This is a new feature which I’d like to try out.\u00a0 I will ask a question about concepts in theoretical physics, and the readers will try to answer it.\u00a0 It may be a straightforward question, or it may be a trick question.\u00a0 I figure this will give me a better idea of what my readership does and does not understand about what I’ve written, and give readers a chance to show off their skills.<\/p>\n
Challenge #0 happened kind of accidentally in the comment section of this post<\/a>, and there it was suggested that this might be a cool regular feature.\u00a0 So let’s give Challenge #1 a whirl, and see what happens.<\/p>\n The metric of special relativity<\/a> in Cartesian coordinates, in units where the speed of light $$c = 1$$, is $$!ds^2 = dx^2 + dy^2 + dz^2 – dt^2.$$<\/p>\n (The d’s are just a calculus notation to indicate that you can use this metric to measure infinitesimal distances between nearby points, something which is very useful in general relativity where the metric is a function of position<\/a>.\u00a0 Here, however the metric is constant in space and time, so you could replace the d’s with $$\\Delta$$’s if you like.)<\/p>\n It has a ten dimensional group<\/a> of symmetries, called the Poincar\u00e9 group, which preserve the metric.\u00a0 These are the set of transformations acting on the t<\/em>, x<\/em>, y<\/em>, and z<\/em> coordinates which preserve the metric.\u00a0 (See the link for details.)<\/p>\n Suppose that instead we want to do nonrelativistic Newtonian mechanics.\u00a0 These are the laws of physics which people believed were true before St. Maxwell and Einstein came along, and which are still valid for describing objects travelling much slower than the speed of light.<\/p>\n 1. What is the appropriate <\/em>metric to use when describing the geometry of spacetime<\/em> in non<\/em>-relativistic physics?<\/p>\n 2. What is the symmetry group of this metric?\u00a0 How many dimensions does it have?<\/p>\n 3. Are these the same as the symmetries of Newtonian physics, which this metric is supposed to describe?\u00a0 Why or why not?<\/p>\n The correct answer to these questions reveals something surprising about the way in which relativity is an improvement on nonrelativistic physics.<\/p>\n You need not answer all of these questions, but the answers to one may help confirm that the answers to the others are correct.\u00a0 Experts (e.g. those with graduate education in physics) are requested to wait a while before attempting an answer, in order to give others a chance to respond. This is a new feature which I’d like to try out.\u00a0 I will ask a question about concepts in theoretical physics, and the readers will try to answer it.\u00a0 It may be a straightforward question, or it may be a … Continue reading
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