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8.033: Relativity Introduction

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8.033: Relativity Introduction 8.033: Relativity Lecturer: Professor Salvatore Vitale Notes by: Andrew Lin Fall 2019 Introduction We’ll first go through the main points from the syllabus. Office hours will be right after class, in this classroom if possible. Today, since there isn’t enough material, there won’t be formal office hours. Everyone besides Professor Vitale also has office hours, listed on the syllabus. The purpose of this class is to study relativity, both special and some general. Special relativity deals with velocities larger than what we deal with in our daily lives: we’ll see that Newtonian equations start to fail us dramatically. Later in the class, we’ll start to deal with gravity and acceleration. We will hopefully see that everything makes sense. That’s what’s great about physics: we may need to use math, but everything will make sense. This can’t necessarily be said about quantum mechanics – the “why?” is a hard question there. 18.02, 8.01, and 8.02 are prereqs. 18.06 as a coreq is recommended for those of us who enjoy thinking about the geometry of what’s going on. There’s one required textbook, listed on the syllabus, as well as other recommended books. Problem sets and mock exams will be on the LMOD website, too. Homework is due weekly on Friday at 5:30pm in the usual boxes, and midterm dates will be confirmed by the end of the week. Note that November 6, the date of the second midterm, is chosen so that we’ll get grades back by drop date. Every teacher says the same thing: don’t wait until the night before to do your psets! We’ll see especially from 8.033 that adiabatic learning will be much better than step-by-step. From a long tradition, we drop the worst pset grade, but use this for real emergencies. Late homework will not be graded unless we arrange specially. Also, this is a physics class, not a philosophy class, so we should keep problem sets as concise as possible. Verbosity is not particularly encouraged. Remark 1. For some reason, relativity brings the literacy out of students... On a more serious note, conciseness is important because at some point, we’ll want to write scientific articles which also need to be short and to the point. Also, we will not use Piazza in this class – questions will be dealt with by asking directly in person or by email. Professor Vitale doesn’t like Piazza very much, because he thinks it’s important for us to know how to express our doubts and questions ourselves. (We can yell; no need to raise our hands.) It’s also great to have questions during class because typically questions are shared – probably three or four people also in the class have the same question. Finally, there is a tutoring service starting week 6 of the class, aimed for students who are struggling to pass. This shouldn’t be our first option if we want to ask for help, though. The 8.033 team all actively works with relativity in their work, so this will be a fun class! 1 Fact 2 Grades are not curved. The strictest grade curves will be A/B = 90, B/C = 80, and so on, and there are many things that all factor into our final grade. Lecture notes will be posted after each class. If they’re not there by the end of the day, we should send Professor Vitale an email. (And these notes exist, too!) 1 September 4, 2019 We’re going to start with a slow introduction to special relativity: we’ll review some definitions and results from 8.01 in a way that will be relevant for the future, and we’ll talk about why we need to study beyond Newtonian mechanics. Definition 3 A reference frame or frame is a three-dimensional coordinate system (e.g. (x; y; z)), where every point has a clock that records time, and all clocks are in sync. It’s important to stress here that we shouldn’t imagine a reference frame as having an observer! Instead, there is an observer at every point with a clock, so whenever an event happens, we can quantify where we are (x; y; z), as well as when we are (t). Definition 4 An event is something that happens somewhere at some time. For example, “Betelgeuse blows up” is an event. All observers will agree that this event happened, so we don’t necessarily need a frame to quantify whether this happens. However, depending on which coordinates we’re using, we may assign different numbers to label the same event! For example, we can say that Betelgeuse blew up at a spot (x; y; z) at time t, and we know this because our coordinate frame gives us a clock at this specific point. This is a bit cumbersome to write down, so let’s come up with some shorthand: if S is the standard Cartesian frame, we’ll write B −! (t; x; y; z): S Notice that if we change frames, for example to spherical coordinates in a frame S0, we still have the same event B. However, we now have some different labels B −! (t; r; θ; φ): S0 In this class, we’re going to use four-vectors to make our notation more concise. we’ve already seen this in the examples above, but we can be a bit more abstract: we’ll define x = (t; x; y; z) to be a four-vector (in bold), where the entries of the vector depend on our frame. In this class, we’ll grab coordinates from our vector via x0 = t; x1 = x; x2 = y; x3 = z: Concisely, this will be denoted x µ. By convention, all Greek letters in this class take on the values 0; 1; 2; 3. If we only want to refer to the space part of the four-vector (so not 0), we will use a Latin letter like i; j; k. 2 Let’s now return to physics by going back to 8.01. Most of that class is about particles that move: something we see often is a particle moving in a path. At any time t, we can (or want to) figure out what point ~r(t) the particle is currently at. Remember that these coordinates depend on our frame, and we’re using the standard frame S here. We’ll write this as ~r(t) = (x(t); y(t); z(t)) = x(t)^ex + y(t)^ey + z(t)^ez ; where the e^ vectors are unit basis vectors. From this, we can find the velocity d~r dx dy dz ~u(t) = = ; ; : dt dt dt dt We’ve been lucky so far in our physics experience, because the derivative of the vector is just the term-by-term derivative of the individual components. We can then find the acceleration in a similar way: d ~u d 2x d 2y d 2z ~a(t) = = ; ; : dt dt2 dt2 dt2 This is as much as we need to do physics with a particle in 8.01: let’s reframe this model for the purposes of 8.033. Let’s think of the particle’s trajectory as a sequence of events, writing it as a path of the form P : the particle is at a specific spot now: This stresses that we don’t need a frame to think of a particle’s trajectory, but eventually we do need one to do calculations for it! If we add a coordinate frame S to it, we can then turn each event Pi ! (ti ; xi ; yi ; zi ) into a specific four-vector. There’s an uncountable number of events, so we usually index our events by a real number τ. This gives us a parameterization P (τ) = (t(τ); x(τ); y(τ); z(τ)) : There’s a few frames that have played important roles in our physics journey so far: Definition 5 An inertial frame is one where F~ = m~a is valid. Remark 6. Note that this is a vector equation, so it’s really three equations: we can technically write it out as 8 x x >F = ma <> F y = mat or as F i = mai . This is one reason why we like to use indices in the way introduced! > :>F z = maz In 8.01, if we are given an inertial frame, we can find infinitely many other inertial frames by starting from the initial frame and using what are called Galilean transformations: 1. rigid translations (move the origin), which can be described by ~x ! ~x 0 = ~x + ~a and t ! t0 = t + T 0, or more simply just as x µ ! x µ0 = x µ + aµ, 2. rigid rotations (in three dimensions), described as ~x ! ~x 0 = R~x for some rotation matrix R (with t held constant), 3. or boosts, where ~x ! ~x 0 = ~x − ~vt and t is held constant. Physically, this just means that our axes are parallel to the original ones, but the origin is moving at some constant velocity. 3 Notably, there’s no “correct” reference frame: F~ = m~a holds in all such inertial frames, so there’s no experiment that can help us distinguish one from another as being correct. In other words, doing physics in inertial frame S versus inertial frame S0 is equivalent, but we can still consider S0 relevant to S. Just remember that it’s perfectly valid to switch between any inertial frames! Let’s verify that these boosts indeed preserve the laws of mechanics: Proposition 7 If F~ = m~a, then F~0 = m~a0 in our boosted frame.
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