Vectors, Tensors and Forms

Abstract

In this chapter we develop the main mathematical concepts used in this book. First vectors, not only as quantities with length and direction, but as differential operators. Then tensors of arbitrary rank are introduced. As a preparation for using Cartan’s formalism we introduce forms, i.e. antisymmetric covariant tensors. This antisymmetric tensor formalism is most effective when we introduce an orthonormal basis field. Using an orthonormal basis co-moving with an observer, i.e. where the time-like vector is equal to the 4-velocity of the observer, simplifies the physical interpretation of the calculations.

Exercises

• 3.1. Four-vectors

• (a) Given three four-vectors

  $$\begin{aligned} & \vec{A} = 4\vec{e}_{t} + 3\vec{e}_{x} + 2\vec{e}_{y} + \vec{e}_{x} , \; \vec{B} = 5\vec{e}_{t} + 4\vec{e}_{x} + 3\vec{e}_{y} ,\quad \vec{C} = \vec{e}_{t} + 2\vec{e}_{x} + 3\vec{e}_{y} + 4\vec{e}_{x} \\ & \vec{e}_{t} \cdot \vec{e}_{t} = - 1,\; \vec{e}_{x} \cdot \vec{e}_{x} = \vec{e}_{y} \cdot \vec{e}_{y} = \vec{e}_{z} \cdot \vec{e}_{z} = 1 \\ \end{aligned}$$

  (3.134)

  Show that $\vec{A}$ is time-like $\left( {\vec{A} \cdot \vec{A} < 0} \right)$, $\vec{B}$ is light-like $\left( {\vec{B} \cdot \vec{B} = 0} \right)$ and $\vec{C}$ is space-like $\left( {\vec{C} \cdot \vec{C} > 0} \right)$.

• (b) Assume that $\vec{A}$ and $\vec{B}$ are two non-vanishing orthogonal vectors, $\vec{A} \cdot \vec{B} = 0$. Show the following

  • If $\vec{A}$ is light-like, then $\vec{B}$ is space-like or light-like.

  • If $\vec{A}$ and $\vec{B}$ are light-like, then they are proportional.

  • If $\vec{A}$ is space-like, then $\vec{B}$ is time-like, light-like or space-like.

  Illustrate this in a 3-dimensional Minkowski diagram.

• (c) A change of basis is given by

  $$\vec{e}_{{t^{\prime } }} = \cosh \theta \vec{e}_{t} + \sinh \theta \vec{e}_{x} ,\quad \vec{e}_{{x^{\prime } }} = \sinh \theta \vec{e}_{t} + \cosh \theta \vec{e}_{x} ,\quad \vec{e}_{{y^{\prime } }} = \vec{e}_{y} ,\quad \vec{e}_{z} = \vec{e}_{z}$$

  (3.135)

  Show that this describes a Lorentz transformation along the x-axis, where the relative velocity v between the reference frames is given by $v = \tanh \theta$. Draw the vectors in a 2-dimensional Minkowski diagram and find what type of curves $\vec{e}_{{t^{\prime } }}$ and $\vec{e}_{{x^{\prime } }}$ describe as $\theta$ varies.

• (d) The 3-vector $\vec{v}$ describing the velocity of a particle is defined with respect to an observer. Explain why the 4-velocity $\vec{u}$ is defined independent of any observer. The 4-momentum of a particle, with rest mass m, is defined by $\vec{p} = m\vec{u} = m{\text{d}}\vec{r}/{\text{d}}\tau$, where $\tau$ is the co-moving time of the particle. Show that $\vec{p}$ is timelike and that $\vec{p} \cdot \vec{p} = - \,m^{2}$. Draw, in a Minkowski diagram, the curve to which $\vec{p}$ must be tangent to and explain how this is altered as $m \to 0$. Assume that the energy of the particle is being observed by an observer with 4-velocity $\vec{u}$. Show that the energy he measures is given by

  $$E = - \vec{p} \cdot \vec{u}.$$

  (3.136)

  This is an expression which is very useful when one wants to calculate the energy of a particle in an arbitrary reference frame.

• 3.2. The tensor product

• (a) Given two 1-forms $\underline{\alpha } = \left( {1,1,0,0} \right)$ and $\underline{\beta } = \left( { - 1,0,1,0} \right)$. Show—by using the vectors $\vec{e}_{0}$ and $\vec{e}_{1}$ as arguments—that $\underline{\alpha } \otimes \underline{\beta } \ne \underline{\beta } \otimes \underline{\alpha }$.

• (b) Find the components of the symmetric and antisymmetric parts of $\underline{\alpha } \otimes \underline{\beta }$.

• 3.3. Symmetric and antisymmetric tensors

• (a) A tensor T of rank 2 in four-dimensional spacetime with Minkowski metric $\eta_{\alpha \beta } = {\text{diag}}\left( { - 1,1,1,1} \right)$ has contravariant components

  $$T^{\alpha \beta } = \left( {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & { - 1} \hfill & 0 \hfill & 2 \hfill \\ 2 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & { - 2} \hfill & 0 \hfill \\ \end{array} } \right).$$

  Find

  • 1. The components of the symmetric tensor $T^{{\left( {\alpha \beta } \right)}}$ and the antisymmetric tensor $T^{{\left[ {\alpha \beta } \right]}}$.

  • 2. The mixed components $T_{\beta }^{\alpha }$.

  • 3. The covariant components $T_{\alpha \beta }$.

• (b) Does it make sense to talk about the symmetric and the antisymmetric parts of a mixed tensor, i.e. a tensor with both vector- and form-arguments? Explain!

• 3.4. Contractions of tensors with different symmetries

Let A be an antisymmetric tensor of rank $\left( \begin{aligned} 2 \hfill \\ 0 \hfill \\ \end{aligned} \right)$, B a symmetric tensor of rank $\left( \begin{aligned} 0 \hfill \\ 2 \hfill \\ \end{aligned} \right)$, C an arbitrary tensor of rank $\left( \begin{aligned} 0 \hfill \\ 2 \hfill \\ \end{aligned} \right)$ and D an arbitrary tensor of rank $\left( \begin{aligned} 2 \hfill \\ 0 \hfill \\ \end{aligned} \right)$. Show that

$$A^{\alpha \beta } B_{\alpha \beta } = 0,\quad A^{\alpha \beta } C_{\alpha \beta } = A^{\alpha \beta } C_{{\left[ {\alpha \beta } \right]}} ,\quad B_{\alpha \beta } D^{\alpha \beta } = B_{\alpha \beta } D^{{\left( {\alpha \beta } \right)}} .$$

(3.137)

• 3.5. Coordinate transformation in an Euclidean plane

In this exercise we shall consider vectors in an Euclidean plane. Let $\left\{ {\vec{e}_{x} ,\vec{e}_{y} } \right\}$ be an orthonormal basis in the plane,

$$\vec{e}_{i} \cdot \vec{e}_{j} = \delta_{ij} .$$

A position vector as decomposed in this basis is

$$\vec{x} = x^{i} \vec{e}_{i} = x\vec{e}_{x} + y\vec{e}_{y}$$

A new coordinate system $\left\{ {x^{\prime } ,y^{\prime } } \right\}$ is related to the $\left\{ {x,y} \right\}$-system by the transformation

$$x^{\prime } = 2x - y,\quad y^{\prime } = x + y$$

• (a) Find $\vec{e}_{{x^{\prime } }}$ and $\vec{e}_{{y^{\prime } }}$ expressed in terms of $\vec{e}_{x}$ and $\vec{e}_{y}$

• (b) Find the basis vectors in the $\left\{ {x^{\prime } ,y^{\prime } } \right\}$-system in terms of $\left\{ {\vec{e}_{x} ,\vec{e}_{y} } \right\}$.

• (c) Find the components of the metric tensor in the $\left\{ {x^{\prime } ,y^{\prime } } \right\}$-system.

• (d) Calculate the line-element in the $\left\{ {x^{\prime } ,y^{\prime } } \right\}$-system.

• (e) We now define a set of basis vectors $\vec{\omega }^{{i^{\prime } }}$ by $\vec{\omega }^{i} = M_{i}^{{i^{\prime } }} \vec{e}_{i}$ with summation over i. The scalar products of these vectors define the contravariant components of the metric tensors.

  Use this to find the contravariant components of the metric tensor in the $\left\{ {x^{\prime } ,y^{\prime } } \right\}$-system.