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, ...
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