Skip to main content

Mathematical Methods of Physics: For Undergraduate

Sunil Karna

Chapter 3 Tensor Analysis

Tensor is a convenient tool to establish relativistic invariance among the laws of nature. It was introduced by G. Ricci and it is used by Einstein to describe a general theory of relativity. Tensor is an entity which is invariant in coordinate transformation. It is a mathematical object that represents a physical quantity which is independent of the choice of coordinate system. Tensors are used to describe many physical phenomena, including the behavior of fluids and gases, the motion of objects under gravity, and the behavior of electromagnetic fields.
Tensors are a natural and logical generalization of vectors. A scalar is simply a number (positive, negative, and zero). The magnitude of a scalar is independent of the reference system if the system of unit is fixed, e.g., the temperature (scalars) at a point in space is fixed irrespective of the reference system but obviously, \(0^{o}C=32^{o}F.\) The magnitude of a vector remains constant irrespective of the frames of reference if the system of unit is fixed. However, the components of a vector depend upon both the frame of reference and system of units. If the system of units is changed the magnitude of a vector also changes, e.g., the magnitude and the direction of acceleration due to gravity at a point in space is fixed irrespective of the reference system but \(g = 9.8m/s^{2} = 32 ft/s^{2}\) (no change in direction). In vector algebra, the sums and products of two vectors can be defined very clearly but the quotient of two vectors has no meaning. Tensors arise physically in this situations. For example, consider a stress system which is defined as a force per unit area, i.e., \(stress = \frac{\vec{F}}{\vec{A}}\text{.}\) However, the division by a vector is undefined; hence in a stress system, force can be defined as a product of a new entity T with a vector area \(A\text{,}\) i.e., a force \(F =A\cdot T\text{,}\) where \(T\) is a tensor associated with at least two directions. Similarly, \(\vec{a} = \frac{\vec{F}}{m}\) and \(\vec{j} = \sigma \vec{E}\) but these relations hold only on isotropic media. In an anisotropic medium, the acceleration, \(\vec{a}\) is not necessarily parallel to the applied force, and the current flows in a direction different from that of electric field. In such a sutuation the above equation can be generalized as -
\begin{equation*} j_{x} = \sigma_{xx} E_{x} + \sigma_{xy} E_{y} + \sigma_{xz} E_{z}, \end{equation*}
\begin{equation*} j_{y} = \sigma_{yx} E_{x} + \sigma_{yy} E_{y} + \sigma_{yz} E_{z}, \end{equation*}
\begin{equation*} j_{z} = \sigma_{zx} E_{x} + \sigma_{zy} E_{y} + \sigma_{zz} E_{z} \end{equation*}
where \(\sigma_{ij} = (i,j = x,y,z)\) are said to be the components of conductivity tensors of the medium. Similar expression can be obtained for mass (or reciprocal mass) tensors of the particle in the medium. The laws of physics which express the nature are independent upon the choice of reference frame. A study of the transformation of a physical quantity from one frame to another leads to the tensor analysis.
Tensors are characterized by their rank, which is the number of indices required to specify all of their components. A scalar quantity, such as temperature, is a tensor of rank zero. A vector, such as velocity or force, is a tensor of rank one. A tensor of rank two, such as the stress tensor or the electromagnetic field tensor, requires two indices to specify all of its components. Tensors are important in relativity theory, where the curvature of spacetime is described by a tensor called the Riemann tensor. In quantum field theory, the properties of particles and fields are described by tensors, such as the spin tensor and the polarization tensor.
Let us consider a group of students who are measuring the maximum height attained by a projectile or an electric field at a point produced by a charge in a laboratory with their own sets of instrument. Each of them has their own frames of reference orientated in different directions independent of one another. Now if one measures the maximum height of the projectile as 2 meters would it be possible that the others may find it to be 5m, 1m, 10m etc? or if one finds the electric field at a certain point to be 40 N/C and directed eastward, is it likely that another would find it to be 30 N/C and directed southward, and the third would measure it to be 60 N/C and directed upward? Finally, would the height of the projectile or the value of electric field change for individuals just because each of them has distinct coordinate systems? The answer is no because we know that 5 ft person is 5 ft long and it does not matter who and how someone measures it. Thus it is clear that the physical phenomena appear to be the same for all observers stationary relative to each other, independent of the position of each observer, and of the orientation of their frames of refernce.
The physical quantites which possess such invarience properties can only be taken into account in the formulation of physical laws. These quantities are scalars, vectors, and in general, tensors. The tensor is useful for describing the properties of substances which vary in directions. It must be noted that a tensor is an entity which is independent of the system of coordinates to which its components are referred. It is only its components which undergo transformation from one coordinate system to another. It is not possible to represent tensors as any geometrical pictures like scalars or vectors, hence tensors can be introduced only through their properties of transformation under coordinate transformations.
Let \((x^{1}{,} x^{2}{,} x^{3}{,} \cdots{,} x^{n})\) and \((\bar{x}^{1}{,} \bar{x}^{2}{,} \bar{x}^{3}{,} \cdots{,} \bar{x}^{n})\) be coordinates of a point in two different frames of reference. Suppose there exists \(n\) independent relations between the coordinates of the two systems as
\begin{equation*} \bar{x}^{1} = \bar{x}^{1}(x^{1}, x^{2}, x^{3}, \cdots, x^{n}), \end{equation*}
\begin{equation*} \bar{x}^{2} = \bar{x}^{2}(x^{1}, x^{2}, x^{3}, \cdots, x^{n}), \end{equation*}
\begin{equation*} \vdots \hspace{2cm} \vdots \end{equation*}
\begin{equation*} \bar{x}^{n} = \bar{x}^{n}(x^{1}, x^{2}, x^{3}, \cdots, x^{n}) \end{equation*}
\begin{align} \therefore \quad \bar{x}^{i} \amp = \bar{x}^{i}(x^{1}, x^{2}, x^{3}, \cdots, x^{n}) \tag{3.0.1}\\ \text{also,}\quad {x}^{i} \amp = {x}^{i}(\bar{x}^{1}, \bar{x}^{2}, \bar{x}^{3}, \cdots, \bar{x}^{n}), \tag{3.0.2} \end{align}
where \(i = 1,2,3, \cdots n\text{.}\)
The eqns. (3.0.1) and (3.0.2) define a transformation of coordinates from one frame to another. Differentiating eqn. (3.0.1), we get -
\begin{equation*} \,d\bar{x}^{i} = \frac{\partial \bar{x}^{i} }{\partial x^{1}}\,dx^{1} + \frac{\partial \bar{x}^{i} }{\partial x^{2}}\,dx^{2} +\cdots +\frac{\partial \bar{x}^{i} }{\partial x^{n}}\,dx^{n} = \sum\limits_{j=1}^{n}\frac{\partial \bar{x}^{i} }{\partial x^{j}}\,dx^{j} \end{equation*}
or, simply,
\begin{equation} \,d\bar{x}^{i} = \frac{\partial \bar{x}^{i} }{\partial x^{j}}\,dx^{j}\tag{3.0.3} \end{equation}
Eqn. (3.0.3) is a coordinate transformation equation.
Notations: A collection of indices (subscripts and superscripts) is used to make the mathematical development of tensor analysis. The superscripts such as \(T^{ij\cdots}\) is a contravarient tensor, the subscripts such as \(T_{ij\cdots}\) is a covarient tensor, and \(T_{l m n\cdots}^{ij\cdots}\) is a mixed tensor.
PrevTopNext