∆t 0 is called the proper time and is equal to the time interval between two events that occur at the same position Only one inertial frame (S’) measures the proper time and it does so with a single clock that is present at both events An inertial reference frame moving with velocity u relative to the proper time frame must use two clocks to measure the time interval: One at the position of the first event and one at the position of the second event By rearranging our earlier equations, the time interval in the frame where two clocks are required is as follows 36
∆t 0 is called the proper time and is equal to the time interval between two events that occur at the same position Only one inertial frame ( S’ ) measures the proper time and it does so with a single clock that is present at both events An inertial reference frame moving with velocity u relative to the proper time frame must use two clocks to measure the time interval: One at the position of the first event and one at the position of the second event By rearranging our earlier equations, the time interval in the frame where two clocks are required is as follows 37
∆t 0 is called the proper time and is equal to the time interval between two events that occur at the same position Only one inertial frame (S’) measures the proper time and it does so with a single clock that is present at both events An inertial reference frame moving with velocity u relative to the proper time frame must use two clocks to measure the time interval: One at the position of the first event and one at the position of the second event By rearranging our earlier equations, the time interval in the frame where two clocks are required is as follows 38
∆t 0 is called the proper time and is equal to the time interval between two events that occur at the same position Only one inertial frame (S’) measures the proper time and it does so with a single clock that is present at both events An inertial reference frame moving with velocity u relative to the proper time frame must use two clocks to measure the time interval: One at the position of the first event and one at the position of the second event By rearranging our earlier equations, the time interval in the frame where two clocks are required is as follows 39
∆t = ∆t 0 / 1 − 𝑣 $ /𝑑 2 = 𝛿 ∆t 0 and thus ∆t ≥ ∆t 0 The stretching out of time of the time interval is called time dilation The equation Above tells two things: Firstly, if it were possible to travel faster than the speed of light then 1 – u 2 /c 2 would be negative and 1 − 𝑣 $ /𝑑 2 would be an imaginary number. We don’t have imaginary time! Secondly, a time dilation plot of ∆t/∆t 0 as a function of relative velocity, u will tend to infinity as u approaches c (or in other words as u/c approaches one) This is illustrated graphically in the following slide 40
∆t = ∆t 0 / 1 − 𝑣 $ /𝑑 2 = 𝛿 ∆t 0 and thus ∆t ≥ ∆t 0 The stretching out of time of the time interval is called time dilation The equation Above tells us two things: Firstly, if it were possible to travel faster than the speed of light then 1 – u 2 /c 2 would be negative and 1 − 𝑣 $ /𝑑 2 would be an imaginary number. We don’t have imaginary time! Secondly, a time dilation plot of ∆t/∆t 0 as a function of relative velocity, u will tend to infinity as u approaches c (or in other words as u/c approaches one) This is illustrated graphically in the following slide 41
∆t = ∆t 0 / 1 − 𝑣 $ /𝑑 2 = 𝛿 ∆t 0 and thus ∆t ≥ ∆t 0 The stretching out of time of the time interval is called time dilation The equation Above tells two things: Firstly, if it were possible to travel faster than the speed of light then 1 – u 2 /c 2 would be negative and 1 − 𝑣 $ /𝑑 2 would be an imaginary number. We don’t have imaginary time! Secondly, a time dilation plot of ∆t/∆t 0 as a function of relative velocity, will tend to infinity as u approaches c (or in other words as u/c approaches one) This is illustrated graphically in the following slide 42
Time ime dila dilatio tion ∆t/∆t 0 = 𝛿 8 As u approaches c, 7 ∆t/∆t 0 = 𝜹 = 1/ √(1− u 2 /c 2 ) 𝜹 approaches 6 infinity 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Speed u relative to the speed of light ( u/c ) 43
Time dilation is sometimes described by saying that moving clocks run slow. This must be interpreted carefully The whole point of relativity is that all inertial frames are equally valid so there is no absolute sense in which a clock is moving or at rest 44
Time dilation is sometimes described by saying that moving clocks run slow. This must be interpreted carefully The whole point of relativity is that all inertial frames are equally valid so there is no absolute sense in which a clock is moving or at rest 45
To illustrate this point, this image shows two firecracker explosions i.e. two events that occur at different positions in the ground frame Assistants on the ground need two clocks to measure the time interval ∆t In the train reference frame however a single clock is present at both events, hence the time interval measured in the train reference is the proper time ∆t 0 46
To illustrate this point, this image shows two firecracker explosions i.e. two events that occur at different positions in the ground frame Assistants on the ground need two clocks to measure the time interval ∆t In the train reference frame however a single clock is present at both events, hence the time interval measured in the train reference is the proper time ∆t 0 47
To illustrate this point, this image shows two firecracker explosions i.e. two events that occur at different positions in the ground frame Assistants on the ground need two clocks to measure the time interval ∆t In the train reference frame however a single clock is present at both events, hence the time interval measured in the train reference is the proper time ∆t 0 48
In this sense the moving clock (the one that is present at both events) ‘runs slower’ than the the clocks that are stationary with respect to both events More generally, the time interval between two events is smallest in the reference frame in which the two events occur at the same position 49
In this sense the moving clock (the one that is present at both events) ‘runs slower’ than the the clocks that are stationary with respect to both events More generally, the time interval between two events is smallest in the reference frame in which the two events occur at the same position 50
In deriving the time dilation equation we made use of a light clock which made our analysis clear and easy The conclusion is about time itself Any clock, regardless of how it operates (e.g. a grandfather clock, a wind-up wristwatch, alarm clock or supper accurate quartz clock (as used in GPS satellites)) behave the same! 51
In deriving the time dilation equation we made use of a light clock which made our analysis clear and easy The conclusion is about time itself Any clock, regardless of how it operates (e.g. a grandfather clock, a wind-up wristwatch, alarm clock or supper accurate quartz clock (as used in GPS satellites)) behave the same! 52
In deriving the time dilation equation we made use of a light clock which made our analysis clear and easy The conclusion is about time itself Any clock, regardless of how it operates (e.g. a grandfather clock, a wind-up wristwatch, digital watch, alarm clock or a super accurate quartz clock) behaves in the same way! 53
Time ime dila dilatio tion ∆t/∆t 0 = 𝛿 8 As u approaches c, 7 ∆t/∆t 0 = 𝜹 = 1/ √(1− u 2 /c 2 ) 𝜹 approaches 6 infinity 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Speed u relative to the speed of light ( u/c ) 56
For 𝛦 t/ 𝛦 t 0 = 7, u/c = 0.990 For 𝛦 t/ 𝛦 t 0 = 8, u/c = 0.992
Fa Faster than the speed of light?
Space is expanding faster than the speed of light. This is because spacetime itself is expanding and is denying us the opportunity to see further than 14 billion light years In water, muons can travel faster then the speed of light. This is known as Cherenkov light which has a distinct blue hue. It can be observed in nuclear reactors. Although this is true nothing can travel faster than the speed of light in a vacuum Neutrinos from super nova explosions arrive at earth before photons do. This is because the photons take a significant amount of time to escape from the exploding star while neutrinos (with near zero mass)escape unhindered We are constantly moving through spacetime at the speed of light in a vacuum. We either experience space or time or a mixture of both 59
Hubble ultra deep field image Galaxies as old as 13 billion years are visible 60
Space is expanding faster than the speed of light. This is because spacetime itself is expanding and is denying us the opportunity to see further than 14 billion light years In water, muons can travel faster than the speed of light. This is known as Cherenkov light which has a distinct blue hue. It can be observed in nuclear reactors. Although this is true, nothing can travel faster than the speed of light in a vacuum Neutrinos from super nova explosions arrive at earth before photons do. This is because the photons take a significant amount of time to escape from the exploding star while neutrinos (with near zero mass)escape unhindered We are constantly moving through spacetime at the speed of light in a vacuum. We either experience space or time or a mixture of both 61
An example of Cherenkov radiation inside a nuclear reactor where muons (heavy electrons) travel faster than photons of light in water 62
Space is expanding faster than the speed of light. This is because spacetime itself is expanding and is denying us the opportunity to see further than 14 billion light years In water, muons can travel faster than the speed of light. This is known as Cherenkov light which has a distinct blue hue. It can be observed in nuclear reactors. Although this is true, nothing can travel faster than the speed of light in a vacuum Neutrinos from super nova explosions arrive at earth before photons do. This is because the photons take a significant amount of time to escape from the exploding star while neutrinos (with near zero mass) escape unhindered We are constantly moving through spacetime at the speed of light in a vacuum. We either experience space or time or a mixture of both 63
Space is expanding faster than the speed of light. This is because spacetime itself is expanding and is denying us the opportunity to see further than 14 billion light years In water, muons can travel faster than the speed of light. This is known as Cherenkov light which has a distinct blue hue. It can be observed in nuclear reactors. Although this is true, nothing can travel faster than the speed of light in a vacuum Neutrinos from super nova explosions arrive at earth before photons do. This is because the photons take a significant amount of time to escape from the exploding star while neutrinos (with near zero mass)escape unhindered We are constantly moving through spacetime at the speed of light in a vacuum. We either experience space or time or a mixture of both 64
Ti Time e Dilati tion n in n na natur ture 65
Image of an exploding supernova in a distant galaxy. Its brightness decays at a certain rate but because it is moving away from us at a substantial fraction of the speed of light, it decays more slowly as seen from earth. The super nova is a ‘moving clock that runs slow.’ 66
High energy cosmic ray protons entering our upper atmosphere interact with the nuclei of N 2 and O 2 generating pions which then decay into muons (heavy electrons) which move off at a speed of 0.994 c The half life of a muon is 2.2 microseconds. After 660 meters half the muons would have decayed but at a speed of 0.994c the half life is 20 microseconds. About 25% of the muons created reach the ground. If there was no time dilation only 1/2 20 muons would reach the earth 67
High energy cosmic ray protons entering our upper atmosphere interact with the nuclei of N 2 and O 2 generating pions which then decay into muons (heavy electrons) which move off at a speed of 0.994c. The half life of a muon is 2.2 microseconds After 660 meters half the muons would have decayed but at a speed of 0.994c the half life is 20 microseconds. About 25% of the muons created reach the ground. If there was no time dilation only 1/2 20 muons would reach the earth 68
High energy cosmic ray protons entering our upper atmosphere interact with the nuclei of N 2 and O 2 generating pions which then decay into muons (heavy electrons) which move off at a speed of 0.994c. The half life of a muon is 2.2 microseconds. After 660 meters half the muons would have decayed but at a speed of 0.994 c the half life is 20 microseconds About 25% of the muons created reach the ground. If there was no time dilation only 1/2 20 muons would reach the earth 69
High energy cosmic ray protons entering our upper atmosphere interact with the nuclei of N 2 and O 2 generating pions which then decay into muons (heavy electrons) which move off at a speed of 0.994c. The half life of a muon is 2.2 microseconds. After 660 meters half the muons would have decayed but at a speed of 0.994c the half life is 20 microseconds. About 25% of the muons created reach the ground If there was no time dilation only 1/2 20 muons would reach the earth 70
High energy cosmic ray protons entering our upper atmosphere interact with the nuclei of N 2 and O 2 generating pions which then decay into muons (heavy electrons) which move off at a speed of 0.994c The half life of a muon is 2.2 microseconds After 660 meters half the muons would have decayed but at a speed of 0.994c the half life is 20 microseconds About 25% of the muons created reach the ground If there was no time dilation only 1/2 20 muons would reach the earth 71
You can build your own muon detector! All you need is a mobile phone with a camera + a strip of black insulation tape For an iPhone download the app from cosmicrayapp.com. For other phones there are equivalent apps Tape up the camera lens and you are ready to go Just follow the app’s instructions
Wh Why y do don’t n’t we expe xperienc nce time di dilation n in n our ur ev everyday lives? 73
The sun with the earth in tow is travelling around the centre of the milky way at a speed of approximately 220 000 m/s At this speed 𝜹 for the earth is only 1.00000027 around the centre of our galaxy At such a low value of 𝜹 , the surface of the earth is to all intents and purposes an inertial reference frame A high velocity rifle bullet has a 𝜹 of only 1.000 000 000 001 It is not surprising that we don’t experience relativity I our everyday lives! 74
The sun with the earth in tow is travelling around the milky way at a speed of 217 261 m/s At this speed 𝜹 for the earth is only 1.000 000 3 as it moves around the centre of our galaxy At such a low value of 𝜹 , the surface of the earth is to all intents and purposes an inertial reference frame A high velocity rifle bullet has a 𝜹 of only 1.000 000 000 001 It is not surprising that we don’t experience relativity I our everyday lives! 75
The sun with the earth in tow is travelling around the milky way at a speed of 217 261 m/s At this speed 𝜹 for the earth is only 1.000 000 3 as it moves around the centre of our galaxy At such a low value of 𝜹 , the surface of the earth is to all intents and purposes an inertial reference frame A high velocity rifle bullet has a 𝜹 of only 1.000 000 000 001 It is not surprising that we don’t experience relativity I our everyday lives! 76
The sun with the earth in tow is travelling around the milky way at a speed of 217 261 m/s At this speed 𝜹 for the earth is only 1.000 000 3 as it moves around the centre of our galaxy At such a low value of 𝜹 , the surface of the earth is to all intents and purposes an inertial reference frame A high velocity rifle bullet has a 𝜹 of only 1.000 000 000 001 It is not surprising that we don’t experience relativity I our everyday lives! 77
The sun with the earth in tow is travelling around the milky way at a speed of 217 261 m/s At this speed 𝜹 for the earth is only 1.000 000 3 as it moves around the centre of our galaxy At such a low value of 𝜹 , the surface of the earth is to all intents and purposes an inertial reference frame A high velocity rifle bullet has a 𝜹 of only 1.000 000 000 001 When bloodhound finally reaches its target speed of 1000 mph, its 𝜹 will only be 1.000 000 000 000 6 78
Ti Time e Dilati tion n in n Practi tice 79
Cathode ray tube in which electrons reach 30% of the speed of light 80
81
Le Length c con ontract ction on
Re Relativity of length We also need to derive a quantitative relationship between lengths in different coordinate systems (i.e. different reference frames) using another thought experiment Once again, we have a train travelling near to the speed of light along a stretch of straight railway track Sarah is travelling in the carriage in reference frame S’ Next to her on the seat is a ruler, a light source and a mirror as illustrated 83
Re Relativity of length We also need to derive a quantitative relationship between lengths in different coordinate systems (i.e. different reference frames) using another thought experiment Once again, we have a train travelling near to the speed of light along a stretch of straight railway track Sarah is travelling in the carriage in reference frame S’ Next to her on the seat is a ruler, a light source and a mirror as illustrated 84
Re Relativity of length We also need to derive a quantitative relationship between lengths in different coordinate systems (i.e. different reference frames) using another thought experiment Once again, we have a train travelling near to the speed of light along a stretch of straight railway track Sarah is travelling in the carriage in reference frame S’ Next to her on the seat is a ruler, a light source and a mirror as illustrated 85
Re Relativity of length We also need to derive a quantitative relationship between lengths in different coordinate systems (i.e. different reference frames) using another thought experiment Once again, we have a train travelling near to the speed of light along a stretch of straight railway track Sarah is travelling in the carriage in reference frame S’ Next to her on the seat is a ruler, a light source and a mirror as illustrated 86
Sarah 87
Sarah Peter 88
By using logic like the derivation of time dilation we get ℓ = ℓ 0 / 𝛿 Length contraction formula In special relativity a length ℓ 0 measured in the frame in which the body is at rest is called a proper length Lengths measured perpendicular to the direction of travel are not contracted (the velocity in the y and z direction is zero) 89
By using logic like the derivation of time dilation we get ℓ = ℓ 0 / 𝛿 Length contraction formula In special relativity a length ℓ 0 measured in the frame in which the body is at rest is called a proper length Lengths measured perpendicular to the direction of travel are not contracted (the velocity in the y and z direction is zero) 90
By using logic like the derivation of time dilation we get ℓ = ℓ 0 / 𝛿 Length contraction formula In special relativity a length ℓ 0 measured in the frame in which the body is at rest is called a proper length Lengths measured perpendicular to the direction of travel are not contracted (the velocity in the y and z direction is zero) 91
Rearranging the previous equation we get ℓ / ℓ 0 = 1/ 𝛿 What this tells us is that observers measure any ruler to contract in length if it moves relative to them To the traveler her ruler will continue to show the proper length ℓ 0 as she is at rest in her reference frame What the equation also tells us is that as a traveler approaches the speed of light her ruler will contract to zero as observed by a stationary observer as shown in the next slide 92
Rearranging the previous equation we get ℓ / ℓ 0 = 1/ 𝛿 What this tells us is that observers measure any ruler to contract in length if it moves relative to them To the traveler her ruler will continue to show the proper length ℓ 0 as she is at rest in her reference frame What the equation also tells us is that as a traveler approaches the speed of light her ruler will contract to zero as observed by a stationary observer as shown in the next slide 93
Rearranging the previous equation we get ℓ / ℓ 0 = 1/ 𝛿 What this tells us is that observers measure any ruler to contract in length if it moves relative to them To the traveler her ruler will continue to show the proper length ℓ 0 as she is at rest in her reference frame What the equation also tells us is that as a traveler approaches the speed of light her ruler will contract to zero as observed by a stationary observer as shown in the next slide 94
Rearranging the previous equation we get ℓ / ℓ 0 = 1/ 𝛿 What this tells us is that observers measure any ruler to contract in length if it moves relative to them To the traveler her ruler will continue to show the proper length ℓ 0 as she is at rest in her reference frame What the equation also tells us is that as a traveler approaches the speed of light her ruler will contract to zero as observed by a stationary observer as shown in the next slide 95
Le Leng ngth th contr trac actio tion ℓ / ℓ 0 = 1/ 𝛿 1 0.9 𝓶 / 𝓶 0 = 1/ 𝛅 = √(1− u 2 /c 2 ) 0.8 0.7 0.6 As u approaches c, 0.5 1/ 𝛅 approaches 0.4 zero 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Speed u relative to the speed of light c (u/c) 96
Tarring roads reduces the distance! An advert seen in Johannesburg international airport A useful relationship to remember: ∆t 0 / ∆t = l/l 0 = 1/ 𝛿
Tarring roads reduces the distance! An advert seen in Johannesburg international airport A useful relationship to remember: ∆t 0 / ∆t = ℓ / ℓ 0 = 1/ 𝛿
Length th con ontr traction action of of a a cu cube as as it it wou ould ld ap appear ar at t var ariou ious s rela lativ tive velocitie locities Measured length Visual Appearance 0.0 c Measured length Visual Appearance Measured length Visual Appearance 0.99 c 0 0.5 c
Length th con ontr traction action of of a a cu cube as as it it wou ould ld ap appear ar at t var ariou ious s rela lativ tive velocitie locities Measured length Visual Appearance 0.0 c Measured length Visual Appearance Measured length Visual Appearance 0.99 c 0 0.5 c
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