Illustration by Jean-Leon Huens, National Geographic Stock

Illustration by Jean-Leon Huens, National Geographic Stock

Classical Mechanics

Newton’s three laws of motion are the foundation of the entire edifice of classical mechanics—that is, the branch of physics which describes the motion and behavior of objects, whether terrestrial or celestial, which travel at speeds much slower than the speed of light and whose mass exceeds that of an atom but are much less than that of a galaxy. Newton’s classical mechanics served as the foundation for the development of thermodynamics.

Initially people thought of thermodynamics as separate from mechanics and thought heat was due to a mysterious liquid named the caloric moving from a hot body to a cold body. Experiments done by James Prescott Joule, Nicolas Carnot and others provided compelling evidence that all quantities and processes related to thermodynamics are fundamentally mechanical phenomena. Temperature is just a measure of how rapidly atoms in an object are "jiggling" back and forth: the difference between hot objects and cold objects is that the atoms oscillate about their equilibrium positions faster and with greater amplitudes. Whenever any object is in thermal contact with another colder object, the atoms in the hot object collide with the atoms in the cold object causing them to jiggle more rapidly increasing the object's temperature: this process is known as heat. 

Thermodynamics served as the blueprint for future engineers who designed the steam engine and internal-combustion engine and precipitated the greatest technological revolution in the history of the human species ushering the first and second industrial revolutions. Newtonian mechanics is also the foundation of every other field in physics: the special and general theories of relativity, quantum theory (which has given birth to the computer and AI revolution which we see unfolding around us today), and so on. Isaac Newton’s Principia is therefore the foundation for modern civilization. In the entirety of our 200,000 year long history since humans first came to be, it is single-handedly the most important book ever written and the greatest achievement in human thought of all time. For it paved the way for future physicists and other scientists who would later formulate thermodynamics, electromagnetism, special and general relativity, and quantum theory all of which, together, have birthed nearly all the modern marvels in technology we see in the world around us today.


Electromagnetism

Maxwell's equations (shown in image on the left) are the foundation of classical electromagnetism and, together with the Lorentz force law, describe all electromagnetic phenomena in the universe. Maxwell's laws of electromagnetism (expressed as Maxwell's equations) and Newton's laws of mechanics were the two pillars of physics all the way until the beginning of the 20th century. Here is a quick summary of Maxwell's equations and the Lorentz force law, as well as their practical significance: Guass's

Law (the top equation in the image) describes how electricity is produced by an electric field that—according to Coulomb's Law, a special case of the more general, Lorentz force law—“pushes” electrons around; Gauss's Law describes how an electric field is produced by a charge distribution with a voltage whereas Faraday's Law (third equation from the top) predicts (correctly) there is a second way to produce an electric field—by generating a changing magnetic flux (these are two different ways of generating an electric field and, hence, electricity); Ampere's Law (bottom equation) describes how a magnetic field is produced by either a current or a changing electric flux; all four equations, collectively, describe how a changing magnetic field and a changing electric field produces an electromagnetic field (which is a combination of both kinds of fields). The electric field and the magnetic field, once thought of as separate, are actually two different manifestations of the same thing: the electromagnetic field. Sometimes, the electromagnetic field generated by charged particles has only electric fields making it up. This, for example, happens when the charged particles are stationary. There are other instances where the electromagnetic field generated by charged particles has only the magnetic field part. This, for example, can happen if charged particles like electrons are moving steadily, at a constant speed, down the length of a copper wire. The force on a charged particle (or collection of charged particles) exerted by any electromagnetic field is given by the Lorentz force law, \(\vec{F}_{EM}=\vec{F}_E+\vec{F}_B\); after calculating this force, Newton’s second law can be used to determine the motion of that charged particle (or collection of charged particles). We can use Maxwell’s equations to determine the electromagnetic field produced by charged particles in a myriad of different possible scenarios; then, by using the Lorentz force law and Newton’s second law, we can determine the effect of the field (and, hence, of the particles that produced the field) on any other group of charged particles. It is difficult to appreciate the generality of this statement; indeed, it encompasses all classical electromagnetic phenomena in the universe. It is the theoretical bedrock for understanding everything from the generation of DC or AC as sources of power, to the operation of motors and generators, to the production of radio waves and x-rays, to an understanding of what light actually is and how its propagates at a uniform speed with respect to all observers. Maxwell's providence of depth, insight and understanding into electromagnetism was the blueprint for later engineers who designed and constructed telecommunication systems, the electrical grid, and numerous other transformative inventions. Understanding the continuity between Maxwellian electromagnetism and its generality and those aforementioned inventions is the key to understanding the cause of the electrification of infrastructure. Maxwell’s ideas on electromagnetism thus played an essential role in the cause of the industrial revolution, along with Newton’s ideas on mechanics and subsequent physicists and engineers who used mechanics to formulate thermodynamics.


Special & General Relativity

Maxwell's equations predict that the speed of any electromagnetic wave in the the Universe is always \(c=3×10^8\text{ m/s}\) and independent of the observer’s frame of reference. For example, if one observer is sitting on a train moving 30 m/s and their glass of water appears stationary on the table (moving 0 m/s relative to them), an observer standing on the ground watching the train go by would say the glass of water is moving 30 m/s. An observer in a spaceship outside of the Earth’s gravitational pull would say that the glass is moving 100’s of meters per second! Since Maxwell’s equations disagreed with Newton’s mechanics, one might have 

incorrectly assumed that Maxwell’s equations were wrong. However, the Michelson-Morley experiment demonstrated that the speed of any electromagnetic wave (light, microwaves, x-rays, etc.) does not depend on the speed of the observer. This means that it does not matter whether you flash a beam of light from your flashlight while standing still or going 99.9% the speed of light in any direction: you will observe the beam of light travel away from you at \(c=3×10^8\text{ m/s}\). This experiment along with many others demonstrated that Maxwell’s equations are right and that Newtonian mechanics needed to be modified. Einstein worked out that the consequences of the speed of EM waves being c for any observer are the equations of special relativity. These equations tell us that time dilation, length contraction, and other bizarre phenomena occur when one frame of reference moves away from us at very high speeds close to that of light. For example, if I were sitting on the Earth and could “watch” a person flying by me in a glass box at 99.9% the speed of light, they would appear to be moving in slow-motion and they and the whole box would appear almost completely flat. If they slowed down and landed on the Earth’s surface right next to me, they would no longer appear to be moving in slow-motion or to be flat (because their speed relative to me would now be pretty much 0). 

Newtonian mechanics asserts that time and space are constant throughout the Universe: all clocks tick at the same rate and a meter stick is the same length no matter where it is. This assumption seemed intuitive and reasonable; however it is wrong because it does not agree with experimental observations. To reconcile theory with experiment, Einstein asserted that time and space are not constant, that clocks tick at different rates and that the length of a meter stick can appear different to different observers. Later experiments showed that planes with clocks flying around the world ticked slightly slower than clocks on Earth by an amount which agreed with Einstein’s equations. Calculations done by GPS systems must take into account both the effects of time dilation and length contraction to pinpoint the location of your vehicle; otherwise the GPS would be off by several meters.


Astronomy & Cosmology

Cosmology is the study of the whole universe on the largest scale. This field goes all the way back to ancient times when humans first speculated on the nature of the heavens. Albert Einstein's publication of his General Theory of Relativity set the foundation for modern cosmology. Studying the cosmos on the grandest scale might seem like an impossible task; but Einstein greatly simplified matters by assuming what he called the Cosmological Principle: the assertion that over immense distances the distribution of matter and energy throughout the universe is more or less uniform. This pivotal assumption is the foundation of modern cosmology and is 

experimentally verified by the Cosmic Microwave Background Radiation (CMBR). 

After making this assumption, Einstein derived the FRW metric, plugged it into his field equations, and derived the FRW equation. He added a fudge factor—which he called the Cosmological Constant—to make the universe static, motionless and unchanging. But Edwin Hubble's observations that all the galaxies are rushing away from us made this assumption untenable. Around that time an amateur physicist named Georges Lemaître used Einstein's theory of gravity to formulate the Big Bang theory. This theory implied that if one were to run the clocks backward in time far enough, there would be a time when all of the matter and energy in the universe was infinitely dense. During the 1940s two researchers named Ralph Alpher and George Gamow calculated—using the Big Bang theory, Einstein's mass-energy equivalence principle, and the rules of nuclear physics—that during the early epochs of the universe's life, conditions were right for hydrogen and helium to form. They published their work in the famous Alpher–Bethe–Gamow paper. About three decades later Fred Hoyle calculated that all the elements in the periodic table heavier than helium (up to uranium) were created in the nuclear furnaces and death-roes of the stars.

Throughout history, there have been many revolutions in cosmology: the transition from the Ptolemaic and Aristotelian Earth-centered universe to the Copernican Sun-centered universe; the static, eternal and unchanging universe from Newton's time all the way into the early twentieth century to a universe that is finite and flying apart. In our time we are perhaps on the cusp of another great revolution in cosmology. Recent experimental and theoretical work done in 2016 by a group of Oxford researchers argue that the motion of the galaxies can be explained by accounting for the slight non-uniformities in matter and energy density throughout the universe—making the postulation of the existense of dark energy unnecessary. There is also a new theory of gravity which has been developed which, if proven correct, could make dark matter unnecessary too. These are exciting times to be alive indeed! 


Quantum Mechanics

Scrodinger's time-dependent equation (top-left of image) is the single most important equation in quantum mechanics. It is used to determine how any state \(|\psi(t)⟩\) of a quantum system changes with time; at all times \(t\), you'll know what \(|\psi(t)⟩\) is. This equation is also used to determine the probability \(P(L,t)\) of measuring any physical quantity \(L\) at any time \(t\). What the two functions \(|\psi(t)⟩\) and \(P(L,t)\) are depends on the total energy of the system (which is associated with the energy operator \(\hat{E}\)) and the initial state 

\(|\psi(0)⟩\) of the system. All you need are these two initial conditions to determine the entire future of the system. In classical mechanics the state of a particle is specified by two numbers—the position \(\vec{R}\) and the momentum \(\vec{p}\) of the particle. Newton's second law—\(\sum{\vec{F}}=m\frac{d^2\vec{R}}{dt^2}\)—can be used to determine the entire future of the state {\(\vec{R}, \vec{p}\)} of the system. The state {\(\vec{R}, \vec{p}\)} encapsulates everything about the system: if you know what \(\vec{R}\) and \(\vec{p}\) are, then you can determine the kinetic and potential energy \(KE\) and \(PE\), the angular momentum \(\vec{L}\), and every other physical quantity associated with the system. According to classical mechanics if we knew the state {\(\vec{R}_i, \vec{p}_i\)} of every particle in the universe, we could in principle use Newton's second law \(\sum{\vec{F}}=m\frac{d^2\vec{R}}{dt^2}\) to determine the entire future of the universe. We could determine the state {\(\vec{R}, \vec{p}\)} of every particle and know every measurable quantity associated with each particle at any time \(t\). We would know everything.

In quantum mechanics if we knew the initial state \(|\psi_i(0)⟩\) of every particle in the universe, we could use the quantum analogue of Newton's second law—namely Schrodinger's time-dependent equation—to determine the future state \(|\psi_i(t)⟩\) of every particle in the universe. But where quantum mechanics differs from classical mechanics is that the state \(|\psi_i(t)⟩\) does not encapsulate everything about the system—rather it encapsulates everything that can be known about the system, which isn't everything. Each particle would have its own wavefunction \(\psi_{i,j}(t)\); in general there would be a probability amplitude associated with any physical measurement. Although the probability function \(P(L,t)\) is deterministic, the measurement of any physical quantity \(L\) is inherently probabilistic. Therefore, there is an inherent randomness built into the cosmos. The physical origin of this built-in randomness is that any measurement or observation is blunt; it involves a photon hitting another particle which always introduces uncertainty into the measurement of any physical quantity. This uncertainty is empirically quantified by the generalized uncertainty principle; a special case of this principle is the famous Heisenberg uncertainty principle (middle-left of image) which states that it is physically impossible to simultaneously measure the position and momentum of a particle. Using elementary algebraic manipulations, the Heisenberg uncertainty principle can be "rewritten" as the time-energy uncertainty principle; we think that the latter played an essential role in the earliest moments of the universe and is what's responsible for the origin of the CMBR. (Although it accounts for the random fluctuations we see present in the CMBR, it does not explain why, overall, the CMBR is so smooth—to one part in \(10^5\). The prevailing explanation of the observed uniformity in the CMBR is told by inflationary theory, although most scientists would concede that the problem is not fully understood.) 


Space Travel & Colonization

We have discussed how the cause of the first two industrial revolutions can be traced all the way back to the discovery of the laws of classical mechanics which were first published by Isaac Newton in his Principia in 1687. Engineers around that time took this knowledge of how the universe works and used it to devise the "blue prints" for the technologies which would shaped the first two industrial revolutions and transform the human condition. By understanding how the universe works, we can manipulate matter and energy in the way we see fit to design and construct new technologies. All industrial revolutions are caused by the convergence of new communications, transportation, and energy technologies. The first industrial revolution was caused by the invention of radio, the locomotive, and steam power; and the second by the development of the telephone, automobiles and the internal combustion engine, and by oil and fossil fuels. Today, we are living in the midst of the third industrial revolution which is also the result of advances in our knowledge of how the universe works. This industrial revolution is being caused by driverless electric vehicles and maglev transportation, the internet, and renewable energy. Whenever this convergence occurs, new infrastructure must be built to support those new technologies. For example, during the first two industrial revolutions, thousands of miles of railroads, telephone cables, and highways had to be built. In our time, we will also need to update our infrastructure to support the new mix of communications, transportation, and energy technologies.

Let's go through a few examples of some of these updates that we'll need to make. Since most renewable energies, like solar and wind, are essentially intermittent, we must update our electrical grid by installing super capacitors and new energy storage technologies into them in order to store and share energy on those days when its cloudy and not very windy outside. We must also continue to expand our IoT infrastructure in order to manage all of the new economic and industrial activity. Also, as electric vehicles gradually phase out vehicles relying on the internal combustion engine, more and more charging stations (all connected to the electrical grid) must be built. If our socioeconomic system rises so as to be commensurate with modern science and technology, then the third industrial revolution will culminate as we transition from a type 0 civilization to a Type 1 planetary civilization. And after this crucial transition is made, humanity will begin to colonize the solar system, the Milky Way galaxy, and beyond.

Humans and all other living creatures use language to make sense of the world around them. The more complex that a language is, the more comprehensively and accurately it can describe the outside universe. This allows humans using say English to more comprehensively understand the outside world than other species which use much simpler languages and forms of communication involving mostly sounds and gestures. Similarly, this is also why humans using the language of mathematics can understand the universe on a much deeper level than humans solely relying on English; English cannot describe the fundamental nature of atoms or black holes and, thus, someone who knows only English cannot understand the universe as comprehensively as someone who knows mathematics. All of the lessons we have covered on physics have been about using mathematics to describe how the universe works. But another major focus on Greg School will be to take our knowledge of how the universe works and see how technologies can be created based on that understanding. For example, we know from the Lorentz force law that if light (oscillating electric and magnetic fields) emitted by the Sun shined against a massive plate (possibly kilometers across), the fields comprising the light would exert electromagnetic forces on the charged particles comprising the atoms of the plate according to the Lorentz force law. We also know that since there is no friction in space then, according to Newton's laws of motion, the plate will start to accelerate and move. How fast? According to calculations, such a plate could exceed 20% the speed of light. This is an example of how our knowledge of the universe was used to design solar sail starships. Such starships would be the basis for interstellar transportation. But we shall also discuss other transportation systems for moving between the planets in a solar system, between galaxies, and we'll eventually even discuss possible ways of traveling between universes.

Transportation is one pillar supporting any technological civilization including the prior industrial civilizations of the eighteenth through the nineteenth centuries and also Type 1, 2, and 3 civilizations. The other pillars are communications and energy technologies. Some of the energy sources which we'll be discussing in this subject that any spacefaring civilization utilize includes energy from stars and lasers to propel solar sail starships, nuclear fusion, negative energy to create wormholes and warp drives to travel to distant galaxies and possibly other universes, and so on. And we'll also discuss how radio and other sophisticated technology can be used for communication. Other infrastructure necessary to support those technologies—such as rotating habitats and cloud cities—will also be covered in this subject.


Calculus

Calculus was independently invented by Gottfried Leibniz and Isaac Newton during the 17th-century to describe the motion of falling bodies such as the apple or the Moon. Long before calculus we were able to study the slope of a function or the area underneath a function using simple algebra so long as the slope (or rate-of-change) of that function is constant; calculus is a big cook book of techniques which we use to find the slope or area whenever the slope (rate-of-change) of a function is changing. Perhaps it struck Newton while sitting idly underneath an apple tree that as an apple falls, the slope (which, in this case, is also the linear velocity) of its vertical position above the ground is constantly changing. Therefore, to solve for its equation of motion given say its acceleration would involve finding the area underneath the velocity function. The mathematics for doing such hadn't, at the time, existed and so Newton invented a new branch of mathematics just to solve problems involving finding the motion of falling bodies. This turned out to be not the only application of calculus. Calculus is necessary whenever we are dealing with quantities that are continuously changing such as the amount of bacteria in a petri dish, the amount of fuel in a rocket, and so on. We'll, for the most part, save discussions of the applications of calculus for separate subjects in science. In this subject, we'll be primarily interested in the mathematical techniques which are used in the study of quantities that are continuously changing. This entire subject will be dedicated to the myriad techniques involved in finding the slopes of curves and the areas underneath them.