## Transforming the Celestial Sphere

A spherical spaceship swooshes by at 99% of the speed of light. What will it look like? Squashed because of Lorentz contraction – like an ellipsoid? No. The outline of a moving sphere will remain spherical. Roger Penrose explained this first in 1958 – 50 years after Einstein’s formulation of the theory of special relativity….

## Joys of Geometry

Creating figures with math software does not feel like fabricating illustrations for science posts. It is more of a meditation on geometry. I want to literally draw every line. I am not using grid lines or rendered surfaces. I craft a parametric curve for every line. A curve is set of equations. Yet, playing with…

## Vintage Covectors

Covectors in the Dual Space. This sounds like an alien tribe living in a parallel universe hitherto unknown to humans. In this lectures on General Relativity, Prof. Frederic Schuller says: Now comes a much-feared topic: Dual vector space. And it’s totally unclear why this is such a feared topic! A vector feels familiar: three numbers…

## Dirac’s Belt Trick

Is classical physics boring? In his preface to Volume 1 of The Feynman Lectures on Physics, Richard Feynman worries about students’ enthusiasm: … They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many…

## Statistical Independence and Logarithms

In classical mechanics you want to understand the motion of all constituents of a system in detail. The trajectory of each ‘particle’ can be calculated from the forces between them and initial positions and velocities. In statistical mechanics you try to work out what can still be said about a system even though – or…

## Integrating the Delta Function (Again) – Dirac Version

The Delta Function is, roughly speaking, shaped like an infinitely tall and infinitely thin needle. It’s discovery – or invention – is commonly attributed to Paul Dirac[*]. Dirac needed a function like this to work with integrals that are common on quantum mechanics, a generalization of a matrix that has 1’s in the diagonal and…

## Delta Function Haiku

I have proved that a Lorentzian bell curve becomes the Dirac Delta Function in the limit. Now I want to look at another representation of the Delta Function. As this is a shorter proof, a haiku will do. ~ Infinite numbers of oscillations added. Need to damp them down Symmetrically attach an exponential for each…

## The Improper Function and the Poetry of Proofs

Later the Delta Function was named after their founder. Dirac himself called it an improper function. This time, the poem is not from repurposed snippets of his prose. These are just my own words to describe a proof: ~ In the limit the Lorentzian becomes the improper function. In the limit of tiny epsilons it…

## Poetry: Dynamical Variables and Observables

The lines of the following poem are phrases selected from consecutive pages of the second chapter of Paul Dirac’s Principles of Quantum Mechanics, Fourth Edition (Revised), Dynamical Variables and Observables. we may look upon the passage for the triple product We therefore make the general rule in spite of this fundamental difference which conforms with…

## Poetry: The Principle of Superposition

The lines of the following poem are phrases selected from consecutive pages of the first chapter of Paul Dirac’s Principles of Quantum Mechanics, Fourth Edition (Revised), The Principle of Superposition. ~ one would be inclined to think There must certainly be some internal motion from general philosophical grounds we cannot expect to find any causal…

## Entropy and Dimensions (Following Landau and Lifshitz)

Some time ago I wrote about volumes of spheres in multi-dimensional phase space – as needed in integrals in statistical mechanics. The post was primarily about the curious fact that the ‘bulk of the volume’ of such spheres is contained in a thin shell beneath their hyperspherical surfaces. The trick to calculate something reasonable is…

## Spheres in a Space with Trillions of Dimensions

I don’t venture into speculative science writing – this is just about classical statistical mechanics; actually about a special mathematical aspect. It was one of the things I found particularly intriguing in my first encounters with statistical mechanics and thermodynamics a long time ago – a curious feature of volumes. I was mulling upon how…

## Learning General Relativity

Math blogger Joseph Nebus does another A – Z series of posts, explaining technical terms in mathematics. He asked readers for their favorite pick of things to be covered in this series, and I came up with General Covariance. Which he laid out in this post – in his signature style, using neither equations nor…