Relativity: Einstein's equations

1. Energy conservation equation /h3>

We already have:

G^μν = 8 πGT^μν

Einstein tried to link the Energy momentum tensor T^μν to the curvature of space. Therefore, the left hand G^μν of the above equation must contain a tensor curvature. This tensor could not be the one of Riemann because it contains three indexes. The right one is then the one of Ricci.

The first attempt was to set R^μν = K T^μν, where K is a constant. But the conservation of the tensor Energy written as:

∂_μT^μν = 0

or better, with the covariant derivative: ∇_μT^μν = 0
will imply ∇_μR^μν = 0 which is not true.

In fact

∇_μ R^μν = (1/2) g^μν ∂_μR

R is the curvature scalar, so we can replace the ordinary derivative ∂_μR by a covariant derivative ∇_μR. Therefore:

∇_μ R^μν = (1/2) g^μν ∇_μR

We have:
∇_μ (g^μν R) = R ∇_μ g^μν + g^μν∇_μR =

Since
∇_μ g^μν = 0

Then ∇_μ (g^μν R) = g^μν∇_μR

Hence ∇_μ R^μν = (1/2) ∇_μ(g^μν R)

That is: ∇_μ[R^μν - (1/2)(g^μν R)] = 0

We set

G^μν = R^μν - (1/2)(g^μν R)

Called Einstein's tensor,

and write: ∇_μG^μν = 0

Energy conservation Equation
∇_μG^μν = 0

We have then

R^μν - (1/2)(g^μν R) = K T^μν
Equation that expresses the conservation of energy.

In vacuum, where there is no matter,
Einstein's equation is G^μν = 0, because T^μν = 0.

So
R^μν - (1/2)(g^μν R) = 0

Contracting with the metric g_μν gives:
g_μν [R^μν - (1/2)(g^μν R)] = 0

That is:
R - (1/2)(g_μνg^μν R) = 0
g_μνg^μν R = δ_μ^ν R = δ_μ^μ R = (1 + 1 + 1 + 1)R = 4 R

Therefore:
R - 2R = - R = 0
That is R = 0

Where there is no matter, there is no curvature.

2. Einstein Equation

Let's solve for the constant K in the equation:
R^μν - (1/2)(g^μν R) = K T^μν

Multiplying by the metric g_μν gives:
g_μν[R^μν - (1/2)(g^μν R)] = K g_μνT^μν

We obtain:
- R = K T^μ_ν = K T

Therefore:
R = - K T, and
R^μν + (1/2)(g^μν K T) = K T^μν
or
R^μν = [T^μν - (1/2)(g^μν T] K

Recall:

G_μν = 8 πGT_μν

Using the Eienstein tensor:

G^μν = R^μν - (1/2)(g^μν R)

We have then the value of the constant K
K = 8 πG

Therefore:

G^μν = R^μν - (1/2)(g^μν R) = 8 πGT^μν

G^μν = R^μν - (1/2)(g^μνR) = 8 πGT^μν

3. Complete Einstein Equation

The Energy conservation Equation :

∇_μG^μν = 0

Is integrated as:
G^μν = constant.

That is :
R^μν - (1/2)(g^μνR) = 8 πGT^μν + Const.

This equation can be written as:
R^μν - (1/2)(g^μνR) + Const = 8 πGT^μν

The constant Const is set as equal to Λg^μν. Λ is called
the cosmological constant

Therefore the complete Einstein's equation is:

G^μν + g^μνΛ = R^μν - (1/2)(g^μνR) + g^μνΛ = 8 πGT^μν

Einstein's Equation:

R^μν - (1/2)(g^μνR) + Λg^μν = 8 πGT^μν


The cosmological constant Λ can be set as follows:

In free space without graviation, T^μν = 0

Contracting :
R^μν - (1/2)(g^μνR) + Λg^μν = 8 πGT^μν
with the metric g_μν gives:
g_μνR^μν - (1/2)(g_μνg^μνR) + Λg_μνg^μν = 0

That is:
R - 2 R + 4 Λ = 0

Therefore
R - 2 R + 4 Λ = 0
Λ = R/4

Λ = R/4

Einstein's Equation takes the form:

R^μν - (1/4)(g^μνR) = 8 πGT^μν