The problem One of the celebrated results of Einstein’s special theory of relativity is that no physical object can travel faster than light, no matter the frame of reference. A popular explanation for this is that as an object approaches the speed of light, its relativistic energy \(E = \gamma mc^2\) 1 tends to infinity. Thus, it would require...

In this post, we will derive the conservation of total mass of a closed system from simpler physical facts, within the framework of classical mechanics. We will first recall the key concepts which will be used to build the argument, namely: The conservation of total linear momentum The principle of relativity Then, we wil...

Let us find the covariant derivative of a rank \((p, q)\) tensor field \(\pmb{\phi}\) by applying the Leibniz law repeatedly: [\nabla_\rho \left( \pmb{\phi} \otimes \pmb{\psi} \right) = \left( \nabla_\rho \pmb{\phi} \right) \otimes \pmb{\psi} + \pmb{\phi} \otimes \left( \nabla_\rho \pmb{\psi} \right)] Note that we will be implicitly using a Le...

Welcome to Part 3 of ‘Algebra Done Tensorially’. It’s been a while since the previous posts, so let us resume our investigation of algebras without further ado. For readers who haven’t read earlier posts in this series yet, I’d recommend you to read Part 1 and Part 2 first :) Parts Topics Part 1 (Bilin...

A Supertask A supertask is a countably infinite sequence of tasks or events, which occurs in a finite amount of time. The word ‘supertask’ was coined by the twentieth-century philosopher James F. Thomson. He went on to provide an example of a supertask that soon became his namesake philosophical puzzle. The problem may be stated as: we are give...

Phase space Consider a one-dimensional harmonic oscillator evolving according to the equation of motion: [m \ddot{x} + kx = 0] It is possible to investigate the essential properties of \(x \left( t \right)\) using a geometric intuition solely by playing with the equation of motion, rather than directly finding the general solution analyticall...

When we first learn mathematics in school, we are taught the four fundamental operations of arithmetic: addition, subtraction, multiplication and division. It is a common practice to teach addition first. After all, we’re adding things all the time in real life: apples to apples, oranges to oranges and so on. Only then are we taught that the rev...

Welcome to another post on, well, you guessed it: scalar field theory. Today, we will be deriving a Lagrangian density for scalar fields that appears almost everywhere in physics (namely in the Klein-Gordon and related theories). Common arguments for assuming the form of the Lagrangian in question either take motivation from too specific system...

In ‘Deriving the Gamma Function from Scratch’, we investigated the analytic origin of the extended factorial. Namely, it comes from the complex solution for the functional equation of the discrete factorial function. Today, we will take a more general route and interpret factorials as invariant points under a certain transformation, in order to...

Once again, let us see how classical scalar field theory naturally gives rise to common natural phenomena. Today, we will derive the evolution of harmonic oscillators. Scalar form Scalar fields Given a scalar field (in flat spacetime) \(\phi \left( x^\rho \right)\) with the following Lagrangian density, [\mathcal{L} = \frac{1}{2} \partial_\...