PRIDE CHEKERA(푑푥^1 )^2 + (푑푥^2 )^2
Remember: 푔휇푣 = { 0, for 휇 ≠푣; 1, for 휇=푣
The object 푔휇푣 defines the metric of space-time and in the world
of manifolds it is called the metric tensor, a dot product of basis
vectors of vectors, 푒⃗⃗⃗휇⃗⃗. 푒⃗⃗⃗푣 = 푔휇푣. This is the covariant form (in
covariant transformations basis vectors act only on ordinary
vectors) of the metric tensor. Its contravariant form (in
contravariant transformations basis vectors act only on
convectors) is 푒⃗⃗⃗휇⃗^ .푒⃗⃗⃗⃗푣^ = 푔휇푣 and mixed form (in mixed form basis
vectors act on both convectors and ordinary vectors) is
푒⃗⃗⃗휇 .⃗푒⃗⃗⃗푣^ = 푔휇푣. The dot product of basis vectors can also be defined
in terms of a mathematical objects known as the Kronecker
delta, 훿푖푗. We can write: 푒⃗⃗⃗휇⃗⃗ .⃗푒⃗⃗⃗푣^ = 훿휇푣. Like the metric tensor, the
Kronecker delta is defined by: 훿휇푣 = {1; for v = μ and 0; for v
≠ 휇. The Kronecker delta is also significant for exchanging
indices on tensors, e.g. 훿푖푗푎푖푟 = 푎푗푟.
Note that, tensors of single index are of order one or rank one,
e.g. 푎푖 and those with two indices are of order two or rank two,
e.g. 푎푖푗.
The metric is very important in the sense that it symmetrical, in
physics terms, when time arrow is reversed the metric remains
invariance; this means laws of physics remain the same even if
we reverse time arrow back into the past. I have dwelled on this