Relativity - Mrs Physics



NATIONAL QUALIFICATIONS CURRICULUM SUPPORT

Physics

Special Relativity

Teacher’s Notes

[HIGHER]

[pic]

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Acknowledgement

Learning and Teaching Scotland gratefully acknowledges this contribution to the National Qualifications support programme for Physics.

© Learning and Teaching Scotland 2010

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Contents

Introduction 4

Relativity – an overview

Relativity before Einstein 5

Special relativity 7

Time dilation and length contraction 8

Thought experiment 1 9

Thought experiment 2 13

Experimental verification 15

Relativistic effects regarding mass 16

Appendix 19

Introduction

These Teacher’s Notes cover more than the minimum required for assessment. Suitable comments regarding assessable material are included. For the derivation of the equations, the algebra is kept to a minimum in order to concentrate on the concepts and correct use of the equations. Teachers can include additional algebraic steps depending on their students. Confusion and errors can arise on the part of students by mixing up t, t’, l and l’, often producing ‘wrong physics’. Students do not need to be able to reproduce the derivations.

Relativity – an overview

Einstein’s work on relativity was published in two stages: Special Relativity was published in 1905 and General Relativity was published in 1916. However, this work is often perceived as being part of ‘modern physics’. In the eyes of students this is a curious label. Furthermore, relativity, as a concept in physics, predates the work of Einstein. It is useful to approach relativity from a Newtonian perspective.

Relativity before Einstein

Students of Higher Physics will be familiar with Newton’s Laws of Motion. These laws allow us to describe the motion of objects, regardless of size or position. They also allow us to predict subsequent motions. Given starting data, equations allow us, in theory, to predict subsequent motion. This leads to a ‘clockwork’-type view of the world. Quantum mechanics, developed in the 1920s, provides another set of rules for the physics of the very small at the atomic and subatomic level. This theory and Einstein’s relativity give a less ‘clockwork’ view of the world, leading to some philosophical implications. Quantum mechanics is not included in Higher Physics but the term can usefully be mentioned.

To return to pre-Einstein time, it may be useful to consider the way people thought about the world and the universe in general. Often time, t, is a time interval in many physics problems and discussions. When asked ‘What time is it?’ there is the implication that if the clocks were accurate enough and set correctly there would be just one answer to this question. Einstein lived in Bern where some clocks chimed the hour at different times! Accurate watch making and clock setting led to thoughts about time.

The Newtonian picture of the universe was built on the idea of absolute space and time – a rigid framework against which all measurements and experiments could be carried out.

This implies that there is constancy to the rate at which time passes. In the Newtonian model, clocks tick at the same rate regardless of their movement. Furthermore, clocks appear to tick at the same rate relative to observers who may have a different motion relative to the clock.

Similarly, absolute space implies a static backdrop against which all movement can be referenced. It is useful to consider examples of everyday experiences of movement relative to a stationary background. Consider the sensation when sitting on a train next to another train, and one train moves. Sometimes it is difficult to be sure whether you are moving or the other train is moving. It is only after your train or the other train has moved off that you can be sure by relating to the background.

One further and hugely important assumption in Newton’s view of the universe is that the laws of physics remain the same whether one is moving steadily or at rest. This is also known as Galilean invariance. This assumption, of the universality of the laws of physics, remains true regardless of the introduction of relativity or quantum mechanics or any other theory. Students may wish to comment on the validity of this assumption. Can we even do physics and astronomy if we do not make this assumption?

In order to relate measurements taken by observers travelling at different velocities, relative to absolute space, a number of simple rules were established. These rules were set out by Galileo and are still known today as Galilean transformations. Students do not need to be familiar with the formal statements of the transformations but they should be familiar with their use to calculate relative velocities. Common examples include passengers walking towards the front or rear of moving vehicles and the relative velocities of two vehicles moving towards each other. Numerical calculations should be restricted to movement in one dimension, using simple vector addition.

It is useful to introduce the term ‘frame of reference’, which refers to any laboratory, vehicle, platform, spaceship, planet etc and often has a relative velocity to another ‘frame of reference’. These are called inertial frames of reference when the movement is in a straight line with a constant velocity. For example, a platform could be one frame of reference and a high-speed train a different frame of reference, where the platform and the train have a constant relative velocity.

The Galilean transformations assume that the speed of waves depends on the motion of the wave source relative to the observer. This is intuitive for a number of wave motions and it was assumed that it would be true for light, which was known to be a wave. It is possible to ‘catch up’ a sound wave or a water wave, and students can consider a number of cases to illustrate how a wave may be seen to have different velocities relative to moving observers.

Special relativity

Special relativity is ‘special’ because it considers only the case where different observers (frames of reference) are in relative motion with a constant velocity. The case of accelerated observers was developed by Einstein 11 years after special relativity and is described in his general relativity.

Einstein very much supported the Newtonian view that the laws of physics should apply everywhere and for every observer. However, his knowledge of the work of James Clerk Maxwell led him to see that there was an apparent contradiction with this principle.

Maxwell was a Scottish physicist who made an enormous leap forward in explaining how light is produced. His theory of electromagnetism showed that light is an electromagnetic wave. Students are not required to consider Maxwell’s equations. However, it is important that they understand that Maxwell’s equations allowed the speed of light to be predicted theoretically. Experiments undertaken by the American physicist Albert Michelson obtained excellent agreement with the theoretically predicted value.

Knowing that light is a wave led some physicists to conclude that light must be passing through something, i.e. ‘something’ must be vibrating, even in the vacuum of space. This something was called the ether and Michelson and his colleague Morley set out to detect the motion of the Earth through the ether. However, their experiment failed to detect any difference in the speed of light when it was measured for different speeds of the Earth in its orbit around the Sun. The details of the classic experiment to detect the ‘ether drift’ are not required for assessment but teachers may find it worthwhile to describe this experiment since it is a very good example of the validity of a ‘null result’ (see Appendix).

Einstein postulated that the speed of light is constant for all observers. The Michelson-Morley experiment did not detect a difference in the speed of light as the Earth moves in different directions and speeds because the speed of light does not depend on the motion of the source. All observers measure the speed of light to be the same, regardless of the motion of the source of light, or the motion of the observer towards or away from the source of light. Einstein knew that this tied in with Maxwell’s work in that the theoretical value of the speed of light made no reference to the motion of the source. Einstein also abolished the concept of the ether and any ether drift.

There are a number of startling consequences of the postulate that the speed of light is measured to be the same regardless of the motion of source or observer. The Newtonian idea of the backdrop of absolute space and time was firmly rejected. Also, observers moving relative to each other with a constant velocity, in different reference frames, would disagree about the measured separation in space and time of events they observed. Two events may be simultaneous for one observer but not for the other observer.

To summarise

Special relativity (which applies to observers/reference frames in relative motion with constant velocity) has two postulates:

1. The laws of physics are the same for all observers in all parts of the universe.

2. Light always travels at the same speed in a vacuum, 3.0 × 108 m s–1 (299,792,458 m s–1 to be more precise).

(Light does slow down inside transparent material such as glass.)

To give a simple example:

A spaceship travelling at 100 million m s–1 approaches a planet considered to be at rest. An observer on the planet sends a light signal to the spaceship. The spaceship will measure the speed as 300 million m s–1 and not as (300 + 100) million m s–1. We cannot apply our usual ‘Newtonian rules’ for relative velocity.

From these two postulates Einstein produced a new theory of motion.

We know that speed = distance/time, and speed = fλ, hence if the speed is to remain constant ‘something’ must happen to the distance and time! This is the essence of special relativity.

Time dilation and length contraction

Students at Higher level should be able to follow the derivation of the equations showing time dilation and length contraction, although the derivations themselves are not required.

Notation:

t and l – time interval (‘event’) or length of object under discussion in a frame of reference (eg on a platform)

t’ and l’ – time interval or length of object ‘measured’ by travellers in a different frame of reference (eg on a train)

v – relative velocity of the two frames of reference

(Note: Recall that no one frame of reference is any more ‘stationary’ or ‘moving’ than any other. There is no ‘absolute rest’. We have chosen the platform to contain the ‘event’.)

Thought experiment 1

Consider a person on a platform who shines a laser pulse upwards, reflecting the light off a mirror. The time interval for the pulse to travel up and down is t (no superscript).

[pic]

A different frame of reference, for example a train moving along the x-axis at high speed v, passes. From the point of view of travellers on board the train, the light travels as shown in the diagram above.

The time taken for the light to travel up and back, as measured by travellers in this frame, is t’ (t dash).

In the time t’ that it takes for the light to travel up and back down the train in this frame, the train has travelled a distance d.

Both observers measure the same speed for the speed of light.

[pic]

|A right- angled triangle can be formed where the vertical side |[pic] |

|is the height, h, of the pulse ([pic]ct), the horizontal side is| |

|half of the distance, d, gone by the train ([pic]vt’) and the | |

|hypotenuse is half the distance gone by the pulse as seen by the| |

|travellers on the train ([pic]ct’). | |

Applying Pythagoras to the triangle gives:

([pic]ct’ )2 = ([pic]ct)2 + ([pic]vt’)2

(ct’ )2 = (ct)2 + (vt’)2

(c2 – v2) t’ 2 = c2 t2

[pic] t’ 2 = t2

[pic] (1)

What assumptions have we made?

(i) The two frames are moving relative to each other along the x-axis, ie the train passes the platform. There is no bending or circular motion involved.

(ii) We require two travellers on the train since the start and finish places are separate. This is fine since two clocks can be synchronised in the same frame of reference.

It is useful to mention to students the significance of the term [pic]. (The reciprocal 1/ [pic] is known as the gamma factor.) This term occurs in relativity equations and its size determines when relativity effects will be observed. At everyday speeds it is almost unity.

Students should be able to interpret the final equation, stating what each of the symbols represent, and what the equation means in terms of the time interval for each observer.

Note: The laser pulse starts and finishes at the same place on the platform. Thus equation (1) is used to calculate the time interval t’ registered in a frame of reference, eg the train, for an event which take place in a different frame of reference from the ‘event’.

For example if v = 0.4c then (v/c)2 = 0.16 and [pic] = [pic] = 0.917.

Let us use this value of v = 0.4c in our thought experiment with a laser pulse time of 8.0 ns.

Thus, if t = 8.0 ns, we can calculate t’, giving t’ = 8.7 ns. A longer time interval is ‘measured’ by travellers on the train. This effect, known as time dilation (dilation = expanding), is a direct result of the postulate that the speed of light is measured to be the same by all observers.

Time dilation leads to observers being unable to agree about simultaneous events. Two events may appear to be simultaneous to one observer, but may not be simultaneous for others.

Some comments

Sometimes the pulse is referred to as the ‘tick of a clock’ and t the period (or twice the period).

(a) Teachers may find a variety of derivations leading to different formats of this equation. The SQA data sheet has equation (1).

(b) It is important to recall that no one frame of reference is any more ‘stationary’ or ‘moving’ than any other. There is no ‘absolute rest’. It is the relative motion that matters. We often think of ourselves here on Earth as ‘stationary’ and solve problems from that point of view. Care is needed to sort out the frames of reference for different situations or problems.

Another example and some effects

Consider a space ship passing Earth at a velocity of 0.5c. It emits a pulse (or on Earth we observe ‘ticks’ of their clock) of duration ΔT = 2.0 ns. We on Earth can ‘measure’ the duration t’ we observe. Note that we are not in the same frame of reference as the ‘event’ so our time interval is t’ not t. The duration of the event, in the frame of the event on the space ship, is t.

Using equation (1) gives us an observed time interval of 1 /0.87 = 2.3 ns.

We consider their clock is running slow, time is passing more quickly for us so they could end up ‘younger’! Although time dilation can give rise to interesting discussions on time travel (into the future), the explanation of the twin paradox requires more consideration since any acceleration may involve general relativity. Also a returning twin would have to ‘change’ frames of reference.

When [pic]is almost unity no effect is noticeable. It is useful to calculate this term (or the reciprocal) for various speeds, for example:

a supersonic plane 422 m s–1 (900 mph); 0.1 × 108 m s–1;

0.3 × 108 m s–1 (10%c); 1.0 × 108 m s–1; 2.0 × 108 m s–1,

2.8 × 108 m s–1 and 99% c.

Students can clearly see that effects in everyday life are not noticeable. We need v > 10%c for any noticeable effects.

Thought experiment 2

For length contraction, a rod of length l is placed in a frame of reference. eg on the platform. Consider the length l’ measured by travellers in a passing train or space ship, a different frame of reference. The relative velocity is v.

|To measure the length of the rod we need to determine the |[pic] |

|start and end points. A mirror is fixed to one end and the | |

|time taken (t) for a light pulse to be reflected from the | |

|mirror and return is recorded. | |

|Time taken in this frame is t | |

|Hence distance is 2 l = ct | |

| |[pic] |

|For the different frame of reference, a light pulse is started| |

|when the left-hand side of the rod is in line, but the | |

|measured time taken to reach the mirror (t1) will be greater | |

|due to the relative velocity. | |

|Distance to mirror is c t1 = l’ + vtt | |

| |[pic] |

|As the pulse returns, the frame of reference is still moving. | |

|The measured time taken for the return distance is t2. | |

|Distance back from mirror is | |

|c t2 = l’ – vt2 | |

The total time taken for the pulse to measure the rod in the different frame is t1 + t2 = l’/(c – v) + l’/(c + v).

Hence the total time in this different frame is t’ = l’ 2c/(c2 – v2)

and the time in the frame with the rod is t = 2l/c.

The time measurements start and finish at the same place in the platform frame.

Notice that the rod is on the platform.

Using t’ = t/(1 – v2/c2) substitute for t’ and t and simplify, giving:

[pic] (2)

We can see that the length l’ is less than l since [pic] is less than 1.

This effect is known as length contraction.

For length contraction the equation is expressed using l for the length in the frame of reference with the actual object and l’ the apparent length measured in the other frame.

Students are expected to be able to interpret and use the equation.

A specific object, eg a metre rule, will have the same length, L say, when measured by a person actually in a frame of reference with the object, eg a person on a spaceship with the actual object would measure length L. (This length L of a specific object is termed the ‘proper length’ of the object.)

Another example

Suppose we on Earth observe a very fast car moving past at 0.3c. The length of the car is 4 m. What length will we observe?

This length l = 4 m (actual length of the car to a person in the car) and l’ is our observed length in our different frame of reference (we are not in the same frame of reference as the car).

With v = 0.3c, [pic] = 0.95.

Using [pic] gives l’ = 4 × 0.95 = 3.8 m. We observe a car length of 3.8 m.

Similarly, a spaceship or passing asteroid would appear shorter to us here on Earth.

Conversely objects on the Earth, will appear ‘shorter’ to observers on passing spaceships. No particular frame of reference is any more ‘stationary’ than any other.

Special Relativity means that the Newtonian view, that space and time are absolute and completely separate, has to be rejected. Special relativity requires us to think of space and time as inextricably linked and our measurements of distance and time depend on the motion of the observer. The effects of time dilation and length contraction are only observed at very high speeds (close to the speed of light). Students should study an experiment in which the effects are confirmed.

Experimental verification

|One verification arises in the study of elementary particles moving close to the speed of| |

|light. Cosmic rays, believed to be produced in deep space, collide with atoms in the | |

|Earth’s upper atmosphere. They produce showers of muons, which are short-lived elementary| |

|particles about 200 times more massive than electrons. When muons are produced in | |

|laboratories, we find that their typical mean lifetime is very short – about two | |

|millionths of a second – before they decay into other particles. The cosmic ray muons are| |

|moving with a speed of about 99.9% of the speed of light. However, even at this speed, a | |

|muon would travel only about half a kilometre in two millionths of a second. Yet | |

|substantial numbers of cosmic ray muons are detected at sea level, about 60 km below the | |

|altitude where the muons are created. This is because time dilation means that, viewed in| |

|our reference frame, the lifetime of the fast-moving muons is considerably longer, ie to | |

|us time runs more slowly than for them because they are moving at speeds very close to | |

|the speed of light relative to us. Of course in the reference frame of the muons, they | |

|are not moving at all – hence their lifetime is still only two millionths of a second. | |

|Also the reason why they are able to reach sea level is that the distance which they | |

|travel – measured to be 60 km in our reference frame – is considerably less in their | |

|reference frame, an example of length contraction. | |

|(Muon mean lifetime = 2.2 μs, and half-life = 1.56 μs.) | |

Students are not expected to be able to quote numerical values.

Another example is accurate time measurements on airborne clocks. The time dilation is small for the relatively low speeds of aircraft, but clock precision can be very high. The differences can be measured.

An aside

The Newtonian equations of motion we have already studied give us accurate results for most of our everyday situations. For high speeds we need to use a different set of equations, namely the relativistic time dilation and length contraction above. Special relativity equations reduce to our familiar equations when the speeds involved are less than 10% of the speed of light.

For situations involving atomic, nuclear and sub-nuclear ‘particles’ we find that neither Newtonian or relativity equations give results that agree with experiment. One of the reasons is that energy is not ‘emitted or absorbed’ as a continuous stream but is quantised into packets. Also there is a wave/particle aspect that must be accepted and included. In these situations we need to use another approach, that of quantum mechanics, which involves another set of equations and different mathematics but gives excellent agreement between theory and experiment. Relativity has been included at this subatomic level to give relativistic quantum mechanics. Details of quantum mechanics are not required for Higher Physics but the photoelectric effect and wave/particle duality are in the Higher Physics: Particles and Waves unit.

Relativistic effects regarding mass

In Newtonian mechanics mass is considered to be conserved and to remain constant, but at high speeds this could lead to different effects for different observers. This could appear to violate our basic assumption that the laws of physics are the same for all observers. For example, consider a small probe making an impact on a planet. Momentum is a quantity that is conserved. The ‘damage’ caused on impact will depend on the momentum of the probe. This momentum should be the same when calculated by a ‘stationary’ person on the planet or by an observer on a fast-moving spaceship near the rocket. Each observer will measure a different velocity for the rocket, but there is only one impact and one lot of damage!

Hence each observer somehow considers the mass to be different, since momentum p = mv.

The observer on the spaceship could consider the inertial mass of the rocket to be greater than the observer on the planet.

Einstein showed that there is a mass energy equivalence, leading to his famous equation E = mc2. With relativity there is not a separate conservation of mass and energy but a single conservation of mass/energy.

The equivalence of mass and energy could be introduced here, but is included in the Particles and Waves unit in the Nuclear Fusion topic and teachers may prefer to teach it in that context.

This apparent increase in inertial mass at high speeds means that it is more and more difficult, requiring a great deal more energy, to increase the speed of objects near the speed of light and impossible for an object to reach the speed of light. The muons mentioned above have an extremely small mass hence they are capable of travelling close to the speed of light.

In the Large Hadron Collider at Cern in Switzerland a great deal of energy is needed to accelerate protons to speeds near to the speed of light.

To summarise

1. Defining t, t’, l and l’ as shown below, equations for time dilation and length contraction can be derived. The equations (1) and (2) are provided on the SQA data sheet.

t and l time interval (‘event’) or length of object under discussion in a frame of reference

t’ and l’ time interval or length of object ‘measured’ by travellers in a different frame of reference

v relative velocity of the two frames of reference.

[pic] (1)

[pic] (2)

2. No object can travel faster than the speed of light.

3. Relativistic effects are negligible when relative velocity is less than10% of the speed of light.

4. Experimental verification is provided by observing the life time of fast-moving muons.

5 There is not a separate conservation of mass, but a combined conservation of mass and energy. A greater energy than expected is required to increase the speed of an object as its speed approaches the speed of light.

Appendix

Michelson-Morley experiment

The apparatus consists of a light source, a beam splitter (a half-silvered glass plate) and two mirrors, M1 and M2, each of which is equidistant from the beam splitter. The beam splitter is at 45o to the incident beam and the return beams pass to the detector, a telescope.

The prevailing theory held that the ether formed an absolute reference frame. The Earth is orbiting the Sun and therefore moving through the ether. Thus, sometimes one light beam may be travelling in the same direction as the ether and the other beam at right angles to the ether. They will have slightly different speeds. The detector should show interference between the two beams travelling perpendicular to each other. The aim was to measure the speed of the Earth relative to the ether.

No discernible fringes were found, despite repeating the experiment at different times in the Earth’s orbit and with different orientation. On the prevailing theory and experimental accuracy, the small destructive interference should have been observed.

This null result had great importance since it could not be explained.

-----------------------

Travellers in this different frame of reference observe the ‘event’ (eg out of the window of the train), which takes place in the platform frame of reference and measure a time t’.

platform

mirror

h

Person in same frame as ‘event’ measures a time t.

Total distance travelled by pulse 2h = ct.

h

v

Platform frame of reference

Train frame of reference

2 h = c t

d = v t’

Total distance travelled by pulse= ct’

Horizontal distance travelled by train = vt’

[pic]ct

[pic]c t’

[pic]v t’

l

l' + vt1

l’

l’

l' – vt2

Muons

Speed = 99.9%c

Distance to us

60 km

Muon distance

[pic]

= 60 × 0.045 = 2.7 km

Life time to us

= 2.7 × 103/0.999c

= 9 aÈzÈ{ȔȕȨȩȼȽÈÆÈÇÈÈÈ}É~ɇɈÉ?É?É’É“ÉÆÉØÖ²Ö²ÖØÖØμs

light source

mirror

M1

mirror

M2

beam splitter

L

L

detector

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