Mathematical and Physical Prerequisites

Abstract

We begin in this chapter with an introductory Sect. 3.1 in which we consider different notions of the concept of space that play a fundamental role in physical and mathematical theories. All theory is usually formulated by using objects with certain algebraic properties that are embedded in a particular concept of “space”. Then, in Sect. 3.2, starting from a fundamental discussion of the properties of sets we define the concept of a function, a commutative ring and a field. We define a linear vector space and discuss the important concept of basis vectors and linear forms (functionals) as mappings between vector spaces. In Sect. 3.3 we study topological spaces which are in essence open sets endowed with some additional properties such as “neighborhood” and “connectivity”. From this we define a metric and the concept of charts, coordinate systems and manifolds. The concept of manifolds as a general notion of “space”, endowed with certain properties necessary to do physics then leads us to tangent and cotangent spaces, one-forms and tensors which are important geometric quantities used abundantly in modern physical theories. From a discussion of metric spaces and a metric connection in Sect. 3.4 we come to a discussion of Riemannian manifolds in Sect. 3.5, which are the fundamental mathematical structures in which the general theory of relativity is formulated. The classical problems of inertia, motion and coordinate systems in Sect. 3.6 lead us to taking a short glimpse on the formulation of relativistic field equations for the description of condensed matter (that is, of fluids and solids) in Sect. 3.7. This chapter ends with Sect. 3.8 in which further literature is listed, followed by many exercises.

Problems

3.1

Sets

Which of the following terms are sets according to the notation introduced in Sect. 3.2?

(a) $\{0,11,15,16,0,3\},\quad$ (b) $(x,y,z),\quad$ (c) $\{A\},\quad$ (d) $\{\{\varnothing \}\},\quad$ (e) $\{\varnothing ,\{1,2\},a\},\quad$ (f) $[4,Z,w],\quad$ (g) $\{1,7,9,m\}.$

3.2

Sets

Let $X=\{1,2\},\, Y=\{2,3,4\}.$ Which of the following statements are correct?

(a) $Y\subset X,\quad$ (b) $Y\supset X,\quad$ (c) $X\ne Y,\quad$ (d) $X=Y,\quad$ (e) $\{2,4\}\subset Y,\quad$ (f) $Y\supset 3,\quad$ (g) $2\in Y,\quad$ (h) $\{\{3,4\},2\}\subset Y.$

3.3

Sets

Let the set X consist of n objects. Prove that the set P(X) has exactly $\left( {\begin{array}{c}n\\ k\end{array}}\right) =\frac{n!}{k!(n-k)!}$ subsets of X of k elements each. Hence, show that P(X) contains $2^n$ members.

3.4

Topology

How many distinct topologies does a finite set having two or three points admit?

3.5

Tensors

Let

$$\begin{aligned} \left( \begin{matrix} 1 &{} 0 &{} -1\\ 0 &{} -1 &{} 1\\ -1 &{} 1 &{} 1 \end{matrix}\right) \end{aligned}$$

(3.149)

the matrix of the coordinates $A_j$ of a mixed tensor $\texttt {t}$ of rank 2 with respect to a basis $\mathbf{B}=\{\varvec{e}_1,\varvec{e}_2,\varvec{x}_3\}$ and the dual basis $\mathbf{B^{\star }}=\{\varvec{e}^1,\varvec{e}^2,\varvec{x}^3\}\;.$ Calculate the coordinates of this tensor upon a chance of basis:

$$\begin{aligned} \bar{e}_1 = \varvec{e}_2+\varvec{e}_3\;,\,\,\bar{e}_2=\varvec{e}_1+\varvec{e}_3\;,\,\,\bar{e}_3=\varvec{e}_2+\varvec{e}_3\;. \end{aligned}$$

(3.150)

3.6

Contraction

Show, that (3.65) is a tensor by transforming the components into a different (primed) coordinate system.

3.7

Affine Connection

Show, that $\varGamma ^a_{bc}$ is not a tensor.

Hint: Write down the transformation law for the three components when changing the coordinate system and use the chain rule of differentiation.

3.8

Christoffel Symbols

Establish (3.91) by assuming that the quantity defined by (3.90) has the tensor character indicated. Take the partial derivative of

$$\begin{aligned} \delta ^{a'}_{c'} = \frac{\partial x^{a'}}{\partial x^{c'}} = \frac{\partial x^{a'}}{\partial x^{d}} \frac{\partial x^{d}}{\partial x^{c'}} \end{aligned}$$

(3.151)

with respect to $x^{b'}$ to establish the alternative form (3.91).

3.9

Metric Connection

Write down the transformation law of $g_{ab}$ and $g^{ab}$. Show directly, that the metric connection (3.106) indeed transforms like a connection.

3.10

Christoffel Symbols in orthonormal coordinates

The line elements of $\mathbf{R^3}$ in Cartesian, cylindrical polar, and spherical coordinates are given respectively by

• 1. $ds^2 = dx^2 + dy^2 + dz^2\;,$

• 2. $ds^2 = dr^2 + r^2d\phi ^2 + dz^2\;,$

• 3. $ds^2 = dr^2 + r^2d\theta ^2 + r^2\sin ^2\theta d\phi ^2\;.$ Calculate in each case $g_{ab}$, $g^{ab}$, $g=\text {Det}(g_{ab})$, $\varGamma ^{a}_{bc}$, $\varGamma _{abc}$.

3.11

Minkowski Coordinates

The Minkowski line elements is given by

$$\begin{aligned} ds^2 = dt^2 - dx^2 - dy^2 - dz^2\;, \end{aligned}$$

(3.152)

with Minkowski coordinates

$$\begin{aligned} (x^a) = (x^0,x^1,x^2,x^3) = (t,x,y,z)\;. \end{aligned}$$

(3.153)

• 1. What is the signature?

• 2. Is the metric non-singular?

• 3. Is the metric flat?

3.12

Schwarzschild Coordinates

Taking the following coordinates

$$\begin{aligned} x^a = (x^0,x^1,x^2,x^3) = (t,r,\theta ,\phi )\;, \end{aligned}$$

(3.154)

the four-dimensional spherically symmetric line element is

$$\begin{aligned} ds^2 = e^{\nu }dt^2-e^{\lambda }dr^2-r^2d\theta ^2-r^2\sin ^2d\phi ^2\;, \end{aligned}$$

(3.155)

with arbitrary functions $\nu =\nu (t,r)$ and $\lambda =\lambda (t,r)$.

• 1. Calculate $g_{ab}$, $g^{ab}$ and g.

• 2. Calculate $\varGamma ^a_{bc}$.

• 3. Calculate $R^a_{bcd}$

• 4. Calculate $R_{ab}$, R.

The Schwarzschild coordinates constitute a spherically symmetric solution of Einstein’s field equations. In fact, this was the first approximate solution of the equations found by Karl Schwarzschild in 1916 [262].

3.13

Levi-Civita Tensor Density

Proof, that the tensor density $\varepsilon _{abcd}$ is a Lorentz-invariant. Hint: Write down the transformation law for the components of ${\varepsilon }$ as a determinant.

3.14

Covariant Derivative

In a plane polar coordinate system $r=x^1\;\theta = x^2$ a vector filed is given by

$$\begin{aligned} v^1=A\cos x^2\;,\,\,\,v^2=-(A/x^1)\sin x^2\;. \end{aligned}$$

(3.156)

Calculate the covariant derivatives.