PRIDE CHEKERA|퐴^ | = √(퐴푥)^2 +(퐴푦)^2 (Pythagorean theorem). This is simple as
it looks. Euclid extended this geometry into a three dimensional
flat space defined with three spatial components, x, y and z.
These three planes are orthogonal to one another. Any vector in
a vector field which resides in such a geometry is defined by
vector components, ⃗퐴⃗⃗⃗⃗푥^ , 퐴⃗⃗⃗⃗푦⃗^ and 퐴⃗⃗⃗⃗푧^. Each vector component is a
dot product of a component of a vector and the basis vector; e.g.
퐴⃗⃗⃗⃗⃗푥^ = 퐴푥.푒⃗⃗⃗푥. With this we can deduce the modulus of a vector 퐴^
in Euclid space as: 퐴^ = √(퐴푥)^2 +(퐴푦)^2 +(퐴푧)^2.
Now we can also extend this geometry into four space-time
dimensions. In such a geometry the arbitrary component of a
basis vector is represented by; 푒⃗⃗휇⃗ for μ = 1;2;3;4. The dot
product of two basis vectors gives a mathematical object
defining the metric of space-time. 푒⃗⃗⃗휇 .푒⃗⃗⃗푣 = 푔휇푣. A dot product of
vector components is: 퐴⃗⃗⃗⃗⃗푥^ .퐴⃗⃗⃗⃗⃗푦^ = 퐴푥⃗푒⃗⃗푥 .퐴푦푒⃗⃗⃗⃗푦 = ⃗푒⃗⃗푥 .푒⃗⃗⃗⃗푦 .퐴푥퐴푦 =
푔푥푦퐴푥퐴푦. Considering the latter expression, there is something
important we need to take note of. Whenever we have such an
expression with repeated indices will introduce Einstein’s
summation convention. These repeated indices as subscripts and
superscripts are called dummy indices; e.g. let consider the
following expressing: ∑푁푖 푎푖푗푏푖푟. In this expression i is the
dummy index, thus using Einstein summation we remove the
sigma notation and sum the expression 푎푖푗푏푖푟 over i to N