Relativity: Metrics & tangent vectors

2. Basis g_μ of tangent vectors

x = (x⁰, x¹, x², x³)
y = (y⁰, y¹, y², y³)

dx = (dx⁰, dx¹, dx², dx³)
dy = (dy⁰, dy¹, dy², dy³)

dy⁰ = (∂y⁰ /∂x⁰)dx⁰ + (∂y⁰ /∂x¹) dx¹
+ (∂y⁰ /∂x²)dx² + (∂y⁰ /∂x³)dx³ = (∂y⁰ /∂x^ν)dx^ν
ν = 0. 1. 2. 3.

Similarly
dy¹ = (∂y¹ /∂x^ν)dx^ν
...
dy^μ = (∂y^μ /∂x^ν)dx^ν

vector g⁰ = ∂/∂⁰ (vector dy)
vector g¹ = ∂/∂¹ (vector dy)
...
vector g^μ = ∂/∂^μ (vector dy)

or
vector g^μ = ∂dy^m /∂^μ
dy^m are all the components of the vecto dy.

Now
The scalar product of the two vectors g^μ and g^ν is:

vector g^μ . vector g^ν =
∂dy^m /∂^μ ∂dyⁿ /∂^ν

Therefore:
ds² = dy dy =
dy^m dyⁿ = g^μ dx^μ g^ν dx^ν =
g^μ . g^ν dx^μ dx^ν

ds² = g^μ . g^ν dx^μ dx^ν

We have already defined the squared line element ds² as:
if
dy^m = (∂y^m /∂x^μ)dx^μ, and dyⁿ = (∂yⁿ /∂x^ν)dx^ν
then ds² is defined as:

ds² = η dy^m dyⁿ

ds² = η dy^m dyⁿ = η_mn ∂y^m /∂x^μ)dx^μ (∂yⁿ /∂x^ν)dx^ν =

η_mn (∂y^m /∂x^μ) (∂yⁿ /∂x^ν) dx^μdx^ν

with
g_μν = η_mn (∂y^m /∂x^μ) (∂yⁿ /∂x^ν)

g_μν = η_mn (∂y^m /∂x^μ) (∂yⁿ /∂x^ν)

Therefore
ds² = g_μν dx^μdx^ν

Equating gives:

ds² = g^μ . g^ν dx^μ dx^ν = g_μν dx^μdx^ν

That is:

g_μν = g^μ . g^ν

g_μν = g^μ . g^ν

2. Raising & lowering indeces

Lorentz Transformations give:

y⁰ = γ(x⁰ - βx¹ )
y¹ = γ(x¹ - β x⁰ )
y² = x²
y³ = x³


x⁰ = γ(y⁰ + βy^{1/sup> )
x1 = γ(y1 + β y0 )
x2 = y2
x3 = y3}

Let's write the vector A with its contravaiant components:
A^μ = (y⁰,y¹, y², y³) =
(∂y^m/∂x^μ) x^μ

With its covaiant components, we have:
A_μ = (y₀,y₁, y₂, y₃) =
(∂x^m/∂y^μ) x_μ

In the Lorentz transforms: the (∂x^m/∂y^μ) factors give the same result as the contravariant factors
(∂y^m/∂x^μ), except - β changes int + β. Therefore:

y₀ = γ(x₀ + βx₁ )
y₁ = γ(x₁ + β x₀ )
y₂ = x₂
y₃ = x₃

What is the relationship between A^μ and A_μ?

Changing x₀ into - x⁰, and letting the others the same, we can write the above transformations as:

y₀ = - γ( x⁰ - βx¹ ) = - y⁰
y₁ = γ(x¹ - β x⁰ )= y¹
y₂ = x² = y²
y₃ = x³ = y³

Therefore:

A_μ = g_μν A^μ with
x_μ = g_μν x^μ

g_μν is the the following tensor metric:



A_μ = g_μν A^μ with
x_μ = g_μν x^μ

Raising indices:
A_μ = g_μν A^ν with x_μ = g_μν x^ν

We can remark g_μν = g_μν^-1
g_μν^-1 is the inverse of g_μν . We write g_μν^-1 = g^νμ

g_μν^-1 = g^μν

so
g_μν = g_νμ

g_μν = g_νμ

Lowering indices:
A^μ = g^μν A_ν with x^μ = g^μν x_ν