.

These bars were later replaced with a more stable platinum-iridium bar.

Measuring Distance

The need to have an even more precise standard of length that did not rely on referencing a metal bar in a vault somewhere brought scientists to basing their definition in the atom.

The metre is now defined as being the distance travelled by light in 1/299792458 of a second.

Note how this requires that the speed of light is precisely known!

1.5.1 einstein and relativity

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