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Chapter 15: Tensor Analysis

Chapter 15: Tensor Analysis

Mathematical Physics, Part 2. Theory, numbered worked examples and 4 previous years’ questions (CSIR NET-JRF, GATE, IIT-JAM and JEST, 2020–2026) with full solutions. Click Show solution under any question.

Theory

Introduction

Physical laws must not depend on the coordinate system used to describe them. The consequences of this requirement lead to tensor analysis, used in general relativity, differential geometry, mechanics, elasticity, hydrodynamics, electromagnetic theory and many other fields.

Spaces of \(N\) Dimensions

A point in three-dimensional space is a set of three numbers, its coordinates, in a chosen frame of reference: for example \((x,y,z)\), \((\rho,\phi,z)\) and \((r,\theta,\phi)\) in rectangular, cylindrical and spherical coordinates. By analogy, a point in \(N\)-dimensional space is a set of \(N\) numbers \((x^1,x^2,\dots,x^N)\); here \(1,2,\dots,N\) are superscripts, not exponents. That we cannot visualize spaces of more than three dimensions has no bearing on their existence.

Coordinate transformations Let \((x^1,x^2,\dots,x^N)\) and \((\bar x^1,\bar x^2,\dots,\bar x^N)\) be the coordinates of a point in two frames of reference, related by \(N\) independent relations \[ \begin{aligned} \bar x^1&=\bar x^1(x^1,x^2,\dots,x^N),\\ \bar x^2&=\bar x^2(x^1,x^2,\dots,x^N),\\ &\ \,\vdots\\ \bar x^N&=\bar x^N(x^1,x^2,\dots,x^N), \end{aligned} \] or briefly \begin{equation*}\tag{15.1} \bar x^k=\bar x^k(x^1,x^2,\dots,x^N),\qquad k=1,2,\dots,N, \end{equation*} where the functions are single-valued and continuously differentiable. Conversely, to each \((\bar x^1,\dots,\bar x^N)\) there corresponds a unique \((x^1,\dots,x^N)\) given by \begin{equation*}\tag{15.2} x^k=x^k(\bar x^1,\bar x^2,\dots,\bar x^N),\qquad k=1,2,\dots,N. \end{equation*} Relations (15.1) and (15.2) define a transformation of coordinates.

Summation convention The sum \(a_1x^1+a_2x^2+\cdots+a_Nx^N=\sum_{j=1}^Na_jx^j\) is written simply \(a_jx^j\): an index (subscript or superscript) repeated in a term is summed from \(1\) to \(N\) unless otherwise specified. This is the summation convention. The letter is immaterial (\(a_jx^j=a_px^p\)); a repeated index is called a dummy or umbral index.

An index occurring only once in a term is a free index and stands for any of \(1,2,\dots,N\), like \(k\) in (15.1) or (15.2), each of which represents \(N\) equations.

Contravariant and Covariant Vectors

Suppose \(N\) quantities \(A^1,\dots,A^N\) in coordinates \(x^k\) are related to \(N\) quantities \(\bar A^1,\dots,\bar A^N\) in coordinates \(\bar x^k\) by \[ \bar A^p=\sum_{q=1}^N\pd{\bar x^p}{x^q}A^q,\qquad p=1,2,\dots,N, \] or, by the summation convention,

\[ \bar A^p=\pd{\bar x^p}{x^q}A^q. \]

Then they are the components of a contravariant vector (contravariant tensor of rank one).

If instead \(N\) quantities \(A_1,\dots,A_N\) transform as \[ \bar A_p=\sum_{q=1}^N\pd{x^q}{\bar x^p}A_q,\qquad p=1,2,\dots,N, \qquad\text{or}\qquad \bar A_p=\pd{x^q}{\bar x^p}A_q. \] they are the components of a covariant vector (covariant tensor of rank one).

Superscripts denote contravariant and subscripts covariant components; the coordinates \(x^k\) are an exception. We often speak simply of “the tensor \(A^p\)” or “the tensor \(A_p\)”.

Contravariant, Covariant and Mixed Tensors

The \(N^2\) quantities \(A^{qs}\) are the contravariant components of a tensor of rank two if they transform as \[ \bar A^{pr}=\sum_{s=1}^N\sum_{q=1}^N\pd{\bar x^p}{x^q}\pd{\bar x^r}{x^s}A^{qs},\qquad p,r=1,2,\dots,N, \qquad\text{or}\qquad \bar A^{pr}=\pd{\bar x^p}{x^q}\pd{\bar x^r}{x^s}A^{qs} \] The \(N^2\) quantities \(A_{qs}\) are the covariant components of a tensor of rank two if \[ \bar A_{pr}=\pd{x^q}{\bar x^p}\pd{x^s}{\bar x^r}A_{qs}. \] and \(A^q_s\) are the components of a mixed tensor of rank two if \[ \bar A^p_r=\pd{\bar x^p}{x^q}\pd{x^s}{\bar x^r}A^q_s. \]

Kronecker delta The Kronecker delta is \[ \delta^j_k=\begin{cases}0&\text{if } j\neq k,\\ 1&\text{if } j=k.\end{cases} \] As the notation indicates, it is a mixed tensor of rank two (Example 15.14).

Tensors of Rank Greater Than Two, Tensor Fields

Higher-rank tensors are defined similarly. For example, \(A^{qst}_{kl}\) are the components of a mixed tensor of rank \(5\), contravariant of order \(3\) and covariant of order \(2\), if \[ \bar A^{ijm}_{rn}=\pd{\bar x^i}{x^q}\pd{\bar x^j}{x^s}\pd{\bar x^m}{x^t}\pd{x^k}{\bar x^r}\pd{x^l}{\bar x^n}A^{qst}_{kl}. \]

Scalars or invariants A function \(\phi\) of the \(x^k\), with value \(\bar\phi\) in new coordinates \(\bar x^k\), is a scalar or invariant if \(\phi=\bar\phi\). A scalar is a tensor of rank zero.

Tensor fields If a definite tensor is assigned to each point of a region of \(N\)-dimensional space, a tensor field is defined; it is a vector or scalar field when the rank is one or zero. A tensor (field) is not the set of its components in one coordinate system but all such sets under every transformation of coordinates.

Symmetric and skew-symmetric tensors A tensor is symmetric (skew-symmetric) in two contravariant, or two covariant, indices if its components are unchanged (change sign) when these indices are interchanged; e.g. \(A^{mpr}_{qs}=A^{pmr}_{qs}\) (\(A^{mpr}_{qs}=-A^{pmr}_{qs}\)). A tensor symmetric (skew-symmetric) in every pair of contravariant and every pair of covariant indices is called symmetric (skew-symmetric).

Fundamental Operations with Tensors

  1. Addition. The sum of tensors of the same rank and type (same numbers of contravariant and of covariant indices) is a tensor of that rank and type: if \(A^{mp}_q\) and \(B^{mp}_q\) are tensors, so is \(C^{mp}_q=A^{mp}_q+B^{mp}_q\). Addition is commutative and associative.

  2. Subtraction. Likewise \(D^{mp}_q=A^{mp}_q-B^{mp}_q\) is a tensor of the same rank and type.

  3. Outer multiplication. The product of two tensors, formed by multiplying components, is a tensor whose rank is the sum of their ranks, called the outer product; e.g. \(A^{pr}_qB^m_s=C^{prm}_{qs}\). Not every tensor is a product of two lower-rank tensors, so division of tensors is not always possible.

  4. Contraction. Setting one contravariant index equal to one covariant index and summing gives a tensor of rank two less. In \(A^{mpr}_{qs}\) (rank \(5\)), \(r=s\) gives \(A^{mpr}_{qr}=B^{mp}_q\) (rank \(3\)); then \(p=q\) gives \(B^{mp}_p=C^m\) (rank \(1\)).

  5. Inner multiplication. Outer multiplication followed by contraction gives an inner product. From \(A^{mp}_q\) and \(B^r_{st}\), the outer product is \(A^{mp}_qB^r_{st}\); \(q=r\) gives the inner product \(A^{mp}_rB^r_{st}\), and \(q=r\), \(p=s\) gives \(A^{mp}_rB^r_{pt}\). Inner and outer multiplication are commutative and associative.

  6. Quotient law. If an inner product of a quantity \(X\) with an arbitrary tensor is a tensor, then \(X\) is a tensor.

Matrices

A matrix \(A\) of order \(m\) by \(n\) is an array of elements \(a_{pq}\) in \(m\) rows and \(n\) columns, denoted by \[ \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix} \qquad\text{or}\qquad \begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{pmatrix} \] or briefly \([a_{pq}]\) or \((a_{pq})\), \(p=1,\dots,m\); \(q=1,\dots,n\); we use \([a_{pq}]\). If \(m=n\), the matrix is square of order \(m\). If \(m=1\) it is a row matrix (row vector); if \(n=1\), a column matrix (column vector).

The elements \(a_{11},a_{22},\dots,a_{nn}\) of a square matrix form its principal (main) diagonal. The unit matrix \(I\) has ones on the principal diagonal and zeros elsewhere; the null matrix \(O\) has all elements zero.

Matrix algebra For matrices \(A=[a_{pq}]\) and \(B=[b_{pq}]\) of the same order (\(m\) by \(n\)):

  1. \(A=B\) if \(a_{pq}=b_{pq}\) for all \(p\) and \(q\).

  2. The sum \(S\) and difference \(D\) of \(A\) and \(B\) are the matrices defined by \[ S=A+B=[a_{pq}+b_{pq}],\qquad D=A-B=[a_{pq}-b_{pq}]. \]

  3. The product of a scalar \(\lambda\) and \(A\) is \(\lambda A=[\lambda a_{pq}]\).

The transpose \(A^T\) is formed by interchanging rows and columns: if \(A=[a_{pq}]\), then \(A^T=[a_{qp}]\).

Matrix multiplication If \(A=[a_{ik}]\) is \(m\times p\) and \(B=[b_{kj}]\) is \(p\times n\), the product \(AB=[c_{ij}]\) is defined: its \(ij\)-entry is row \(i\) of \(A\) times column \(j\) of \(B\), \[ c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ip}b_{pj}=\sum_{k=1}^pa_{ik}b_{kj}. \] Matrices whose product is defined are called conformable.

Determinants The determinant of an \(n\)-square matrix \(A=[a_{ij}]\), written \(\abs A\), \(\det A\), \(\abs{a_{ij}}\) or \(\det[a_{ij}]\), is \[ \abs A=\sum_{\sigma\in S_n}(\operatorname{sgn}\sigma)\,a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)}. \] where \(S_n\) is the set of permutations \(\sigma\) of \(\{1,2,\dots,n\}\) and \(\operatorname{sgn}\sigma=\pm1\) for even or odd \(\sigma\).

Proposition If \(P=AB\), with \(A\), \(B\) \(n\)-square, then \[ \det P=(\det A)(\det B). \]

Inverses The inverse \(A^{-1}\) of a square matrix \(A\) satisfies \(AA^{-1}=A^{-1}A=I\). It exists if and only if \(\det A\neq0\); \(A\) is singular if \(\det A=0\) and nonsingular otherwise.

Line Element and Metric Tensor

In rectangular coordinates the arc length satisfies \(\dd s^2=\dd x^2+\dd y^2+\dd z^2\); in general curvilinear coordinates \((u_1,u_2,u_3)\) this becomes \[ \dd s^2=\sum_{p=1}^3\sum_{q=1}^3g_{pq}\,\dd u_p\,\dd u_q. \] Such spaces are three-dimensional Euclidean spaces.

In \(N\) dimensions the line element \(\dd s\) is defined by the quadratic form called the metric form or metric, \[ \dd s^2=\sum_{p=1}^N\sum_{q=1}^Ng_{pq}\,\dd x^p\,\dd x^q, \] or, using the summation convention,

\[ \dd s^2=g_{pq}\,\dd x^p\,\dd x^q. \]

If some coordinate transformation \(x^j\to\bar x^k\) reduces the metric to \((\dd\bar x^1)^2+\cdots+(\dd\bar x^N)^2=\dd\bar x^k\,\dd\bar x^k\), the space is \(N\)-dimensional Euclidean; in general it is Riemannian.

The \(g_{pq}\) are the components of a covariant tensor of rank two, the metric or fundamental tensor, which we always choose symmetric.

Conjugate or reciprocal tensors Let \(g=\abs{g_{pq}}\neq0\) and define \[ g^{pq}=\frac{\text{cofactor of }g_{pq}}{g}. \] Then \(g^{pq}\) is a symmetric contravariant tensor of rank two, the conjugate or reciprocal tensor, and

\[ g^{pq}g_{rq}=\delta^p_r. \]

Associated Tensors

New tensors are obtained by raising or lowering indices. Raising \(p\) in \(A_{pq}\) gives \(A^{p\cdot}_{\cdot q}\), the dot marking the original position of the moved index; raising \(q\) also gives \(A^{pq}_{\cdot\cdot}\), written \(A^{pq}\) when no confusion arises. These are inner products with \(g_{pq}\) or \(g^{pq}\); for example, \[ A^{p\cdot}_{\cdot q}=g^{rp}A_{rq},\qquad A^{pq}=g^{rp}g^{sq}A_{rs},\qquad A^{p\cdot\cdot}_{\cdot rs}=g_{rq}A^{pq}_{\cdot\cdot s},\qquad A^{qm\cdot tk}_{\cdot\cdot n\cdot\cdot}=g^{pk}g_{sn}g^{rm}A^{q\cdot st}_{\cdot r\cdot\cdot p}. \] Read multiplication by \(g^{rp}\) as “replace \(r\) by \(p\) in what follows and raise it”, and by \(g_{rq}\) as “replace \(r\) by \(q\) and lower it”.

Tensors obtained from a given tensor by inner products with \(g_{pq}\) and \(g^{pq}\) are its associated tensors. Thus \(A^p\) (contravariant components) and \(A_p\) (covariant components) are associated, with

\[ A_p=g_{pq}A^q\qquad\text{or}\qquad A^p=g^{pq}A_q. \]

In rectangular coordinates \(g_{pq}=\delta_{pq}\), so \(A_p=A^p\); this is why earlier chapters did not distinguish the two kinds of components.

Christoffel's Symbols

The symbols

\[ [pq,r]=\frac12\Bigl(\pd{g_{pr}}{x^q}+\pd{g_{qr}}{x^p}-\pd{g_{pq}}{x^r}\Bigr),\qquad \Chr s{pq}=g^{sr}[pq,r] \]

are the Christoffel symbols of the first and second kind. Other notations for \(\Chr s{pq}\) are \(\{pq,s\}\) and \(\Gamma^s_{pq}\); the latter wrongly suggests a tensor.

Transformation laws of Christoffel's symbols With bars denoting quantities in \(\bar x^k\), \begin{gather*} \overline{[jk,m]}=[pq,r]\pd{x^p}{\bar x^j}\pd{x^q}{\bar x^k}\pd{x^r}{\bar x^m} +g_{pq}\pd{x^p}{\bar x^m}\frac{\partial^2x^q}{\partial\bar x^j\partial\bar x^k},\\ \overline{\Chr n{jk}}=\Chr s{pq}\pd{\bar x^n}{x^s}\pd{x^p}{\bar x^j}\pd{x^q}{\bar x^k} +\pd{\bar x^n}{x^q}\frac{\partial^2x^q}{\partial\bar x^j\partial\bar x^k} \end{gather*} (Example 15.46). The Christoffel symbols are therefore not tensors unless the second terms vanish.

Length of a Vector, Angle between Vectors, Geodesics

The inner product \(A^pB_p\) is a scalar, the analogue of the scalar product. The length \(L\) of the vector \(A^p\) or \(A_p\) is defined by \[ L^2=A^pA_p=g^{pq}A_pA_q=g_{pq}A^pA^q. \] and the angle \(\theta\) between \(A^p\) and \(B^p\) by \[ \cos\theta=\frac{A^pB_p}{\sqrt{(A^pA_p)(B^qB_q)}}. \]

Geodesics The distance between the points \(t_1\) and \(t_2\) of a curve \(x^r=x^r(t)\) in a Riemannian space is \[ s=\int_{t_1}^{t_2}\sqrt{g_{pq}\od{x^p}t\od{x^q}t}\;\dd t. \] A curve that makes this distance a minimum (more generally, stationary) is a

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