## The 10 Commandments of Index Expressions and Tensor Calculus

Having more than 10 years of experience in teaching vector and tensor calculus and special and general relativity, I have noted that many people have the same problems when trying to manipulate index expressions. The problem often lies in being familiar with the very basic ideas, such as repeated indices implying a sum (the summation […]

## Coordinate Dependent Statements in an Expanding Universe

In this Insight we will discuss the general Robertson–Walker (RW) universe, in which a set of co-moving observers with proper time ##t## observe the universe to be homogeneous and isotropic. In such a universe, the line element takes the form $$ds^2 = dt^2 – a(t)^2 [dr^2 + f(r)^2 d\Omega^2] \equiv dt^2 – a(t)^2 dS^2,$$ where […]

## The Birth of a Physics Textbook

It is said that you should accomplish three things in life: Plant a tree. Have a child. Write a book. Out of the three, I have completed at least one. At the time of writing this Insight, my textbook “Mathematical Methods for Physics and Engineering” just hit the virtual online shelves. This Insight will […]

## Knut Lundmark and the Prehistory of Dark Matter

Look in your standard cosmology textbook (in my case, the closest one at hand is Barbara Ryden’s Introduction to Cosmology) and go to the section which describes experimental discovery of dark matter. Two of the things you will likely find in there are the following pieces of information: The virial theorem applied to the Coma cluster by […]

## A Geometrical View of Time Dilation and the Twin Paradox

Based on the amount of questions we receive on the topic at Physics Forums, there is a lot of confusion in the general public about how time dilation works and the resolution of the twin paradox. More often than not, the confusion is based upon misunderstandings about how time dilation works and an unawareness of […]

## What is relativistic mass and why it is not used much?

It happens that the term relativistic mass is used, in particular in introductory text on special relativity. It should be noted that whether or not to use relativistic mass to a large extent is a matter of convention, convenience, and semantics as long as it is used properly and does not have any impact on […]

## Plus/minus What? How to Interpret Error Bars

People some times find themselves staring at a number with a ± in it when a new physics result is presented. But what does it mean? The aim of this Insight is to give a fast overview of how physicists (and other scientist) tend to present their results in terms of statistics and measurement errors. […]