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Derivation of the Schwarzschild Metric

The coordinates used are (t,r,θ,ϕ)(t, r, \theta, \phi), where θ\theta is the colatitude and ϕ\phi is the longitude. The rr coordinate will be explained in the next section.

The general Einstein field equations are given by:

Rμν−12Rgμν+Λgμν=8πTμν.R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = 8 \pi T_{\mu\nu}.

The metric is static - independent of the time coordinate:

gμν,t=0.g_{\mu\nu,t} = 0.

The solution is vacuum (Tμν=0T_{\mu\nu} = 0) and with zero cosmological constant:

Rμν−12Rgμν=0,Rμνgμν−12Rgμνgμν=0,R−12Rδμμ=0,R−2R=0,R=0, \begin{align*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 0, \\ R_{\mu\nu} g^{\mu\nu} - \frac{1}{2} R g_{\mu\nu} g^{\mu\nu} = 0, \\ R - \frac{1}{2} R \delta^{\mu}_{\mu} = 0, \\ R - 2 R = 0, \\ R = 0, \end{align*}

simplifying the Einstein field equations:

Rμν=0.R_{\mu\nu} = 0.

The metric is in the form:

gμν=[g00g01g02g03g10g11g12g13g20g21g22g23g30g31g32g33] g_{\mu\nu} = \begin{bmatrix} g_{00} & g_{01} & g_{02} & g_{03} \\ g_{10} & g_{11} & g_{12} & g_{13} \\ g_{20} & g_{21} & g_{22} & g_{23} \\ g_{30} & g_{31} & g_{32} & g_{33} \end{bmatrix}

The spacetime is spherically symmetric, meaning it is invariant under rotations, mirroring and time reversals. Taking the mirror image for the coordinate σ≠1\sigma \neq 1 (rr cannot be negative) means:

x~σ=−xσ,\tilde{x}^{\sigma} = -x^{\sigma},

and the metric is transformed as follows (where μ≠σ\mu \neq \sigma):

g~μσ=∂xα∂x~μ∂xβ∂x~σgαβ=−∂xα∂x~μgασ=−gμσ. \begin{align*} \tilde{g}_{\mu \sigma} &= \frac{\partial x^{\alpha}}{\partial \tilde{x}^{\mu}} \frac{\partial x^{\beta}}{\partial \tilde{x}^{\sigma}} g_{\alpha\beta} \\ &= -\frac{\partial x^{\alpha}}{\partial \tilde{x}^{\mu}} g_{\alpha\sigma} \\ &= -g_{\mu\sigma}. \end{align*}

Since the metric is invariant under this transformation, this implies:

gμσ=0,σ≠μ,g_{\mu\sigma} = 0, \qquad \sigma \neq \mu,

thus, the metric is diagonalized:

gμν=[g000000g110000g220000g33] g_{\mu\nu} = \begin{bmatrix} g_{00} & 0 & 0 & 0 \\ 0 & g_{11} & 0 & 0 \\ 0 & 0 & g_{22} & 0 \\ 0 & 0 & 0 & g_{33} \end{bmatrix}

When θ\theta and ϕ\phi are constant, g00g_{00} and g11g_{11} should only depend on rr (by symmetry):

gμν=[−A(r)0000B(r)0000g220000g33] g_{\mu\nu} = \begin{bmatrix} -A(r) & 0 & 0 & 0 \\ 0 & B(r) & 0 & 0 \\ 0 & 0 & g_{22} & 0 \\ 0 & 0 & 0 & g_{33} \end{bmatrix}

and g22g_{22} and g33g_{33} should use metric for a sphere:

gμν=[−A(r)0000B(r)0000r20000r2sin⁡2θ] g_{\mu\nu} = \begin{bmatrix} -A(r) & 0 & 0 & 0 \\ 0 & B(r) & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end{bmatrix}

and when r→∞r \to \infty, the metric should be the same as the metric for a sphere:

lim⁡r→∞gμν=[−1000010000r20000r2sin⁡2θ] \lim_{r \to \infty} g_{\mu\nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end{bmatrix}

The inverse metric is equal to:

gμν=[−1A(r)00001B(r)00001r200001r2sin⁡2θ] g^{\mu\nu} = \begin{bmatrix} -\frac{1}{A(r)} & 0 & 0 & 0 \\ 0 & \frac{1}{B(r)} & 0 & 0 \\ 0 & 0 & \frac{1}{r^2} & 0 \\ 0 & 0 & 0 & \frac{1}{r^2 \sin^2 \theta} \end{bmatrix}

The general equation for Christoffel symbols is as follows:

Γλμσ=12gλκ(gκμ,σ+gκσ,μ−gμσ,κ),\Gamma^{\lambda}{}_{\mu\sigma} = \frac{1}{2} g^{\lambda\kappa} (g_{\kappa\mu,\sigma} + g_{\kappa\sigma,\mu} - g_{\mu\sigma,\kappa}),

The inverse metric is diagonal, so the equation simplifies:

Γλμσ=12gλλ(gλμ,σ+gλσ,μ−gμσ,λ),\Gamma^{\lambda}{}_{\mu\sigma} = \frac{1}{2} g^{\lambda\lambda} (g_{\lambda\mu,\sigma} + g_{\lambda\sigma,\mu} - g_{\mu\sigma,\lambda}),

Splitting the equation, for the time coordinate, we have:

Γ0μσ=−12A(r)(g0μ,σ+g0σ,μ−gμσ,0)=−12A(r)(g0μ,σ+g0σ,μ),Γ00σ=Γ0σ0=−12A(r)g00,σ=−12A(r)g00,σ,Γ0μν=[0A′(r)2A(r)00A′(r)2A(r)00000000000] \begin{align*} \Gamma^0{}_{\mu\sigma} &= -\frac{1}{2 A(r)} (g_{0\mu,\sigma} + g_{0\sigma,\mu} - g_{\mu\sigma,0}) \\ &= -\frac{1}{2A(r)} (g_{0\mu,\sigma} + g_{0\sigma,\mu}), \\ \Gamma^0{}_{0\sigma} = \Gamma^0{}_{\sigma0} &= -\frac{1}{2A(r)} g_{00,\sigma} \\ &= -\frac{1}{2A(r)} g_{00,\sigma}, \\ \Gamma^0{}_{\mu\nu} &= \begin{bmatrix} 0 & \frac{A'(r)}{2A(r)} & 0 & 0 \\ \frac{A'(r)}{2A(r)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \end{align*}

for the rr coordinate, we have:

Γ1μσ=12B(r)(g1μ,σ+g1σ,μ−gμσ,1),Γ10σ=Γ1σ0=−12B(r)g11g0σ,1,Γ11σ=Γ1σ1=12B(r)g11,σ,Γ12σ=Γ1σ2=−12B(r)g2σ,1,Γ13σ=Γ1σ3=−12B(r)g3σ,1,Γ1μν=[A′(r)2B(r)0000B′(r)2B(r)0000−rB(r)0000−rsin⁡2θB(r)] \begin{align*} \Gamma^1{}_{\mu\sigma} &= \frac{1}{2 B(r)} (g_{1\mu,\sigma} + g_{1\sigma,\mu} - g_{\mu\sigma,1}), \\ \Gamma^1{}_{0\sigma} = \Gamma^1{}_{\sigma0} &= -\frac{1}{2 B(r)} g^{11} g_{0\sigma,1}, \\ \Gamma^1{}_{1\sigma} = \Gamma^1{}_{\sigma1} &= \frac{1}{2 B(r)} g_{11,\sigma}, \\ \Gamma^1{}_{2\sigma} = \Gamma^1{}_{\sigma2} &= -\frac{1}{2 B(r)} g_{2\sigma,1}, \\ \Gamma^1{}_{3\sigma} = \Gamma^1{}_{\sigma3} &= -\frac{1}{2 B(r)} g_{3\sigma,1}, \\ \Gamma^1{}_{\mu\nu} &= \begin{bmatrix} \frac{A'(r)}{2 B(r)} & 0 & 0 & 0 \\ 0 & \frac{B'(r)}{2 B(r)} & 0 & 0 \\ 0 & 0 & -\frac{r}{B(r)} & 0 \\ 0 & 0 & 0 & -\frac{r \sin^2 \theta}{B(r)} \end{bmatrix} \end{align*}

for the θ\theta coordinate, we have:

Γ2μσ=12r2(g2μ,σ+g2σ,μ−gμσ,2),Γ20σ=Γ2σ0=−12r2g0σ,2,Γ21σ=Γ2σ1=12r2(g2σ,1−g1σ,2),={−12r2g11,2σ=1,12r2g22,1σ=2,Γ22σ=Γ2σ2=12r2g22,σ,Γ23σ=Γ2σ3=−12r2g3σ,2,Γ2μσ=[0000001r001r00000−sin⁡θcos⁡θ] \begin{align*} \Gamma^2{}_{\mu\sigma} &= \frac{1}{2 r^2} (g_{2\mu,\sigma} + g_{2\sigma,\mu} - g_{\mu\sigma,2}), \\ \Gamma^2{}_{0\sigma} = \Gamma^2{}_{\sigma0} &= -\frac{1}{2 r^2} g_{0\sigma,2}, \\ \Gamma^2{}_{1\sigma} = \Gamma^2{}_{\sigma1} &= \frac{1}{2 r^2} (g_{2\sigma,1} - g_{1\sigma,2}), \\ &= \begin{cases} -\frac{1}{2 r^2} g_{11,2} & \sigma = 1, \\ \frac{1}{2 r^2} g_{22,1} & \sigma = 2, \\ \end{cases} \\ \Gamma^2{}_{2\sigma} = \Gamma^2{}_{\sigma2} &= \frac{1}{2 r^2} g_{22,\sigma}, \\ \Gamma^2{}_{3\sigma} = \Gamma^2{}_{\sigma3} &= -\frac{1}{2 r^2} g_{3\sigma,2}, \\ \Gamma^2{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & -\sin \theta \cos \theta \end{bmatrix} \end{align*}

and finally, for the coordinate ϕ\phi:

Γ3μσ=12r2sin⁡2θ(g3μ,σ+g3σ,μ−gμσ,3)=12r2sin⁡2θ(g3μ,σ+g3σ,μ),Γ30σ=Γ3σ0=0,Γ31σ=Γ3σ1=12r2sin⁡2θg3σ,1,Γ32σ=Γ3σ2=12r2sin⁡2θg3σ,2Γ33σ=Γ3σ3=12r2sin⁡2θg33,σΓ3μσ=[00000001r000cot⁡θ01rcot⁡θ0] \begin{align*} \Gamma^3{}_{\mu\sigma} &= \frac{1}{2 r^2 \sin^2 \theta} (g_{3\mu,\sigma} + g_{3\sigma,\mu} - g_{\mu\sigma,3}) \\ &= \frac{1}{2 r^2 \sin^2 \theta} (g_{3\mu,\sigma} + g_{3\sigma,\mu}), \\ \Gamma^3{}_{0\sigma} = \Gamma^3{}_{\sigma0} &= 0, \\ \Gamma^3{}_{1\sigma} = \Gamma^3{}_{\sigma1} &= \frac{1}{2 r^2 \sin^2 \theta} g_{3\sigma,1}, \\ \Gamma^3{}_{2\sigma} = \Gamma^3{}_{\sigma2} &= \frac{1}{2 r^2 \sin^2 \theta} g_{3\sigma,2} \\ \Gamma^3{}_{3\sigma} = \Gamma^3{}_{\sigma3} &= \frac{1}{2 r^2 \sin^2 \theta} g_{33,\sigma} \\ \Gamma^3{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & \cot \theta \\ 0 & \frac{1}{r} & \cot \theta & 0 \end{bmatrix} \end{align*}

The Riemann tensor is equal to:

Rρμσν=Γρνμ,σ−Γρσμ,ν+ΓρσλΓλνμ−ΓρνλΓλσμ.R^{\rho}{}_{\mu\sigma\nu} = \Gamma^{\rho}{}_{\nu\mu,\sigma} - \Gamma^{\rho}{}_{\sigma\mu,\nu} + \Gamma^{\rho}{}_{\sigma\lambda} \Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\sigma\mu}.

The Ricci tensor is equal to zero:

Rμν=Rρμρν=Γρνμ,ρ−Γρρμ,ν+ΓρρλΓλνμ−ΓρνλΓλρμ=0.R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu} = \Gamma^{\rho}{}_{\nu\mu,\rho} - \Gamma^{\rho}{}_{\rho\mu,\nu} + \Gamma^{\rho}{}_{\rho\lambda} \Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\rho}{}_{\nu\lambda} \Gamma^{\lambda}{}_{\rho\mu} = 0.

The R00R_{00} is equal to:

R00=Γρ00,ρ−Γρρ0,0+ΓρρλΓλ00−Γρ0λΓλρ0=Γ100,1+Γρρ1Γ100−Γρ00Γ0ρ0−Γρ01Γ1ρ0=Γ100,1+Γ001Γ100+Γ111Γ100+Γ221Γ100+Γ331Γ100−Γ100Γ010−Γ001Γ100=Γ100,1+Γ111Γ100+Γ221Γ100+Γ331Γ100−Γ100Γ010=2A′′(r)B(r)−2A′(r)B′(r)4(B(r))2+B′(r)2B(r)A′(r)2B(r)+1rA′(r)2B(r)+1rA′(r)2B(r)−A′(r)2B(r)A′(r)2A(r)=A′′(r)B(r)2(B(r))2−A′(r)B′(r)2(B(r))2+A′(r)B′(r)4(B(r))2+A′(r)rB(r)−(A′(r))24A(r)B(r)=0,⋅4rA(r)(B(r))2,0=2rA(r)A′′(r)B(r)−2rA(r)A′(r)B′(r)+rA(r)A′(r)B′(r)+4A(r)B(r)A′(r)−rB(r)(A′(r))2=2rA(r)A′′(r)B(r)−rA(r)A′(r)B′(r)+4A(r)B(r)A′(r)−rB(r)(A′(r))2. \begin{align*} R_{00} &= \Gamma^{\rho}{}_{00,\rho} - \Gamma^{\rho}{}_{\rho0,0} + \Gamma^{\rho}{}_{\rho\lambda} \Gamma^{\lambda}{}_{00} - \Gamma^{\rho}{}_{0\lambda} \Gamma^{\lambda}{}_{\rho0} \\ &= \Gamma^1{}_{00,1} + \Gamma^{\rho}{}_{\rho1} \Gamma^1{}_{00} - \Gamma^{\rho}{}_{00} \Gamma^0{}_{\rho0} - \Gamma^{\rho}{}_{01} \Gamma^1{}_{\rho0} \\ &= \Gamma^1{}_{00,1} + \Gamma^0{}_{01} \Gamma^1{}_{00} + \Gamma^1{}_{11} \Gamma^1{}_{00} + \Gamma^2{}_{21} \Gamma^1{}_{00} + \Gamma^3{}_{31} \Gamma^1{}_{00} - \Gamma^1{}_{00} \Gamma^0{}_{10} - \Gamma^0{}_{01} \Gamma^1{}_{00} \\ &= \Gamma^1{}_{00,1} + \Gamma^1{}_{11} \Gamma^1{}_{00} + \Gamma^2{}_{21} \Gamma^1{}_{00} + \Gamma^3{}_{31} \Gamma^1{}_{00} - \Gamma^1{}_{00} \Gamma^0{}_{10} \\ &= \frac{2 A''(r) B(r) - 2 A'(r) B'(r)}{4 (B(r))^2} + \frac{B'(r)}{2 B(r)} \frac{A'(r)}{2 B(r)} + \frac{1}{r} \frac{A'(r)}{2 B(r)} + \frac{1}{r} \frac{A'(r)}{2 B(r)} - \frac{A'(r)}{2 B(r)} \frac{A'(r)}{2 A(r)} \\ &= \frac{A''(r) B(r)}{2 (B(r))^2} - \frac{A'(r) B'(r)}{2 (B(r))^2} + \frac{A'(r) B'(r)}{4 (B(r))^2} + \frac{A'(r)}{r B(r)} - \frac{(A'(r))^2}{4 A(r) B(r)} \\ &= 0, \qquad \cdot 4r A(r) (B(r))^2, \\ 0 &= 2r A(r) A''(r) B(r) - 2r A(r) A'(r) B'(r) + r A(r) A'(r) B'(r) + 4 A(r) B(r) A'(r) - r B(r) (A'(r))^2 \\ &= 2r A(r) A''(r) B(r) - r A(r) A'(r) B'(r) + 4 A(r) B(r) A'(r) - r B(r) (A'(r))^2. \end{align*}

The R11R_{11} is equal to:

R11=Γρ11,ρ−Γρρ1,1+ΓρρλΓλ11−Γρ1λΓλρ1=Γ111,1−Γ001,1−Γ111,1−Γ221,1−Γ331,1+Γρρ1Γ111−Γ01λΓλ01−Γ11λΓλ11−Γ21λΓλ21−Γ31λΓλ31=−Γ001,1−Γ221,1−Γ331,1+Γ001Γ111+Γ111Γ111+Γ221Γ111+Γ331Γ111−Γ010Γ001−Γ111Γ111−Γ212Γ221−Γ313Γ331=−Γ001,1−Γ221,1−Γ331,1+Γ001Γ111+Γ221Γ111+Γ331Γ111−Γ010Γ001−Γ212Γ221−Γ313Γ331=−2A(r)A′′(r)−2(A′(r))24(A(r))2+1r2+1r2+A′(r)2A(r)B′(r)2B(r)+1rB′(r)2B(r)+1rB′(r)2B(r)−A′(r)2A(r)A′(r)2A(r)−1r1r−1r1r=−A′′(r)2A(r)+(A′(r))22(A(r))2+A′(r)B′(r)4A(r)B(r)+B′(r)rB(r)−(A′(r))24(A(r))2=−A′′(r)2A(r)+(A′(r))24(A(r))2+A′(r)B′(r)4A(r)B(r)+B′(r)rB(r)=0,⋅4r(A(r))2B(r),0=−2rA(r)B(r)A′′(r)+rB(r)(A′(r))2+rA(r)A′(r)B′(r)+4(A(r))2B′(r). \begin{align*} R_{11} &= \Gamma^{\rho}{}_{11,\rho} - \Gamma^{\rho}{}_{\rho1,1} + \Gamma^{\rho}{}_{\rho\lambda} \Gamma^{\lambda}{}_{11} - \Gamma^{\rho}{}_{1\lambda} \Gamma^{\lambda}{}_{\rho1} \\ &= \Gamma^1{}_{11,1} - \Gamma^0{}_{01,1} - \Gamma^1{}_{11,1} - \Gamma^2{}_{21,1} - \Gamma^3{}_{31,1} + \Gamma^{\rho}{}_{\rho1} \Gamma^1{}_{11} \\ &- \Gamma^0{}_{1\lambda} \Gamma^{\lambda}{}_{01} - \Gamma^1{}_{1\lambda} \Gamma^{\lambda}{}_{11} - \Gamma^2{}_{1\lambda} \Gamma^{\lambda}{}_{21} - \Gamma^3{}_{1\lambda} \Gamma^{\lambda}{}_{31} \\ &= -\Gamma^0{}_{01,1} - \Gamma^2{}_{21,1} - \Gamma^3{}_{31,1} + \Gamma^0{}_{01} \Gamma^1{}_{11} + \Gamma^1{}_{11} \Gamma^1{}_{11} + \Gamma^2{}_{21} \Gamma^1{}_{11} + \Gamma^3{}_{31} \Gamma^1{}_{11} \\ &- \Gamma^0{}_{10} \Gamma^0{}_{01} - \Gamma^1{}_{11} \Gamma^1{}_{11} - \Gamma^2{}_{12} \Gamma^2{}_{21} - \Gamma^3{}_{13} \Gamma^3{}_{31} \\ &= -\Gamma^0{}_{01,1} - \Gamma^2{}_{21,1} - \Gamma^3{}_{31,1} + \Gamma^0{}_{01} \Gamma^1{}_{11} + \Gamma^2{}_{21} \Gamma^1{}_{11} + \Gamma^3{}_{31} \Gamma^1{}_{11} \\ &- \Gamma^0{}_{10} \Gamma^0{}_{01} - \Gamma^2{}_{12} \Gamma^2{}_{21} - \Gamma^3{}_{13} \Gamma^3{}_{31} \\ &= -\frac{2 A(r) A''(r) - 2 (A'(r))^2}{4 (A(r))^2} + \frac{1}{r^2} + \frac{1}{r^2} + \frac{A'(r)}{2 A(r)} \frac{B'(r)}{2 B(r)} + \frac{1}{r} \frac{B'(r)}{2 B(r)} + \frac{1}{r} \frac{B'(r)}{2 B(r)} \\ &- \frac{A'(r)}{2 A(r)} \frac{A'(r)}{2 A(r)} - \frac{1}{r} \frac{1}{r} - \frac{1}{r} \frac{1}{r} \\ &= -\frac{A''(r)}{2 A(r)} + \frac{(A'(r))^2}{2 (A(r))^2} + \frac{A'(r) B'(r)}{4 A(r) B(r)} + \frac{B'(r)}{r B(r)} - \frac{(A'(r))^2}{4 (A(r))^2} \\ &= -\frac{A''(r)}{2 A(r)} + \frac{(A'(r))^2}{4 (A(r))^2} + \frac{A'(r) B'(r)}{4 A(r) B(r)} + \frac{B'(r)}{r B(r)} \\ &= 0, \qquad \cdot 4r (A(r))^2 B(r), \\ 0 &= -2r A(r) B(r) A''(r) + r B(r) (A'(r))^2 + r A(r) A'(r) B'(r) + 4 (A(r))^2 B'(r). \end{align*}

The R22R_{22} is equal to:

R22=Γρ22,ρ−Γρρ2,2+ΓρρλΓλ22−Γρ2λΓλρ2=Γ122,1−Γ332,2+Γρρ1Γ122−Γ12λΓλ12−Γ22λΓλ22−Γ32λΓλ32=Γ122,1−Γ332,2+Γ001Γ122+Γ111Γ122+Γ221Γ122+Γ331Γ122−Γ122Γ212−Γ221Γ122−Γ323Γ332=−B(r)−rB′(r)(B(r))2+csc⁡2θ+A′(r)2A(r)−rB(r)+B′(r)2B(r)−rB(r)+1r−rB(r)+1r−rB(r)−−rB(r)1r−1r−rB(r)−cot⁡2θ=−1B(r)+rB′(r)(B(r))2−rA′(r)2A(r)B(r)−rB′(r)2(B(r))2−2B(r)+2B(r)+1=−1B(r)+rB′(r)2(B(r))2−rA′(r)2A(r)B(r)+1=0,⋅2A(r)(B(r))2,0=−2A(r)B(r)+rA(r)B′(r)−rB(r)A′(r)+2A(r)(B(r))2 \begin{align*} R_{22} &= \Gamma^{\rho}{}_{22,\rho} - \Gamma^{\rho}{}_{\rho2,2} + \Gamma^{\rho}{}_{\rho\lambda} \Gamma^{\lambda}{}_{22} - \Gamma^{\rho}{}_{2\lambda} \Gamma^{\lambda}{}_{\rho2} \\ &= \Gamma^1{}_{22,1} - \Gamma^3{}_{32,2} + \Gamma^{\rho}{}_{\rho1} \Gamma^1{}_{22} \\ &- \Gamma^1{}_{2\lambda} \Gamma^{\lambda}{}_{12} - \Gamma^2{}_{2\lambda} \Gamma^{\lambda}{}_{22} - \Gamma^3{}_{2\lambda} \Gamma^{\lambda}{}_{32} \\ &= \Gamma^1{}_{22,1} - \Gamma^3{}_{32,2} + \Gamma^0{}_{01} \Gamma^1{}_{22} + \Gamma^1{}_{11} \Gamma^1{}_{22} + \Gamma^2{}_{21} \Gamma^1{}_{22} + \Gamma^3{}_{31} \Gamma^1{}_{22} \\ &- \Gamma^1{}_{22} \Gamma^2{}_{12} - \Gamma^2{}_{21} \Gamma^1{}_{22} - \Gamma^3{}_{23} \Gamma^3{}_{32} \\ &= -\frac{B(r) - r B'(r)}{(B(r))^2} + \csc^2 \theta + \frac{A'(r)}{2 A(r)} \frac{-r}{B(r)} + \frac{B'(r)}{2 B(r)} \frac{-r}{B(r)} + \frac{1}{r} \frac{-r}{B(r)} + \frac{1}{r} \frac{-r}{B(r)} \\ &- \frac{-r}{B(r)} \frac{1}{r} - \frac{1}{r} \frac{-r}{B(r)} - \cot^2 \theta \\ &= -\frac{1}{B(r)} + \frac{r B'(r)}{(B(r))^2} - \frac{r A'(r)}{2 A(r) B(r)} - \frac{r B'(r)}{2 (B(r))^2} - \frac{2}{B(r)} + \frac{2}{B(r)} + 1 \\ &= -\frac{1}{B(r)} + \frac{r B'(r)}{2 (B(r))^2} - \frac{r A'(r)}{2 A(r) B(r)} + 1 \\ &= 0, \qquad \cdot 2 A(r) (B(r))^2, \\ 0 &= -2 A(r) B(r) + r A(r) B'(r) - r B(r) A'(r) + 2 A(r) (B(r))^2 \\ \end{align*}

R00+R11R_{00} + R_{11} is equal to:

R00+R11=2rA(r)A′′(r)B(r)−rA(r)A′(r)B′(r)+4A(r)B(r)A′(r)−rB(r)(A′(r))2+(−2rA(r)B(r)A′′(r))+rB(r)(A′(r))2+rA(r)A′(r)B′(r)+4(A(r))2B′(r)=2rA(r)A′′(r)B(r)−rA(r)A′(r)B′(r)−rB(r)(A′(r))2+4A(r)B(r)A′(r)+(−2rA(r)A′′(r)B(r))+rA(r)A′(r)B′(r)+rB(r)(A′(r))2+4(A(r))2B′(r)=4A(r)B(r)A′(r)+4(A(r))2B′(r)=0,0=B(r)A′(r)+A(r)B′(r)=(A(r)B(r))′, \begin{align*} R_{00} + R_{11} &= 2r A(r) A''(r) B(r) - r A(r) A'(r) B'(r) + 4 A(r) B(r) A'(r) - r B(r) (A'(r))^2 \\ &+ (-2r A(r) B(r) A''(r)) + r B(r) (A'(r))^2 + r A(r) A'(r) B'(r) + 4 (A(r))^2 B'(r) \\ &= 2r A(r) A''(r) B(r) - r A(r) A'(r) B'(r) - r B(r) (A'(r))^2 + 4 A(r) B(r) A'(r) \\ &+ (-2r A(r) A''(r) B(r)) + r A(r) A'(r) B'(r) + r B(r) (A'(r))^2 + 4 (A(r))^2 B'(r) \\ &= 4 A(r) B(r) A'(r) + 4 (A(r))^2 B'(r) \\ &= 0, \\ 0 &= B(r) A'(r) + A(r) B'(r) = (A(r) B(r))', \end{align*}

implying:

A(r)B(r)=C,A(r) B(r) = C,

where CC is an arbitrary constant. As r→∞r \to \infty, −A(r)→−1  ⟺  A(r)→1-A(r) \to -1 \iff A(r) \to 1 and B(r)→1B(r) \to 1 since CC is constant everywhere, the following applies:

lim⁡r→∞A(r)B(r)=1⋅1=1,A(r)B(r)=1,B(r)=1A(r),B′(r)=−A′(r)(A(r))2. \begin{align*} \lim_{r \to \infty} A(r) B(r) &= 1 \cdot 1 = 1, \\ A(r) B(r) &= 1, \\ B(r) &= \frac{1}{A(r)}, \\ B'(r) &= -\frac{A'(r)}{(A(r))^2}. \end{align*}

Substituting into R22R_{22}:

0=−2A(r)B(r)+rA(r)B′(r)−rB(r)A′(r)+2A(r)(B(r))2=−2A(r)1A(r)−rA(r)A′(r)(A(r))2−r1A(r)A′(r)+2A(r)(1A(r))2=−2−rA′(r)A(r)−rA′(r)A(r)+21A(r)=−2−2rA′(r)A(r)+21A(r),0=−2A(r)−2rA′(r)+2,0=−A(r)−rA′(r)+1,rA′(r)=1−A(r),rdAdr=1−A(r),1rdrdA=11−A(r),drr=dA1−A(r),∫drr=−∫dAA(r)−1,ln⁡r=−ln⁡(A(r)−1)+C0,ln⁡r=ln⁡(1A(r)−1)+ln⁡C,ln⁡r=ln⁡(CA(r)−1),A(r)−1=Cr,A(r)=1+Cr=r+Cr,A′(r)=−Cr2,B(r)=(1+Cr)−1=rr+C,B′(r)=C(r+C)2, \begin{align*} 0 &= -2 A(r) B(r) + r A(r) B'(r) - r B(r) A'(r) + 2 A(r) (B(r))^2 \\ &= -2 A(r) \frac{1}{A(r)} - r A(r) \frac{A'(r)}{(A(r))^2} - r \frac{1}{A(r)} A'(r) + 2 A(r) \left(\frac{1}{A(r)}\right)^2 \\ &= -2 - r \frac{A'(r)}{A(r)} - r \frac{A'(r)}{A(r)} + 2 \frac{1}{A(r)} \\ &= -2 - 2r \frac{A'(r)}{A(r)} + 2 \frac{1}{A(r)}, \\ 0 &= -2 A(r) - 2r A'(r) + 2, \\ 0 &= -A(r) - r A'(r) + 1, \\ r A'(r) &= 1 - A(r), \\ r \frac{dA}{dr} &= 1 - A(r), \\ \frac{1}{r} \frac{dr}{dA} &= \frac{1}{1 - A(r)}, \\ \frac{dr}{r} &= \frac{dA}{1 - A(r)}, \\ \int \frac{dr}{r} &= -\int \frac{dA}{A(r) - 1}, \\ \ln r &= -\ln (A(r) - 1) + C_0, \\ \ln r &= \ln \left(\frac{1}{A(r) - 1}\right) + \ln C, \\ \ln r &= \ln \left(\frac{C}{A(r) - 1}\right), \\ A(r) - 1 &= \frac{C}{r}, \\ A(r) &= 1 + \frac{C}{r} \\ &= \frac{r + C}{r}, \\ A'(r) &= -\frac{C}{r^2}, \\ B(r) &= \left(1 + \frac{C}{r}\right)^{-1} \\ &= \frac{r}{r + C}, \\ B'(r) &= \frac{C}{(r + C)^2}, \\ \end{align*}

The metric now looks like this:

gμν=[−(1+Cr)0000(1+Cr)−10000r20000r2sin⁡2θ] g_{\mu\nu} = \begin{bmatrix} -\left(1 + \frac{C}{r}\right) & 0 & 0 & 0 \\ 0 & \left(1 + \frac{C}{r}\right)^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end{bmatrix}

and the Christoffel symbols like this:

Γ0μν=[0−C2r(r+C)00−C2r(r+C)00000000000]Γ1μν=[−C(r+C)2r30000C2r(r+C)0000−(r+C)0000−sin⁡2θ(r+C)]Γ2μσ=[0000001r001r00000−sin⁡θcos⁡θ]Γ3μσ=[00000001r000cot⁡θ01rcot⁡θ0] \begin{align*} \\ \Gamma^0{}_{\mu\nu} &= \begin{bmatrix} 0 & -\frac{C}{2r (r + C)} & 0 & 0 \\ -\frac{C}{2r (r + C)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \\ \Gamma^1{}_{\mu\nu} &= \begin{bmatrix} -\frac{C (r + C)}{2r^3} & 0 & 0 & 0 \\ 0 & \frac{C}{2 r (r + C)} & 0 & 0 \\ 0 & 0 & -(r + C) & 0 \\ 0 & 0 & 0 & -\sin^2 \theta (r + C) \end{bmatrix} \\ \Gamma^2{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & -\sin \theta \cos \theta \end{bmatrix} \\ \Gamma^3{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & \cot \theta \\ 0 & \frac{1}{r} & \cot \theta & 0 \end{bmatrix} \end{align*}

The geodesic equations are given by:

d2xσdτ2+Γσμνdxμdτdxνdτ=0.\frac{d^2 x^{\sigma}}{d\tau^2} + \Gamma^{\sigma}{}_{\mu\nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = 0.

We will make it match to Newtonian gravity:

τ→t,dtdτ→1,dxidτ→0,gμν→ημν+hμν, \begin{align*} \tau &\to t, \\ \frac{dt}{d\tau} &\to 1, \\ \frac{dx^i}{d\tau} &\to 0, \\ g_{\mu\nu} &\to \eta_{\mu\nu} + h_{\mu\nu}, \end{align*}

where hμνh_{\mu\nu} is some error due to the weak gravity and ημν\eta_{\mu\nu} is the Minkowski metric in cartesian coordinates:

ημν=[−1000010000100001]hμν→0. \begin{align*} \eta_{\mu\nu} &= \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ h_{\mu\nu} &\to 0. \end{align*}

Substituting into the geodesic equation (considering only space coordinates - d2tdt2=0\frac{d^2 t}{dt^2} = 0):

d2xidt2+Γi00(dtdt)2=0,d2xidt2=−Γi00, \begin{align*} \frac{d^2 x^i}{dt^2} + \Gamma^i{}_{00} \left(\frac{dt}{dt}\right)^2 &= 0, \\ \frac{d^2 x^i}{dt^2} &= -\Gamma^i{}_{00}, \end{align*}

d2xidt2\frac{d^2 x^i}{dt^2} should match Newtonian gravitation:

d2xidt2=−∂φ∂xi,\frac{d^2 x^i}{dt^2} = -\frac{\partial \varphi}{\partial x^i},

implying:

Γi00=∂φ∂xi.\Gamma^i{}_{00} = \frac{\partial \varphi}{\partial x^i}.

Solving for Γi00\Gamma^i{}_{00}:

Γi00=12giκ(gκ0,0+gκ0,0−g00,κ)the metric is diagonal and time independent=−12giig00,i=−12(ηii+hii)(η00,i+h00,i)=−12h00,i=−12∂h00∂xi=∂φ∂xi \begin{align*} \Gamma^i{}_{00} &= \frac{1}{2} g^{i\kappa} (g_{\kappa0,0} + g_{\kappa0,0} - g_{00,\kappa}) \qquad \textrm{the metric is diagonal and time independent} \\ &= -\frac{1}{2} g^{ii} g_{00,i} \\ &= -\frac{1}{2} (\eta^{ii} + h^{ii}) (\eta_{00,i} + h_{00,i}) \\ &= -\frac{1}{2} h_{00,i} \\ &= -\frac{1}{2} \frac{\partial h_{00}}{\partial x^i} = \frac{\partial \varphi}{\partial x^i} \end{align*}

Solving for h00h_{00}:

−12∂h00∂xi=∂φ∂xi,∂h00∂xi=−2∂φ∂xi,∂h00∂xi=−2∂φ∂xi,h00=−2φ+φ0,h00=2Mr+φ0, \begin{align*} -\frac{1}{2} \frac{\partial h_{00}}{\partial x^i} &= \frac{\partial \varphi}{\partial x^i}, \\ \frac{\partial h_{00}}{\partial x^i} &= -2 \frac{\partial \varphi}{\partial x^i}, \\ \frac{\partial h_{00}}{\partial x^i} &= -2 \frac{\partial \varphi}{\partial x^i}, \\ h_{00} &= -2 \varphi + \varphi_0, \\ h_{00} &= \frac{2M}{r} + \varphi_0, \end{align*}

The g00g_{00} is equal to:

g00=−1+2r+φ0.g_{00} = -1 + \frac{2}{r} + \varphi_0.

As r→∞r \to \infty, the g00g_{00} component should reduce to −1-1:

lim⁡r→∞g00=lim⁡r→∞−1+2Mr+φ0=−1+φ0=−1, \begin{align*} \lim_{r \to \infty} g_{00} &= \lim_{r \to \infty} -1 + \frac{2M}{r} + \varphi_0 \\ &= -1 + \varphi_0 = -1, \end{align*}

implying φ0=0\varphi_0 = 0. The g00g_{00} component is equal to:

g00=−1+2Mr=−(1−2Mr), \begin{align*} g_{00} &= -1 + \frac{2M}{r} \\ &= -\left(1 - \frac{2M}{r}\right), \end{align*}

g00g_{00} is the same in cartesian and spherical coordinates.

Solving for CC:

g00=−(1−2Mr)=−(1+Cr),2Mr=−Cr,C=−2M=−rs, \begin{align*} g_{00} = -\left(1 - \frac{2M}{r}\right) &= -\left(1 + \frac{C}{r}\right), \\ \frac{2M}{r} &= -\frac{C}{r}, \\ C &= -2M = -r_s, \end{align*}

rs=C=2Mr_s = C = 2M is called the Schwarzschild radius. It is the radius of the event horizon of a black hole. In SI units this is equal to:

rs=2GMc2.r_s = \frac{2GM}{c^2}.

I will modify the units such that M=1M = 1 - unit converter. The metric and christoffel symbols are equal to:

gμν=[−(1−rsr)0000(1−rsr)−10000r20000r2sin⁡2θ]Γ0μν=[0rs2r(r−rs)00rs2r(r−rs)00000000000]Γ1μν=[rs(r−rs)2r30000−rs2r(r−rs)0000−(r−rs)0000−sin⁡2θ(r−rs)]Γ2μσ=[0000001r001r00000−sin⁡θcos⁡θ]Γ3μσ=[00000001r000cot⁡θ01rcot⁡θ0] \begin{align*} \\ g_{\mu\nu} &= \begin{bmatrix} -\left(1 - \frac{r_s}{r}\right) & 0 & 0 & 0 \\ 0 & \left(1 - \frac{r_s}{r}\right)^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end{bmatrix} \\ \Gamma^0{}_{\mu\nu} &= \begin{bmatrix} 0 & \frac{r_s}{2r (r - r_s)} & 0 & 0 \\ \frac{r_s}{2r (r - r_s)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \\ \Gamma^1{}_{\mu\nu} &= \begin{bmatrix} \frac{r_s (r - r_s)}{2r^3} & 0 & 0 & 0 \\ 0 & -\frac{r_s}{2 r (r - r_s)} & 0 & 0 \\ 0 & 0 & -(r - r_s) & 0 \\ 0 & 0 & 0 & -\sin^2 \theta (r - r_s) \end{bmatrix} \\ \Gamma^2{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & -\sin \theta \cos \theta \end{bmatrix} \\ \Gamma^3{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & \cot \theta \\ 0 & \frac{1}{r} & \cot \theta & 0 \end{bmatrix} \end{align*}
rs=0 mr_s = 0\ m

We can simplify the geodesic equations by rotating the coordinate system such that θ\theta is constant at π2\frac{\pi}{2} (dθdλ=d2θdλ2=dθ=0\frac{d\theta}{d\lambda} = \frac{d^2\theta}{d\lambda^2} = d\theta = 0 ). The metric and Christoffel symbols simplify to:

ds2=−(1−rsr)dt2+(1−rsr)−1dr2+r2dϕ2,Γ0μν=[0rs2r(r−rs)00rs2r(r−rs)00000000000]Γ1μν=[rs(r−rs)2r30000−rs2r(r−rs)0000−(r−rs)0000−(r−rs)]Γ2μσ=[0000001r001r000000]Γ3μσ=[00000001r000001r00] \begin{align*} ds^2 &= -\left(1 - \frac{r_s}{r}\right) dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 d\phi^2, \\ \Gamma^0{}_{\mu\nu} &= \begin{bmatrix} 0 & \frac{r_s}{2r (r - r_s)} & 0 & 0 \\ \frac{r_s}{2r (r - r_s)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \\ \Gamma^1{}_{\mu\nu} &= \begin{bmatrix} \frac{r_s (r - r_s)}{2r^3} & 0 & 0 & 0 \\ 0 & -\frac{r_s}{2 r (r - r_s)} & 0 & 0 \\ 0 & 0 & -(r - r_s) & 0 \\ 0 & 0 & 0 & -(r - r_s) \end{bmatrix} \\ \Gamma^2{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \\ \Gamma^3{}_{\mu\sigma} &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{1}{r} & 0 & 0 \end{bmatrix} \end{align*}

The general geodesic equations are as follows:

d2xσdλ2+Γσμνdxμdλdxνdλ=0.\frac{d^2 x^{\sigma}}{d\lambda^2} + \Gamma^{\sigma}{}_{\mu\nu} \frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} = 0.

Substituting into equation for σ=2\sigma = 2:

d2θdλ2+Γ2μνdxμdλdxνdλ=0,2rdrdλdθdλ=0,2rdrdλ⋅0=0,0=0, \begin{align*} \frac{d^2 \theta}{d\lambda^2} + \Gamma^2{}_{\mu\nu} \frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} &= 0, \\ \frac{2}{r} \frac{dr}{d\lambda} \frac{d\theta}{d\lambda} &= 0, \\ \frac{2}{r} \frac{dr}{d\lambda} \cdot 0 &= 0, \\ 0 &= 0, \end{align*}

meaning it is a valid solution. The geodesic equations are in the form:

d2tdλ2+rsr(r−rs)dtdλdrdλ=0,d2rdλ2+rs(r−rs)2r3(dtdλ)2−rs2r(r−rs)(drdλ)2−(r−rs)(dϕdλ)2=0,d2ϕdλ2+2rdrdλdϕdλ=0, \begin{align*} \frac{d^2 t}{d\lambda^2} + \frac{r_s}{r(r - r_s)} \frac{dt}{d\lambda} \frac{dr}{d\lambda} = 0, \\ \frac{d^2 r}{d\lambda^2} + \frac{r_s(r - r_s)}{2r^3} \left(\frac{dt}{d\lambda}\right)^2 - \frac{r_s}{2r(r - r_s)} \left(\frac{dr}{d\lambda}\right)^2 - (r - r_s) \left(\frac{d\phi}{d\lambda}\right)^2 = 0, \\ \frac{d^2 \phi}{d\lambda^2} + \frac{2}{r} \frac{dr}{d\lambda} \frac{d\phi}{d\lambda} = 0, \end{align*}

or when using the geometrized units where M=1M = 1 (rs=2r_s = 2):

d2tdλ2+2r(r−2)dtdλdrdλ=0,d2rdλ2+r−2r3(dtdλ)2−1r(r−2)(drdλ)2−(r−2)(dϕdλ)2=0,d2ϕdλ2+2rdrdλdϕdλ=0, \begin{align*} \frac{d^2 t}{d\lambda^2} + \frac{2}{r(r - 2)} \frac{dt}{d\lambda} \frac{dr}{d\lambda} &= 0, \\ \frac{d^2 r}{d\lambda^2} + \frac{r - 2}{r^3} \left(\frac{dt}{d\lambda}\right)^2 - \frac{1}{r(r - 2)} \left(\frac{dr}{d\lambda}\right)^2 - (r - 2) \left(\frac{d\phi}{d\lambda}\right)^2 &= 0, \\ \frac{d^2 \phi}{d\lambda^2} + \frac{2}{r} \frac{dr}{d\lambda} \frac{d\phi}{d\lambda} &= 0, \end{align*}

and the metric:

ds2=−(1−2r)dt2+(1−2r)−1dr2+r2dϕ2.ds^2 = -\left(1 - \frac{2}{r}\right) dt^2 + \left(1 - \frac{2}{r}\right)^{-1} dr^2 + r^2 d\phi^2.