Relativity: Schwarzschild metric

1. Line element for an infinitesimal displacement
in the spherical coordinates system

1.1. Line element ds

If r has the components x, y and z in the Cartesian coordinates system, we can express then as follows:

x = r sinθsin φ
y = r sinθcos φ
z = r cosθ

Therefore

dx = dr sinθsinφ + r cos θ sinφ dθ + r sinθcosφdφ
dy = dr sinθcosφ + r cos θcosφ dθ - r sinθsinφdφ
dz = dr cos θ - r sin θ dθ

Hence

(ds)² = (dx)² + (dy)² + (dz)² =
(dr sinθsinφ)² + (r cos θ sinφ dθ)² + 2dr sinθsinφr cos θ sinφ dθ + (r sinθcosφdφ)² + 2r sinθcosφdφ(dr sinθsinφ + r cos θ sinφ dθ)
+ (dr sinθcosφ)² + (r cos θcosφ dθ)² + 2dr sinθcosφr cos θcosφ dθ+ (r sinθsinφdφ)² - 2r sinθsinφdφ(dr sinθcosφ + r cos θcosφ dθ) +
(dr cos θ)² + (r sin θ dθ)² - 2 dr cos θr sin θ dθ
= (dr)² + r²dθ² + r² sin²θ dφ²

ds² = dr² + r²dθ² + r² sin²θ dφ² =

dr² + r²dΩ

With

dΩ = dθ² + sin²θ dφ²

1.2. Line element ds in Schwarzschild Geometry

ds² = g_ttdt² - g_rrdr² - g_θθdθ² - g_φφdφ²

The components of the Schwarzschild element ds do not change by rotation, that is dΩ remains constant. Therefore:

g_θθ = r² and
g_φφ = r² sin²θ

We want then derive the expression of the components g_θθ and g_φφ.

2. Ricci tensor

The simple solution of the Einstein's field equations is the metric of Schwarzschild.
We derive this metric in a spherical coordinates system (t, r, θ, φ) with the following conditions:
The metric is spherically symmetric (isotropic). That is the components of the metric do not change with the rotations θ to - θ and φ to - φ
The space-time is static that is the components of the metric do not depend on time: ∂_tg_μν = 0
The sphere where gravity is responsible of the curvature of the space-time is not charged., outside this central sphere, there is just a vacuum.
Therefore the energy-momentum tensor T_μν is null, and taking the cosmological constant Λ = 0, we will then solve the zero Ricci tensor R_μν = 0.

R_μν = 0

Setting Ricci tensor R_μν = 0, will involves the Riemann tensor R_αβμν = 0. We will then deals with the Christoffel symbols that involve the metric, according to the following relationships:

R_αμ = g^βνR_αβμν

and

R^α_νμβ = Γ^α_νδΓ^δ_μβ - Γ^α_μδΓ^δ_νβ + ∂_νΓ^δ_μβ - ∂_μΓ^δ_νβ

with

Γ^k_ij = (1/2) g^kl [dg_il/∂x^j + dg_jl/∂xⁱ - dg_ij/∂x^l]

3.1. Ricci tensor: Component R_tt

R₀₀ = ∂_μΓ^μ₀₀ - ∂₀Γ^μ_0μ + Γ^μ₀₀Γ^ν_μν - Γ^ν_0μΓ^μ_ν0

Since ∂₀ = 0

R₀₀ = ∂_μΓ^μ₀₀ - Γ^μ₀₀Γ^ν_μν - Γ^ν_0μΓ^μ_ν0

μ = 0
R₀₀ = - Γ^ν₀₀Γ⁰_ν0
ν = 0 : 0
ν = 1 : - Γ¹₀₀Γ⁰₁₀
ν = 2 : 0
ν = 3 : 0


μ = 1
R₀₀ = ∂₁Γ¹₀₀ + Γ¹₀₀Γ^ν_1ν - Γ^ν₀₁Γ¹_ν0

ν = 0 : ∂₁Γ¹₀₀ + 0
ν = 1 : Γ¹₀₀Γ¹₁₁
ν = 2 : Γ¹₀₀Γ²₁₂
ν = 3 : Γ¹₀₀Γ³₁₃

μ = 2
R₀₀ = ∂₂Γ²₀₀ + Γ²₀₀Γ^ν_2ν - Γ^ν₀₂Γ²_ν0

ν = 0 : 0
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


μ = 3
R₀₀ = ∂₃Γ³₀₀ + Γ³₀₀Γ^ν_3ν - Γ^ν₀₃Γ³_ν0

ν = 0 : 0
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


Finally,

R₀₀ = - Γ¹₀₀Γ⁰₁₀ + ∂₁Γ¹₀₀ + Γ¹₀₀Γ¹₁₁ + Γ¹₀₀Γ²₁₂ + Γ¹₀₀Γ³₁₃

We have:
g₂₂ = r². Then ∂₁g₂₂ = 2r
g₃₃ = r²sin²θ. Then ∂₁g₃₃ = 2rsin²θ


Therefore

Γ¹₀₀ = ∂₁g₀₀/2g₁₁
Γ⁰₁₀ = ∂₁g₀₀/2g₀₀
Γ¹₁₁ = ∂₁g₁₁/2g₁₁
Γ²₁₂ = ∂₁g₂₂/2g₂₂ = 2r/2r² = 1/r
Γ³₁₃ = ∂₁g₃₃/2g₃₃ = 2rsin²θ/2r²sin²θ = 1/r

Hence:

R₀₀ = - [∂₁g₀₀]²/4g₀₀g₁₁ + ∂²₁g₀₀/2g₁₁ - ∂₁g₀₀∂₁g₁₁/4[g₁₁]² + ∂₁g₀₀/rg₁₁

R₀₀ = R_tt = - [∂₁g₀₀]²/4g₀₀g₁₁ + ∂²₁g₀₀/2g₁₁ - ∂₁g₀₀∂₁g₁₁/4[g₁₁]² + ∂₁g₀₀/rg₁₁

3.2. Ricci tensor: Component R_rr

R₁₁ = ∂_μΓ^μ₁₁ - ∂₁Γ^μ_1μ + Γ^μ₁₁Γ^ν_μν - Γ^ν_1μΓ^μ_ν1

R₁₁ = ∂_μΓ^μ₁₁ - ∂₁Γ^μ_1μ + Γ^μ₁₁Γ^ν_μν - Γ^ν_1μΓ^μ_ν1

μ = 0

R₁₁ = 0 - ∂₁Γ⁰₁₀ + Γ⁰₁₁Γ^ν_0ν - Γ^ν₁₀Γ⁰_ν1 = - ∂₁Γ⁰₁₀ + Γ⁰₁₁Γ^ν_0ν - Γ^ν₁₀Γ⁰_ν1 ν = 0 : - ∂₁Γ⁰₁₀ + 0 - (Γ⁰₁₀)² = - ∂₁Γ⁰₁₀ - (Γ⁰₁₀)²
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


μ = 1
R₁₁ = ∂₁Γ¹₁₁ - ∂₁Γ¹₁₁ + Γ¹₁₁Γ^ν_1ν - Γ^ν₁₁Γ¹_ν1 = Γ¹₁₁Γ^ν_1ν - Γ^ν₁₁Γ¹_ν1

ν = 0 : Γ¹₁₁Γ⁰₁₀
ν = 1 : 0
ν = 2 : Γ¹₁₁Γ²₁₂
ν = 3 : Γ¹₁₁Γ³₁₃

μ = 2
R₁₁ = ∂₂Γ²₁₁ - ∂₁Γ²₁₁ + Γ²₁₁Γ^ν_2ν - Γ^ν₁₂Γ²_ν1

ν = 0 : - ∂₁Γ²₁₂
ν = 1 : 0
ν = 2 : - Γ²₁₂Γ²₂₁ = - [Γ²₁₂]²
ν = 3 : 0


μ = 3
R₁₁ = ∂₃Γ³₁₁ - ∂₁Γ³₁₃ + Γ³₁₁Γ^ν_3ν - Γ^ν₁₃Γ³_ν1

ν = 0 : - ∂₁Γ³₁₃
ν = 1 : 0
ν = 2 : 0
ν = 3 : - [Γ³₁₃]²


Finally,

R₁₁ = - ∂₁Γ⁰₁₀ - (Γ⁰₁₀)² + Γ¹₁₁Γ⁰₁₀ + Γ¹₁₁Γ²₁₂ + Γ¹₁₁Γ³₁₃ + - ∂₁Γ²₁₂ - (Γ²₁₂)² - ∂₁Γ³₁₃ - (Γ³₁₃)²

We have:
g₂₂ = r². Then ∂₁g₂₂ = 2r
g₃₃ = r²sin²θ. Then ∂₁g₃₃ = 2rsin²θ


Therefore

Γ⁰₁₀ = ∂₁g₀₀/2g₀₀
Γ¹₁₁ = ∂₁g₁₁/2g₁₁
Γ²₁₂ = ∂₁g₂₂/2g₂₂ = 2r/2r² = 1/r
Γ³₁₃ = ∂₁g₃₃/2g₃₃ = 2rsin²θ/2r²sin²θ = 1/r

Hence:

R₁₁ = - ∂₁²g₀₀/2g₀₀ + [(∂₁g₀₀/2g₀₀]² + ∂₁g₁₁ ∂₁g₀₀/4g₀₀g₁₁ + ∂₁g₁₁/g₁₁r

R₁₁ = R_rr - ∂₁²g₀₀/2g₀₀ + [(∂₁g₀₀/2g₀₀]² + ∂₁g₁₁ ∂₁g₀₀/4g₀₀g₁₁ + ∂₁g₁₁/g₁₁r

3.3. Ricci tensor: Component R_θθ

R₂₂ = ∂_μΓ^μ₂₂ - ∂₂Γ^μ_2μ + Γ^μ₂₂Γ^ν_μν - Γ^ν_2μΓ^μ_ν2

R₂₂ = ∂_μΓ^μ₂₂ - ∂₂Γ^μ_2μ + Γ^μ₂₂Γ^ν_μν - Γ^ν_2μΓ^μ_ν2

μ = 0

ν = 0 : 0
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


μ = 1

ν = 0 : ∂₁Γ¹₂₂ + Γ¹₂₂Γ⁰₁₀
ν = 1 : Γ¹₂₂Γ¹₁₁
ν = 2 : Γ¹₂₂Γ²₁₂
ν = 3 : Γ¹₂₂Γ³₁₃

μ = 2
ν = 0 : 0
ν = 0 : - Γ¹₂₂Γ²₁₂
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


μ = 3
ν = 0 : 0
ν = 1 : 0
ν = 2 : 0
ν = 3 : - Γ³₂₃Γ³₃₂


Finally,

R₂₂ = ∂₁Γ¹₂₂ + Γ¹₂₂Γ⁰₁₀ + Γ¹₂₂Γ¹₁₁ + Γ¹₂₂Γ²₁₂ + Γ¹₂₂Γ³₁₃ - Γ¹₂₂Γ²₁₂ - Γ³₂₃Γ³₃₂

R₂₂ = R_θθ = - 1 + 1/g₁₁ - r∂₁g₁₁/2g₁₁² + r∂₁g₀₀/2g₀₀g₁₁

3.4. Ricci tensor: Component R_φφ

R₃₃ = ∂_μΓ^μ₃₃ - ∂₃Γ^μ_3μ + Γ^μ₃₃Γ^ν_μν - Γ^ν_3μΓ^μ_ν3

Since the components are independent of φ, we have:
∂₃Γ^μ_3μ = 0

R₃₃ = ∂_μΓ^μ₃₃ - Γ^μ₃₃Γ^ν_μν - Γ^ν_3μΓ^μ_ν3

μ = 0

ν = 0 : 0
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


μ = 1

ν = 0 : ∂₁Γ¹₃₃ + Γ¹₃₃Γ⁰₁₀
ν = 1 : Γ¹₃₃Γ¹₁₁
ν = 2 : Γ¹₃₃Γ²₁₂
ν = 3 : 0

μ = 2
ν = 0 : ∂₂Γ²₃₃
ν = 0 : 0
ν = 1 : 0
ν = 2 : 0
ν = 3 : 0


μ = 3
ν = 0 : 0
ν = 1 : - Γ¹₃₃Γ³₁₃
ν = 2 : - Γ²₃₃Γ³₂₃
ν = 3 : 0


Finally,

R₃₃ = ∂₁Γ¹₃₃ + Γ¹₃₃[Γ⁰₁₀ + Γ¹₁₁ + Γ²₁₂] + ∂₂Γ²₃₃
- Γ¹₃₃Γ³₁₃ - Γ²₃₃Γ³₂₃

R₁₁ = R_φφ = sin²θ R₂₂

In the special case we consider, that is with all the approximations made, we will have: R_tt = R_rr = R_θθ = R_φφ = 0

4. The Schwarzschild metric

We want to determine the expressions of the components g₀₀ and g₁₁.

We will use:

R₀₀ = R_tt = 0

∂₁²g₀₀ = 2g₁₁{ + [∂₁g₀₀]²/4g₀₀g₁₁ + ∂₁g₀₀∂₁g₁₁/4[g₁₁]² - ∂₁g₀₀/rg₁₁ }

R₁₁ = R_rr = 0

Eliminating ∂₁²g₀₀ between R₀₀ and R₁₁ leads to:

∂₁²g₀₀ = 2g₀₀ { + [(∂₁g₀₀/2g₀₀]² + ∂₁g₁₁ ∂₁g₀₀/4g₀₀g₁₁ + ∂₁g₁₁/g₁₁r}

Simplifying gives:

- ∂₁g₀₀/r = ∂₁g₁₁ g₀₀/g₁₁r

That is

g₁₁∂₁g₀₀ + g₀₀∂₁g₁₁ = 0

This equation can be written as:

∂₁(g₀₀g₁₁) = 0
That is
g₀₀g₁₁ = constant

Now, far from the sphere of gravity, that is when r tends toward infinity, the metric of Schwarzschild will coincide with the metric of Minkowski, which is ds² = dt² - dx² - dy² -dz². Therefore g₀₀ will tends toward 1 and g₁₁ will also tends toward 1.

Let's then write:

lim g₀₀ = 1
r → ∞
lim g₁₁ = 1
r → ∞

g₀₀g₁₁ = 1

Using the equation:
g₁₁∂₁g₀₀ + g₀₀∂₁g₁₁ = 0 in the expression of R₂₂, we find the equation of Schwarzschild:

R₂₂ = R_θθ = 0
- 1 + 1/g₁₁ - r∂₁g₁₁/2g₁₁² + r (- ∂₁g₁₁/g₁₁) /2g₁₁ = 0

That leads to:

- 1 + 1/g₁₁ - r ∂₁g₁₁/2g₁₁² + - r ∂₁g₁₁/2g₁₁² = 0

or
r ∂₁g₁₁ = - g₁₁² + g₁₁

r ∂₁g₁₁ = g₁₁(1 - g₁₁)

Integrating this equation gives:

∂g₁₁/[g₁₁(1 - g₁₁)] = ∂r /r

1/[g₁₁(1 - g₁₁)] can be decomposed in simple elements as:

1/[g₁₁(1 - g₁₁)] = 1/g₁₁ + 1/(1 - g₁₁)

Therefore

∫ ∂r /r = ∫ ∂g₁₁/[g₁₁(1 - g₁₁)] = ∫ ∂g₁₁/g₁₁ - ∫ ∂g₁₁/(g₁₁ - 1)

That is

ln (r) = ln(g₁₁ ) - ln(g₁₁ - 1) =
ln(g₁₁/(g₁₁ - 1)) + Co

Then

ln(g₁₁/(g₁₁ - 1)) = ln(r) + C
g₁₁/(g₁₁ - 1) = r exp{Co} = r Co
Co and C are constant.

Hence

g₁₁(r) = 1/(1 - 1/rC) = (1 - 1/rC)^-1

Since g₀₀g₁₁ = 1, we have
g₀₀ = (1 - 1/rC)

g₀₀(r) = (1 - 1/rC)
g₁₁(r) = (1 - 1/rC)^-1

The Schwarzschild metric is then written as

ds² = (1 - 1/rC)dt² - (1 - 1/rC)^-1dr²

- r²(dθ² + sin²θdφ²)

5. Expression of the Schwarzschild metric

Using the weak-field limit approximation, we have found in th lecture Newtonian limit:

g₀₀ = 1 + 2Φ = 1 + 2(Φ/c²)

where
Φ = - GM/r is the Newtonian gravitational potential.

Hence the component (1 - 1/rC) in the Scwarzschild metric becomes:
(1 - 1/rC) = g₀₀ = 1 + 2Φ = 1 + 2(Φ/c²) = 1 - 2(GM/rc²).

That is:

1/C = 2(GM/c²)

We have then the expression of the constant:

C = c²/2GM

Let 1/C = r_s the Scwwarzschild radius, called the horizon of events, so

r_s = 2GM/c²

The Schwarzschild metric take then the following form:

ds² = (1 - r_s /r)dt² - (1 - r_s /r)^-1dr²

- r²(dθ² + sin²θdφ²)

5. The Schwarzschild metric with the
cosmological constant Λ

If the cosmological constant Λ is not zero, the Einstein's equation, in the vacuum (T_μν = 0), becomes:

R_μν = Λ g_μν


We have found the following Schwarzschild equation by setting
R_θθ = R₂₂ = 0

r ∂₁g₁₁ = g₁₁(1 - g₁₁)

Having g₂₂ = g_θθ = r², the above equation becomes:

r ∂₁g₁₁ = g₁₁(1 - g₁₁) + r²Λ

r ∂₁g₁₁ = g₁₁(1 - g₁₁) + Λr²

The Schwarzschild equation :
r ∂₁g₁₁ = g₁₁(1 - g₁₁)
can be written:

r ∂₁g₁₁/g₁₁² = (1/g₁₁) - 1

Therefore, adding the cosmological constant contribution, we have:

r ∂₁g₁₁/g₁₁² = (1/g₁₁) - 1 + Λr²

Let's remark that
- d(1/g₁₁)/dr = [dg₁₁/dr]/g₁₁²
So the above equation becomes:

- r d(1/g₁₁)/dr = (1/g₁₁) - 1 + Λr²
Or
r d(1/g₁₁)/dr + (1/g₁₁) = 1 - Λr²

The homogeneous equation is written as:

r d(1/g₁₁)/dr + (1/g₁₁) = 0

Let's denote 1/g₁₁ by f(r), so
r f' + f = 0
or
f'/f = - 1/r

Integration gives:
ln(f) = ln(1/r) + C
C is a constant.

That is
f = K . (1/r)
K is a constant.

Using the varying constant method yields:

f' = K'/r - K/r²

Pluging this in the general equation with the cosmological constant Λ :
r f' + f = 1 - Λr²

That gives:

K' - K/r + K/r = 1 - Λr²
That is
K' = 1 - Λr²

Integrating gives
K = r - (1/3)Λr³ + A
A is a constant.

Therefore

f = K/r = 1 - (1/3)Λr² + A/r

The expression of g₁₁ becomes:

f = 1/g₁₁ = 1 - (1/3)Λr² + A/r
Or

g₁₁ = [1 - (1/3)Λr² + A/r]^-1

We already know the expression of the second component of the Scwarzschild metric g₁₁.

g₁₁ = (1 - r_s /r)^-1

Hence:

1 - r_s /r = 1 - (1/3)Λr² + A/r
With Λ = 0, we have:

A = - r_s

Therefore:

g₁₁ = [1 - (1/3)Λr² - r_s/r]^-1

g₁₁ = [1 - (1/3)Λr² - r_s/r]^-1

The expression of the Scwarzschild metric with a cosmological constant is:

ds² = [1 - (Λ/3)r² - r_s/r] dt² - [1 - (Λ/3)r² - r_s/r]^-1 dr² - r²(dθ² + sin²θdφ²)