A Hypothetical Introductory Chapter

TL;DR: This is an example of how a hypothetical first chapter for introductory physics might look. There is no distinction between algebra/trig-based and calculus-based courses because computation removes that boundary. I have been heavily influenced by Noether, Einstein, Arons, Chabay and Sherwood, Feynman, Hobson, Landau and Lifshitz, Thorne and Blandford, and many others. 

(I began working on this two years ago and haven’t done as much with it as I had hoped to because of other obligations. It’s silly. It’s rough. It’s not really presentable, and I’m only making it visible to so someone in particular can see it. Feedback is welcome, but it’s much less than a work in progress. It’s more of a first attempt at getting some ideas onto the page and it won’t stay visible for long.)

For a very long time, I have wanted to write a companion to a physics textbook. I suppose this may be a fleeting goal of every physics teacher at some point or another because we all probably think we can “do it better” than others who came before us. I make no such claim, but there are things I think can be improved upon, especially at the introductory level, where there is a great deal of hegemony and gatekeeping over the topics included (and excluded) and the mathematical level of the presentation. There are concepts and topics I feel should be introduced much earlier. There is mathematics I feel should also be introduced much earlier. I don’t accept the notion that this will make physics “harder” for students who struggle. In fact, it may do the opposite. Perhaps we need to do a better job of mainstreaming the “difficult” material in order to stimulate research into better ways of presenting it. Students themselves tell us they want to learn about the exotic things like black holes and gravitational waves, but I think it is useless to do so on a Newtonian foundation. Why not create a new (even though it’s over a hundred years old and not really new) foundation to start with and let it take students where they really want to go? No, it won’t be easy, but I think it’s worth considering.

Here is my first attempt at such a foundation.

Introduction
Chapter 0
0.1 It begins with a gedankenexperiment.
0.2 Let’s change the original gedankenexperiment.
0.2.1 You exist and are alone.
0.2.2 You exist and are not alone.
0.2.3 You decide whether or not to interact.
0.3 What do you know at this point?
0.3.1 You are learning physics.
0.3.2 Here’s what you currently know.
0.4 You can do more now.
0.4.1 Space and time are real.
0.4.2 You can choose a frame of reference.
0.4.2.5 You can choose a coordinate system.
0.4.3 Let’s elaborate a bit more on space and time.
0.4.4 Introducing symmetry.
0.4.5 Examples of symmetry.
0.4.6 Symmetry and conservation theorems.
0.4.7 Let’s elaborate a bit more on fields.
0.5 Where are we going with all this?

Introduction

Gentle reader,

There are many ways to introduce physics to students. An instructor may have a preference for one particular way, perhaps the way she invented or perhaps the way physics was introduced to her as a student. There are conceptual approaches and there are mathematical approaches. It is, however, important to understand that physics relies on both conceptual and mathematical understanding. Some instructors deny this and insist on keeping mathematics separate from physics. I understand their reasons for wanting to do this, but I do not agree that it is possible, mainly because so much of mathematics was invented specifically for understanding, and quantifying, physics. Calculus itself was invented, at least in part, for understanding planetary motion (and really all motion resulting from classical gravitational theory). Dirac invented the delta function as part of the mathematical framework for quantum mechanics. Vectors, in their traditional form seen in introductory physics, were first invented by Hamilton and modified by Gibbs and Heaviside. My point is that any separation of physics and mathematics is artificial and potentially dangerous, especially in introductory courses. Every introduction to a new discipline must begin somewhere, and must necessarily begin with certain assumptions of the student by the instructor. For you, I want this list of assumptions to be as short as possible. At least just to get us started, make it consist of something you probably do every day. Specifically, I will assume that you know how to sense your environment. You see things. You hear things. You smell things. You touch things. You taste things. That’s all I will assume for the moment. If and when that changes, I will tell you.

Let’s start from the very beginning. Here we go.

Chapter 0

0.1 It begins with a gedankenexperiment.

There is a German word, gedankenexperiment, that basically means using your imagination to visualize certain scenarios that aren’t easily realized in, say, a classroom or even in a laboratory. Einstein made extensive use of the process of imagining such scenarios, each instance of which we will call a gedankenexperiment (German for “thought experiment”), in formulating much of our contemporary understanding of the physics and mathematics that describe our Universe. You will now begin your understanding of physics with a gedankenexperiment.

Suppose you close your eyes. You may think you “see” darkness, but you actually see nothing (that’s a weird thing to say). Now, imagine that the entire Universe is like that. Nothing. There are no planets, stars, galaxies, or atoms from which any of these things could form. You experience no sensations at all. You are not breathing (that requires air, which isn’t there). You see nothing (that requires light, which isn’t there). You hear nothing (that requires a medium in which mechanical disturbances propagate, neither of which is there). You smell nothing (that requires molecules, which aren’t there). You touch nothing (that requires something to touch, which isn’t there). You taste nothing (there is nothing to taste). You sense no motion (you may not understand why, but you will soon). Our Universe contains things we can’t actually see called fields (examples are electromagnetic fields and gravitational fields). We can’t see them but they have observable properties about which you will learn later. We may get a bit philosophical and say that you may indeed not even be conscious to recognize that you are witnessing this Universe. There is simply nothing, and that is difficult for most people to imagine. Furthermore, there is no preferred direction. In fact, the concept of “direction” doesn’t exist because regardless of which way face, the Universe’s appearance is the same. A truly completely darkened room looks the same regardless of which wall you face, right?

Exercise 1. Carry out the gedankenexperiment described above, in which you attempt to witness a Universe consisting of absolutely nothing.

Now, there is a fundamental problem with what I just asked you to do, namely that it isn’t possible for you to imagine and witness a completely empty Universe because if it were truly empty, you could not, by definition, be there to witness it! One of the first things in physics you must get used to is that we frequently need to imagine situations or scenarios that cannot realistically exist. Why? Because, accept it or not (and you eventually must), these otherwise difficult situations illustrate much of the physics you will learn. It is sometimes the goal of experimental physicists to create these situations in a laboratory to the greatest extent possible.

0.2 Let’s change the original gedankenexperiment.

0.2.1 You exist and are alone.

There are at least two ways you can extend your first gedankenexperiment to make it somewhat more meaningful for you. First, you can imagine that the Universe contains one and only one thing: you. You can simply allow yourself to exist in the otherwise empty Universe. This conveniently gets around the impossibility of the original gedankenexperiment, and allows you to formally experience (in your imagination) your conditions. More importantly, it promotes you to the role of an observer. You can be there to witness something in case it happens, but for the moment let’s assume nothing is happening. Oh, and let’s not complicate the simplicity of the gedankenexperiment by worrying about how you breathe in an otherwise empty Universe. Let’s also introduce the assumption that your body works in this gedankenexperient as it normally would in any other situation (e.g. everything functions normally). Physics is about simplicity after all.

Exercise 2. Carry out the first modified gedankenexperiment, in which you allow yourself to exist in an otherwise empty Universe. Literally close your eyes and try to imagine how you would experience such a situation.

0.2.2 You exist and are not alone.

Second, you can imagine that the Universe contains you as described as above, and another entity. Assuming you know about them, this entity could be a proton, neutron, electron, atom, molecule, star, planet, galaxy, or an electromagnetic field (you don’t yet know what that is, but you will). Wait a minute! You may object to the other entity being an atom on the grounds that a human can’t see an individual atom. Okay I can fix that. Supposed the other entity is something else, in fact it can be anything you want it to be. But let’s suppose you are so far away from it that you can’t tell what it is. It displays no discernible structure. It displays no discernible shape. It has no discernible orientation. It displays no disernible size. You may object that the assumption that you can see this entity means that it either emits its own light or reflects light from some other source. Well, if you and this entity are the only two entities in the Universe, the second case is disallowed and we must therefore assume that the entity indeed emits some light, but let’s not worry about that light’s color. We can, at least for now, simply call it “white light” for no good reason other than that is how it appears against the black background of the Universe.

Exercise 3. Carry out the second modified gedankenexperiment, in which you allow yourself and another entity to exist in an otherwise empty Universe. Literally close your eyes and try to imagine how you would experience such a situation.

0.2.3 You decide whether or not to interact.

Okay, so now you have another choice to make (it’s getting more interesting). You must now decided whether or not you, the observer, are allowed to interact with any other entities present in your Universe. If you disallow this, your presence as an observer has no effects whatsoever on the behaviors of any other entities that may be present. This is not true if you allow yourself to experience interactions with other entities. Either way, the choice is yours and yours alone. By the way, you can also decide whether or not the entities in your Universe, other than you, can interact with each other.

0.3 What do you know at this point?

0.3.1 You are learning physics.

You may think you haven’t “done” or learned any physics at this point, but I claim otherwise. Your perception may come from the fact that we have not used any “fancy” terminology. Always remember that physics is about simplicity, and sometimes our intuition tricks us into thinking that simplicity is complexity; we aren’t used to it. We need to retrain our intuition, and that’s not always easy to do.

0.3.2 Here’s what you currently know.

Anyway, let me (attempt to) convince you that you have indeed learned some important foundational physics. Loosely speaking, we can define “physics” as the study of the entire Universe, its properties, and its behavior. Well, from the gedankenexperiments above you know the Universe is basically the totality of everything there is that exists. It’s that simple, and for our purposes that is a perfectly fine definition and we will accept it as such. In some subdisciplines of physics, Universe may have other meanings! That’s hard to explain, so we won’t go there yet.

Regardless of where you are in the Universe, it looks the same. There is no preferred location. A physicist would say the Universe is homogeneous and can refer to the Universe’s homogeneity.

Regardless of which direction you face, the Universe looks the same. There is no preferred direction. A physicist would say the Universe is isotropic and can refer to the Universe’s isotropy (that’s a funny word).

An observer must be present to “do” physics. That observer must be there to see what’s going on, to make measurements of entities’ properties, and to make predictions about how those entities will behave. You are the observer.

Because the Universe is, by our agreed upon definition, all there is that exists, you sometimes only want to focus on one or more entities in the Universe, a selected subset of all there is that exists. This selected subset is called your system. Everything else is called the surroundings. The system boundary separates the system from the surroundings. This boundary need not have any particular shape, but it must completely enclose the system. It is also not necessarily a material boundary. Most of the time it is a mathematical boundary, meaning it can’t actually be seen but serves some mathematical purpose. It can’t have any gaps or holes in it. You are free to choose the boundary’s shape to be as arbitrary or as contrived as you wish, with the only requirement being that it must completely surround your chosen system. The goal is to understand, and predict, the behavior of the system, not of the entire Universe. You can indeed choose the Universe itself to be your system, but that isn’t necessary most of the time. Once you choose your system, it is then assumed that the entities inside the system can interact with each other, but you can decide whether or not entities outside the system can interact with entities inside the system. If they can, your system is said to be an open system. If they cannot, your system is said to be a closed system. This choice often determines the mathematical complexity of describing the system’s behavior. Choice of system is up to the person “doing” the physics, and the same entities can be studied as parts of different choices of systems. Again, this most often affects the mathematical complexity of the physics. By strategically choosing your system, you can make a given problem either very simple or very complex. There are advantages to either strategy.

The conceptually simplest entities in the Universe have no discernible structure, shape, orientation, or size. From now on, I will refer to these entities as particles. You will soon learn that there are other physical entities that simply can’t be called particles; they are called fields. They are mysterious at first, but as your intuition changes you will become more comfortable with working with them and talking about them. There is also a deep connection between particles and fields. Specifically, particles are excitations in fields. That doesn’t mean anything to you at this point, but you will eventually understand it.

Exercise 4. Find someone who knows nothing about physics (this may be difficult). Practice explaining each of the following terms to that person without using any mathematics, fancy language, complex terminology, or anything other than simple, experiential explanations: Universe, homogeneity, isotropy, observer, system, surroundings, system boundary, open system, closed system, particle, field. Practice your explanations with different persons until you can present them without having memorized definitions or jargon and and until you can do it without referring to this or any other source.

Exercise 5. Fields are somewhat mysterious, but nonetheless very real. You have likely encountered them every day without even realizing it. Do you have a wireless phone? If so, think about what happens when you talk to someone or text someone. You create information either by speaking into the phone or by typing it into the phone. To get to your friend on the other end (who also has a phone), that information must first exist within your phone. Then after a brief time interval, that same information shows up on your friends phone. How did it get there? There are no wires, so it couldn’t have gotten there through wires. You may say it got there by traveling through the air, but your phones would work perfectly well above Earth’s atmosphere where there is no air. The answer is that the information got to the other phone by propagating through the electromagnetic field. You can’t see it (this is a slight lie that will be corrected later), touch it, smell it, taste it, or hear it, but the electromagnetic field is there all the time. We think it always was, and we think always will be. Now, suppose you go outside on a clear night and look up at the sky. You see a red star. Think of that star as a friend with a phone texting you, “I am a red star.” How does that information get to your eyes?

0.4 We can do more now.

0.4.1 Space and time are real.

So you find yourself in a Universe consisting of you, as an observer, and a particle. What can you do now? Start by simply looking at the particle. What is it doing? There are really only two possibilities: it is stationary or it is moving. You must be careful from here on to say that it is either stationary or moving relative to you, the observer. This is because there is no such thing as absolute motion. Why? Well, it’s simple to understand in a gedankenexperiment. Imagine a Universe containing only you, the observer. In such a Universe, there is no motion. Why? Because of the properties of space itself. Oh wait! What is space? Well, I didn’t tell you this until now because I’m trying to keep things as simple as possible, but now we really need this concept. So, space is the entity that separates all the other entities in the Universe. If there is you and a particle, space is what separates you and the particle. I know what you’re thinking. How can space, which we can’t see or really sense in any conventional way, be a physical entity? Well, it can be, and in fact is it just that. It has properties that can be measured. It can expand. It can also propagate disturbances we can detect in a laboratory. Start retraining your intuition now to think of space itself as a physical entity. Oh, and shortly we will include time as a physical entity too. What? Yes! Time is a physical entity with properties that can be measured. It can expand. It can contract. More intuitively, time can separate you and some other entity, like a particle. Less intuitively, you will eventually see that space and time are really two aspects of the same thing called spacetime. I know, that sounds absolutely silly. Still, it is true. Onward.

Exercise 6. How would you measure space? Think of something you’ve probably used hundreds of times.

Exercise 7. How would you measure time? Again, think of something you’ve probably used hundreds of times.

Exercise 8. Perform a gedankenexperiment in which space, not time, separates you and a particle. You need not think of an exotic situation. Just think of something simple. The simpler, the better. What items would you need to carry out this observation in the room you are in right now?

Exercise 9. Perform a gedankenexperiment in which time, not space, separates you and a particle. You need not think of an exotic situation. Just think of something simple. The simpler, the better. What items would you need to carry out this observation in the room you are in right now?

0.4.2 You can choose a frame of reference.

Okay, back to the concept of motion. In a Universe containing only you, there can be no motion. Space is homogeneous and isotropic, so it there is no way to distinguish “here” from “there” or “that direction” from “this direction.” If you have a device for measuring time, you can indeed distinguish between “now” and “then” but that won’t help you in this situation. So once again, in a Universe containing only you, there can be no motion.

Now, let’s do a new gedankenexperiment. This time, imagine a Universe containing you and a particle. Assume you have a time measuring device with you. Choose an arbitrary moment in time, and call it “now” for no really good reason. At that very moment, let’s call the place in space where the particle exists “here” for, again, no really good reason. So you have just observed the particle to be “here” at time “now.” Seems simple, and it really is. You have given labels to otherwise arbitrary points in space and time. No other points have these same labels. Now, no matter what you do your time measuring device will keep indicating that time is passing, and there is nothing you can do to prevent this. After some arbitrary amount of time has passed, look at the particle again. Where is it? If it is still at the point in space you previously labeled “here” then you can correctly say that the particle “has not moved.” If it is at any other point in space, let’s arbitrarily call it “there” for no good reason (other than “here” is already taken and no two points in space can have the same label), you can correctly say the particle “has moved.” It has gone from “here” to “there” and the time at which it is “there” you can call “then” for, once again, no good reason (other than “now” is already taken and no two points in time can have the same label). The particle has gone from “here” and “now” to “there” and “then.” The simplest way of saying this, and yes it’s rather obvious, is that the partcle has moved. Be careful though! It has moved according to, or relative to, you (the observer). If you shift the point of view to particle a physicist would say you have chosen a new frame of reference (or reference frame). Later, after a bit of sophistication, we will formulate a slightly better definition of reference frame. One helpful analogy is to perhaps think of a reference frame as one room in a house of many rooms. Going from room to room changes your immediate surroundings and your description of your surroundings. Different people standing in different rooms will give you different instructions for getting to the kitchen, but the kitchen itself is always where it’s supposed to be.

0.4.2.5 You can choose a coordinate system.

If you’ve taken an algebra course before, you’ve most probably dealt with coordinate systems. Some sources treat coordinate systems and frames of reference as the same thing, but in physics they are different. If you think of a frame of reference as a room in a house, a coordinate system is a way of uniquely locating, or labeling, every item in that room. In physics, we don’t have to limit this conceptual labeling to material objects. We can label every point in space and time with a unique label we call the point’s coordinates. These coordinate can take any useful form, but it must be the case that every unique point must have a unique set of coordinates shared with no other points. Every coordinate system has an origin, and you can make that origin anywhere you want it to me. In our room analogy, you can put the origin at a corner for convenience, but you can also put it anywhere else for any reason whatsoever. Coordinates can be linear distances or even angles, and they can also be rates of change of these quantities. Nature doesn’t care what coordinates we use or where we put our origins. Nature doesn’t “know” about coordinates any more than it “knows” about reference frames.

0.4.2.6 You don’t need reference frames or coordinate systems.

Here’s the most fascinating thing about reference frames and coordinate systems: we don’t need them to do physics! That sounds strange, but it’s true. There are mathematical frameworks that let us predict and describe aspects of physics that are the same in all coordinate systems and reference frames. For example, when you observe a particle there are at least two physical quantities that can be predicted. One is the particle’s mass. The other is the reading on the clock that the particle carries with it, and has carried with it since the particle was created. Don’t take this idea of a clock too literally, but think of it like this. Most people carry a watch with them, or at least a handheld phone with a clock. We frequently look at our watch to read the time. We sense that time passes as a certain rate, and that feels natural and intuitive to us. Similarly, the particle experiences the passage of time at a certain rate, and the cool thing is that all observers will predict (calculate) and measure the that particular rate regardless of the frame of reference in which they observe the particle.

0.4.3 Let’s elaborate a bit more on space and time.

Let me elaborate a bit on space and time. We think nothing in our everyday lives of an object being “here” “now” and “there” “then” because we are intuitively used to motion. We are even not bothered by an object being “here” “now” and “here” “then” because this just means the object didn’t go anywhere, right (think about it)? Time, however, is different in two important respects. The first is that an object can’t be “here” “now” and “there” “now.” Why? Well, that means that no time would elapse and as I hinted above, you can’t stop time from passing. Time passes and there’s nothing we can do to stop that. The second is that for every “now” there must be a “then” that happens after our “now.” In other word, time passes in a “forward” direction. Why is this? The simple answer is that no one knows! It’s true. We don’t really understand why time only passes in what we call the “forward” direction.

Exercise 10. How many unique directions are there in space? Give each one you can think of a name. How are they related to each other geometrically?

Now, I’m going to introduce a brand new concept that was first introduced by Einstein and Minkowski. You have most probably heard of the term “perpendicular” before.

Exercise 11. Explain, as accurately as you can, what it means for two directions to be perpendicular. This will be challenging!

Okay, here is the new concept: time is perpendicular to space. Calm down! What in the world does that even mean? Well, that is precisely what I want you to think about. There were some hints above.

Exercise 12. Explain, as accurately as you can, what is means for time to be perpendicular to space. You should refer to Exercise 9. This will be challenging!

Let me see if I can tie everything together with a new gedankenexperiment for you. This time, imagine a Universe containing you, a particle, and another observer (a person of your choice). Assume you and the other observer have identical time measuring devices that measure time at the same rate and give the same readings at every request.

Exercise 13. Suppose that you look at the particle and say it is “here” “now” and “here” “then” after some amount of time has passed. The other observer, however, says the particle is “here” “now” and “there” “then” after the same amount of time has passed. You and the other observer agree on “here,” “now,” and “then” but do not agree on “there.” How can this possibly be? In other words, how can two different observers give different reports on what the particle is doing? Explain this as carefully as you can. This will be challenging!

I won’t divulge the solution directly, but I will say that there is something of great importance here. Your report on the particle’s behavior is correct relative to you. The other observer’s report on the particle’s behavior is correct relative to the other observer. When you ask a particular observer for her report on the particle’s behavior, the report you get is correct only for that observer. It may be correct for other observers too, but it doesn’t necessarily have to be. In physics, we say that choosing a particular observer to report the particle’s behavior to us is choosing a frame of reference.

Exercise 14. We have basically established that motion is the act of going from “here” to “there” while a time measuring device goes from “now” to “then.” Discuss the extent to which this definition of motion does or does not make sense to you.

Exercise 15. Here is another example. Consider a gedankenexperiment in which you and another observer watch a particle. From your frame of reference, which is just the physics way of saying “as observed by you,” you report that the particle is “here” “now” and “here” “then.” A physicist would say the particle is stationary. I claim that if you were to ask the other observer to describe the particle’s behavior from her frame of reference, which is just the physics way of saying “as observed by the other observer,” there are not one, but two, possible reports the other observer could make. Can you figure out what they must be? Note that this is not a matter of personal opinion. There are two possible descriptions the other observer can give that would be consistent with your description of the particle’s behavior.

Exercise 16. Okay I may have slightly misled you in the previous exercise. There may actually be another possible scenario. Can you figure out what it is? As a hint, consider that your description only mentioned the particle and not the other observer. Would including a description of the other observer’s behavior change anything about how the other observer would describe the particle’s behavior? This may be challenging!

Exercise 17. Many times in this chapter you have seen the terms “assume” and “assumption.” Does this bother you, given this is ostensibly an introduction to physics, which most people think is remarkably concrete?

Here’s something else about time and space. Einstein taught us, and you will learn, that time passes at different rates as measured from different reference frames. Measurements of space also vary when measured from different reference frames. However, in any given reference frame, time and space always combine in such a way as to equal the reading on the clock that a particle carries around with it.

There are physical quantities that share certain mathematical properties as time and space. Energy and momentum are such quantities, with energy playing the role of time and momentum playing the role of space. Energy and momentum vary when measured from different reference frames, but in any given reference frame they combine in such a way as to equal the particle’s mass.

0.4.4 Introducing symmetry.

There is a very important concept in physics called symmetry. It is so important that it is considered a foundational concept despite not being realistically included in most introductory courses. We will employ it extensively in our treatment. A symmetry is present when you change some physical aspect of a situation but some physical quantity remains unchanged.

0.4.5 Examples of symmetry.

Consider a particle moving in the same arbitrary direction, neither getting faster or getting slower, for an arbitrarily long duration of time.

The particle’s behavior will be the same regardless of where in the Universe it is located. Changing the particle’s location doesn’t change its state of motion.

The particle’s behavior will be the same regardless of the time at which the motion began. Changing the moment at which the particle’s motion began doesn’t change its state of motion.

The particle’s behavior will be the same regardless of the direction in which it is moving. Changing the particle’s initial direction doesn’t change its state of motion.

These three examples alone are enough to derive almost all of the branch of physics called classical mechanics, which means physics as defined by Newton.

0.4.6 Symmetry and conservation theorems.

Emmy Noether is a person you have probably not heard of, but she is potentially the most important physicist and mathematician of the twentieth century. Among her many professional accomplishments, she discovered an intimate relationship between symmetries and conservation theorems. She discovered that to every symmetry there corresponds a quantity whose numerical value stays the same before and after some physical process or event happens. The quantity whose numerical value stays the same is said to be conserved. Each of the three symmetries in the previous section corresponds to a conserved physical quantity. They are, respectively, momentum, energy, and angular momentum. Now, if you already know anything about physics or have previously taken a physics course these words may be familiar to you. That doesn’t mean you understand them at any deep level, and indeed even knowing how to calculate numerical values for these three quantities doesn’t mean you understand them either. The point is we are going to assume no prior understanding of these three quantities so we can build up a new intuition for them.

0.4.7 Let’s elaborate a bit more on fields.

I want to elaborate a bit on these mysterious fields I mentioned earlier. Some particles are endowed with a property called electric charge, and as far as we can tell there are only two kinds of electric charge in our Universe. They are arbitrarily called positive and negative for no particular reason, and these names were first coined by Benjamin Franklin. The Universe contains an entity called the electromagnetic field (there’s only one of them). It surrounds us and permeates the entire Universe. Among other things, it allows two very important phenomena to happen. It allows our eyes, along with our brains, to create the sensation of vision and it allows a wireless phone conversation to exist between two people with devices not mechanically connected to each other. The electromagnetic field has two “sides” to it that we historically call an “electric field” and a “magnetic field.” You may or may not have heard these terms before, but I will not assume that you have.

Here’s the deal. This electromagnetic field really exists everywhere in the Universe, but a particle with electric charge enhances the measurable properties of the electromagnetic field near you, the observer. These measurable properties can be consistent with an electric field or with a magnetic field, depending on the frame of reference from which the properties are measured. In other words, the electromagnetic field can be measured to be either an electric field, a magnetic field, or a combination of both! This is weird! Like any field, the electromagnetic field can’t move. Fields don’t move; they simply exist at points in space (e.g. “here” or “there”). Their properties can change, but fields don’t move. Observers, on the other hand, can move relative to something else. Observers in different reference frames will report that the electromagnetic field is an electric field, a magnetic field, or some combination of both. But regardless of reference frame, it’s always the same electromagnetic field!

Einstein developed the special theory of relativity by thinking about the fact that although fields don’t move, a disturbance in a field can indeed move by propagating through space and time (it can be “there” “now” and “here” “then”). To his astonishment, he reasoned that all such disturbances in the electromagnetic field propagate through the same amount of space in the same amount of time regardless of the frame of reference from which the disturbances is observed. This particular speed occurs so frequently that is has a special symbol, c. Oh, and by the way, a disturbance in the electromagnetic field has a name you have surely heard before: light.

0.5 Where are we going with all this?

As you progress through your study of physics, you will build on these very fundamental concepts and idealizations. You will learn how to describe them mathematically, which literally involves inventing new ways of thinking about mathematics. It’s not just about adding, subtracting, multiplying, and dividing numbers although those will no doubt be important. You will learn about mathematical objects whose properties match those of the physical entities (e.g. particles and fields) they represent. Most importantly, you will learn to use these mathematical objects to make predictions about the Universe itself, and that is part of the essence of physics.

If you tell me the metric tensor for your spacetime, I can show you how to calculate distances and angles in any reference frame.

If you tell me the duration of a tick-tock on a clock you carry around with you, I can tell you how space and time combine in any reference frame.

If you tell me a particle’s mass, I can tell you I can tell you everything about how that particle’s energy and momentum combine in any reference frame.

If you tell me something about the geometric relationship between electric fields and magnetic fields in your frame, I can tell you how electric and magnetic fields combine in any reference frame.

Knowing the speed of causality, I can tell you something about how you move through time and space in any reference frame.