NYJC EJC Special Relativity Notes

9814 H3 Physics Special Relativity Lecture Notes 1 H3 Topic 3: Special Relativity Content • Michelson-Morley experiment • Inertial frames and universal light speed • Lorentz transformations • Length contraction and time dilation • Velocity addition • Energy–momentum relation Learning Outcomes Candidates should be able to: (a) discuss qualitatively the results of the Michelson–Morley interferometer experiment and its implications on the ether theory (knowledge of the details of the experiment is not required) (b) state the postulates of the special theory of relativity, that in all inertial frames, the laws of physics are the same and the speed of light in free space is the same regardless of the motion of the light source or observer (c) appreciate the failure of Galilean transformation equations when applied to a moving source of light (d) discuss the concept of simultaneity (e) show an understanding of the terms proper time and proper length (f) apply the Lorentz transformation equations to solve one-dimensional problems (g) derive the time dilation formula and length contraction formula, making use of the Lorentz factor (h) apply the time dilation formula and the length contraction formula in related situations (e.g. the lifetime of fast-moving muons) or to solve problems (i) use the one -dimensional relativistic velocity addition formula to calculate velocities in different inertial frames or to solve problems (j) use the relativistic energy –momentum relation E 2 = (pc)2 + (mc2)2 to solve problems, and show that it reduces, in the appropriate limits, to: 1. E = pc (for massless particles); or 2. E = mc2 + 1 2 mv2 (for particles moving at low speeds v << c).

9814 H3 Physics Special Relativity Lecture Notes 2 1 Speed of Light and the Michelson-Morley’s Experiment 1.1 Speed of Light 1.1.1 The Measurement of the Speed of Light The speed of light (exactly 299 ,792,458 m s −1, usually denoted c) is an important fundamental constant in physics that is known to great precision today. In fact, one of the base units, the metre, is defined based on the speed of light. Up until late 1600s, however, the speed of light was believed to be infinite. Galileo Galilei (1564 – 1642) was one of the first to question that. On a clear night in the year 1638 (or thereabout), in the Italian countryside, Galileo conducted the first experiment to measure the speed of light with the help of an assistant. Each c arrying a lantern supplied with shutters, Galileo and his assistant placed themselves as far apart as they could stand yet still able to see each other (about 1.5 km from each other). Galileo’s instruction to his assistant was to flash his lantern as soon as he saw Galileo’s lantern flash. Knowing their physical separation, Galileo had hoped that the time delay between his lantern flash and his observation of the lantern flash of his assistant would allow him to calculate the speed of light. The result of this experiment was , however, quite inconclusive since we now know that the expected delay of about one hundred- thousandth of a second is way below the reaction time of human beings. More than two centuries later, a French Physicist – Armand Hippolyte Fizeau – repeated a much- improved version of Galileo’s experiment. His experimental set-up is shown in Fig. 1.1. The setup consisted of a pair of cogwheels set at opposite ends of a long axis. The wheels are arranged such that the cogs of one were opposite the intercog openings of the other, i.e. the light beam from the source on the right cannot be seen by the eye on the left no matter how the cogwheels are positioned. However, when the wheels are set in fast rotation such that during the time taken for the light to propagate from one wheel to the other, the cogwheels have turned half the distance between neighbou ring cogs, light was expected to pass through the assembly unobtrusively. Fizeau concluded that the velocity of light was 3 x 10 8 m s−1, a value that coincided with that obtained three decades after Galileo’s death by the Danish astronomer Olaus Roemer based on his observations on the apparent delay of lunar eclipses of Jupiter’s moons when Jupiter was at different distances from Earth. Fig. 1.1 Fizeau’s method for measuring the velocity of light To a distant mirror

9814 H3 Physics Special Relativity Lecture Notes 3 1.1.2 Maxwell’s Equations Seemingly unrelatedly, by 19 th century, physicists have started to make progress in the field of electromagnetism. They began to recognize the relationship between electricity and magnetism. Through the works of Hans Christian Oersted, Andre Marie Ampere and Michael Faraday, it was already clear that moving charges (electric current) sets up a magnetic field and a changing magnetic field induces an emf which would drive a current. In 1865, Scottish mathematician James Clerk Maxwell formalized these observations with a set of four equations which we now call Maxwell’s Equations. From these equations, Maxwell was able to show that oscillating charges set up a propagating transverse wave traveling at a speed of 1 00εµ − = 3.0 × 108 m s−1. A young admirer of Faraday, Maxwell believed that the closeness of this constant to the speed of light was more than mere coincidence, especially since Faraday demonstrated in 1845 that a magnetic field produces a measurable effect on a beam of light passing through glass. Furthermore, experimental evidences such as Young’s Double Slit interference and polarization of light proved beyond any doubt, the transverse wavelike nature of light. These prompted him to speculate that light involves transverse oscillations of electric and magnetic field lines. This conclusion was a grand synthesis of the hitherto separate fields of optics and electromagnetism and is considered a triumph of mind equal to that of Newton in mechanics. Its experimental verification by H. Hertz in 1887 and its commercial exploitation by M. G. Marconi et al led to our present radio, TV, and satellite communications. 1.1.3 Luminiferous Ether This new understanding of the nature of light immediately led to new problems. All the other wave phenomena known at the time, sound waves, water waves and waves in strings, are mechanical waves, whose propagation necessitates a propagating “medium” . Furthermore, the speeds of the waves are constant relative to their respective medium . For example, the speed of sound is about 340 m s −1 relative to its medium, the air. But it wasn’t clear what medium electromagnetic waves travel in. To reconcile this, p hysicists of that time believed that there exists something that fills up all transparent matter and space. It serves as the substratum for the propagation of light and other electromagnetic waves. By the end of 19th century, the existence of such a medium, which was named the “luminiferous ether”, was firmly established in physicists’ mind. The presumed properties of the luminiferous ether are such that it is perfectly transparent, and does not hinder the motion of, or interact with, planets, stars as well as all earthly objects that have been moving through it for billions of years. To detect it experimentally, physicists made use of the fact that the speed of light is presumably constant relative to it : the luminiferous ether reference frame will be an inertial frame in which the speed of light is a constant value c. However, if an observer were to be moving with respect to the luminiferous ether frame with velocity v , this observer would measure a different speed for light, ranging from c + v to c – v, depending on the direction of relative motion (recall what you learned in the chapter on inertial frames) . There is, thus, a ‘simple’ experiment which we can now carry out to establish the existence of t