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Relativity Notes Lecturer Phys1201. Advanced Physics 2. 2008 (version 1) Relativity Notes Lecturer: Craig Savage. [email protected]. www.anu.edu.au/Physics/Savage These notes should be read before the relevant lectures, and studied in depth after them. Contents 1. Overview 2. Simulation 3. Light Clocks – time dilation 4. Light Clocks – length contraction and the relativity of simultaneity 5. The Lorentz Transformations 6. The Meaning of the Lorentz Transformations 7. The Twin Paradox 8. Four-dimensional Geometry 9. 4-vectors 10. Relativistic Optics 11. The relativity of simultaneity revisited Appendix – Taylor expansion References 1. Overview One of the great themes in physics is unification: taking things Summary that look different and discovering that they are really parts of a • Relativity unifies space and greater whole. Newton did this when he realised that apples fall time into “space-time”. and planets orbit the Sun for the same reason – gravity. Maxwell • It is based on two postulates. and others realised that electricity, magnetism, and light were aspects of electromagnetism, summarised by Maxwell’s equations. Einstein and Minkowski made the next big unification when they realised that space and time were parts of a greater four-dimensional “space-time”. This unification is called the “special theory of relativity” and will be our major subject [1]. From it a number of other unifications followed, such as the unification of energy and mass expressed by the equation E = mc2 [2]. Special relativity is based on two postulates out of which Einstein built the entire theory by logical deduction [1,3]. The relativity principle. The result of every experiment is independent of its speed. The constancy of the speed of light. Every measurement of the speed of light in a vacuum gives the same result: c = 3x108 m/s. The relativity principle is an extension of Galileo’s relativity principle for mechanics to all of physics. However, the second postulate is radical, and seems crazier the more you think about it. It means you can never catch up with a light wave. It says that if PHYS1201 Relativity Notes, 2008. 1 you chase a pulse of light in a rocket, travelling at nearly the speed of light, you will measure its speed to be the same as if you had stayed at home. Crazy but true! The same year, 1905, Einstein also kick-started quantum mechanics by showing that light had a unified wave and particle nature [4]. In 1915 he continued unifying by showing that gravity was not a force, but rather a property of space-time. The physics of this is called the “general theory of relativity”. Fundamental physics has continued unifying since 1915. Important successes are: • quantum mechanics (1925-30), which unified particles and fields. • electro-weak theory (1970), which unified electromagnetism and the weak nuclear interaction (responsible for radioactive decay, and for nuclear fusion in the Sun) Currently, some speculative ideas, such as string theory, are being pursued with a view to unifying gravity with both the electro-weak interaction and the strong nuclear interaction (which holds nuclei together against the electrical repulsion of the protons). 2. Simulation Contents Computer simulations are used to conduct investigations that might otherwise be difficult or impossible. For example Summary modelling the causes of climate change, or the effects of changes The relativistic world cannot in economic policy. At ANU we have developed a simulation of be directly experienced, but it special relativistic physics called “Real Time Relativity”. You can be simulated. will do a lab using it, and need to get some practice before the lab. You can play with it on the computers in the tutorial room or download it and run it yourself, if you have a suitable computer [5]. 3. Light clocks – time dilation Contents There are many different ways to develop the special theory of relativity. One is to look at certain crucial experiments and Summary deduce it from them – this is called the “inductive” approach. It • The essential physics of is described in detail in a technical paper by H. Roberston [6]. relativity follows from Three experiments from which special relativity follows are: the analysing light clocks. Michelson-Morely, the Kennedy-Thorndike, and the Ives- • Time dilation is the slowing Stilwell. The first two are optical interferometry experiments, of the ticks of moving clocks. while the last measures time dilation using the frequency of light emitted by fast moving hydrogen atoms. We will take the deductive approach, as Einstein did [1]. This postulates certain truths and then uses logic to deduce consequences. The idea is that we do not doubt the postulates, unless we find a consequence that disagrees with a reliable experiment. The critical postulates are the two given in section 1 of these notes: the relativity principle and the constancy of the speed of light. We shall deduce special relativity starting from an analysis of light clocks. These are a conceptually simple kind of clock that no one has ever actually built. However they are beautifully adapted to deriving the consequences of the relativity postulates. PHYS1201 Relativity Notes, 2008. 2 A light clock uses the path of a pulse of light over a fixed distance as its unit of time. In relativity, and astronomy, it is convenient to measure distance in units of “light- distance”, such as light-seconds or light-years. A light-second is the distance light travels in a second, 3 × 108 m, slightly less than the distance to the Moon. A light- nanosecond is 0.3 m. So a clock with a path length of a light-nanosecond would be about 30 cm long, or 15 cm if we use a round trip. A light clock consists of a light source and a light detector next to each other, with a mirror situated such that the emitted light is reflected back to the detector; see diagram. When the detector detects the light pulse it immediately triggers the source to emit another pulse. Each time a pulse is emitted the clock ticks off a nanosecond (for a 30 cm path). You can image mirror a digital display counting off the ticks. The light clock works much the same way as any clock does – it counts the number of some boringly repetitive physical phenomenon known to have a constant repeat time. In your watch it may be the mechanical vibration of a quartz crystal. The light clock has two advantages for us: 1) it is simple, so we can understand exactly what’s going on, 2) its reliability is based directly on one of the postulates of relativity – the constancy of the speed of light. We could use a more complicated clock for our source & analysis, but that would require a lot more work, and much more complicated detector arguments. A light clock. What would a light clock look like if it were moving past us at close to the speed of light? Let’s start by assuming that the light path is oriented perpendicular to the clock’s motion, and that the length of the light path does not change – we’ll prove this later. Then the path taken by the light pulse is shown in the diagram. It is longer than for a stationary clock. Remember that we are postulating that “every measurement of the speed of light in a vacuum gives the same result”, so the speed of light has its usual value of c = 3 × 108 m/s. It therefore takes the light longer to travel the longer distance. Since the clock ticks are determined by the light pulses, a moving clock ticks slower than a stationary clock – that is, ticks take longer. A moving light clock. What does this mean? Are light clocks crazy clocks? We can answer that by using the relativity principle. Attach your favourite clock to the top of a light clock. Perform the experiment of checking whether they stay synchronised – they do, because both are “good” clocks. Now repeat the experiment with the clocks moving, and you moving along with them. The relativity principle, which we are postulating to be true, says that: the result of every experiment is independent of its speed. Hence we deduce that you will again find that the light clock and your favourite clock agree. If I’m watching the clocks move past at close to the speed of light, because they agree for you they agree for me Why? Think about putting their digital readouts next to each other. Consequently, any moving clock will slow down in exactly the same way as the light clock does. If every clock slows down then time itself slows down – for what is time but what clocks measure (for a physicist anyway). PHYS1201 Relativity Notes, 2008. 3 Time itself slows down! Go over the argument again. Where’s the trick? … there is none! If we assume the relativity principle and the constancy of the speed of light to be true then time slows down for moving objects. So if you don’t like the conclusion you have to object to at least one of the postulates. Can we do experiments to check this? Yes. Countless experiments have verified this result, called “time dilation”. The GPS, or Global Positioning System, uses a set of satellites to determine the position of a receiver to within a few metres [7]. GPS receivers do this by precisely timing radio signals from satellites, and then determining distances by dividing by the speed of light. With distances to 4 satellites you can fix your position. Why 4 and not 3? See [7]. This system relies on precise timing, so the satellites contain atomic clocks – clocks that count the oscillations of electrons in atoms.
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