PHYS1001 Relativity - Research School Of Physics
PHYS1001RelativityCRAIG SAVAGE & JOHN ASLANIDES
PrefaceSee the next section for how to get the most out of thiseText.You will learn better if you attempt the questions asyou read along. Do what you can before looking at thehints or answer.Please contact Craig with any comments or feedbackabout this eText: email Craig.This eText is adapted from Chapter 11 of Physics andTechnology for Future Presidents by Richard Muller.LHC images are from the photo-book LHC by P. Ginter, R.-D. Heuer and Franzobel. For copyright reasonsit is exclusively for use in ANU physics courses. Theadaption was by Craig Savage and John Aslanides.Version: 30/3/12The Large Hadron Collider - the world’s largest scientificinstrument. Protons are accelerated to close to the speedof light in the vacuum pipe.
S ECTION 1Using this eText on theiPadGlossary. Some bold words have glossary entries that appear if you tap them. Double tapping any word looks it up inthe dictionary or glossary.Adding highlights and notes. Touch and drag to highlight text. Tapping a word bring up a menu of options: Define it,highlight it, attach a note, Search for it in the text. Notes are also available as Study Cards.Red text are hyperlinks to external web pages. Touching onewill take you out of iBooks into the Safari web browser. To getback, double click the Home button and touch the iBooks iconat the bottom of the screen.Learn more about how to use this iBooks textbook.Navigation. Tapping the screen brings up the navigation bar.Tapping the Contents icon bring up the contents. Swipingtakes you through the chapters. Select the section from the bottom thumbnail images.The Notes icon brings up the highlighting and notes functions. Swipe, or tap the page edges, to turnthe page. Double tapping an object may open it in full screenmode.Questions. There are questions throughout the text. Attempting them will consolidate what you have learnt from reading.Most people will find the green questions more challengingthan the blue ones.2
S ECTION 2Objectives Be able to calculate relative velocities using the relativisticformula. Understand why the speed of light is invariant. Understand what rest mass m0 and relativistic mass mare. Understand how the formulae E mc2 explains kinetic energy. Understand how the low speed formula for kinetic energyE (1/2)m0v2 arises from the relativistic energy.The relativity section of Foundations of Physics is mainlyabout concepts. The objectives of this section on relativity arefor you to: Understand the meaning of time as a fourth dimension. Be able to apply time dilation for high speed travel. Understand how acceleration determines who time dilation applies to. Understand what a zero rest mass particle is. Know what a neutrino is and why they must have restmass. Understand how tachyons might be accommodated withinrelativity. Know that the simultaneity of spatially separated events isrelative. Be able to calculate the Einstein factor and its inverse, theLorentz factor. Know that the Einstein factor differs from its everyday valueof 1 only close to the speed of light. Be able to apply Lorentz contraction.3
S ECTION 3Useful numbers & stuffEnergy units.1 eV 1.6 10-19 J.Elementary particle charge.The charge on an electron or proton has magnitude1.6 10-19 C.Metric prefixes.nano (n): 10-9This section contains numbers and other information thatmight be useful to you, especially in answering the reviewquestions.Speed of light in different units.c 3 108 m/sc 1.08 108 km/hrRest masses of particles.micro ( ): 10-6milli (m): 10-3kilo (k): 103mega (M): 106giga (G): 109tera (T): 1012A proton has a rest mass is 1.67 10-27 kg or 938 MeV.An electron has a rest mass is 9.11 10-31 kg or 0.511 MeV.In particle physics energy is measured in units called“eV”, which is the kinetic energy an electron gains whenis accelerated through 1 Volt of potential difference. Onemillion eV is denoted by MeV, and a billion eV by GeV.4
C HAPTER 1WhyrelativitymattersA medical linac in an Australian hospital.Relativity plays a small, but sometimes critical, role inmost peoples’ lives.Medical linacs, like the one shown opposite, producerelativistic electrons for treating tumors.The GPS navigation system uses relativistic time dilation.Devices that accelerate particles to relativistic speeds,such as the LHC, are essential for learning about thefundamental physical nature of things.Elementary particles called muons are travelingthrough your body right now. They can only do this because of the relativistic effect called “time dilation”.Linacs accelerate electrons to near the speed of light.The electrons are then either directed onto tissue to killit, or directed onto a metal target to produce x-rays,which are used to kill tissue.
S ECTION 1GPS navigationThe GPS satellite orbits.The GPS, or Global Positioning System, works by measuringthe time radio signals take travelfrom GPS satellites to the receiver.Since radio signals travel at thespeed of light, this gives the distance to the satellite. Each satellitealso transmits its position to the receiver. Knowing the positions andGPS receiver.distances to four satellites is enoughto determine the receiver’s position.This accuracy of the location depends on the accuracy of thesignal timing. Hence each satellite has an atomic clock to keepprecise time to within nano-seconds. Relativistic time dilationhas to taken into account in designing these clocks, otherwisethe GPS system would stop being useful within a few days.6
S ECTION 2Particle acceleratorsThe ANU Research School of Physics and Engineeringshowing the white nuclear physics accelerator tower inthe centre.Accelerators may accelerate electron, protons, or heavy ions.The largest one in Australia is the Australian synchrotron inMelbourne. It accelerates electrons to close to the speed oflight. The electrons are used to produce x-rays for scientific research.The Australian SynchrotronThe ANU has a large accelerator for heavy ions that is usedprimarily for nuclear physicsresearch. ANU also has a number of small accelerators formaterials research and sampledating. In these acceleratorsthe heavy particles usuallyonly reach only a few percentof the speed of light.Inside the accelerator tower.7
S ECTION 3The LHCThe LHC found surprisingly little else during 2011. In particular it found no evidence for supersymmetry which is an essential part of string theory. Supersymmetric particles are also aleading candidate for dark matter. Hopefully, 2012 will bringnew discoveries.Some basic information about the LHC. The LHC has a circumference of 27 km. In operation the LHC has two counter-rotating beams madeup of 2808 bunches of 1.15 1015 protons each.The LHC, or Large Hadron Collider, is an international physics machine used to investigate the most fundamental structure of matter and its interactions. It is a large undergroundcircular structure, 27 km in circumference, near Geneva.It can accelerate either protons or lead nuclei up to very closeto the speed of light. These are in counter-circulating beamsthat may be smashed together. It is these collisions that givethe information. Relativity is essential in its design and in interpreting the meaning of the collision debris.The LHC ran during 2011 and started running again in March2012. The detector experiments have reported preliminarysigns of a much anticipated new particle - the Higg’s boson. Ifthese signs hold up its existence will be confirmed this year.The Higg’s boson is important because it is predicted by theHigg’s mechanism that is the origin of the rest mass of the elementary particles. The LHC accelerates protons to a total energy of 7 GeV. This corresponds to a speed of 0.9999999911 c. That’s eight9s followed by two 1s. Expressed differently it’s 99.99999911%of the speed of light. The associated Einstein factor is1.33 10-4, and the Lorentz factor is γ 7500.Schematic diagram of the underground LHC layout.In reality the separation between the vacuum tubes containingthe two counter-circulating proton beams is only a few centimeters.8
G ALLERY 1.1 LHC. Images are from the photo-book LHC byP. Ginter, R.-D. Heuer and Franzobel. Swipe to see new mages.Tap for larger versions.I NTERACTIVE 1.1 A schematic of the LHC layout.GenevaDetectorSPSLHCATLAS is the largest of the four LHC detectors: 46 m long,25 m high, and weighing 7000 tonnes. It is a general purpose detector.1234The blue ring is where the particle beams countercirculate. They collide at the four locations where shaftscan be seen. Large machines called “detectors” surroundthe collision points and are used to characterise the debrisfrom the collisions.9
S ECTION 4Atmospheric muonsMuons are elementary particles, like heavy electrons. Theylive for about 2.2 micro-seconds, which is why we don’t thinkmuch about them in everyday life. However they are constantly being produced in the upper atmosphere from collisions of high energy particles, such as protons, with the nucleiof atmospheric molecules.They travel at speeds close to that of light, 3 108 m/s. So in2.2 micro-seconds they can cover a distance of:(3 108 m/s) (2.2 10-6 s) 660 m.However, somehow many of them make it the 50 km or so tothe surface of the Earth. Here they are responsible for a significant part of the our background environmental radiationdose.Cosmic rays from space hit molecules in the top of theatmosphere, about 50 km up. These produce showers ofelementary particles, some of which decay into muonsthat reach the surface.You should understand how this is possible once you havestudied time dilation.10
C HAPTER 2TimeMore than 10,000 scientists from around the worldwork together peacefully at CERN, the home of the LHC.People of different beliefs or form countries that are inconflict. Here, they all work together in peace and harmony.The nature of time and space is at theheart of the theory of relativity.
S ECTION 1A DialogueHow can you say that when you don’t know what the motionof time is?Because we can measure relative rates. We can make timeslow down in the laboratory. We see it in the stars.Can we travel in time?Sure. We’re doing it right now. We are both going forward intime.I meant, can we go back in time?Nothing in physics prevents that. But I don’ t believe we can.The student’s words are in plain text. Richard Muller’s responses are in italics.What is time?Believe? I thought we were discussing physics.Nothing in physics prevents going backwards in time. Butbackward time travel violates my belief in my own free will.I don’ t know.What determines the direction of time?Someone told me it was the fourth dimension.Some people will tell you it is determined by entropy. Butthat is controversial, and not proven. There is no way to testthe idea, so it too is more a belief than it is solid physics.That’s just a physicist’s way of confusing you. It’s true, but itis a much less deep statement than you would guess.Time moves. I know that. But I don’t understand it. Whatdoes it mean that time moves?I don’ t know.Does time ever slow down?Yes.Nothing about time is obvious. Yet, given the mysterious nature of time, you may be surprised at some of the things we doknow about it. For example, we know that if two twins are exactly the same age and one travels while the other stays athome, then when they are brought back together, the movingtwin will have experienced less time than the other twin!12
There is nothing odder about time than that. Yet Albert Einstein gave us a formula that tells us precisely how much lesstime the moving twin experienced. And that fact has been experimentally verified with very sensitive clocks flown on airplanes. Even radioactive atoms, when they move, experienceless time than those that are stationary. That fact is verifiedevery day at accelerator laboratories where such atoms aresent near the speed of light, and physicists note that their radioactive decays slow down.Einstein created the theory of relativity in the early 1900s.The theory of relativity consists of two parts. The first is called“the special theory of relativity” and it has to do with the nature of time, space, energy, and momentum. It was in thiswork, published in 1905, that Einstein presented his famousequation E mc2. The second part was published in 1916 andis called the “General Theory of Relativity.” It is really a theoryof gravity. It “explains” all of gravity as due to a bending ofspace and time. This theory is needed to understand some ofthe recent discoveries in cosmology about the nature of theUniverse.The theory of relativity is important to physicists, to philosophers, to those who plan trips to other planets, and to anyonewho wants to have their mind stretched.“A human being is part of a whole, called by us the Universe,a part limited in time and space. He experiences himself, histhoughts and feelings, as something separated from the rest a kind of optical delusion of his consciousness. This delusionis a kind of prison for us, restricting us to our personal desiresand to affection for a few persons nearest us. Our task mustbe to free ourselves from this prison.” Albert Einstein.13
S ECTION 2Events - and the “fourthdimension”Time is often called the “fourth dimension.” That turns out tobe a useful definition, not an observable fact. And it is notsomething super mysterious or deeply abstract. In fact, whenused in that way, the word “dimension” is being used in a verytechnical and narrow way: the “dimension” of a quantity is thenumber of different numbers you need to describe it.pose I were to tell you that there is an event at my house at 8pm tonight. Then there is no confusion; you might not knowwhat is going to happen, but you have located it in both timeand space. The event can have a name, such as “Elizabeth’sbirthday party” or “Melinda goes to bed.” But to be unique(Elizabeth has a birthday each year, and Melinda goes to bedalmost every night) you also specify the time. Events are specified by four numbers. So we say that events are four dimensional. That’s not deep. It is trivial. That’s is the entire meaning of saying that time is the fourth dimension.That is not what is interesting about time. What is interestingis that the amount of time can change depending on the velocity that an object is moving in the three dimensions of space.That idea is deep, and requires some explanation.Suppose you wish to specify a location on the Earth. You cando that with three coordinates, such as latitude, longitude,and altitude. Or you could use a system with x, y, and z. Thekey thing is that you need only three numbers. Any two objects that have the same set of three numbers must be at thesame location. In math, we say that a location is a 3- dimensional number. That’s all that the fancy word dimension actually means. Space is three dimensional.If you want to specify an event, rather than a location, then itis sufficient to give the location and the time of the event. Sup14
Q UESTION 2.1 Your friend is not convinced by this four di-mensions stuff. She says that there are only three dimensions,because you can specify an event with only three numbers: latitude, longitude, and time. Is she:Q UESTION 2.2 : One event occurs in Canberra at 12:00 p.m.on Friday, and another event occurs in Sydney at 12:10 p.m. thesame day. Is it possible for me to be present at both events?How fast would I have to go? Compare this speed to the speedof light.A. Right, the the fourth coordinate can be calculated from theother three.B. Right, because relativity isn’t needed for everyday life.C. Wrong, because she’s forgotten about height.D. Wrong, because latitude and longitude aren’t valid coordinates.15
S ECTION 3Time dilationIn the opening of this chapter we saw how two twins can experience different amounts of time. That seems to violate common sense. How can it be true? The answer is that the effect isvery small unless the velocity is very fast--that’s why younever notice it. Common sense is based on experience, andthat kind of time dilation is not part of our normal lives, so itviolates our common sense. It also violated the common senseof ancient men to think that the sun has a million times thevolume of the earth. Does that violate your common sense?Sometimes, all it takes to incorporate something into yourcommon sense is to hear it many times, or to gain familiaritywith it. Maybe after you have read this chapter, time dilationwill start to become part of your common sense.Time dilation is so small that it’s difficult even to measure unless the velocity is near the speed of light. For airplanes moving at 1000 km/h, the effect is about 5 10–13. That means thatif you traveled at this speed for one day, you would lose 43nanoseconds. (We get this number by multiplying 5 10–13 bythe number of seconds in a day. The number of seconds in aday is 24 60 60 86,400. That is the time it would takelight to travel about 13 m.) If you fly for a year, you will experience 16 microseconds less time than your twin who doesn’ttravel.This small effect becomes large if the velocity approaches thespeed of light. At 60% of the speed of light, the time dilationfactor is 0.8! Let me show you how to do the calculation yourself. Suppose one object is moving at a velocity v. Let thespeed of light be called c. In science fiction, the ratio of v to cis called the “lightspeed.” If you are moving at 60% the speedof light, your lightspeed is 0.6. In physics, we usually call thelightspeed “beta” and use the Greek letter β (which looks likea B with a tail).β v/c light speedEinstein gave an exact formula for calculating this. Althoughthe term is not usually given a name, I like to call it the Einstein factor. Time will slow down byThe Einstein Factor:Let’s get back to our example. If the light speed β 0.6, thenthe equation gives the Einstein factor to be16
If a man named John stays at home, and his fraternal twinMary travels at 0.6 lightspeed, then her time will go slower ata rate that is only 0.8 that of John’s time. If John ages 1 year,then Mary will age only 0.8 years. When Mary returns, andthey compare ages, John will be 0.2 years older than Mary(i.e. a little more than 2 months older). Yet they are twins,born at the same time.The effect gets much more dramatic as Mary’s velocity increases. Suppose she travels at light speed 0.99999, i.e. at99.999% the speed of light. If you plug that into the time equation, you’ll find Mary’s time progresses at a rate only 0.0045the rate of John’s time. If John ages 1 year, Mary will age0.0045 years. To convert that to days, multiply by the numberof days in a year: 0.0045 years 365 days/yr 1.6 days.Not only that, but she will experience only 1.6 days, whileJohn experiences a full year. If they began as freshmen, Marywill still be a freshman, but John will be a sophomore.The fastest that any astronaut has ever traveled is approximately Earth escape velocity, about 11 km/s. This is equivalentto light speed β 0.0037. Plug this into the time equation(use a calculator) and you’ll find that the astronaut time goesat a rate that is 0.99999933 slower than Earth time. That isn’ta big change (since the number is so close to 1). If he travelsfor 1 year (that is, 365 days 24 h 3600 seconds per hour 3.16 107 s), then he will experience 0.02 seconds less than ifhe stayed at home. That’s not enough for him to notice unlesshe is carrying a very accurate clock.Q UESTION 2.3 If Mary travels at 0.8c for 10 years of John’stime, how much older is Mary when John has aged 10 years?We have sent radioactive atoms at velocities close to the speedof light, and their radioactivity does slow down, by exactly thepredicted factor.Suppose we go faster than the speed of light, for example, wetry light speed β 2. Try plugging it into the equation and seewhat happens. We’ll discuss faster-than-light particles later inthis chapter.I’ve worked out the value of the time factor for
This eText is adapted from Chapter 11 of Physics and Technology for Future Presidents by Richard Muller. LHC images are from the photo-book LHC by P. Gin-ter, R.-D. Heuer and Franzobel. For copyright reasons it is exclusively for use in ANU physics courses. The adaption was by Craig Savage and John Aslanides. Version: 30/3/12
1 RELATIVITY I 1 1.1 Special Relativity 2 1.2 The Principle of Relativity 3 The Speed of Light 6 1.3 The Michelson–Morley Experiment 7 Details of the Michelson–Morley Experiment 8 1.4 Postulates of Special Relativity 10 1.5 Consequences of Special Relativity 13 Simultaneity and the Relativity
Introduction Special Relativity General Relativity Curriculum Books The Geometry of Special Relativity Tevi
Sean Carroll, “Spacetime and Geometry” A straightforward and clear introduction to the subject. Bob Wald, “General Relativity” The go-to relativity book for relativists. Steven Weinberg, “Gravitation and Cosmology” The go-to relativity book for particle physicists. Misner, Thorne and Wheeler, “Gravitation”
Theory of Relativity. Einstein's General Theory of Relativity by Asghar Qadir. Einstein'sGeneralTheoryofRelativity ByAsgharQadir . Relativity: An Introduction to the Special Theory (World Scientific 1989) or equivalent, but do not have a sound background of Geometry. It can be used
The theory of relativity is split into two parts: special and general. Albert Einstein came up with the spe-cial theory of relativity in 1905. It deals with objects mov-ing relative to one another, and with the way an observer's experience of space and time depends on how she is mov-ing. The central ideas of special relativity can be formu-
entitled La fisica del '900: Henri Poincaré e la relatività, delivered at the Seminari di Storia delle Scienze, Almo Collegio Borromeo, Pavia 1995, on 30 March 1995. Partial results of this historiographical inquiry were discussed in: Henri Poincaré and the rise of special relativity, in Quanta Relativity
Topics in the foundations of general relativity and Newtonian gravitation theory / David Malament. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-50245-8 (hardcover : alkaline paper) ISBN-10: 0-226-50245-7 (hardcover : alkaline paper) 1. Relativity (Physics) 2. Gravitatio
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