Special Relativity –Experiments, Thought
Experiments and Examples
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Number 150
Factsheet
Physics
Special Relativity is about the idea that there can be no absolute motion, you can only measure motion, length, mass and time dependent upon your position as an observer.
Einstein’s Postulates of Special Relativity:
1. The Laws of Physics apply equally in all inertial frames of reference
2. The speed of light in a vacuum is measured as the same in all inertial frames
Evidence that supports the Theory of Special Relativity
••••• The Michelson-Morley experiment shows that the speed of light is constant regardless of your apparent motion.
The Michelson-Morley Interferometer Experiment
The ether was believed to be a stationary substance that was thought to permeate space and was the medium through which light waves travelled.
Michelson and Morley set out to prove the existence of the ether and measure the absolute speed of the Earth as it moves through space.
Fig 1a. The Michelson-Morley Experiment
Light source
A
M
O
B
The experiment is set up with two mirrors at A and B and half silvered mirror at M. When light hits a half silvered mirror, some reflects off and some passes through. The observer watches the screen at O.
Light travels from the source and the beam is split by M. Some passes through to the mirror at B while some is reflected up to A.
Light source
A
M
B Direction of motion through ether
Light source
A
M
O
B Direction of motion through ether
A and B reflect the light beam back towards the half silvered mirror at M. Some light from A passes through M and travels towards the screen at O, while some of the light from B is reflected to O.
Light source
A
M
O
B Direction of motion through ether
interference pattern
As the two beams travel from M to O, they interfere and create an interference pattern. The observer can see the pattern on the screen at O. If the experiment really was moving through the ether as Michelson and Morley thought, then in the time it takes the light to travel from M to A and back, the experiment will have moved a short distance. This would mean that the light would have to travel a greater distance along path A, as shown:
A
Direction of motion through ether
Fig 1b.
Fig 1c.
Fig. 1e Fig 1d.
150. Special Relativity –Experiments, Thought Experiments and Examples
Physics Factsheet
The light following the distance M to B and back travels the expected distance.
[Note that, due to the movement through the ether, the distance travelled M to B is different from the distance B to M, however the total distance is still twice the distance between B and M. Along arm A, the total distance travelled is greater than twice the distance between M and A]
If the experiment was rotated through 90° it becomes the light following path B that needed to travel a little further, while path A remains as expected.
Light source
A M O
B
Direction of motion through ether
If their idea was correct this would mean that the path difference between the A path and B path would change when they rotated it. As a result they expected to see the interference pattern change.
Remember that the interference pattern depends on the path
difference
The Result
When rotated, the pattern did not shift at all! The speed of light and the distance it travelled along each arm did not seem to be affected by how it moved through the ether.
This means that:
• the speed of light is measured as the same regardless of the motion of the observer.
• the ether either does not exist or cannot be detected.
• without the ether to measure against there can be no idea of absolute velocity. Movement can only be described relative to other things.
Inertial Frames of Reference - Examples
Imagine standing in a lift. There is no way to see outside of the lift. When it accelerates upwards you feel heavier and as it decelerates you would feel lighter. As the lift is not moving at constant speed it cannot be an inertial frame of reference.
If the lift moves upwards at a constant speed, you feel exactly the same as if you were standing outside on solid ground. If you performed any experiment in the lift you would get exactly the same result as if you did it outside. For example, a pendulum would swing at the same rate whether the lift was moving or stationary (note; it would swing faster if the lift accelerated upwards).
When choosing examples of inertial frames, select situations that are capable of moving at constant speeds. Boats, cars, buses etc are far from ideal due to the constant bumps so you have to make assumptions. For instance, for a bus to be an inertial frame it must: • Not alter speed
• Not change direction
• Be travelling along a perfectly smooth, flat road
Inside an inertial frame of reference all objects will obey Newton’s Laws of motion.
Example Exam Question 1
(a) Give an advantage and a disadvantage of using a boat as an inertial reference frame [2 marks] (b) (i) Standing on the Earth’s surface is a good approximation to an inertial frame of reference. Describe another good approximation to an inertial frame of reference. [1 mark] (ii) Explain your choice [1 mark] (iii) Explain one draw back of your idea [1 mark] (iv) State and explain one drawback of the Earth’s surface as an inertial frame of reference. [2 marks] (c) An aeroplane accelerating for take off is not an inertial frame of reference. Explain how a passenger would be able to prove this. [2 marks]
Answers
(a) Advantage: it is possible for the water to be completely calm which would mean it was perfectly level when it moved. 3
Disadvantage: waves would cause the boat to rock and move
up and down which means it would be accelerating. 3 (b) (i) Of course there are several answers to part b. Here is a
good example:
A space shuttle in deep space, far from any massive objects, with no engines or thrusters in use. 3
(ii) The engines are not in use so it will not be changing speed or direction. Therefore any experiment performed on the shuttle would have the same result as if it were stationary. 3 (iii) Being far from large objects means the gravitational
attraction is minimal, but there will be some acceleration due to the gravity, no matter how small.
(iv) The Earth is rotating 3 so any point on the surface is moving in a circular path and will be experiencing a centripetal acceleration.3 (note that mentioning gravity here is not correct. An inertial frame on the surface would
be sitting in a gravitational field but would not be accelerating due to it).
(c) An object such as a ball placed on the floor will appear to accelerate backwards as the aeroplane accelerates forwards. 3 From the reference point of the inside of the aeroplane, the ball appears not to obey Newton’s Laws of motion. 3 Therefore it cannot be an inertial frame of reference.
Time Dilation - Thought Experiment
A bulb inside a train carriage is switched on. There is a mirror attached to the ceiling.
Light from the bulb will travel up to the mirror, M, and be reflected back down to the bulb.
Consider an observer on the platform watching the train pass by at a constant speed.
As the light beam travels from bulb to mirror and back, the train will also move forwards. This results in the beam taking a path as shown in Fig 2b
bulb
M
In this case, the light has travelled a greater distance than in Fig 2a. There are two possible explanations:
1. The light has travelled faster in figure 1b in order to cover the greater distance in the same time.
or
2. If the light has travelled at the same speed in both pictures, then time would have to travel more slowly in fig 1b to allow the light to cover the extra distance.
The Postulates of Relativity say that light is measured as the same speed in all inertial frames of reference. Therefore the only possible explanation is that time moved slower inside the moving train when viewing it from the platform.
Two stop clocks are started at the same time. One is placed in the train window, the other given to an observer standing in the platform. The observer views both clocks simultaneously. The one on the train will show less time has passed than the clock on the platform.
Time moves more slowly in an inertial frame that is moving with respect to the observer.
Example Exam Question 2
Eric and Martha are twins. Martha travels to Alpha Centauri at close to the speed of light and immediately starts the return journey. Eric estimates that because Martha was moving, she will have experienced time dilation and therefore be younger than he is. Martha believes that the two situations are reversible. She argues that from her perspective, the Earth and Eric moved away and then returned so it will be Eric who has aged less and will be younger. (a) Using the idea of inertial frames of reference, explain why
Martha’s logic is flawed. [3 marks] (b) If, instead of returning, Martha travels to Alpha Centauri and
stays there:
Fi g 2a
Fi g 2b
Answer
(a) The two situations are not reversible. Martha travelled to Alpha Centauri and then returned. She was travelling in two different directions and therefore occupied two different reference frames.3 Eric’s velocity remained constant meaning he only ever occupied one frame of reference. 3 Therefore we cannot say that Eric’s journey was the same as Martha’s. 3 (b) (i) From Martha’s point of reference, Eric was moving. Time
would move slower for him, so she would age faster. (ii) From Eric’s point of reference, Martha was moving. Time
would move slower for her, so he would age faster. 3 (iii) This is not a paradox because the twins cannot be directly
compared as they are no longer in the same place.3 Note: they simply get different results due to their point of view. Even if they were to communicate with each other, it is still not possible to directly compare them as it would take time for the signal to reach the other twin.
This problem is known as the Twin Paradox. You should learn the answers very carefully.
Muon Detection – Evidence for Time Dilation
• Muons are unstable particles created as cosmic rays strike the upper atmosphere.
• They travel at high speed towards the Earth’s surface where they can be detected.
• Being unstable they have a very small half life, so many do not reach the Earth before they decay.
• Far more Muons are detected than should be possible. • The only explanation is that due to time dilation, the time for the
journey as measured from the Earth is less.
• As it takes less time, less muons have chance to decay. More muons will be detected at the Earth’s surface.
This requires use of the equations;
λ is the decay constant given by:
N is the number of particles at time t N0 is the number of particles at time t=0
t is the time (s)
Where t1/2 is the half life (s)
Combined these give:
and
Where t0 is the proper time (the dilated, smaller time)
t is the time measured at the Earth’s surface
v is the speed of the muons (m s-1)
c is the speed of light, 3x108 m s-1
t 0 N e N −λ = 1 2 ln( 2 ) t λ = 1 2 ln( 2 )t t 0 N e N − =
(
02 2)
t t 1 v c = −Example Exam Question 3
Muons are created in the upper atmosphere. A high altitude detector is attached to a balloon flying at an altitude of 2×103 m. It detects
3×106 muons.
(a) If the particles are travelling vertically at 0.995c, how long will it take them to reach the Earth’s surface as measured by an observer on the Earth? [1 mark] (b) If the half life is 1.53×10-6 s, how many muons would you expect
to detect at sea level? [2 marks] (c) (i) Show that the dilated time for the journey is about 0.7ìs. [2 marks]
(ii) The actual number detected at sea level is 2.19×106
demonstrate that this is evidence for time dilation [3 marks]
Answers
(a) t = distance/speed = 2×103/(0.995 × 3×108) = 6.70×10-6s = 6.70µs 3 (b) The number of muons expected depends on the time the muons take to reach the Earth’s surface as measured at the Earth’s surface:
150. Special Relativity –Experiments, Thought Experiments and Examples
3(correct substitution) 3 (correct answer)
(c) (i) The time as measured from the Earth is 6.70ìs, this is t. We need to find t0.
0.669ìs = 0.7ìs to 1 decimal place.
(c) (ii) In order to show that this is evidence for time dilation, we need to calculate the time that it would take to reach this number of muons and compare it to the dilated time.
3(correct substitution) 3(correct answer)
0.69ìs is similar to 0.7ìs in the previous part. This suggests that the time experienced by the muons is time dilated 3
Length Contraction – Thought Experiment
Before following this thought experiment you must remember two important ideas which come from the Postulates of Relativity: • The speed of light is measured as c in every inertial frame of
reference.
• The speed of light is unaffected by the speed of the object which emits it.
This thought experiment explains how a length will contract when it is moving in relation to the observer.
A single pulse of light is fired along a road after a moving car. The distance along the road that it has travelled is measured by the driver and the flashlight operator.
3(equation correctly rearranged)
The first picture shows the car moving forward with respect to the flashlight and the road.
The light is flashed at A.
In the time taken for the pulse of light to reach the car, the car will travel forward a small distance.
The beam of light reaches the car at B. The beam travels a distance from A to B.
Imagine that we consider the same situation from the frame of reference of the car.
In this frame, the car is stationary and it is the road and observer that are moving backwards.
A
V
Light flashed
A B V
Light reaches car distance traveled by beam
Fig 3a
A'Distance travelled by beamB'
Light reaches car Light flashed
V A
V
V V
The light will be flashed at A’.
The flashlight and the road will then move backwards, away from A’, but the car will remain stationary.
The speed of light remains c just like the first time. Remember this is not affected in anyway by the movement of the flashlight.
The light then travels and hits the car at B’. The pulse of light travels from A’ to B’.
Compare the two distances that the light pulse had to travel. It is clear that the distance A to B is much greater than A’ to B’.
How does this result fit in with the theory?
The distance A to B is the proper length (l0).
The proper length is the length as measured when at rest to it. This is the distance along the road that the pulse travelled when measured by the flashlight observer who is at rest to the road. Distance A’ to B’ is the contracted length (l).
The contracted length is the length as measured when it is moving. This is the distance travelled along the road by the pulse as measured by the driver when the road is moving with respect to the driver.
Any length is longer when measured at rest to it and contracted when the length is moving with respect to the observer. Fig 3b
Physics Factsheet
6 1 6 2 ln( 2 )t ln( 2 ) 6.7 10 t 6 1.53 10 5 0 N N e 3 10 e 1.44 10 − − − × × − × = = × × = ×(
02 2)
0(
2 2)
t t t t 1 v c 1 v c = ⇒ = × − −(
2 2)
6.7 1 ( 0.995c ) c 0.669 s = × − = µ 1 2 ln( 2 )t 1 t 0 2 0 0 1 2 N ln t N N N ln( 2 )t e ln t N N t ln( 2 ) − ⎛⎜ ⎞⎟× ⎛ ⎞ ⎝ ⎠ = ⇒ ⎜ ⎟= − ⇒ = − ⎝ ⎠ 6 6 6 7 2.19 10 ln 1.53 10 3 10 t 6.9 10 0.69 s ln( 2 ) − − ⎛ × ⎞ × × ⎜ × ⎟ ⎝ ⎠ = − = × = µPractice Questions
1. Describe a simple experiment that could be performed to show that an accelerating lift is not an inertial reference frame. [2 marks] 2. (a) What evidence is there to support the theory of time dilation. [4 marks]
(b) Explain, without use of calculations, a thought experiment to demonstrate time dilation. [5 marks]
3. There is a clock on an aeroplane and a clock on the runway. The aeroplane flies over the runway at constant speed. (a) The pilot says that he will see the clock on the ground moving slower than his own clock. Is he correct? [1 mark] (b) What should an observer on the ground expect to see if they could see both clocks at the same time? [1 mark] 4. Explain a thought experiment that demonstrates the principle of length contraction without performing any calculations.
Answers
1. There are several simple answers. One is to weigh a known mass. 3 If the weight is measured as more or less than would be given by the equation:
Weight = mass × 9.81 N kg-1
then the lift must be accelerating. 3
A more complex answer is to time the period of a pendulum of known length and calculate acceleration due to gravity. 3 If the answer was not close to 9.81 ms-2 then the lift is accelerating. 3
2. (a) Muons have a very short half life and are created high in the atmosphere.3 They travel towards the Earth’s surface at close to the speed of light. A much higher proportion of particles are detected at the Earth’s surface than would be expected given the time it takes for them to cover the distance. 3 This suggests that it actually takes less time for them to complete the journey.3 This smaller time is accounted for by the time dilation due to the particles’ high speed.3
(b) The train thought experiment should be described here. Marks are awarded for:
• sketching the beam of light in both frames of reference. 3 3
• showing that the beam is takes a longer path when viewed from the embankment. 3
• applying the Postulate of Relativity that states the speed of light is constant in any frame of reference. 3
• therefore coming to the conclusion that time must move slower in the train carriage to allow the beam of light to cover the extra distance. 3
3. (a) Yes. 3
(b) The ground observer would see the clock in the aeroplane moving slower than the clock on the runway. 3 4. The racing car explanation should be used here.
Marks are awarded for:
• sketching the path of the beam of light in each frame of reference, showing the motion of the objects involved (the car and the observer). 3 3
• demonstrating that the beam has to travel further when the distance is measured in the frame of reference of the observer. 3 • point out that the same distance is being measured in each case 3
• draw the conclusion that one of the length measurements must have contracted in order for this to be true. 3
Acknowledgements:
