Theory of Everything

A short introduction to the history of physics, and string theory as a Theory of Everything

M-theory and dualities

The fine tuning problem and the hierarchy problem




The first article explains what string theory is and why many physicists study it. The second article explains what the first superstring revolution and the second superstring revolution are in laymens terms. I also explain the significance of my calculation of string coupling constant. The last article derives a simple mathematical relation that is very important in string theory. fel



History of chemistry up until the beginning of 20th century

The structure of atoms and the periodic table



In these two articles, I explain the basics of the basics of chemistry as chemistry partly shares history with physics up until the early 20th century. Of course, I am not an expert on chemistry, but having taken AP chemistry, I know chemistry just enough to explain the basics of its basics.



Evidences that the Earth is not flat, but round


History of astronomy up until the mid 17th century

History of astronomy from the late 17th century to the early 21st century



In the first article, I provide the evidences that the Earth is round, as it is alarming that the Flat Earth proponents are gaining momentum these days. They are idiots who reject scientific evidences. They dont deserve to enjoy modern scientific innovations. In the second article, I provide a brief overview of history of astronomy from ancient time to the time of Johannes Kepler and Galileo Galilei. In the third article, I provide a brief overview of history of astronomy from the time of Sir Isaac Newton to the present. Physics was essentially applied to and received feedback from astronomy during the time period treated in this article.


Bosons and Fermions

Supersymmetry: an exposition for laymen

Bosons, Fermions, and Paulis exclusion principle

Pauli's exclusion principle, color charge of quarks, asymptotic freedom and confinement

Bosons, Fermions and the statistical properties of identical particles

Bosons, Fermions in quantum mechanics picture



These five articles explain what bosons and fermions are, and how the concept of these categories of particles is applied in physics. The first article explains how supersymmetry relates bosons and fermions to one another. Supersymmetry is a very important topic in the contemporary particle physics community. First theorized in the early 1970s, many expect that it will soon be supported by experiments at the LHC in Geneva, Switzerland. The second article can be read independently of the first. The third, fourth, and fifth articles can be read independently of one another, but you may have to read the second article in order to understand them. The third article explains how the idea of color charge of quarks was introduced to reconcile the theory with Pauli's exclusion principle, and goes on to explain some interesting characteristics of the strong force. The fourth article explains the microscopic statistical properties of identical particles which may seem bizarre to macroscopic human beings. The fifth article supplies another derivation of Pauli's exclusion principle via an approach alternative to that taken in the second article.


Black hole

Why time goes more slowly at a lower place and what a black hole is



Most people have probably heard that nothing can get out of a black hole once it has been sucked into it. The article here explores this assertion in more detail qualitatively. For a quantitative treatment, read By how much does time go more slowly at a lower place? after reading the above article.


Newtons Inverse Square Law of Gravity

Conic sections and Newtons law of gravity

The inverse square law and the 3-dimensional world



These two articles can be read more or less independently of one another.


Some basic mathematics and statistics

What is a vector?


Expectation and Standard deviation


Asymptotic behavior of polynomials

Exponential function vs polynomial function

Scalar field and vector field

Proof by induction



Vector is a very important basic mathematical concept in physics. To learn more about vectors, read the sections Dot product and Linear Algebra and Quantum Mechanics. The article on vector is the only prerequisite to Electricity and Magnetism, the first part, which in turn is the only prerequisite to the section Special Relativity. Logarithm is often used in physics and mathematics. It serves as a prerequisite for What is entropy? From a microscopic point of view and the articles in the section Loop Quantum Gravity (for laymen). Expectation and Standard deviation serves as a prerequisite for A short introduction to quantum mechanics XII: Heisenbergs uncertainty principle. Exponential function vs polynomial function will play an important role in understanding my research on black hole entropy, which is reviewed in our article Approximation of the naïve black hole degeneracy.


Energy and Entropy

The conservation of energy

What is entropy? From a microscopic point of view

What is entropy? From a macroscopic point of view



Energy and entropy are very important concepts in physics. The second article explains what entropy is and why it always increases. The third article explains what entropy is from a macroscopic point of view. It can be read independently of the first one. These articles serve as prerequisites for the section Statistical Mechanics.


Loop Quantum Gravity

Discrete area spectrum and the black hole entropy I: Loop quantum gravity and the Bekenstein-Hawking entropy

Discrete area spectrum and the black hole entropy II: Domagala-Lewandowski-Meissner formula

Discrete area spectrum and the black hole entropy III: Newer variables

Discrete area spectrum and the Hawking radiation spectrum I

Discrete area spectrum and the Hawking radiation spectrum II: Single unit area deduction

Discrete area spectrum and the Hawking radiation spectrum III: Maxwell-Boltzmann

Discrete area spectrum and the Hawking radiation spectrum IV: Single-partition black hole



These seven articles try to explain, in laymens terms, the research I have done on the black hole partially with Brian Kong. These articles assume that readers are familiar with the mathematical expression log, which is explained in my article Logarithm. You may skip the second and the third articles if its too hard since you dont need it to understand the other articles. To understand the sixth article, you will need to know the concept of identical particles as treated in Bosons, Fermions and the statistical properties of identical particles. We will explain our research again in experts terms in later articles.


Basic high school physics without calculus, the first part

Newtons first law

Newtons second law

The free fall



I have written Newtons first law so that those who have never learned physics can have a chance to learn about the wonderful ideas of the two great physicists, Galilei and Newton. To understand Newtons second law, you have to understand what a vector is. The last article explains how one can relate the distance and the time an object falls due to gravity on Earth. It serves as a prerequisite for the articles listed in the section Some basic ideas in General Relativity.


Trigonometric Functions

What are trigonometric functions?

Polar coordinate and trigonometric functions defined over all values

Rotation in the Cartesian coordinate system

Addition and subtraction rules for trigonometric functions


What is Fourier series?

Inverse trigonometric functions



Even though you may naively imagine that trigonometric functions are important only for geometry, they are actually very important and widely used in mathematics. For example, trigonometric functions are important in describing waves in physics. As quantum mechanics is about particles and waves, knowledge of trigonometric functions is essential in understanding quantum mechanics.


The purpose of the article Rotation in Cartesian coordinates is to serve as a basis to make an interesting comparison with the Lorentz transformation in my article on that topic in the section For students learning high school physics.


Complex Numbers

Complex numbers

Complex conjugate

Complex numbers and the trigonometric functions

Fundamental theorem of algebra



Complex numbers is a prerequisite to quantum mechanics. Everything in quantum mechanics is calculated in terms of complex numbers, even though at the end of these calculations one must obtain an answer in terms of the real numbers to compare with experimental results, since complex numbers are not real and cant be the values of measurements. The last article explains the beautiful relationship between complex numbers and the trigonometric functions, which you may appreciate better after reading Eulers formula listed in A crash course in calculus, the second part.


Electricity and Magnetism, the first part

Electric charge and Coulomb force


Magnetic field

Electric current makes compass needle turn

Magnet exerts force on wire through which electric current passes

The force between two parallel wires through each of which electric current passes



These six articles provide the prerequisites for the articles explaining the relation between Einsteins theory of special relativity and electromagnetism presented in the section For students learning high school physics. Nevertheless, they could be interesting on their own as those who did not take physics in high school do not usually know about the phenomena presented in these articles. What is a vector? listed in the section Some very basic high school physics is the prerequisite to Electric charge and Coulomb force.


Special Relativity

Why is the speed of light constant?

Time dilation in Einsteins theory of special relativity

Electromagnetic forces and time dilation in special relativity

Lorentz-Fitzgerald contraction: Why is a moving object shortened?

Origin of the magnetic force from the perspective of special relativity

The relativity of simultaneity

Lorentz transformation

Lorentz transformation and Rotation, a comparison

Light cone



This section groups together eight articles on special relativity, in a suggested order of reading. (You may also want to read pages 177-209 of The Evolution of Physics.) While the third and fifth articles require the basic knowledge on electricity and magnetism, as covered in the section Electricity and Magnetism, the first part, others can be read without these prerequisites. The third and fifth articles show that Einstein's theory of special relativity is consistent with electromagnetism. Of course, this should be the case since light is an electromagnetic wave, and Einstein's theory of special relativity is based on the fact that the speed of light in vacuum is always constant. Nevertheless, it is exciting to check these consistencies between electromagnetism and special relativity in easy ways, since understanding them from the original construction of special relativity requires advanced knowledge. Moreover, it is always exciting to see that you arrive at the same conclusion even from very different perspectives.


The third article makes use of the relationship between electric current and magnetic force. Later, in the fifth article, this relationship is partly derived by using the special theory of relativity, but the relevant facts are stated in the third article and can be taken for granted for the purpose of understanding the point of the third article. If you dont wish to take it for granted, you may want to read the articles in the section Electricity and Magnetism, the first part, which deal with the interplay between electric fields and magnetic fields.


Unlike other articles, the last three articles related to Lorentz transformation are mathematical to some extent. Even though you only need to know the concept of square root to comprehend them, the derivation of Lorentz transformation might be complicated. Lorentz transformation and Rotation, a comparison assumes prior knowledge on "Rotation in Cartesian coordinates" covered in the section "Trigonometric functions."


Paradoxes in Special Relativity

Twin paradox

Length contraction paradox

Ki yungs misunderstanding



Theory of relativity is quite confusing and unintuitive. At first glance, it seems wrong. Therefore, many people could come up with a lot of paradoxes that seemed to show that theory of relativity was wrong. However, it isnt, since these paradoxes can be resolved. These articles deal with such paradoxes.


Electricity and Magnetism, the second part

Electric field

Gausss law

Why is the electric field inside spherical shell zero?

Gravitational field on the surface of the Earth

Magnetic monopole and Gausss law



The prerequisites to Electric field are What is a vector and Electric charge and Coulombs law. The prerequisites to Gausss law are Electric field and The inverse square law and the 3-dimensional world. These articles do not use any complicated math, but we will formulate Gausss law mathematically in our later article Revisiting Gauss law and the derivation of Coulombs law listed in the section Maxwells equations.


Basic high school physics without calculus, the second part

Constant acceleration in 1-dimension

Projectile motion

Work and kinetic energy

Potential energy and conservation of energy


Center of mass

Newtons third law and conservation of momentum

Elastic collision in 1-dimension

Elastic collision in 1-dimension in center of mass frame

Elastic collision in 2-dimension

Normal force

Motions on an inclined plane

Centripetal force

Centrifugal force



Moment of inertia

Angular momentum

Hookes law and harmonic oscillator

Harmonic oscillator and circular motion

Electric potential



There are two aims for this section. The first one is completing the course on very basic high school physics that began in an earlier section. The second one is presenting the elastic collision problem in 2-dimension, which plays an important role in our later article Compton scattering.


Electricity and Magnetism, the third part

Faradays law of induction

Faradays law of induction from the point of view of magnetic force on moving charge



Faraday experimentally discovered his law of induction in the 19th century. The first article explains Faradays law while the second article explains the origin of Faradays law; Faradays law nicely fits with the picture presented in that article. These articles serve as prerequisites for Faradays law of induction in Maxwells equations, in which we present the mathematical equations for the material described qualitatively in the above articles.


Dark Matter and Modified Newtonian Dynamics

Conformal Gravity, a proposed alternative to the dark matter theory, falsified



In the above article, I explain my own research on conformal gravity. To this end, I explain here what the dark matter theory is. In our later article Problems with Mannheims conformal gravity program I will fully explain my own research on conformal gravity with all the technical details.


Dimensional Analysis, the first part

The power of dimensional analysis

Natural units



Dimensional analysis is widely used in advanced physics. Although it may not be apparent at first glance, dimensional analysis is essential to understanding nature. There is a sequel to these articles in the section Dimensional Analysis, the second part.



The symmetry of physical laws: the CPT theorem for laymen

CP violation and the Nobel Prize for Physics 2008



These two articles explain what Kobayashi and Maskawa won the Nobel Prizes for. These articles could be a little bit difficult to understand for beginners.


Neutrino oscillation

Neutrino oscillation



This article explains at a laymen level what is the neutrino oscillation for which Nobel Prize in Physics was awarded in 2015. We will revisit it fully in our later article Neutrino oscillation, clarified as neutrino oscillation seems so bizarre and cannot be understood without quantum mechanics.



Travelling wave

Standing wave

Huygens principle

Snells law

Youngs interference experiment

Interference from thin films



Familiarity with trigonometric functions is the prerequisite to the above articles. They serve as prerequisites to Bohr model, De Broglies matter waves, and Schrödinger equation listed in Historical introduction to quantum mechanics.


A Crash Course in Calculus, the first part

What is a limit?

What are continuous functions?

Intermediate value theorem

Derivatives, velocity, and acceleration

Addition, subtraction rules for differentiation and Leibnizs rule

Derivatives of the polynomials

The sandwich theorem and the limit of sin theta/theta

Derivatives of the trigonometric functions

Velocity re-visited and extremum

The chain-rule

Quotient rule

LHôpitals rule

Exponential functions and natural logs

Differential and infinitesimal change

Implicit differentiation



Calculus is an essential tool for physics. Its a very basic area of mathematics which you cannot avoid with if you truly want to understand physics. Calculus was first discovered by Newton to apply it for physics, even though it is widely believed that Leibniz discovered it independently, despite the fact that Newton accused Leibniz of plagiarism. Calculus is composed of two parts: differentiation and integration. Interestingly, integration is anti-differentiation. If I make an analogy, integration is to differentiation as subtraction is to addition and division to multiplication. In the first part of the crash course, I explain differentiation, which is also called taking the derivative.


A Crash Course in Calculus, the second part

Integration and the fundamental theorem of calculus

Taylor series

Eulers formula and hyperbolic functions

Integration by parts

Even functions and odd functions

Integration by substitution



In the second part, I explain integration and other topics which are usually covered in the second semester of calculus.


Applications of Calculus, the first part

Area inside a circle, surface area of a sphere and volume inside a sphere

The derivation of Snells law using calculus

Convexity, concavity and the point of inflection

Center of mass of a half-ball

Fourier transformations

Expectation values in quantum field theory (1)

Probability density function

Gaussian distribution

Gamma function



We already covered Snells law in the section Wave, but we provide an alternative derivation using calculus in the second article. To appreciate Fourier transformations, you must know linear algebra well, but it is simple enough to understand without it. I included the article on Fourier transformations here because it is not usually covered in freshman mathematics while its construction is interesting. For more motivations behind Fourier transformations, please read What is Fourier series? listed in the section Trigonometric functions. Actually, the whole book Who is Fourier? treats this topic, along with all its prerequisites such as trigonometric functions, complex numbers, calculus, and vector. I highly recommend the book. Expectation values in quantum field theory (1) prepares you to understand Feynman diagram. The last two articles explain some statistics.


The products of vectors

The law of cosines

The dot product

The dot product and the law of cosines

The cross product



There are two different ways to multiply two three-dimensional vectors: Dot product and cross product. These articles assume the familiarity with vectors as treated in What is a vector? Unlike cross product, dot product can be defined in any dimensions, and plays a very important role in physics, math and engineering.


A Crash Course in Calculus, the third part

Partial derivatives and the chain-rule

Differential equations

Partial differential equations

Separation of variables method in partial differential equations

Multiple integrals



In the third and fourth parts, I cover the content of the third semester of calculus, which most science and engineering college students who have taken a year of calculus in high school take in the first semester of their freshman year. The first four articles are important in understanding quantum mechanics.


Some basic ideas in General Relativity

Equivalence principle

By how much does time go more slowly at a lower place?



These articles survey two ideas Einstein discovered in 1907 and found very useful for his final discovery of the theory of general relativity in 1915. Both articles assume very basic high school physics knowledge as covered in The free fall and Newtons second law. The second article assumes the knowledge of the first article, familiarity with the constancy of the speed of light, in addition to very basic high school physics knowledge. For an alternative approach to the second article, please read Why time goes more slowly at a lower place and what a black hole is listed in the section Black hole.


Mathematical Introduction to Physics

Newtons first and second laws

Newtons third law and the conservation of momentum with calculus

Kinetic energy and Potential energy in one dimension

Harmonic oscillator

Simple pendulum

Kinetic energy and Potential energy in three dimensions, Line Integrals and Gradient

Earths Gravity near its surface

Centripetal force revisited with calculus

Newtons law of universal gravitation and Keplers third law

Newtons proof of Keplers second law

Kinetic energy, Potential energy and angular momentum in polar coordinate

Re-visiting angular momentum conservation in central force

Reduced mass

Moment of Inertia revisited

Rolling down on an inclined plane

A pencil rolling down on an inclined plane, an International Physics Olympiad problem

Equilibrium of forces

Reflection and transmission of travelling wave

Youngs interference experiment, revisited




I wrote the first four articles for those who haven't taken high school physics to read before going on to my articles on quantum mechanics. The articles in listed in this section assume familiarity with calculus at the level of the section A Crash Course in Calculus, the third part. For those who have read the articles in our earlier sections Some very basic high school physics the first three articles, Earths Gravity near its surface and Centripetal force revisited with calculus will be a review. Nevertheless, it is good to study the same material for the second time, from a slightly different approach using calculus. In addition to some physics, Kinetic energy and Potential energy in three dimensions, Line Integrals and Gradient explains some basic concepts in multivariable calculus using physics as an example. The last two articles are about waves and require the section Wave as prerequisites.



Lorentz force

Magnetic force on a current-carrying wire

Magnetic dipole

Electron magnetic moment

Relativistically moving charges around a square loop, an International Physics Olympiad problem



The main purpose of these articles is to introduce the concept of the magnetic dipole which will turn out to be crucial in our later article Stern-Gerlach experiment. Nevertheless, if you are impatient and want to jump into Stern-Gerlach experiment as soon as possible, you can go ahead and read it without reading these four articles as I explain there the concept of magnetic dipole needed for Stern-Gerlach experiment in a light manner. Nevertheless, if you want a firm basis, you would need to read Magnetic dipole.


Historical Introduction to Quantum Mechanics

Plancks relation

Rydberg formula

Bohr model

De Broglies matter waves

Schrödinger equation


Infinite potential well

Stern-Gerlach experiment



I wrote these articles to complement my articles on quantum mechanics listed in the section Linear Algebra and Quantum Mechanics, because I worry that those articles may be too mathematical and allow readers to lose the physical picture. However, those articles can be read more or less independently as well without knowing these articles much. The first two articles do not assume any knowledge of physics. Bohr model assumes knowledge of high school physics, such as centripetal force and Coulomb force. Plancks relation and De Broglies matter waves should help to understand my article, A short introduction to quantum mechanics X: comparison with de Broglies matter waves and time-dependent Schrödinger equation. Schrödinger equation, Tunneling, and Infinite potential well assume knowledge of differential equations.


Enumerative Geometry

Conic sections in Cartesian coordinate

What is enumerative geometry?

Conic sections in polar coordinate



College students or even math majors dont usually learn enumerative geometry, but I included the second article to give readers a sense of what kind of mathematics is used in string theory. The first article helps understanding the second article. The third article is a prerequisite for Keplers laws, revisited.


Dimensional Analysis, the second part

The Planck length, the Planck time, the Planck mass and the fine structure constant



Building on the ideas in our earlier article Natural units we explain the important concepts.