Tensors YouTube

Introduction to tensors

Tensors can be defined in several ways. For our purposes, we shall consider tensors as objects with components that transform in a way that is Lorentz invariant.

Tensors can have different rank, corresponding to the number of indices - and hence the number of components - needed to describe them.

Tensors can be contravariant, covariant or mixed - a combination of contravariant and covariant.

Crucially, equations that are written in terms of tensors are Lorentz invariant, thus we can write the laws of physics in terms of them, since that is one of the basic requirements of relativity.

Rank-zero tensors - scalars

A scalar is a rank-zero tensor, meaning that it requires zero indices to describe it.

A scalar field, $\phi = \phi(t,x,y,z)$, say, does not have any components at all, so doesn't change under a change of coordinates. For example, if the temperature of a moving body is measured at certain locations (events) in its own reference (proper) frame, then we wouldn't expect that a transformation to some other (coordinate) frame will affect the temperatures at those locations (events).

Other examples of scalars are spatial distance in Euclidean space and proper time/length in space-time, since they remain invariant under changes of coordinates whereas components of four-vectors are not scalars, since they do change under coodinate transformations.

It is also true that equations involving only scalars remain invariant under a change of coordinates.

For example, suppose that the spatial distance between two between particles - in meters, say - happened to be the same value as the average temperature of the two particles - in degrees fahrenheit, say.

Would this depend on the choice of coordinates? No, since both quantities are scalar quantities hence any expression only involving them will also be scalar quantities.

We shall see important examples of scalar fields - waves - later in the course.

Rank-one tensors - four-vectors

A four-vector is a rank-one tensor, meaning that it requires one index (hence four components) to describe it.

Recall that we can write Lorentz invariant transformations as matrices - either space-time boosts, $\mathbf{L}$ or spatial rotations, $\mathbf{R}$. We can easily convert these to index notation. We let $L_\nu^\mu$ represent the sixteen quantities of a Lorentz invariant transformation matrix, where the superscript $\mu$ ranges over the rows of the matrix, and $\nu$ ranges over the columns.

Then, the coordinates, $X^\mu$, of a position four-vector in a particular frame, can be transformed into coordinates (of the same position) in a different frame, $X^{\mu'}$, by contracting indices with the transformation matrix, $$X^{\mu'} = L_\nu^{\mu'} X^\nu$$ where the placement of the indices - superscript/subscript - is the only possible one that transforms components of contravariant four-vectors into components of other contravariant four-vectors.

Nb. We don't need to start thinking of $L_\nu^\mu$ as a (rank-two) tensor because that would mean that we were considering how it behaved under a Lorentz invariant transformation, but the $L_\nu^\mu$ describe the components of the transformation itself, not the components of quantities under the transformations.

We define contravariant four-vectors to be rank-one tensors that transforms in the same way as position four-vectors.

That is, $A^\mu$ are the components of a contravariant four-vector, if for any Lorentz invariant transformation $L_\nu^\mu$, the components, $A^{\mu'}$, of the same four-vector in the new frame are given by $$A^{\mu'} = L_\nu^{\mu'} A^\nu$$

Covariant four-vectors are also rank-one tensors, and, so far, we have defined their components only in terms of their contravariant counterparts. Thus, $A_\mu$ are the covariant components of four-vector with contravariant components, $A^\mu$, if $$\begin{align*}A_0 &= A^0\\A_1 &= -A^1\\A_2 &= -A^2\\A_3 &= -A^3\end{align*}$$

This allowed us to define scalar invariants associated with four-vectors, by contracting the indices of contravariant and covariant forms.

For example, we defined the proper time, $\tau$, associated with an event (or position four-vector), $X^\mu \rightarrow (t,x,y,z)$ by the expression $$\tau^2 = X_\mu X^\mu$$

We defined the proper length of an arbitrary (contravariant) four-vector, $A^\mu$ to be $$\sqrt{A_\mu A^\mu}$$

Furthermore, another scalar quantity that transforms invariantly is the dot product of two (contravariant) four-vectors, $A^\mu,B^\mu$, $$\begin{align*}A_\mu B^\mu &= A_0 B^0 + A_1 B^1 + A_2 B^2 + A_3 B^3\\&= A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3\end{align*}$$

Nb. For our purposes, the relationship between contravariant and covariant forms of a four vector is one that allows for a convenient notation and one that allows for the contraction of indices. But, we could have started describing four-vectors by taking the partial derivatives of scalar fields as an archetypal set of quantities, and then defining covariant four-vectors by demanding that they transform in the same way as this set of derivatives.

An even more convenient notation to describe the relationship between contravariant and covariant four-vectors is to define a matrix of values $$\eta_{\mu \nu} \rightarrow \begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$$ where we have place the indices so that we can write expressions like $$A_\mu = \eta_{\mu \nu} A^\nu$$

By the symmetry of $\eta_{\mu \nu}$, $$A_\mu A^\mu = \left(\eta_{\mu \nu} A^\nu\right) A^\mu = \eta_{\mu \nu} A^\mu A^\nu = A^\mu \left(\eta_{\mu \nu} A^\nu\right) = A^\mu A_\mu$$

Nb. The set of quantities, $\eta_{\mu \nu}$ are the covariant components of a rank-two tensor called the metric, which, in special relativity, has the same components in all frames and the same components regardless of whether we are taking the contravariant or covariant or mixed forms of the tensor. Thus, $\eta_{\mu' \nu'} = \eta_{\mu \nu}$ and $\eta_{\mu \nu} = \eta_\nu^\mu = \eta^{\mu \nu}$.