Relativity: Metric tensor

1. Product of two vectors

We have seen the components of a contravariant transformed vector A^μ(y) of a vector A^μ(x) is;

A^μ(y) = (∂y^μ/∂x^ν) A^ν(x)

and the components of a covariant transformed vector A^μ(x) of a vector A^μ(y) is:

A_ν(x) = (∂x_ν/∂y_μ) A_μ(y)

Now, let's take two contravariant vectors A^m(x) and Bⁿ(x). Their product is the tensor formed by two vectors :

T^mn(x) = A^m(x) Bⁿ(x)

The transform of T^m(x) is T^m(y) which can be written as the product of the two transforms A^m(y) and Bⁿ(y) of the two vectors A^m(x) and Bⁿ(x):

T^mn(y) = A^m(y) Bⁿ(y)

We have:
A^m(y) = ∂y^m/∂x^r A^r(x)
Bⁿ(y) = ∂yⁿ/∂x^s B^s(x)

So

T^mn(y) = A^m(y) Bⁿ(y) = = ∂y^m/∂x^r A^r(x) ∂y^y/∂x^s B^s(x)

= ∂y^m/∂x^r ∂yⁿ/∂x^s A^r(x) B^s(x)

= ∂y^m/∂x^r ∂yⁿ/∂x^s T^rs(x)

T^mn(y) = (∂y^m/∂x^r)(∂yⁿ/∂x^s) T^rs(x)

Similarly, the tensor formed by two covariant vectors A_m (y) and B_n(y) transformed in A_m(x) andB_m(x) is:

T_mn(x) = (∂x_m/∂y_r)(∂x_n/∂y_s) T_rs(y)

2. Metric tensor

We denote the little displacement dA^m(x) of the contravariant vector A^m(x) = (x⁰, x¹, x², ... x^m) by ds, that we square to obtain:

(ds)² = [A^m(x)]<sup>2/sup> = (x0)2 + (x1)2 + (x2)2
+ ... + (xm)2 = Σ (dxm)2 = (dxm)2 =
dxm dxnδ_mn(x)

δ_mn(x) is the Kronecker m n in the coordinate reference frame "x".

So

(ds)2 = δmn(x) dxm dxn

We have:

dxm = ∂xm/∂yr dyr
and
dxn = ∂xn/∂ys dys

dyμ are the components of the vector Am(y) in the coordinate reference frame "y" whish is the transformed of Am(x) in the coordinate reference frame "x".

So
dxm dxn = ∂xm/∂yr dyr ∂xn/∂ys dys =
∂xm/∂yr ∂xn/∂ys dyr dys

Hence:
(ds)2 = δmn(x) ∂xm/∂yr ∂xn/∂ys dyr dys

The part δmn(x) ∂xm/∂yr ∂xn/∂ys is nenoted by g_rs(y) and called metric tensor .

and

(ds)2 = g_rs(y) dyr dys

The tensor g_rs(y) is the transform of
the tensor δmn(x) in the cartesian coordinate reference frame .

Indeed, we can write with a covariant transform:
g_rs(y) = ∂xm/∂yr ∂xn/∂ys δ_mn (x)

Metric tensor:
g_rs(y) = δ_mn(x) ∂m/∂yr ∂n/∂ys

(ds)2 = g_rs(y) dyr dys

The tensor g_rs(y) is the transform of the tensor δ_mn(x) (which is a tensor in the cartesian coordinate reference frame</sup>