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Transcript

Einstein’s Special Relativity This presentation was adapted from one I prepared for my students doing the DS 3.1 in 2004 in the pilot physics program. It was given at STAVCON Nov 2004. This version has been reduced a little to keep its download size moderate. It is copyright but made available for teachers free use in the classroom. Feel free to adapt it to your purposes. Please note that it does not cover the whole curriculum. The first part was directed to teachers at STAVCON and could be omitted or modified for students. There are also resources for teachers shown at the end. Enjoy teaching the Relativity DS! Keith Burrows, Nov 2004 PS – see the notes pages for some added comments. The drawings from Heinemann Physics 12 are by Linus Lane Einstein’s Special Relativity in VCE Physics Keith Burrows STAVCON 2004 Why Relativity? What is physics really all about? What physics really is all about: Newton realised that we assume that time is ‘straight’ – no loops allowed! and that time and space are unrelated. These illustrations from Hawking: Universe in a Nutshell Einstein said that we should not assume this. He said that travel through time and space were intimately related. Special relativity is about that relationship. These illustrations from Hawking: Universe in a Nutshell So why study relativity? Relativity represents a ‘giant step’ in the story of physics. Why leave out the climax of the story? It is an excellent illustration of the process and nature of physics. Through it we, and our students, can get a feel for real physics. “Imagination is more important than knowledge” Albert Einstein So why study relativity? Relativity is about questioning common assumptions and finding new ways of looking at a situation. What the world needs now… is a reasoned, thoughtful, and questioning approach to the social, political and environmental challenges we face. Which is just what we see in the ‘big picture’ story of physics. Albert Michelson (1898): “While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of the principles to all the phenomena which come under our notice.” The view in the early 1900’s In 1900 the mechanical world view seemed capable of explaining just about everything. Did this lead to the materialism and economic rationalism of the twentieth century? Relativity could change the way we think! “When the ideas involved in relativity have become familiar, as they will do when they are taught in schools, certain changes in our habits of thought are likely to result, and to have great importance in the long run.” Bertrand Russell ‘ABC of Relativity’ Relativity for all This is not (just) for the specialists, it is for future... Journalists Teachers Politicians Lawyers Hairdressers Mothers and Fathers Citizens Developing the story 1. Two principles Einstein did NOT want to give up 2. Einstein's crazy idea 3. Time is not as it seems: Time Dilation 4. If time is strange, what about space? 5. Faster than light? Momentum, Energy and E = mc² Summary of 1: Two principles Einstein did NOT want to give up The principle of relativity seems universal Maxwell’s equations suggested light is an electromagnetic wave and has a fixed speed which was assumed to be the speed through the aether Michelson and Morley could not detect the aether. The principle of relativity seemed inconsistent with the predictions of Maxwell’s equations! 1. Two principles Einstein did NOT want to give up (1) Galilean/Newtonian “principle of relativity”: Nothing special about a velocity of zero Velocity can only be measured relative to some other frame of reference No absolute velocity Force changes velocity The laws of physics are the same in any inertial frame. 1. Two principles Einstein did NOT want to give up (1) The principle of relativity 1. Two principles Einstein did NOT want to give up (2) Maxwell and the speed of light In the 1830’s Michael Faraday suggested that light may be some sort of electromagnetic wave phenomenon. In the 1860’s James Clerk Maxwell developed his famous electromagnetic equations. 1. Two principles Einstein did NOT want to give up (2) Maxwell and the speed of light The equations suggested the possibility of electromagnetic waves travelling through space from an accelerated charge. 1. Two principles Einstein did NOT want to give up (2) Maxwell and the speed of light Maxwell’s equations predicted that electromagnetic waves would travel at a speed given by a simple expression involving electric and magnetic constants. c1 o o 3 10 m / s 8 1. Two principles Einstein did NOT want to give up (2) Maxwell and the speed of light But this expression suggested that electromagnetic waves would travel at this fixed speed in any frame of reference. Most physicists, including Maxwell, thought this must be wrong and that perhaps this speed was relative to the aether... the aether being a hypothetical medium which filled all space. 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether Was the speed of electromagnetic waves relative to an absolute frame of reference – an aether? But this would be contrary to the principle of Galilean relativity! Michelson and Morley decided to look for evidence of the Earth’s motion through the aether. 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether The principle of their experiment – an analogy: 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether The principle of their experiment: 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether They couldn’t measure the speed accurately enough, but they could compare speeds in two perpendicular directions very accurately. They knew that these speeds should be a little different if the Earth was speeding through the aether at 30 km/sec. 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether A water analogy: The boat travels at 5 m/s in a river flowing at 3 m/s 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether It travels at 4 m/s across the river and so takes 2000/4 = 500 sec to complete a two way trip 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether But it travels at 2 m/s upstream and 8 m/s downstream and so takes: 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether 1000/2 + 1000/8 = 500 + 125 = 625 sec for the two way trip parallel to the water. 1. Two principles Einstein did NOT want to give up Michelson and Morley look for the aether When they rotated their apparatus they found NO DIFFERENCE in the speed of light in the two perpendicular directions! 1. Two principles Einstein did NOT want to give up Einstein knew about the Michelson-Morley experiment, but he does not seem to have been particularly interested in it… his main interest was in Maxwell’s equations and their predictions about the speed of light and it’s relativity. 1. Two principles Einstein did NOT want to give up Einstein thought the principle of relativity was so fundamental it should apply in all areas of physics – including electromagnetism He also thought Maxwell’s equations were so elegant they, and their prediction about the speed of electromagnetic waves, had to be true. But how could these two great ideas be reconciled? Surely the speed of light (like every other speed) should depend on one’s frame of reference! Summary of 2: Einstein’s crazy idea Einstein thought about light and decided that it should be impossible to catch up to it. He also thought the aether did not make sense and scrapped it. This led to his two postulates: I II No law of physics can identify a state of absolute rest. The speed of light is the same for all observers. The implication of taking these postulates at face value is that time seems to be relative! 2. Einstein's crazy idea (1) Einstein had spent a lot of time wondering about the nature of light… 2. Einstein's crazy idea (2) ... and concluded that you could not catch up to light – or the electromagnetic waves would be ‘frozen’ in space. This was something that had never been seen – and seemed impossible. 2. Einstein's crazy idea (2) Einstein did all his experiments in his head – Gedanken experiments. (Much neater than messy apparatus!) This might sound odd – but many of the greatest discoveries in physics were through Gedanken experiments… Including Newton’s three laws! 2. Einstein's crazy idea (2) Einstein realised that the principle of relativity was an extremely ‘elegant’ principle. The real world would be very messy without it! 2. Einstein's crazy idea (2) If there was a way to find an absolute velocity (a ‘veelo’) – whose system is absolute? It seemed more likely that the universe was truly democratic! 2. Einstein's crazy idea (2) All motion is relative – there is no absolute frame of reference. However changes of velocity are absolute – acceleration is absolute. 2. Einstein's crazy idea (2) If this is the case how could light move in some aether which permeated all space? Einstein decided that it can not be possible to use light to determine an absolute velocity. 2. Einstein's crazy idea (2) He therefore scrapped the idea of the aether… and concluded that any measurement of the speed of light must give the same result: 3 x 108 m/s 2. Einstein's crazy idea (3) I He decided to keep these two great principles and so put forward two postulates which embodied them: No law of physics can identify a state of absolute rest. II The speed of light is the same for all observers. 2. Einstein's crazy idea (4) At first sight these seem quite simple and straightforward – except that in classical physics they are inconsistent! Einstein said the aether was unnecessary, but this still left the problem of the relative velocity of light in different frames. 2. Einstein's crazy idea (5) There was another problem. Einstein realised that there was a problem with ‘simultaneity’ if all observers saw light with the same speed... Ana and Ben see the light hit the ends at the same time... 2. Einstein's crazy idea (5) but Chloe sees the light travelling at c as well and so it takes longer to reach the front of the train. So what Ana and Ben saw as simultaneous, Chloe saw as separate events! 2. Einstein's crazy idea (6) This can only mean that there is something strange about time! Time appears to be relative. Salvador Dali – The Persistence of Memory Summary of 3: Time is not as it seems We can put numbers into this flash-in-the-train situation and find out how different the times are. The light clock enables us to generalise. We find that time in a moving frame appears to run slow (to the stationary observers) The time dilation equation: t where 1 2 v 1 c2 = γto 3. Time is not as it seems (1) We can look at the train situation qualitatively: For Ana & Ben the time for the light to get from the centre and back again is 2l/c. The maths of Chloe’s view. Time taken for light to get to (either) end of carriage and back again: Tc = t1 + t2 = l/(c + v) + l/(c – v) = l(c – v + c + v)/(c² – v²) = 2lc/(c² – v²) = 2lc/[c²(1– v²/c²)] = 2l/[c(1 – v²/c²)] = 2l/c 1/(1 – v²/c²) So: Tc = TA γ² where γ = 1/√(1 – v²/c²) (But note that we cancelled the l’s in the two expressions to obtain this!) 3. Time is not as it seems (2) For Chloe it is 2l/c 1/(1 – v2/c2). Remember that Chloe sees the light travelling at c, not c ± v. We can write this as Tc = TA γ² where 1 v 1 2 c2 3. Time is not as it seems (3) We need to note here that to obtain this result we cancelled the two l’s in the expressions for the times in the two frames of reference. 3. Time is not as it seems (4) As gamma, γ = 1/√(1 – v²/c²) must always be greater than one, we see that time as measured from the rest frame is greater than that within the moving frame. Chloe sees Ana and Ben’s clocks going slowly! 3. Time is not as it seems (5) To be sure about this ‘time dilation’ we need to be a little more careful about the way we measure time. A ‘light clock’ overcomes the problems of possible changes in lengths because it uses light itself. We will call TA the time for a one way trip of the light pulse. 3. Time is not as it seems (6) As we watch a moving clock the light travels further between bounces – but still at speed c! TC is the time for one ‘zig’ in this frame. The rocket travels vTC in the time for one ‘zig’. The maths of the light clock: For A: d = cTA For C: Pythagoras tells us (cTc)2 = (vTc)2 + d2 Eliminate d gives: (cTc)2 = (vTc)2 + (cTA)2 A little reorganising gives: (Tc/TA)2 = c2/(c2 – v2) Or Tc/TA = γ = 1/√(1 – v²/c²) 3. Time is not as it seems (7) We find Tc/TA = γ = 1/√(1 – v²/c²) Or more generally t = γto where t is the time as measured from a stationary frame and to is the time as measured in the moving frame. Einstein’s time dilation equation: 3. Time is not as it seems (8) Earlier we wrote Tc = TA γ² for the time ratio. Why the difference? To obtain this we cancelled the length of the train as seen in one frame with that seen in the other. This assumed they were the same! But if time behaves strangely so maybe does space... Summary of 4: Space is strange too! We cancelled the l’s as seen by A&B on the one hand and by C on the other. But the l Chloe saw had shrunk! – by a factor of γ The length contraction equation: l = lo/γ Moving objects appear shorter because of their motion. This is because space itself contracts, not the object. The twins ‘paradox’ illustrates the strange nature of spacetime. 4. Space is strange too! (1) Einstein’s time dilation equation is correct! The extra gamma in the earlier equation is due to the fact that the l’s were different by the same gamma factor. In fact we see a moving space ship contracted in the direction of its motion. 4. Space is strange too! (2) Einstein’s train was also contracted in the direction of its motion – this is why the time appeared to have slowed by more than γ. In fact the light pulse didn’t go as far as we thought. 4. Space is strange too! (3) Lengths are contracted by the gamma factor Einstein’s length contraction equation: l = lo/γ 4. Space is strange too! (4) Remember that this contraction is all relative From Mr Tompkins in Wonderland by George Gamow 4. Space is strange too! (4) Remember that this contraction is all relative 4. Space is strange too! (4) A two to three dimensional analogy for a three to four dimensional situation: How far is it from Sydney to Perth? 4. Space is strange too! (4) It is hard to picture space that is not ‘straight’ But do our X-Y-Z axes eventually bend around and join up again? ... just as a 2-d grid does on the Earth’s surface. 4. Space is strange too! Einstein’s ‘Twins Paradox’ illustrates the strange relationship between space and time. Imagine one twin travels to Vega, 25 l.y. away at 99.5% of c (γ = 10) His trip will take 25.1 years as measured from Earth (although we won’t get his signal for 50.2 years) (5a) 4. Space is strange too! Although, from Earth, the traveller takes 25.1 years to get to Vega ... the traveller will only experience 2.5 years – because his time, as we see it, is going slowly. But we both agree on what his clock says, that is, 2.5 years. (5b) 4. Space is strange too! (5c) Note that he does not feel that time has slowed down. From our point of view his time has slowed. But what he sees is that the space he is travelling through is contracted by 10 times and so it only takes him 1/10 of the time we calculate. 4. Space is strange too! When he gets to Vega he doesn’t like the Vegans and so turns around and comes straight back, taking another 2.5 years for the return trip. (He is a Gedanken traveller and doesn’t get squashed by the acceleration (5d) 4. Space is strange too! (5e) But what took him 5 years, we saw over 50 years! He returns 5 years older but his brother is 50 years older! This is not a paradox – it is true! Clocks flown around the Earth, and in satellites have confirmed it. From Hawking: Universe in Nutshell Summary of 5: Momentum & Energy As the speed of an object gets greater γ approaches infinity. Time slows to a stop and length contracts to nothing. Why can’t we accelerate past c? Momentum also increases with γ, which makes it appear that mass does also. Hence more impulse increases momentum – but the m, not the v. Total energy also increases in a similar way, but there is a ‘rest mass’ component: E = mc² Energy has mass. 5. Momentum & Energy (1) The Lorentz factor (gamma) approaches infinity as the velocity approaches the speed of light 1 2 v 1 c2 v/c γ 1% 1.00005 10% 1.005 90% 2.29 99% 7.09 99.9% 22.4 99.999% 224 5. Momentum & Energy So what happens to time and length? Time slows down to a standstill! (2) 5. Momentum & Energy So what happens to time and length? Time slows down to a standstill! Length contracts to nothing! (2) 5. Momentum & Energy (3) But why can’t we just keep accelerating beyond the speed of light? 5. Momentum & Energy Einstein showed that momentum was also affected by the Lorentz factor po p (4) 2 v 1 c2 or p = γpo As the speed increases it is as though the mass increases toward infinity: mv = γmov 5. Momentum & Energy (5) Only massless photons can travel at the speed of light. For photons time has slowed to nothing and length to zero. A photon crosses the universe in no time – in its frame of reference. (Which is why it lasts forever!) 5. Momentum & Energy (6) Einstein also showed that the kinetic energy of a mass is given by: Ek = (γ – 1)moc² This looks odd, but (with the help of the binomial theorem) does reduce to Ek = ½mv² at normal speeds. (Remember the v is in the γ.) Reorganising the expression gives: γmoc² = Ek + moc² which he said was the ‘total energy’. 5. Momentum & Energy (7) The ‘total energy’: γmoc² = Ek + moc² So what is the moc² ? Einstein said it is the energy associated with the mass of an object. In fact energy and mass are different manifestations of the same thing: ‘massenergy’ So E = mc² (or γmoc²) actually represents the total energy of an object (including kinetic). Usually, however, the kinetic energy is a 5. Momentum & Energy (8) But it is the kinetic with which we are normally concerned as we can’t change the mass into energy. But, in some nuclear reactions the energy released is so great that there is a significant decrease of mass: When uranium splits into ‘fission fragments’ the mass of the fragments is about 1% less. When hydrogen fuses to produce helium, the helium has less mass than the hydrogen. We find that the energy released is just equal to E = Δmc² where Δm is the lost mass. 5. Momentum & Energy (9) However it is important to realise that this mass has not been ‘converted’ into energy! This is a common misconception created by popular accounts of the meaning of E = mc² Energy and mass are different ‘manifestations’ of the same thing. The energy associated with bonding (whether nuclear or chemical) has mass and it is this mass that is ‘lost’ when energy is released by a reaction. 5. Momentum & Energy (10) If we could contain an atom bomb in a (very strong!) box, would it get lighter after the explosion? The answer is… No – the energy is still in the box, and so is its mass. But as the hot box radiated energy away it would lose mass. The relevance of relativity (1) Is it all just something of concern for space travellers? Of course not! It tells us something very fundamental about the nature of our universe. It takes us beyond the ‘clockwork universe’ picture. It has many practical consequences. Mostly though, it shows us the power of human reason and gives us cause to wonder in awe at the mysteries of the universe in which we live. The relevance of relativity (2) Some of the practical consequences include: The GPS system. The Synchrotron – which could be a few centimetres in diameter if it weren’t for the huge increase in mass of the electrons at 99.9999% of the speed of light. Nuclear energy The relevance of relativity (3) Curiously enough, magnetism can only be understood properly with relativity. In the late 1800’s it was realised that there was a problem with the theory of magnetism. The relevance of relativity (3) A moving charged particle is deflected by a magnetic field. This is the origin of the force that drives all electric motors. Faraday’s original ‘motor’ A modern AC Induction motor The relevance of relativity (3) A moving charged particle is deflected by a magnetic field... whether in a wire or in free space The relevance of relativity (4) A moving charged particle is deflected by a magnetic field... but what if we observe it from a frame of reference moving at the same speed? In this frame it is not moving (although still in a magnetic field) and so... should not experience a force! This can’t be true – if it experiences a force in one frame it must also in another. The relevance of relativity (5) In fact Einstein’s paper was called “On the Electrodynamics of Moving Bodies” It starts with the question of how a force could be velocity dependent and not contravene the ‘principle of relativity’. The answer is that it can’t! So what about F = qvB? The relevance of relativity (5) Relativity says that magnetism and electricity are aspects of the same force. The relevance of relativity (6) We often think of relativity in terms of high speeds. But actually it is needed to explain the magnetic force between currents moving at millimeters per second! But the electrons in one wire are at rest with respect to those in the other wire! The relevance of relativity (6) If we look at the situation from the point of view of the electrons, the positives are moving but the electrons are at rest and should not experience a magnetic force (F = qvB) The moving positives create a magnetic field, but as v = 0 there should be no force on the electrons! The relevance of relativity (6) Actually there is a balance between the forces between the electrons and the positive atoms in the wires Separately, these two forces are HUGE! … but slightly different because: The relevance of relativity (6) The positive atoms are in motion relative to the electrons and are therefore Lorentz contracted – that is, their density increases and the electrons see more positives than negatives And are therefore attracted to the other wire! The relevance of relativity (6) The magnetic attraction in one frame of reference is simply the electrostatic attraction in another frame of reference. Magnetism and electrostatics are the same after all! Resources Resources Resources Resources Resources Activities “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.” Albert Einstein