Special Relativity and Time Dilation 5th December You will need paper, pen and a calculator Use the chat pod to ask questions

Relativity – pre Einstein Absolute space and time Newton’s Laws of Motion. These laws allow us to describe the motion of objects, regardless of size or position. This leads to a ‘clockwork-type’ view of the world. Clocks tick at the same rate regardless of their movement 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.

Relativity- pre Einstein The laws of physics remain the same whether one is moving steadily or at rest. ‘frame of reference’ 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. frame of reference’

Special Relativity Special because it considers only the case where different observers are in relative motion with a constant velocity. Einstein supported the Newtonian view that the laws of physics should apply everywhere Work of James Clerk Maxwell led to an apparent contradiction with this principle

James Clerk Maxwell Great Scottish Physicist His theory of electromagnetism showed that light is an electromagnetic wave Maxwell’s equations allowed the speed of light to be predicted theoretically. Experiments undertaken by Michelson was in agreement with predicted value Einstein concluded that the speed of light is constant for all observers. Maxwells’s equation didn’t take speed or direction into account.

Consequences Newtonian idea of absolute space and time was rejected 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. So, two events may be simultaneous for one observer but not for the other observer.

Summary 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 inside their frame of reference in all parts of the universe. 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). It is always the same for all observers irrespective of their relative velocities. (Light does slow down inside transparent material such as glass.)

Quick check 1 1. A river flows at a constant speed of 0.5 m s−1 south. A canoeist is able to row at a constant speed of 1.5 m s−1. a) Determine the velocity of the canoeist relative to the river bank when the canoeist is moving upstream. b) Determine the velocity of the canoeist relative to the river bank when the canoeist is moving downstream.

Quick check 2 The steps of an escalator move at a steady speed of 1.0 m s−1 relative to the stationary side of the escalator. a) A man walks up the steps of the escalator at 2.0 m s−1. Determine the speed of the man relative to the side of the escalator. b) A boy runs down the steps of the escalator at 3.0 m s−1. Determine the speed of the boy relative to the side of the escalator.

Quick check 3 In the following sentences the words represented by the letters A, B, C, D, E, F and G are missing: Match each letter with the correct word from the list below: acceleration different Einstein’s fast lengthened Newton’s same slow speed of light velocity zero shortened In _____A____ Theory of Special Relativity the laws of physics are the _____B____ for all observers, at rest or moving at constant velocity with respect to each other i.e. _____C____ acceleration. An observer, at rest or moving at constant _____D____ has their own frame of reference. In all frames of reference the _____E____, c, remains the same regardless of whether the source or observer is in motion. Einstein’s principles that the laws of physics and the speed of light are the same for all observers leads to the conclusion that moving clocks run _____F____ (time dilation) and moving objects are _____G____ (length contraction).

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.

Explanation 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.

Quick check 4 A spaceship is travelling away from the Earth at a constant speed of 1.5 × 108 m s−1. A light pulse is emitted by a lamp on the Earth and travels towards the spaceship. Find the speed of the light pulse according to an observer on: a) The Earth; b) The spaceship. 3x 10 8 as speed of light is the same in all directions

Time Dilation For two observers in the same frame of reference with identical accurate clocks the rate at which time elapses on these clocks will depend on their relative motion. If one observer is stationary in that frame of reference and the other is moving, the clock of the moving observer appears to be going slower than the stationary observer. Neither clock can be said to be wrong. If either clock is used to measure the speed of light for that observer they will arrive at the correct value of 3 X 108 m s-1. This is referred to as time dilation.

Where the equations come from 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

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).

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. Pulse as seen by the travellers

A right- angled triangle can be formed where the vertical side is the height, h, of the pulse ( ct), the horizontal side is half of the distance, d, gone by the train ( vt’) and the hypotenuse is half the distance gone by the pulse as seen by the travellers on the train ( ct’).

Applying Pythagoras to the triangle gives:

Assumptions (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.

The twin paradox A space ship passes Earth at a velocity of 0.5c. It emits a pulse of duration ΔT = 2.0 ns. We on Earth can ‘measure’ the duration t’ we observe. 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 gives us an observed time interval of 1 /0.87 = 2.3 ns. We would say their clock is running slow, time is passing more quickly for us so they could end up ‘younger’! So could we 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 is almost unity no effect is noticeable. It is useful to calculate this term 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. The effects in everyday life are not noticeable. We need v > 10%c for any noticeable effects.

Example Calculate the time dilation for a stationary observer relative to an observer travelling at 100000 m s-1 who measures the time elapsed as 24 hours. Time elapsed for moving observer, t = 24 × 60 × 60 = 86400 s. Time for stationary observer = t'

So we can see the time dilation is extremely small even for what appears to be a high velocity. If we repeat the calculation for an object travelling at 2 × 108 m s-1 we can see the difference. Time elapsed for moving observer, t = 24 × 60 × 60 = 86400 s. Time for stationary observer = t'           The time elapsed for the stationary observer is 115918 s. Measurements show that time elapses at different rates for different observers depending on their speed within a frame of reference.

Quick check 5 In the table shown, use the relativity equation for time dilation to calculate the value of each missing quantity (a) to (f) for an object moving at a constant speed relative to the Earth. Dilated time Proper time Speed of object / m s 1 (a) 20 h 1·00 × 108 (b) 10 year 2·25 × 108 1400 s (c) 2·00 × 108 1.40 × 104 s (d) 84 s 60 s (e) 21 minutes 20 minutes (f) (a) 21·2 h (b) 15·1 year (c) 1043 s (d) 1·32 × 104 s (e) 2·10 × 108 m s 1 (f) 9·15 × 107 m s 1