The Physics Post

In the previous tutorial, we explored the metric tensor and basis. In this tutorial, we will delve into the concept of tensors and their properties related to coordinate transformations.  A tensor is a multidimensional array that obeys certain transformation rules under coordinate transformations. A vector is an example of a tensor, specifically one with rank 1. One of the key properties of a vector is that it possesses both magnitude and direction. The significance of direction lies in the fact that a vector can be transformed under a coordinate transformation.  Let's consider the basis vectors denoted by $g_i$, which can be expressed as $g_i = \frac{\partial \vec{r}}{\partial x^i}$. We also have another basis, denoted by $g'_i=\frac{\partial \vec{r}}{\partial x'^i}$.  By applying the chain rule, we can establish the following relationship between the two bases:  $\frac{\partial \vec{r}}{\partial x^j} = \frac{\partial \vec{r}}{\partial x'^k} \frac{\partial x'k}{\pa...

Greetings, readers! Welcome to the second installment of our tutorial series on the fascinating realm of general relativity. In this post, we will delve deeply into the essential concepts of basis vectors and the metric tensor, shedding light on their fundamental roles in understanding the structure of spacetime.  What is a basis? In mathematics, we are familiar with vectors. In 2D Cartesian space, we can express an arbitrary vector as $\vec{A} = a_1 \hat{x} + a_2 \hat{y}$. These unit vectors, which form an orthogonal basis, serve as the fundamental building blocks for describing vectors. More generally, we can write a vector as $\vec{A} = a^1 \hat{e}_1 + a^2 \hat{e}_2$ or $\vec{A} = \sum_\mu a^\mu \hat{e}_\mu$. For simplicity, the summation is often omitted for the same indices, which is referred to as 'Einstein convention'. The product between upper and lower indices denotes the inner product. Here the set of $\{ \hat{e}_\mu \}$ is called a basis set.  In general, ...

Hello, readers! Welcome to the first installment of our series on general relativity, a captivating pillar in modern cosmology. In this posting, we embark on a journey to explore the depths of general relativity, unraveling its historical significance, fundamental concepts, and the mathematical treatment rooted in tensor analysis. Together, we will delve into the fascinating world of spacetime curvature and the profound implications it holds for our understanding of gravity and the universe at large. So, without further ado, let us commence this enlightening tutorial on general relativity!  From special relativity to general relativity  In 1905, Albert Einstein's proposal of special relativity propelled him into the spotlight as a rising star in the scientific community. The cornerstone of special relativity lies in the concept of coordinate transformation. To grasp this idea, let us consider a scenario involving a moving horse with a 30 miles per hour. Now, imagine two indivi...