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, basis vectors possess an inverse property: $\hat{e}_\mu \cdot \hat{e}^\mu = I$. With both basis sets, we can express vectors in two ways: 1) expansion with basis, where $\vec{A} = \sum_\mu A^\mu e_\mu$, and 2) contraction with basis, where $\vec{A} = \sum_\mu A_\mu e^\mu$. The former yields components $A^\mu = \vec{A} e^\mu$, known as contra-variant vectors, while the latter gives components $A_\mu = \vec{A} e_\mu$, known as covariant vectors. In an orthogonal basis, there is no need to distinguish between them, but in a non-orthogonal basis, the two bases can differ (see Fig 1).
Fig.1: Examples of basis and inverse basis in orthogonal and non-orthogonal basis.
Covariant vectors change their components according to the transformation of the coordinate system. If we switch to a different coordinate system, the components of a covariant vector will change accordingly. On the other hand, contravariant vectors change their components in the opposite way. When we switch to a different coordinate system, the components of a contravariant vector transform in the opposite direction compared to the coordinate system transformation.
So, to summarize, covariant and contravariant vectors describe how the components of a vector change when we switch between different coordinate systems. Covariant vectors change in the same direction as the coordinate system transformation, while contravariant vectors change in the opposite direction.
For more detailed exploration of these concepts, we will delve into them further in subsequent discussions.
The Metric Tensor: Revealing Curved Space
Using the basis, we can define the metric tensor, which plays a crucial role in describing the properties of space. The metric tensor is defined as $g_{\mu \nu} \equiv e^\mu e_\nu$. Note that this is not an inner product, so the metric tensor has more components than the basis. Using the inverse basis set, we can also define the inverse metric tensor as $g^{\mu \nu} \equiv e^\mu e^\nu$. Both metric tensors satisfy $g_{\mu \nu} g^{\mu \nu} = e^\mu e_\nu e^\mu e^\nu = I$. This may seem abstract, so let's consider an example.
In 3D Cartesian coordinates, the length of the line element can be expressed as $ds^2 = dx^2 + dy^2 + dz^2$. Using matrix representation, we can rewrite it as $ds^2 = (dx, dy, dz) I (dx, dy, dz)^T$, where $I$ denotes the identity matrix and the upper index $T$ represents 'transpose'. The 3x3 identity matrix represents the metric tensor in 3D Cartesian coordinates. More generally, the metric tensor is denoted by $g_{\mu \nu}$. Simplifying further, we can express it as $ds^2 = g_{\mu \nu} dx^\mu dx^\nu$, highlighting the concise component notation (more elegant).
As mentioned in my previous post, special relativity considers coordinate transformations, including the time component. The flat space for 1 time + 3 space is called Minkowski spacetime. In that case, the line element can be written as $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$. Notice the shape of the metric tensor? That's right! It is written as $g_{\mu \nu} = \text{diag}(-1, 1, 1, 1)$, where $\text{diag}(i,j,k,l)$ indicates that the matrix has only diagonal components of $i, j, k$, and $l$.
As you can see, Euclidean space and Minkowski spacetime have a constant metric tensor. In other words, flat space is described by a constant metric tensor, which implies that the basis vector in flat space is constant. However, in curved space, the basis is not a constant; it becomes a function of coordinates. This is an important distinction between flat and curved space. Consequently, we should treat the metric tensor as a function of spacetime in general relativity. For more details, we will explore the generalization of the metric tensor's properties in curved spacetime in our next post.
Fig2. In curved space, the basis becomes a function of coordinates
Summary:
In this tutorial, we delved into the fundamental concepts of basis and the metric tensor in the context of general relativity. Basis vectors serve as the building blocks for describing vectors, while the metric tensor plays a crucial role in describing the properties of space. We learned about contravariant and covariant vectors, as well as the inverse metric tensor. Understanding these concepts is essential for comprehending the curved nature of spacetime in general relativity. In our next tutorial, we will further explore the properties of the metric tensor in curved spacetime, taking a deeper dive into the fascinating world of general relativity. Stay tuned for an exciting journey into the intricacies of spacetime curvature!
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