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, ...