Special Relativity: Physics

What does special relativity apply to?

• Objects moving at a constant velocity.

What is an inertial frame of reference?

• Those which move at a constant velocity relative to each other.

What are the two assumptions (postulates) that Einstein made? Explain them.

1) Physical laws have the same form in all inertial frames. This means that same rules apply for all objects moving at a constant velocity (as long as it is NOT accelerating).

2) The speed of light is invariant, meaning 3 X 10^8 all the time.

Imagine a ray of light that is bouncing between a mirror on a train. Explain how time dilation works in the frame of reference of the person on the train:

• In the frame of reference of a person on the train, they would see the ray simply moving up & down.

Imagine a ray of light that is bouncing between a mirror on a train. Explain how time dilation works in the frame of reference of a stationary person on the platform:

• But in the reference of a person on a platform (who is stationary), they would see the ray travelling diagonally, and end up at a central point when the mirror moves, then bouncing back down diagonally.

How much distance would the light ray travel in the frame of the stationary user? Why?

• They would see the ray of light travel a larger distance in a longer amount of time.

• If it was in the same time to the moving persons frame of reference, this means it would have to travel faster which does not obey one of the postulates.

Overall, they run slower according a stationary observer’s clock.

What is proper time? (t)

Time as measured by the stationary person.

What is t0?

Time measured by the moving person.

How can you calculate the proper time in the perspective of a moving user e.g. they’re on a train?

• v = s/t (if object is moving horizontally)

• When moving up & down: to = 2l/c

How do you calculate the time taken for the light ray to travel in the perspective of a stationary user? e.g. they’re watching from a platform

• Split the diagram into a triangle

• Using v = s/t & Pythagoras theorem

• t = 2s/c = 2/c X root(vt/2)² + L²

What do we do to the two time equations we’ve made for both the perspective of the stationary person & moving person?

• Because the speed of light is invariant

• .. we will combine those two equations

• .. which make: t = to/root (1 - v²/c²)

What experimental evidence is there for time dilation?

• Detecting muons i.e. muon decay

How does muon decay show effects of time dilation?

• Muons enter the atmosphere at very high speeds

• ..and so experience significant time dilation,

• ..which affects how quickly they decay

How can you measure muon decay?

• You must place one detector at a high altitude and one much further down to measure the change in muon count rate,

• You will also have to measure the distance between the detectors (d) and the speed that they muons are travelling at (v).

How can you calculate the time taken for the muons to travel from the first detector to the second detector? (from the second detectors point of view i.e. observatory)

• t = s/v

How can you calculate the number of half - lives that the muons have undergone during their travel from the first detector to the second detector?

• Time calculated through s/v / muon half - life value.

What is a muon half - life time value?

• 1.5 × 10^-6 s.

What do you expect the number of half lives to (classically) be for these muons?

t = 4.46 (t1/2)

How can you calculate the expected count rate at the 2nd detector in the classical way (without acknowledging special relativity)?

Count rate at 1st detector x (1/2)^ no of half - lives

• where no of half - lives should = 4.46

How do we know that the classical way of calculating the amount of muons detected is wrong?

• We expect that the number of half - lives that the muon will go through is classically 4.46.

• So the percentage of muons that we should detect at the 2nd detector would be (when using the classical equation)

• .. ½ ^4.46 X 100 = 4.5%

• However, 80% of muons were detected at the 2nd detector, thus showing that this equation was wrong & relativity is involved.

Explain why 80% of muons were detected instead of 4.5%

• Muons decay less frequently when traveling close to the speed of light due to the relativistic effect known as time dilation.

• From the perspective of an observer on Earth, the muons are moving very fast.

• According to special relativity, the time experienced by the muon (its "proper time") runs slower compared to the time experienced by the stationary observer on Earth.

• This means that the muon's internal clock (which determines its decay) is ticking slower from the perspective of the Earth observer.

What is the correct way of calculating the expected count rate of the muons at the 2nd detector, when acknowledging special relativity? 1st method:

1) First, calculate the time taken for the muons to move between the detectors from the external frame of reference (laboratory), using : s/v = t

2) Then, find the proper time (t0 ), which is actually experienced by the muons i.e. t0 = t x root(1/v²/c²)

3) Then calculate the number of half - lives the muons will be expected to take i.e. proper time/ muon half life time.

4) Then calculate the expected count rate at the 2nd detector i.e. number of counts at the first detector x 1/2^no of half - lives

What value of number of half - lives (using the special relativity) method do you expect to get?

• 0.4

What is the correct way of calculating the expected count rate of the muons at the 2nd detector, when acknowledging special relativity? 2nd method:

1) Another way is by calculating the time - dilated half - life

• .. through using the time dilation equation & subbing the muon half life time value as t0.

2) You can then find out the number of half - live (like before) i.e. time travelled by muons from detector 1 to 2 (s/v =t) /time dilated half life

3) Then you can calculate the percentage/value of muons expected at the 2nd detector i.e. number of counts at first detector x 1/2^number of half - lives.

How does the correct way of calculating the number/percentage of detected muons confirm relativity?

• Expect to get a value of 80%

• .. which is what was detected.