Relativity, decays and electromagnetic fields

Euclidean space

Vectors and metrics

Vector

A vector in the three-dimensional euclidean space can be written, in terms of its components, as



where the superscript distinguishes the coordinates and it does not represent an exponentiation. We have chosen to distinguish the components of a vector in euclidean space with superscript rather than subscript to immediately introduce the formalism of special relativity. In fact, when we talk about four-vector (four-component generalization of a vector) we will see that we must distinguish components with index at the top (superscript) and components with index at the bottom (subscript).

So we can write the i-th component as



Scalar product

The scalar product between two vectors is written as



or



The scalar product of a vector with itself is





and corresponds to the squared norm of the vector.

Metric

The formula for the scalar product of a vector with itself



can be written also as a combination



with constant coefficients equal to 1, i.e.



A generalization of the latter formula is



with



in fact with this choice the terms which include mixed products of components that do not appear in the scalar product are null.

The coefficients of the combination of the coordinates of the vector can be interpreted as the elements of a diagonal matrix (elements that are not null only if the indices are equal). This 3-dimensional matrix in the euclidean space takes the form



and we can write the square of a vector as a product between vectors and matrix in the following way