PHYSICS 9HB: SPECIAL RELATIVITY & THERMAL/STATISTICAL PHYSICS Tom Weideman University of California, Davis UCD: Physics 9HB – Special Relativity and Thermal/Statistical Physics This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. 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This text was compiled on 09/23/2021 TABLE OF CONTENTS Physics 9HB is the second course in the honors introductory physics series. It covers two natural continuations from the previous course in classical mechanics – special relativity and thermal/statistical physics. 1: FOUNDATIONS OF RELATIVITY Starting with what at first might seem like the most innocuous (and reasonable) assumptions, Albert Einstein upended the physics world with a theory so shockingly counterintuitive that it took decades to gain full acceptance. Before we develop the more formal elements of the theory of relativity, we will have a look at the most basic conceptual implications of the theory. 1.1: THE RELATIVITY PRINCIPLE 1.2: THE NATURE OF TIME 1.3: MORE THOUGHT EXPERIMENTS 1.4: PARADOXES 2: KINEMATICS AND DYNAMICS We now begin to develop the basic formalism of special relativity wherein we introduce mathematical tools for dealing with the effects that relative motion has on any phenomena. In classical mechanics, we first studied how to talk about motion (kinematics), and later followed up with what underlies the causes of these motions, namely force, energy and momentum (dynamics). We follow a similar path here 2.1: SPACETIME DIAGRAMS 2.2: LORENTZ TRANSFORMATION 2.3: VELOCITY ADDITION 2.4: MOMENTUM CONSERVATION 2.5: ENERGY CONSERVATION 3: SPACETIME It's clear from the Lorentz transformations that space and time "mix" into one another in ways that they do not in the Galilean transformation. Here we will explore the notion of placing space and time on equal footings, and a greater insight into what relativity is about is the result. 3.1: VECTOR ROTATIONS 3.2: POSITION 4-VECTOR 3.3: VELOCITY AND ACCELERATION 4-VECTORS 3.4: MOMENTUM 4-VECTOR 3.5: GENERAL RELATIVITY BACK MATTER INDEX GLOSSARY 1 9/23/2021 CHAPTER OVERVIEW 1: FOUNDATIONS OF RELATIVITY Starting with what at first might seem like the most innocuous (and reasonable) assumptions, Albert Einstein upended the physics world with a theory so shockingly counterintuitive that it took decades to gain full acceptance. Before we develop the more formal elements of the theory of relativity, we will have a look at the most basic conceptual implications of the theory. 1.1: THE RELATIVITY PRINCIPLE Our journey begins with Einstein's big idea, and the greatest null experiment of all time. 1.2: THE NATURE OF TIME We continue in the same manner as Einstein – with what he called "thought experiments." These are simply logical arguments that lead to inescapable conclusions if the relativity principle is correct. Some of the most startling of these conclusions are related to the nature of time. 1.3: MORE THOUGHT EXPERIMENTS Considering how universally important the measurement of time is throughout physics, it shouldn't be too surprising that there many more interesting results to coax from thought experiments. 1.4: PARADOXES When the counterintuitive results of relativity are pushed to their limits, the result can be apparent paradoxes. In the early days of relativity, these were used as "proofs" that it was an incorrect theory. Resolving them, as we will do here, gives even greater insight into this subtle theory. 1 9/23/2021 1.1: The Relativity Principle Inertial Frames Back in our studies of classical mechanics, we spent a very brief period of time learning about how to relate the measurements of position and time between two observers in relative motion (go here for the LibreText reminder of this topic). Actually "relative motion" in this context is imprecise. We restricted our study to a very specific kind of relative motion – that for which the two observers maintain a constant relative velocity. For what is to come, we will restrict these frames of reference even further – we will insist that every observer makes its measurements from an inertial frame. This kind of frame is one in which Newton's first law assures that objects will not spontaneously begin to accelerate. That is, if we are in such a frame, and we eliminate all of the real forces present on a stationary object, then the object remains at rest in that frame. The simplest way to understand inertial frames is to consider what kind of frame is not inertial. Suppose you are in spaceship (far away from all gravitational sources), and it is accelerating forward. You hold a pencil in your hand, which is at rest in your frame, but you are exerting a force on it with your fingers, so to test to see if you are in an inertial frame, you release it. As soon as you do, it continues with whatever speed it had at the moment of release, while you and your spaceship continue to accelerate. From your perspective, it is the pencil that accelerates, which tells you that you are not in an inertial frame. [One might wonder why an inertial frame is an additional restriction beyond what we did in 9HA. Certainly the intention was to deal with inertial frames back then, but technically two frames with equal acceleration vectors (but unequal velocities) will also satisfy the Galilean transformation.] Postulate(s) We have a simple experiment is for testing whether our frame is inertial, but it doesn't tell us whether our frame is stationary or moving in a straight line at a constant speed, because when we release the pencil under these circumstances, it remains stationary from our perspective in both cases. So what can of experiment will tell us this? Albert Einstein pondered this very thought, and came up with no answer. Eventually, he felt compelled to assert it as a fundamental aspect of our universe, and the relativity principle was born: No experiment can be performed within an inertial frame that determines whether it is moving or at rest. This is also known as the first postulate of the theory of special relativity. One way that we can express this is in terms of an "argument" between two observers. Figure 1.1.1 – All Observers in Inertial Frames Can Claim to Be Stationary [These kinds of diagrams, where the perspectives of two observers in relative motion, will have some common elements. First, we will always define their relative motion to be parallel to their common x-axes. Second, we will define the primed frame to be moving in the +x direction relative to the unprimed frame. Oh, and most importantly, we will always use three-letter names that begin with the first two letters of the alphabet to label the observers!] Calling this the "first postulate" implies that there is a second postulate, and there is, though one could argue that it follows directly from the first postulate and therefore doesn't need to be stated separately. It is this: Every observer measures the velocity of a light beam to be the same. Tom Weideman 1.1.1 9/9/2021 https://phys.libretexts.org/@go/page/21802 The reason this "second postulate" can be considered a consequence of the first postulate is that the theory from which we derive the speed of light contains no provisions for the motion of the observer (or rather, it predicts the same speed for all observers).