A Derivation of =

LINKS TO A LIFE Additional study material Meitner Day Contents Page 2 Meitner Day additional material 4 Introduction to Lise Meitner 5 The road to Meitner’s discovery 7 The published note in Nature 9 Special Relativity 10 Preliminary considerations 11 The two postulates of Special Relativity 12 Inertial reference frames 14 The invariant space-time interval 17 The Galilean transformation 19 The Lorentz transformation 24 The Lorentz velocity transformation 28 Momentum 37 Bibliography and further reading 1 Meitner day additional material This additional material is to be studied on and after the day itself. The purpose is to provide interested students with an understandable and accessible derivation of the famous E=mc2 formula in terms of mathematics that is appropriate to those students studying A level Physics and/or A level Maths. As an A level student myself, I remember my own frustrating attempts to teach myself the maths and physics of relativity theory. I looked at a variety of books and found that they were either of the popular science kind, with no mathematical content at all, or else they were academic texts that took too much of the maths for granted and as a consequence I stumbled at the point of trying to derive the Lorentz transformation. This material is an attempt to remedy this by presenting a single continuous mathematical argument leading from first principles to the famous formula E=mc2 itself. Mathematical content and requirements. It is my view that the average well prepared sixth form student who studies for an A-level in mathematics will find that the Special Theory of Relativity is not significantly more difficult to learn than the methods of the standard practice exercises in the textbooks that are taught in secondary schools every day of the week. The only difference is that the mathematical tools learned during GCSE and A level must be applied together with some additional concepts to form one continuous mathematical argument. My hope is that this material will bridge the gap between the thinking of A- level mathematics and the level of thinking required in university. It presents one continuous mathematical argument which uses many of the tools learnt in school mathematics but no more. This derivation is a composite of various sources. It relies heavily on that given in the excellent book “Beyond the mechanical universe” by Olenic, 2007. I have tried to present the material in such a way that I balance two conflicting criteria; first I assume the minimum of prerequisite mathematical knowledge and only a level that uses the maths and physics taught in secondary school; and second I try to avoid watering down the subject matter. Students are not required to have already studied in detail 2 electro-magnetism, the theory of electromagnetic waves or any physics beyond A-level maths and physics. I have included only those physical theories and concepts that are directly related to the mathematical narrative. The mathematical derivation presented here is fairly close to the way in which it was first understood by physicists at the turn of the 20th century. This is an algebraic approach based on the Lorentz transformation to convert from one observer’s coordinates to those of another who moves at a fixed speed with respect to the first. The derivation will lead you along the following route. First the need for special relativity will be outlined and the relevant parts of Newtonian physics will be explained. The mathematical argument will then pass through the derivation of the Lorentz transformation and the gamma factor followed by the velocity transformation, the considerations of the conservation of momentum and the conservation of mass and finally by use of the binomial theorem to the E=mc2 formula itself. You are not expected to understand it all at once. It may well take you all the way to the end of year 13 and into university to fully get it all. As you learn something about each of the topics mentioned above you should be able to get a little further. Maths takes practice and you need to develop a way of embracing the difficulties that you encounter along the way. To all students, good luck and if the going gets tuff my advice is to write it out by hand and check out additional material on the internet or from other sources or even ask your teacher. Hopefully this little booklet will spur you on to ask more questions in class at the very least. I am indebted to many people for their help, direct and indirect, in the preparation of this material. I would like to especially thank colleagues at QE Academy Trust School in Crediton, Devon for their help and guidance during the preparation of the text. 3 Lise Meitner An Introduction Lise Meitner, was the first person to account for and to explain the atomic fission of uranium. During the early part of the 20th century, in collaboration with others she participated in the development of a series of experiments designed to bombard uranium with neutrons. The expectation was that some of the uranium nuclei would absorb one or more of the incoming slowly moving neutrons and form heavier elements new to science. They expected to find elements with a greater atomic mass than uranium, known as transuranes. Instead they found barium, an element roughly half as massive as uranium. Lise Meitner provided the staggering explanation that the uranium atom had split in two roughly equal parts, releasing energy as it did so. It is important to remember that up until this moment (the winter of 1938) no one had thought that it would be possible to do anything more than chip a small piece, perhaps an alpha particle or two off an atomic nucleus. Meitner calculated that as each uranium nucleus is split into two, a large amount of energy was released. This was a momentous discovery, a scientific breakthrough of the first magnitude. Meitner had discovered a method of accessing a source of energy that was new to science. It was a moment that would change the world. I and many others believe she never received the credit she deserved and as a consequence she is largely unknown to this day. We are still living with the consequences of what the world has done with the discovery that Meitner made. On the one hand we have nuclear weapons of mass destruction capable of releasing nuclear energy in such gigantic amounts and with such catastrophic after effect that our very existence on the planet may be threatened if ever they are used again in an act of war. On the other hand the discoveries of Lise Meitner may yet turn out to be part of the solution to the equally dangerous threat of global warming that looms large over us all. This is because Lise Meitner’s discovery of nuclear fission constitutes the basis of the technology that is 4 behind the generation of electricity by the nuclear power industry. It may yet come to be the case that Lise Meitner’s discovery plays its part in saving us all from the consequences of continued use of fossil fuels by providing the whole world with base level electricity supply to enable us the time to develop and deploy the full range of carbon neutral reusable resources that are required to power a decarbonised world economy. Scientists are optimists by nature and Lise Meitner should be regarded as foremost amongst them. It will fall to the next generation of physicists to solve these problems. That is you! I hope that you read on and are inspired to follow in Lise Meitner’s footsteps and go on to study maths and physics at A level and university. The road to Meitner’s discovery Lisa Meitner is unusual for making her discovery at the fairly advanced age of 60. This was after a long career in physics working at the Kaiser Wilhelm Institute for Chemistry in Berlin. Like most scientists she worked as part of a team, most notably with Otto Hahn, with whom she collaborated for many years. Fritz Strassmann and Otto Frisch also played key roles in her scientific life. Hahn was a specialist in chemistry, Meitner was a specialist in physics and together they joined forces in their study of radioactive elements. In 1938 they wanted to discover and understand the effects of bombarding uranium with neutrons. Hahn would use his chemistry expertise to separate and isolate the resulting products and Lise would design and prepare the experimental apparatus and provide the physical explanation and interpretation. Their work drew on the discoveries of other great experimentalists including J.J. Thomson, Marie-Curie, Rutherford and Chadwick. It also draws on the theoretical and mathematical advances attributable to Newton, Leibnitz, Coulomb, Maxwell and Einstein. All of these are represented in the set of cards given to you as part of the Meitner day. The published work of contemporaries such as Fermi and Chadwick provided some of the experimental methods and fine- tuned their approach. The basis of the experimental technique was to irradiate uranium with neutrons from a neutron source provided by a sealed mixture of radium 5 and beryllium surrounded by a layer of solid paraffin. The radium, an alpha emitter effectively knocked neutrons out of the beryllium nuclei and the paraffin moderator acted to slow the emitted neutrons down. The slow neutrons were then capable of being captured by the uranium nuclei placed adjacent to the source. The resulting products were then separated and their chemical properties determined by a complex series of experiments involving known carrier elements.

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