HSC Physics – Core Module 1 – Space

4. Current and emerging understanding about time and space has been dependent upon earlier models of the transmission of light

Outline the features of the Aether model for the transmission of light.

In early19th century it was proposed that light was in the form of a wave. Having concluded that light moves as a waveform, nineteenth-century physicists turned to other wave motions in order to better understand light. All other known waveforms at the time required a medium, thus it was also believed that light required a medium through which to travel.

Nobody could find such a medium but belief in its existence was so strong that it was given a name, the ‘luminiferous aether’ and its properties were identified. The aether:

- Filled all of space, had low density and was perfectly transparent- Permeated all matter and yet was completely permeable to material objects.- Had great elasticity to support and propagate the light waves

This list may seem odd in hindsight, but nineteenth century physicists were attempting to explain an unknown phenomenon.

The search for aether was to occupy physicists for several decades before it was finally accepted that:

1) Aether does not actually exist.

2) Electromagnetic waves (including light) are unique in that they do not require a medium of any sort in order to move.

Describe and evaluate the Michelson-Morley attempt to measure the relative velocity of the Earth through Aether

If you were in a boat, how could you tell whether the boat was moving? You might simply look over the side to see if the water is flowing past. If you wanted to be certain, you might feel the flow of water against your hand.

Similarly, if the aether did exist, our Earth, moving through space at about 30 km/s as it orbits the Sun, should be moving through the aether. Thus we should experience a flow of aether past us, known as aether wind. However, the aether was thought to be extremely tenuous, so any aether wind would be hard to detect.

The definitive experiment to detect the aether wind was performed by A. Michelson and E. Morley in 1887 for which the received the Nobel Prize in 1907.

Consider the analogy below:

Two boats are going to have a race, where both boats will complete a 4km circuit at a speed of 5 km/h.

If the objective of the race was to determine the speed of the current, then it could be calculated from the difference in arrival times of the two speedboats.

Also by repeating the race with boats interposed – A heading across the river and B heading along the river – any difference between the boats could be eliminated as a cause for the time difference as boat A should now win by the same margin.

This is essentially what Michelson and Morley did – they raced two light rays over two courses, one into the supposed aether wind and one across it, then swung the apparatus through 90 degrees to interpose the rays. They were looking for a difference between the rays as they finished their race, from which they could calculate the value of aether wind.

The method of comparing the light rays involves a very sensitive effect called interference, and hence this apparatus is referred to as the interferometer. Essentially, when looking into the telescope a pattern of light and dark bands will be seen. If aether wind exists, so that one light ray is indeed faster than the other, then when the apparatus is rotated, so that the rays are interposed, the interference pattern should be seen to shift. However, no such shift was observed.

The experiment was repeated many times by Michelson and Morley at different times of the day and year, but no evidence of an aether wind was ever found.

This resulted in the idea of aether being adapted and changed, but these failed close scrutiny. The Michelson-Morley experiment was repeated many time since 1887 by different groups with more and more sensitive equipment, and no evidence of an aether has ever been found.

Yet, it was not until 1905, when Albert Einstein showed that aether was not necessary, was the idea let go.

Discuss the role of the Michelson and Morley experiments in making determinations about competing theories

Michelson and Morley believed that because the Earth was moving through the Aether, a light beam on the earth would experience aether wind and the speed on the direction of motion with respect to the wind.

The ‘null result’ for the Michelson and Morley experiment could only mean that there was no aether and that the speed of light in a vacuum was independent of the speed of the source and the observer.

That is the speed of light is a universal constant for all observers independent of their velocity.

The Michelson-Morley experiment was crucial because it enabled one of the main predictions of aether model to be experimentally tested.

Experiments like this can be used to determine which of two competing theories is correct. In this case, whether the aether exists or not.

Outline the nature of inertial frames of reference

Galilean Theory of Relativity – “The law of mechanics are the same in all inertial frames of reference”

An inertial frame of reference is one at rest or moving with a constant velocity.

OR

“An inertial frame of reference is one in which Newton’s first law holds”.

An inertial frame of reference is a non-accelerated environment. Only steady motion or no motion is allowed. A non-inertial frame of reference experiences acceleration.

Discuss the principle of relativity

Three hundred years before Einstein, Galileo posed a simple idea, now called the ‘principle of relativity’, which states that all steady motion is relative and cannot be detected with out reference to an outside point.

This idea can be found built into Newton’s First Law of Motion as well.

Simply, if you are travelling inside a vehicle you cannot tell if you are moving at a steady velocity or standing still without looking out the window.

Consider two trains: When sitting in a train and an adjacent train begins to roll – at first you may thing that your own train is moving until you look out of a window at the other side of the carriage.

However, there are two points that must be reinforced.

- The principle of relativity applies only for non-accelerated steady motion; that is, standing at rest or moving with a uniform velocity. This is referred to as an inertial frame of reference. Situations that involve acceleration are referred to as non-inertial frames of reference.

- This principle states that within an inertial frame of reference you cannot perform any mechanical experiment or observation that would reveal to you wether you were moving with uniform velocity or standing still.

An example of this can be seen if we hold a pendulum in a train. Initially the pendulum would hang vertical, but as the train begins to accelerate, the pendulum would no longer be vertical but rather it would swing backwards. This would continue until the train reached a constant speed, where the pendulum would again hold its initial vertical position.

In the late nineteenth century, belief in the aether posed a difficult problem for the principle of relativity, because the aether was supposed to be stationary in space and light was supposed to have a fixed velocity relative to the aether. This meant that if a scientist set up equipment to measure the speed of light from the back of a train carriage to the front, and it turned out that the light was slower than it should be, the train must be moving into the aether.

Describe the significance of Einstein’s assumption of the constancy of the speed of light.

At the turn of the twentieth century, Albert Einstein puzzled over the apparent violation of the principle of relatively posed by the aether model. He had an ability to reduce a problem down to its simplest form and present it as a thought experiment.

In this case the question posed was: If I were travelling in a train at the speed of light an I held up a mirror, would I be able to see my own reflection?

If the aether model was right, light could go no faster than the train. It could never catch up with the mirror to return as a reflection.

The principle of relatively is thus violated because seeing one’s reflection disappear would be a way to detect motion.

On the other hand, if the principle of relativity were not to be violated, the reflection must be seen normally, which means that it is moving away from the mirror holder at normal speed.

However, this would mean than an observer on the embankment next to the train would see that light travelling at twice its normal speed.

This was a considerable dilemma but Einstein decided that the principle of relativity must not be violated and the reflection in the mirror must always be seen.

This in turn meant that aether did not exist.

As a way out of the dilemma, he also decided that the train rider and the person on the embankment must both observe the light travelling at its normal speed.

Einstein realised that if both observers were to see the same speed of light and since speed = d/t , then the distance and time witnessed by both observers must be different.

Hence, the significance of Einstein’s assumption of the constancy of the speed of light was that it showed two things;

1) Aether does not exist and is not necessary to explain of light

2) Time and space (distance) are relative quantities to the speed of the observer.

These ideas were explicitly states in Einstein’s 1905 paper, which presented:

1) A first postulate: The laws of physics are all the same in all frames of reference; that is, the principle of relativity always holds

2) A second postulate: The speed of light in empty space always has the same value, c, which is independent of the motion of the observer; that is everyone always observers the same speed of light regardless of their motion.

3) A statement: The luminiferous aether is superfluous; that is, it is no longer needed to explain the behaviour of light.

Identify that if c is constant then space and time become relative

In Newtonian Physics, distance and velocity can be relative terms, but time is an absolute and fundamental quantity.

For example, again consider the situation of the train rider, the mirror and the observer on the embankment.

The velocity of the light is characterised by two events – the light leaving his face and the light arriving at the mirror. Remembering that the train is travelling at the speed of light, Newtonian physics says that the observer on the embankment outside the train records precisely double the distance of journey of the light compared with that recorded by the train rider; however they both record the same time.

Since v = , this means that the observer on the embankment would measure a velocity

of light twice that measured by the train rider.

However, Einstein’s theory says that this will not occur. Rather, both the observer on the embankment and the train rider will measure precisely the same value for the velocity of light, called ‘c’. He realised that this could only be true if the observer and the rider observed different times as well as different distances in such a way that distance divided by time always equals the same value, c.

Einstein radically altered the assumptions of Newtonian physics so that now the speed of light is absolute, and space and time are both relative quantities that depend upon the motion of the observer.

In other words, the measured length of an object and the time taken by an event depend entirely upon the velocity of the observer.

Further to this, since neither space nor time are absolute, the theory of relativity has replace them with the concept of the space-time continuum. Any event then has four dimensions (three space co-ordinates plus a time co-ordinate) that fully define its position within its frame of reference.

The Lorentz Transformation

In order to reinforce the need for the space-time concept, consider a more general form of the problem mentioned earlier.

A train traveller holds up a mirror to look at is reflection. The train is travelling in the positive x direction. At the very moment (that is time t=0) that the traveller passes an observer on the embankment outside, a light ray leaves the face of the traveller and makes its way to the mirror. A short time later, the light reaches the mirror.

The arrival of the light at the mirror can be marked by the observer within is stationary frame of reference by noting three space co-ordinates, x, y and z, as well as one time co-ordinate, t.

The traveller will similarly note four co-ordinates, this time within his moving frame of reference, as x’, y’, z’ and t’.

According to the classical physics adhered to prior to Einstein, these two sets of co-ordinates will relate together according to the following set of equations, known as the “Galilean transformation equations”.

This simple set of set of equations works quite well in explaining our everyday experiences of motion where velocities are small compared with the speed of light; however, it has problems. It states that time is identical in both frames and implies that the traveller and observer see light travelling at different speeds.

The theory of relativity states explicitly that the speed of light is identical in both frames. If this condition is expressed mathematically, the transformation equations change into:

This set of equations became known as the Lorentz transformation equations’

Lorentz (the deriver) admitted, he was still trying to apply them to the aether and did not realise their true significance, as Einstein did.

Note: if v is small, these equations simplify down to the Galilean equations.

Looking at the first and fourth equations in the set of Galilean transformation equations we can see that the moving traveller records a different distance, x, and time, t, to hat recorded by the observer on the embankment; however, they both observe light moving at the same speed, c.

More over, the expression of distance depends upon time, and the expression for time depends upon distance. It is this entanglement of space and time that shows that they must be considered together as a single concept, now known as space-time.

Discuss the concept that standards are defined in terms of time in contrast to the original metre standard.

The metre as a unit of length was first defined in 1793 when the French government decreed it to be 1x10^-7 times the length of the Earth’s quadrant passing through Paris.

This arc was surveyed and then three platinum standards and several iron copies were made. When it was discovered that the quadrant survey was incorrect, the metre was defined was the distance between two marks on a bar. In 1875 the SI of units was set up so the definition became more formal: a metre was the distance between two lines scribed on a single bar of platinum-iridium alloy. Copies were made for the dissemination of the standard. There is always a need for the accuracy of a unit of measure to keep pace with improvement in technology and science, so the metre has been redefined twice.

The current definition of the metre uses the constancy of the speed of light in a vacuum (299 792 458 m/s) and the accuracy of the definition of a second (9 129 631 770 oscillations of the 133 Cs atom), to achieve a definite that is both highly accurate and consistent with the idea of space and time.

One metre is now defines as the length of path travelled by light in a vacuum during the time interval of 1/299 792 458 of a second.

The term ‘light year’ is a similar distance unit, being the length of path travelled by light in a time interval of one year. One light year is approximately equal to 9.46728 x 10^12 km.

explain qualitatively and quantitatively the consequence of special relativity in relation to:

– the relativity of simultaneity – the equivalence between mass and energy – length contraction – time dilation – mass dilation

The relativity of simultaneity

As a way of better understanding how time is affected by relativity, Einstein analysed our perception of simultaneous events. He pointed out that when we state the time of an event, we are, in fact, making a judgement about simultaneous events. For example, if we say “school bell rings at 9am”, then we are really saying that the ringing of a certain bell and the appearance of 9am on a certain clock are simultaneous events

Einstein contended that if an observer sees two events to be simultaneous then any other observer, in relative motion to the first, generally will not judge them to be simultaneous. In other words, simultaneous events in one frame of reference are not necessarily observed to be simultaneous in a different frame of reference.

This is known as the relativity of simultaneity.

Consider, Einstein’s thought experiment:

An operator of a lamp rides in the middle of a train carriage. The doors at either end of the carriage are light operated. At an instant in time when the operator happens to be alongside the observer on the embankment the operator switches on the lamp which, in turn, opens the doors.

The operator of the lamp will see the two doors opening simultaneously. The distance of each door from the lamp is the same and light will travel at the same speed (c) for both forward and backward so that each door receives the light at the same time and they open simultaneously.

The observer on the embankment however, sees the situation differently. After the lamp is turned on, but before the light has reached the doors, the train has moved so that the front door is know further away and the back door is closer. He sees the light travelling both forward and backwards at the same speed (c) but the forward journey is now longer than the backward journey, so that the back door is seen to open before the front door. They are most definitely not judged to be simultaneous events.

It is tempting to ask who is correct – the operator in the train or the observer on the embankment. The answer is that they both are. Both observers judged the situation correctly from their different frames of reference and this is a direct consequence of the constancy of the speed of light.

The relativity of time

For this experiment, we return back to the train scenario favoured by Einstein. Imagine a traveller, seated in a speeding train. The light clock is arranged vertically, with the lamp at the ceiling and the mirror that the floor.

An observer is watching from the embankment outside the train.

First consider the situation in the rest frame.

If L is the height of the carriage, for the total journey we can say that:

Distance = 2L = c t0

where

t0 is the time taken as seen by the travellerL is the height of the carriage.

Therefore:

Examine now the situation as seen by the observer on the embankment.

From outside the train the observer sees the light travelling long a much longer journey, and its length can be calculated using Pythagoras’ Theorem:

Clock at rest

L

Clock moving at v

ct

vt

Two identical light clocks: one at rest, one moving relative to us. The light blips in both travel at the same speed relative to us, the one in the moving clock goes further, so must take longer between clicks.

t is the time from one mirror to the other.

Note: that to is the time taken for the clock to go click as observed by the train traveller, while tv is the time taken as observed by the person on the embankment.

Looking at the last expression above, we can see that the term is always less than one, so that tv s always greater than t0

This means that the clock takes longer to clock as observed by the person on the embankment, or put another way, the outside observer hears the light clock clicking slower than does the train traveller. Time is passing more slowly on the train as observed by the person outside the train.

This effect is known as time dilation and can be generally stated as follows:

“The time taken for an event to occur within its own rest frame is called the proper time t0. Measurements of this time, tv, made from any other inertial reference frame in relative motion to the first are always greater. The degree of time dilation varies with velocity.”

It can be most simply stated as: moving clocks run slow.

This conclusion has been experimentally verified by comparing atomic clocks that have been flown over long journeys with clocks that have remained stationary for the same period. These experiments are possible only because of the extreme accuracy of atomic clocks built over the last few decades, even though Einstein predicted this effect about 100 years ago.

Further supporting evidence has been found in the abundance of mesons striking the ground after having been created in the upper atmosphere by incoming cosmic rays. The surprising factor here is that mesons have a speed of about 0.996c and at this speed it should take them 16 to travel through the atmosphere. However, when measured, mesons have an approximate life of 2.2 . This anomaly can be explained by the fact that time dilation is the result of them being able to reach the ground.

The relativity of length

As a consequence of perceiving time differently, observers in differing frames of reference also perceive length differently; that is, lengths that are parallel to the direction of motion.

In order to understand how this occurs we will construct a thought experiment.

This time our train traveller has arranged the light clock so that it runs the length of the train, with the lamp and sensors located on the back wall and the mirror on the front wall. As the train passes the observer on the embankment, the light clock emits a light pulse which travels to the front wall and then returns to the back wall which is picked up by the sensor.

This journey is observed by both, but what is the length of the journey that each perceives?

Begin with the situation as seen by the train traveller

The situation seen by the observer at the side of the track is somewhat different because the train is moving at the same time, lengthening the forward leg of the light pulse’s journey and shortening the return leg.

If t1 is the time taken for the forward part of the journey, then:

Similarly, t2 is the time taken for the return so that:

The time taken for the while journey as seen by the observer on the embankment is:

It is now crucial to appreciate that each observer perceives time differently. To take that into account, we need to equate the time dilation equation to the one just derived:

Returning to the thought experiment, this equation means that, since the term is always less than one, the length of the train as observed by the person on the embankment is less than that observed by the person inside the train. The person outside the train has seen the train shorten, and the faster it goes the shorter it gets.

The effect is called length contraction and can be stated as:

“The length of an object measured within its rest frame is called its proper length, L0, or rest length. Measurements of this length, LV made from any other inertial frame of reference frame in relative motion parallel to that length, are always less.”

It can be most simply stated as: Moving objects shorten in the direction of their motion.

Another result of relativity is the observation of the contraction length as the velocity increases.

Note, that as the velocity approaches the speed of light, the observed length approaches 0.

The equivalence between mass and energy + Mass Dilation

Mass of an object also appears to change, as measure by someone in a different frame of reference. The term relativistic mass is used and plotted against velocity below:

The figure shows that as velocity increases the relativistic mass does not vary much from the rest mass until the velocity begins to approach the speed of light. After this, the mass quickly become infinite.

It is this increase in mass that prevents any object from exceeding the speed of light, because as it accelerates to higher velocities its mass increases, which means that further accelerations will require ever greater force. This is further complicated by time dilation because, as speeds increase to near light speed, any applied force has less time in which to act. The combined effect is that as mass becomes infinite and time dilates, an infinite force would be required to achieve any acceleration. Sufficient force can never be supplied to accelerate beyond the speed of light.

If a force is applied to an object, then work is done on it. This also means that energy is given to the object. This would usually take the form the increased kinetic energy as the object speeds up. But at near light speed, the object does not speed up. The applied force is giving energy to the object and the object does not acquire the kinetic energy as expected. Instead, it acquires extra mass.

Einstein inference was that mass (or inertia) of the object contained the extra energy.

Relativity results in a new definition of energy as follows:

Note: When an object is stationary, kinetic energy = 0. But it still has some energy due to its mass. This . energy is called its mass energy or rest energy and is given by:

This has been proven experimentally, most dramatically as the energy released by a nuclear bomb.

Mass Dilation is given by the formula:

Discuss the implications of mass increase, time dilation, and length contraction for space travel

Example:

Designers of a new kind of space craft known as a light sail make the remarkable claim that these crafts could

journey to Proxima Centauri, our closest star, at the speed of 0.1c. This is far in excess of current achievable velocities. Assuming it to be true, how long would such a journey take?

The distance to Proxima Centauri is 4 light years or 3.7869 x 10^13 km. At the speed of 0.1c (1.08 x 10^8 km/h). The time taken is calculated as:

When distances and speeds are this large, a simpler calculation results of the distance is expressed in light years, and the speed is expressed in terms of c:

However this will be time taken as observed from Earth. The space travellers within the spacecraft will according to relativity, record a slightly different travel time. There are two ways to calculate it.

This example illustrates the influence that relativity can have upon space travel when speeds become ‘relavisitic’, which usually means 10% of the speed of light or faster. When speeds are less than this, the effects are almost negligible.

However, when speeds get closer to speed of light – eg. Greater than 0.9c, then the effects sharply increase.

Astronauts in orbit, or even modern speedsters (100000km/h) will not notice any changes. But when the speeds get close to light speed, trips to proxima Centuri seem probable.

For example: If a spacecraft, were to travel at 99.99% of the speed of light, in one earth day (ie. As observed from Earth), would compare to only 20 minutes observed by the astronauts. Furthermore, lengths have contracted to just 1.4% of their original lengths, and the four year light trip to Proxima Centuri would be completed in just over twenty days according to the ships clock.

This poses the question, a trip that takes light four years to travel can theoretically now be done in 20 days. How?

The answer is that the trip still takes four years as observed from Earth; however the clocks that run on the star ship both electronic and biological are running slow so that by their reckoning only 20 days pass.

It should be mentioned that the energy costs of achieving these types of speeds would be prohibitive, even assuming that such speeds were technically possible.

Acceleration is always the most energy costly phase of a space mission.

As mentioned earlier, the effect of mass accumulation and time dilation to require accelerations beyond 0.9c involve ever greater forces and energy input for only marginal increases.

The Twin Paradox

Einstein himself suggested one of the strangest effects of relativity. His idea was that a living organism could be placed in a box and taken on a relativistic flight and returned to its starting place almost without ageing, while similar organism that had remained behind had long since died of old age.

Consider this problem using twins, Martha and Arthur.

Martha steps aboard a star ship that travels at 99.99% light speed. She travels to proxima Centuri. Taking a journey of 20.5 days. But for Arthur, watching closely though telescope, four years have passed. Martha immediately turns the ship around and returns. Upon arrival she is only 41 days older than when she left; however, Arthur has aged eight years waiting for her.

This problem is often considered as a paradox, because the principle of relativity demands that no inertial frame of reference be preferred over others. In other words, relativity’s effects should be reversible simply by looking at them from a different view point.

If this problem is viewed differently, then it is Martha who sees Arthur disappearing away with the Earth at 0.999c and then returning so Arthur should be younger. The apparent paradox is this: if both points of view are valid then each sibling will see the other as older than themselves.

However, this particular problem is not reversible. Martha’s frame has not remained inertial – it has accelerated and decelerated, turned around and then repeated its accelerations. Hence the two frames of reference are not equivalent and there is no paradox because Martha will definitely be younger and Arthur older.