Hello, how are you? Welcome to this book. I am assuming that you want to become a theoretical physicist, but—for one reason or another—you do not want to go to a school. I am okay with that, I never went to a school either, except to assist professors and researchers in their teaching and in their research. Most likely you want to know more about what theoretical physics is and how theoretical physicists do whatever it is that they do. I will proceed under the stated assumption in any case.
What you want to do is a long and hard road—there is no way around that fact. If you want the book, “Theoretical physics in ten minutes,” you will never be able to find it. We are starting at the very beginning, here. Along the way we will cover a lot of interesting things that you might not know. Even those of you who are experts might find some surprises. We will explore different questions and ideas. There are several goals that I want to achieve here:
Rewire your brain to think like a physicist.
Rewire your brain to think mathematically.
Show you the underlying principles of physics as we know them now.
Show you how to use those principles to invent physics for yourself. I will help by inventing parts of it as examples of the techniques that I will present to you.
Show you how to represent physics mathematically.
Show you how to represent physics by the use of the computer software system Mathematica. Note that these sections can be ignored if you do not have access to Mathematica.
Show you that you do not need to go to a university to study theoretical physics.
Finally, and most importantly, show you that theoretical physics is one of the greatest cultural achievements of mankind. That it is your right and privilege to engage in it.
Every once in a while, I will request that you stop reading and do something. I recommend that you do what is requested in every case. At first, I will give you some complete answers to guide you in these tasks, then I will limit it to clues and guidance, and then I will just tell you what to do and leave all of the details to you.
I also suggest that you actively keep a notebook. How do you actively keep a notebook? Here is one way: Given a notebook, draw a straight line down the page you are currently working on, place the line so you still have three-quarters of the page as a working surface. Why draw such a line? You can make notes on your notes on the quarter remaining. As you finish each chapter, review all of your notes up to the current point in the book. As ideas occur to you, write them into the open area on the page. Another way is to use Mathematica and keep your notebook on the computer (though you will want to make backups, and maybe printed backups). I recommend that you use one of the report formats that are included with Mathematica. Play around with the styles until you find those that you like.
Jot down each idea as it is presented. Try to word the idea so you can understand it. If you do not understand the idea, note that lack of understanding and move on. It is impossible to write a book the way the brain thinks; we must write down one topic, then the next, and so on—as if the topics were rungs on a ladder. The brain does not work that way! You might completely understand one section and be completely lost in the next. Don’t sweat it, that’s completely normal. Write down, in your notebook, that you didn’t understand the idea and move on. You will likely get it at some point in the future, possibly when reviewing your notes.
I recommend that you initially write the answers to the Stop Reading and Do Something tasks on a note pad or in loose sheets of paper at first, when you have completed them and checked the answers then write them into your notebook, keep your working notes in a file folder or something similar. As you work through the material, connections to previous topics will occur to you, write those down too—possibly in the margins you have created. Sometimes such connections can lead to interesting research ideas. When this happens, stop reading and explore the idea for a little while. Try to invent tests to see if you are correct. After a few minutes of this, note where you are leaving this idea and move on. You don’t want to get so sidetracked that you stop working through the material!
I recommend that you perform at least one of the projects suggested throughout the material. There is nothing wrong with doing all of them. I also recommend that you do more than one of them. Since these tend to be long projects, do them outside of your reading time. Record the sessions in your notebook as you proceed. I recommend that when you have completed a project that you write up your results outside of your notebook as if you were presenting a research paper.
“How do you present a research paper?” Give your paper a title followed by your name, then write a 100 word (or smaller) summary of the work—this is called an abstract. Then write an introduction where you explain why you were interested enough to try the project, and note major developments done by others in the field that you found productive (it is a good idea to find outside resources, many are free through the World Wide Web). Then explain in one or more sections how you did the work and what results you got. Finish the paper with a section listing all of the sources of information you used by the name of the author, the date of publication, the title, and the publisher (even if this is a web site).
If you follow this entire course you will note that there are a number of topics. Within each topic there are a number of studies. I recommend that you note the number in parentheses next to each study, and then cover each in order—beginning with all those marked as (1). When you have done that, proceed to those marked as (2). Then go on to those subjects of interest marked as (3). Each study is divided into sections.
After you have worked through a few sections make a review of those sections—a synthesis of your notes. Do this every four or five sections. This will involve noting what you think are the most important points covered in your notes up to the current point. When you are done with any study write a final review where you do this for all of the reviews done throughout the study.
A few months after you are done with a study go through your notes again. Make sure that you keep the notebook, you will return to it again as you progress on your path to learning to be a theoretical physicist.
Stop Reading and Do Something #1: Acquire a physical notebook or create a Mathematica file (also called a notebook—See Appendix 1: Introduction to Mathematica if you do this).
What is physics? At this point we have no real way to define physics without assuming a knowledge of physics. If I were to say that physics is the study of matter, energy, and their interactions I would be technically correct. Unfortunately, this definition of physics is meaningless without knowing what matter, energy, and interactions are; and to understand those things requires a knowledge of physics. Another definition is that physics is what physicists study. This is sort of like saying that physics is physics, what we call a circular definition, where we use the word we are trying to define in the definition. So maybe we can’t define physics exactly, maybe we will just have to understand what we mean when we use the word physics. Such ideas are called undefined terms, or we could call them technical terms. As your understanding of the material becomes more sophisticated, the meanings of terms will evolve, too. When we encounter undefined terms I will number and note them, for example:
Term 1: Undefined/Technical Term
A word or phrase used to describe an idea whose definition cannot be specified without resorting to a circular definition.
I suggest that you copy all such technical
terms, along with your understanding of them,
into your notebook.
Term 2: Circular Definition
An attempt to define a word where the word is used in its definition. This use may be explicit—where the word is actually used in the definition, or it may be implicit—where a phrase amounting to the word is used in the definition.
So, let’s get back to the question at hand, what is physics? We can say that physics is a science. But what is a science? Science is a word. We can think of it as being two words. The first word, science, is a noun and represents the facts and ideas developed by scientists. The second word, science, is a program of discovery that produces the facts and ideas of the noun. This program can be called the scientific method, but that is somewhat misleading—every scientist develops their own collection of methods. You begin by observing the real world. From such observations you develop an idea about a pattern in nature. You then assume the perceived pattern to be true and predict the consequences of the assumption in various situations. You then compare those predictions with the real world—either confirming or refuting the idea. Here is a program to apply this idea to physics:
Use observations to gather enough information to find a pattern.
Analyze this pattern to develop an idea.
Make measurements of what you cannot derive mathematically or computationally.
Use the concepts and methods of theoretical physics to see if the idea is consistent with what we already know about physics.
Use the concepts and methods of mathematical physics to convert the idea into a problem in mathematics, solve that problem, and then return the result to its physics context.
Use the concepts and methods of computational physics to convert the idea into a problem in computation, solve that problem, and then return the result to its physics context.
A body of work established by steps 1-6 is called a theory.
Use the concepts and methods of experimental physics to verify predictions made using the theory.
Term 3: Theory
An established body of work in science.
At its most basic level physics has established some physical quantities that can be measured:
Term 4: Position
Where something is located.
Term 5: Size
An objects length, width, height, surface area, and volume.
Term 6: Weight
The result of weighing an object.
Term 7: Temperature
The result of taking a thermometer reading.
Physics concerns itself with several large problems:
The problem of motion. Or, trying to predict how an object will move given some initial data.
The problem of gravity. Or, trying to determine how one body attracts another.
The problem of what matter is.
The problem of what heat is.
The problem of what electricity is.
The problem of what magnetism is.
The problem of what light is.
We will explore each of these in due time.
Term 8: Physics
The science relating to motion, gravity, matter, heat, electricity, magnetism, and light.
Now that we have some vague idea of what physics is, we turn to mathematics. The first question is, what is mathematics?
Stop Reading and Do Something #2: Try to answer this question. Do not take more than five minutes to do this then go ahead and read on.
Mathematics is the study of specific abstract objects, and the relationships between those objects. We call such objects abstract because there is no need to specify what they are outside of mathematics. We need only list their general properties and how they relate to one another. In fact, the central question of all mathematics is: “What are the properties of specific objects and their relationships with each other?”
We can use that idea to see that mathematics can also be viewed as a kind of language for expressing ideas in a pure form. The objects become the nouns and the relations adverbs. We can use symbols to represent the objects and their relationships without much of the ambiguity associated with most languages.
It is also true that mathematics is a way
of getting the results from calculations.
So you can use mathematics as a tool for
getting specific results. In physics these
results are compared to measurements made
in experiments for verification of the theory.
Now I want to address the almost mythical quality of the so-called unreasonable effectiveness of mathematics. This quality describes the wonder that we experience when we make mathematical predictions that are extremely close to what we can observe in reality. The fact that abstract ideas can be applied to the world around us seems magical at first glance, but these abstract ideas came from concrete applications involving—at the beginning of mathematics—the measurement of land areas and the exchange of currency. It is not surprising that from these humble beginnings the vast array of modern mathematical applications are good at describing the world—that is where the abstractions that led to the creation of these applications came from in the first place. In this regard mathematics can be seen as a sequence of abstractions from specific ideas to general principles to new specific applications, and so on.
As we proceed we will examine the objects of different branches of mathematics, then their relationships. Sometimes we will examine how we can use these objects and relationships as language, and sometimes we will examine how to use them to make actual calculations.
Term 9: Mathematics
The study of specific abstract objects, and the relationships between those objects. This study allows us to construct a language using the objects as nouns and the relationships as adverbs. These ideas allow us a way of getting the results from calculations. These ideas evolve through cycles of application and generalization.
The ancient Greek philosophers, such as Aristotle (384 BC—322 BC), had the mistaken idea that gravitation was a natural tendency for objects to be attracted to a mystical place in the world. This place was the Center of The Earth. The heavier an object was the more strongly attracted it would be to that center. In other words, their weight determined their proper place and they all settled into that place. Today scientists laugh at that idea, but what is it that makes this idea wrong? What is the right idea?
Term 10: Aristotle’s Gravitation
The natural tendency for objects to be attracted to the center of the Earth, based on their weight—the heavier an object is, the more strongly it will be drawn to the center.
The fact that Aristotle’s idea of gravity was wrong took a long time to be realized. It was the medieval scientist Galileo Galilei (1564—1642) that put the proverbial “nail in the coffin” of Aristotle’s idea. His argument went something like this; note—I will enumerate the arguments so they are easier to follow (this will be a standard procedure for proofs and derivations):
We will assume that an object that is heavy falls faster than a lighter object as they are each trying to get to their proper places in the world. This explains why it was possible to pick up light objects, but not buildings or mountains; the latter being in their proper places.
What happens when we strap a light object to a heavy one? There are two possibilities; either the combined object acts like a single object, or it does not. This idea is an example of the law of the excluded middle from basic logic. Something either is or it is not, there is no middle point.
If the combination forms a single object, that single object is heavier than either of the two components. By the assumption in step 1 the single heavy object must fall faster than the heavier of the two component objects.
If the combination does not form a single object, then, by the assumption made in step 1, the light object will fall slower than the heavy object. Since they are connected by the strap (which is likely even lighter than the light object), the light object will slow the rate of fall of the heavy object, so the combination will not fall as fast as the single heavy object.
These arguments lead to the prediction that the same combination of objects fall both faster and slower than the heavier of the two component objects. In a sense both answers are wrong since they oppose each other. A situation where every argument leads to a false outcome is called a contradiction. No assertion that leads to a contradiction can be true. This method of proof is proof by contradiction, or reductio ad absurdum. Generally, let us say that you are trying to prove an assertion. The first step in a proof by contradiction is to assume your assertion to be false. You then show that this falsehood leads to a contradiction. Since no assertion leading to a contradiction can be true, the falsehood is then itself false. This proves that your original assertion cannot be false. By the law of the excluded middle, if it cannot be false it must be true. This completes a proof by contradiction.
In this case we have proved that Aristotle’s assertion that objects fall at a rate according to their weight is false; this is the same as proving that objects fall in a way that is independent of their weight. In fact, this principle is called the law of falling bodies. To state this law explicitly, objects fall under thee influence of gravity independent of their weight. This implies that the influence of gravity is the same for all objects.
Having made the prediction that objects fall independently of their weights, experiments were performed that confirmed this result.
This is a fantastic example of the method of theoretical physics that we call a thought experiment. In this instance we have an established idea, predicted that this idea produced results that were contradictory, thus formulated a new hypothesis and confirmed it by both logical reasoning and physical experiment.
Term 11: Gravitation
The natural tendency for objects to be attracted to The Center of The Earth, independently of their weight—all objects are drawn towards The Center equally.
While theoretical physics is rooted deeply in physical intuition, mathematical physics turns a question about physics into a question about mathematics. We solve that mathematics problem, and then turn that mathematical solution into a physics solution. How do we do this? We choose a suitable mathematical object to represent our physical situation. Then we study the properties of that structure in the light of the physical object. We check to make sure the physical situation obeys each of the properties of the mathematical object, this gives us confidence in that representation of the physical situation. Then we begin making predictions that can be tested. This is the process of making a mathematical model. You must guard yourself from getting too attached to such a model, no model is ever more than a shadow of reality.
Term 12: Mathematical Model
The direct application of mathematics to a problem in order to produce predictions that can be compared to reality.
As one might expect, computational physics turns a question about physics into question about computation. Computation has been around at least as long as numbers. What makes it a separate area from mathematics is the advent and insurgence of computers into everything in the modern world. A modern smart phone is more powerful than a room-sized supercomputer of twenty five years ago. Instead of converting a physical phenomena into a mathematical object, we choose a computational object. Then we apply a method of solving a computational problem, what we call an algorithm, to get an answer to the computational problem. Then we convert that answer back into a physics answer through the use of graphical outputs, tables, or mathematical representations. In this book we will use the system called Mathematica as our computational engine.
Term 13: Algorithm
A method of solving computational problems.