1.1 Preliminary Remarks on Vectors 7
1.1.4 Vectors for Special Relativity.
Vectors with four components are used in special relativity theory. The basic vector is
composed of the three components of the position vector; the fourth one is the time,
multiplied by the speed of light. The components of this 4-vectorchange according
to theLorentz transformationwhen the original coordinate system is replaced by a
coordinate system moving with constant velocity with respect to the original one. The
time is also changed in this transformation. Physical quantities with four components,
which transform with the same rule, are referred to asLorentz vectors. Properties
of Lorentz vectors and their application in physics, in particular in electrodynamics,
are discussed in the last chapter of this book.
1.2 Preliminary Remarks on Tensors.
For the first time, students hear about tensors in connection with themoment of inertia
tensorlinking the rotational angular momentum with the rotational velocity of a
rotating solid body. In such a linear relation, two vectors are not just parallel to each
other. Tensors also describe certain orientational dependencies in anisotropic media.
Examples are electric and magnetic susceptibility tensors, mobility and diffusion
tensors. To be more precise, these are tensors of rank 2. Vectors are also referred to
as tenors of rank 1. There are second rank tensors which are physical variables of
their own, like the stress tensor and the strain tensor. The linear relation between two
second rank tensors is described by a tensor of rank 4. An example is the elasticity
tensor linking the stress tensor with the strain tensor. Tensors of different ranks are
used to characterize orientational distributions.
Definitions, properties and applications of tensors represented by their compo-
nents in a 3D coordinate system are discussed in detail in the following sections.
Here just a brief, preliminary answer is given to the question:What is a tensor?
The rule which links the components of the position vector in a rotated coordinate
system with the components of the original one involves a transformation matrix,
referred to asrotation matrix. The product ofcomponents of the position vector
needs the product ofrotation matrices for their interrelation between the rotated
system and the original one.
A tensor of rankis a quantity whose components are transformed upon a rotation
of the coordinate system with the-fold product of the rotation matrix.
In this sense, scalars and vectors are tensors of rank=0 and=1. Often,
tensors of rank=2 are just referred to as tensors. Properties and applications of
second rank tensors, as well as of higher rank tensors, are discussed in the following
sections.