CosmicVoid

Schwarzschild metric

Using the Schwarzschild metric we will compute all the Christoffel symbols.

The line element of the Schwarzschild metric is given by:

ds2=−(1−2Mr)dt2+(1−2Mr)−1dr2+r2dΩ2.\begin{equation*} ds^2=-\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1} dr^2 +r^2d\Omega^2. \end{equation*}

Where,

dΩ2=dθ2+sin2⁡θdϕ2.\begin{equation*} d\Omega^2 = d\theta^2 + \sin^2{\theta} d\phi^2. \end{equation*}

The matrix form:

gij=(−(1−2Mr)0000(1−2Mr)−10000r20000r2sin2⁡θ)\begin{equation*} g_{ij}= \begin{pmatrix} -\left(1-\frac{2M}{r}\right) & 0 & 0 & 0 \\ 0 & \left(1-\frac{2M}{r}\right)^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2{\theta} \end{pmatrix} \end{equation*}

The inverse of the matrix:

gij=(−(1−2Mr)−10000(1−2Mr)00001r200001r2sin2⁡θ)\begin{equation*} g^{ij}= \begin{pmatrix} -\left(1-\frac{2M}{r}\right)^{-1} & 0 & 0 & 0 \\ 0 & \left(1-\frac{2M}{r}\right) & 0 & 0 \\ 0 & 0 & \frac{1}{r^2} & 0 \\ 0 & 0 & 0 & \frac{1}{r^2 \sin^2{\theta}} \end{pmatrix} \end{equation*}

From the metric it is clear that the only the terms with i=j are non-zero. Hence the other terms will vanish throughout the calculations.

The formula for Christoffel symbols is given below:

Γβγα=gαρ2[gβρ,γ+gγρ,β−gβγ,ρ]\begin{equation*} \Gamma^\alpha_{\beta\gamma} = \frac{g^{\alpha\rho}}{2} \left[ g_{\beta\rho,\,\gamma} + g_{\gamma\rho,\,\beta} – g_{\beta\gamma,\,\rho} \right] \end{equation*}

There are forty possible connections for a 4×4 matric.

1)Γ000=g0ρ2[g0ρ,0+g0ρ,0−g00,ρ]=g002[g00,0+g00,0−g00,0]=g002[g00,0]=−(1−2mr)−1[d(1−2mr)dt]2Γ000=0Γttt=0\begin{align*} 1)\quad \Gamma^0_{00} &= \frac{g^{0\rho}}{2} \left[ g_{0\rho,\,0} + g_{0\rho,\,0} – g_{00,\,\rho} \right] \\[1em] &= \frac{g^{00}}{2} \left[ g_{00,\,0} + {g_{00,\,0}} – {g_{00,\,0}} \right] \\[1em] &= \frac{g^{00}}{2} \left[ g_{00,\,0} \right] \\[1em] &= \frac{-\left( 1 – \frac{2m}{r} \right)^{-1} \left[ \frac{d\left(1 – \frac{2m}{r}\right)}{dt} \right]}{2} \\[1em] \Gamma^0_{00} &= 0 \\ \quad \Gamma^t_{tt} &= 0 \end{align*}

As the term inside the derivative of time was independent of time hence the first connection becomes zero.

2)Γ001=g1ρ2[g0ρ,0+g0ρ,0−g00,ρ]=g112[g01,0+g01,0−g00,1]=(1−2Mr)2[0+0−d(−(1−2Mr))dr]Γttr=−12(1−2Mr)(0+2M(−1)1r2)=Mr2(1−2Mr)\begin{align*} 2)\quad \Gamma^1_{00} &= \frac{g^{1\rho}}{2} \left[ g_{0\rho,\,0} + g_{0\rho,\,0} – g_{00,\,\rho} \right] \\[1.5em] &= \frac{g^{11}}{2} \left[ g_{01,\,0} + g_{01,\,0} – g_{00,\,1} \right] \\[1em] &= \frac{\left( 1 – \frac{2M}{r} \right)}{2} \left[ 0 + 0 – \frac{d\left(-\left(1 – \frac{2M}{r}\right)\right)}{dr} \right] \\[1em] &\quad \Gamma^r_{tt} =-\frac{1}{2}\left(1 – \frac{2M}{r}\right)\left(0 + 2M(-1)\frac{1}{r^2}\right) \\ &\quad\qquad = \frac{M}{r^2}\left(1 – \frac{2M}{r}\right) \end{align*}
3)Γ002=g2ρ2[g0ρ,0+g0ρ,0−g00,ρ]=g222[g02,0+g02,0−g00,2]=12r2[0+0−d(−(1−2Mr))dθ]=12r2(d(1−2Mr)dθ)=12r2(0)=0Γttθ=0\begin{align*} 3)\quad \Gamma^2_{00} &= \frac{g^{2\rho}}{2} \left[ g_{0\rho,\,0} + g_{0\rho,\,0} – g_{00,\,\rho} \right] \\[1.5ex] &= \frac{g^{22}}{2} \left[ g_{02,\,0} + g_{02,\,0} – g_{00,\,2} \right] \\ &= \frac{1}{2r^2} \left[ 0 + 0 – \frac{d\left(-(1-\frac{2M}{r})\right)}{d\theta} \right] \\[1.5ex] &= \frac{1}{2r^2} \left( \frac{d\left(1-\frac{2M}{r}\right)}{d\theta} \right) = \frac{1}{2r^2}(0) = 0 \\[1.5ex] \Gamma^\theta_{tt} &= 0 \end{align*}
4)Γ003=g3ρ2[g0ρ,0+g0ρ,0−g00,ρ]=g332[g03,0+g03,0−g00,3]=12r2sin2⁡θ[0−dr2sin2⁡θdϕ]=0Γttϕ=0\begin{align*} 4)\quad \Gamma^3_{00} &= \frac{g^{3\rho}}{2} \left[ g_{0\rho,\,0} + g_{0\rho,\,0} – g_{00,\,\rho} \right] \\[1.5ex] &= \frac{g^{33}}{2} \left[ g_{03,\,0} + g_{03,\,0} – g_{00,\,3} \right] \\[1.5ex] &= \frac{1}{2r^2\sin^2\theta} \left[ 0 – \frac{d\,r^2\sin^2\theta}{d\phi} \right] = 0 \\[1.5ex] \Gamma^\phi_{tt} &= 0 \end{align*}
5)Γ010=g0ρ2[g0ρ,1+g1ρ,0−g01,ρ]=g002[g00,1+g10,0−g01,0]=g002[g00,1+0]Γtrt=gtt2[ddrgtt]Γtrt=−12(1−2Mr)ddr(−(1−2Mr))=+12(1−2Mr)ddr(1−2Mr)=12(1−2Mr)[0−2Mddr(1r)]=12(1−2Mr)(−2M)(−1)(r−2)=M(1−2Mr)1r2=Mr2(1−2Mr)−1\begin{align*} \text{5)}\quad \Gamma_{01}^0 &= \frac{g^{0\rho}}{2} \left[ g_{0\rho, 1} + g_{1\rho, 0} – g_{01, \rho} \right] \\ &= \frac{g^{00}}{2} \left[ g_{00, 1} + g_{10, 0} – g_{01, 0} \right] \\ &= \frac{g^{00}}{2} \left[ g_{00, 1} + 0 \right] \\ \Gamma_{tr}^t &= \frac{g^{tt}}{2} \left[ \frac{d}{dr} g_{tt} \right] \\[1em] \Gamma_{tr}^t &= -\frac{1}{2\left(1-\frac{2M}{r}\right)} \frac{d}{dr} \left(-\left(1-\frac{2M}{r}\right)\right) \\[1em] &= +\frac{1}{2\left(1-\frac{2M}{r}\right)} \frac{d}{dr} \left(1-\frac{2M}{r}\right) \\[1em] &= \frac{1}{2\left(1-\frac{2M}{r}\right)} \left[0 – 2M \frac{d}{dr}\left(\frac{1}{r}\right)\right] \\[1em] &= \frac{1}{\cancel{2}\left(1-\frac{2M}{r}\right)} (-\cancel{2}M) (-1) \left(r^{-2}\right) \\[1em] &= \frac{M}{\left(1-\frac{2M}{r}\right)} \frac{1}{r^2} \\[1em] &= \frac{M}{r^2} \left(1-\frac{2M}{r}\right)^{-1} \end{align*}
6)Γ011=g1P2[g0P,1+g1P,0−g01,P]=g112[g01,1+g11,0−g01,1]=g112[0+g11,0−0]=grr2(ddt(grr))=[grr212(1−2Mr)][ddt(1−2Mr)−1]Γgtrr=0\begin{align*} \text{6)}\quad \Gamma_{01}^{1} &= \frac{g^{1P}}{2} \left[ g_{0P,\,1} + g_{1P,\,0} – g_{01,\,P} \right] \\ &= \frac{g^{11}}{2} \left[ g_{01,\,1} + g_{11,\,0} – g_{01,\,1} \right] \\ &= \frac{g^{11}}{2} \left[ 0 + g_{11,\,0} – 0 \right] \\ &= \frac{g^{rr}}{2} \left( \frac{d}{dt}(g_{rr}) \right) \\ &= \left[ \sout{\frac{g_{rr}}{2}} \frac{1}{2} \left( 1 – \frac{2M}{r} \right) \right] \left[ \frac{d}{dt} \left( 1 – \frac{2M}{r} \right)^{-1} \right] \\ \Gamma_{gtr}^{r} &= 0 \end{align*}
7)Γ012=g2P2[g0P,1+g1P,0−g01,P]=g222[g02,1+g12,0−g01,2]=0\begin{align*} \text{7)}\quad \Gamma_{01}^{2} &= \frac{g^{2P}}{2} \left[ g_{0P,\,1} + g_{1P,\,0} – g_{01,\,P} \right] \\ &= \frac{g^{22}}{2} \left[ g_{02,\,1} + g_{12,\,0} – g_{01,\,2} \right] \\ &= 0 \end{align*}
8)Γ013=g3P32[g03,2+g13,0−g01,3]Γ013=0\begin{align*} \text{8)}\quad \Gamma_{01}^{3} &= \frac{g^{3\cancel{P}3}}{2} \left[ g_{03,\,2} + g_{13,\,0} – g_{01,\,3} \right] \\ \Gamma_{01}^{3} &= 0 \end{align*}
9)Γ020=g0p2[g0p,2+g2p,0−g02,0]=g002[g00,2+g20,0−g02,0]=g002[g00,2]=−12(11−2mr)[ddθ(−(1−2mr))]\begin{align*} \text{9)}\quad \Gamma_{02}^{0} &= \frac{g^{0p}}{2} \left[ g_{0p,\,2} + g_{2p,\,0} – g_{02,\,0} \right] \\ &= \frac{g^{00}}{2} \left[ g_{00,\,2} + \cancel{g_{20,\,0}} – \cancel{g_{02,\,0}} \right] \\ &= \frac{g^{00}}{2} \left[ g_{00,\,2} \right] \\ &= -\frac{1}{2} \left( \frac{1}{1 – \frac{2m}{r}} \right) \left[ \frac{d}{d\theta} \left( -(1 – \frac{2m}{r}) \right) \right] \end{align*}
Γtθt=0 \begin{equation*} \Gamma_{t\theta}^{t} = 0 \end{equation*}
10)Γ021=g1p112[g01,2+g21,1−g02,1]\begin{equation*} \text{10)}\quad \Gamma_{02}^{1} = \frac{g^{\cancel{1p} 11}}{2} \left[ g_{01,\,2} + g_{21,\,1} – g_{02,\,1} \right] \end{equation*}

Here, all the terms inside are zero hence:

Γtθr=0\begin{equation*} \Gamma_{t\theta}^{r} = 0 \end{equation*}
11)Γ022=g222[g02,2+g22,0−g02,2]=gθθ2[gθθ,t]=12r2ddt(r2)Γtθθ=0\begin{align*} \text{11)}\quad \Gamma_{02}^{2} &= \frac{g^{22}}{2} \left[ g_{02,\,2} + g_{22,\,0} – g_{02,\,2} \right] \\ &= \frac{g^{\theta\theta}}{2} \left[ g_{\theta\theta,\,t} \right] \\ &= \frac{1}{2r^2} \frac{d}{dt} (r^2)\\ \Gamma_{t\theta}^{\theta} &= 0 \end{align*}
12)Γ023=g332[g03,2+g23,0−g02,3]Γtθϕ=013)Γ030=g002[g00,3+g30,0−g03,0]=g002[g00,3]=gtt2[ddϕ(11−2Mr)]Γtϕt=014)Γ031=g112[g01,3+g13,0−g03,1]=015)Γ032=g222[g02,3+g32,0−g03,2]=016)Γ033=g332[g03,3+g33,0−g03,3]=g332[g33,0]=ddt(r2sin2⁡θ)=0\begin{align*} 12)\quad \Gamma^3_{02} &= \frac{g^{33}}{2} \left[ g_{03,2} + g_{23,0} – g_{02,3} \right] \\[1.5ex] \Gamma^\phi_{t\theta} &= 0 \\[2ex] 13)\quad \Gamma^0_{03} &= \frac{g^{00}}{2} \left[ g_{00,3} + g_{30,0} – g_{03,0} \right] \\ &= \frac{g^{00}}{2} \left[ g_{00,3} \right] \\ &= \frac{g^{tt}}{2} \left[ \frac{d}{d\phi} \left({\frac{1}{1-\frac{2M}{r}}} \right) \right] \\[1.5ex] \Gamma^t_{t\phi} &= 0 \\[2ex] 14)\quad \Gamma^1_{03} &= \frac{g^{11}}{2} \left[ g_{01,3} + g_{13,0} – g_{03,1} \right] \\ &= 0 \\[2ex] 15)\quad \Gamma^2_{03} &= \frac{g^{22}}{2} \left[ g_{02,3} + g_{32,0} – g_{03,2} \right] \\ &= 0 \\[2ex] 16)\quad \Gamma^3_{03} &= \frac{g^{33}}{2} \left[ g_{03,3} + g_{33,0} – g_{03,3} \right] \\ &= \frac{g^{33}}{2} \left[ g_{33,0} \right] \\ &= \quad \frac{d}{dt} \left( {r^2 \sin^2\theta} \right)\\ &=0 \end{align*}
17)Γ110=g002[g10,1+g10,1−g11,0]=g002[−g11,0]=−12(1−2Mr)ddt[−10(1−2Mr)]Γrrt=0 \begin{align*} 17)\quad \Gamma_{11}^0 &= \frac{g^{00}}{2} \left[ g_{10,1} + g_{10,1} – g_{11,0} \right] \\ &= \frac{g^{00}}{2} \left[ – g_{11,0} \right] \\ &= \frac{-1}{2 \left( 1 – \frac{2M}{r} \right)} \frac{d}{dt} \left[ -\frac{1^0}{\left( 1 – \frac{2M}{r} \right)} \right]\\ \Gamma_{rr}^t &= 0 \end{align*}
18)Γ111=g112[g11,1+g11,1−g11,1]=(1−2Mr)2[ddr[1(1−2Mr)−1]]=12(1−2Mr)(−1)(1−2Mr)−2(0−2M(−1)r2)=12(1−2Mr)(−1)1(1−2Mr)2[2Mr2]Γrrr=−Mr2(1−2Mr)−1\begin{align*} 18)\quad \Gamma_{11}^1 &= \frac{g^{11}}{2} \left[ g_{11,1} + g_{11,1} – g_{11,1} \right] \\ &= \frac{\left( 1 – \frac{2M}{r} \right)}{2} \left[ \frac{d}{dr} \left[ 1 \left( 1 – \frac{2M}{r} \right)^{-1} \right] \right] \\ &= \frac{1}{2} \left( 1 – \frac{2M}{r} \right) (-1) \left( 1 – \frac{2M}{r} \right)^{-2} \left( 0 – \frac{2M(-1)}{r^2} \right) \\ &= \frac{1}{\cancel{2}} \left( 1 – \frac{2M}{r} \right) (-1) \frac{1}{\left( 1 – \frac{2M}{r} \right)^{\cancel{2}}} \left[ \frac{\cancel{2}M}{r^2} \right]\\ \Gamma_{rr}^r &= -\frac{M}{r^2} \left( 1 – \frac{2M}{r} \right)^{-1} \end{align*}
19)Γ112=g222[g12,10+g12,10−g11,2]=g222[−g11,2]=0\begin{align*} 19)\quad \Gamma_{11}^2 &= \frac{g^{22}}{2} \left[ \cancel{g_{12,1}}^0 + \cancel{g_{12,1}}^0 – g_{11,2} \right] \\ &= \frac{g^{22}}{2} \left[ – g_{11,2} \right]\\ &=0 \end{align*}
20)Γ113=g332[g13,10+g13,10−g11,3]=g332[−g11,2]=−g332ddϕ(r2(1−2Mr)−1)Γrrϕ=0\begin{align*} 20)\quad \Gamma_{11}^3 &= \frac{g^{33}}{2} \left[ \cancel{g_{13,1}}^0 + \cancel{g_{13,1}}^0 – g_{11,3} \right] \\ &= \frac{g^{33}}{2} \left[ – g_{11,2} \right]\\ &= \quad- \frac{g^{33}}{2} \frac{d}{d\phi} \left( r^2 \left(1-\frac{2M}{r}\right)^{-1} \right)\\ \Gamma_{rr}^{\phi} &= 0 \end{align*}
21)Γ120=g002[g10,20+g20,10−g12,00]Γrθt=0\begin{align*} 21)\quad \Gamma_{12}^0 &= \frac{g^{00}}{2} \left[ \cancel{g_{10,2}}^0 + \cancel{g_{20,1}}^0 – \cancel{g_{12,0}}^0 \right] \\ \Gamma_{r\theta}^{t} &= 0 \end{align*}
22)Γ121=g112[g11,2+g21,10−g12,10]=g112[g11,2]=g112ddθ(1−2Mr)−1Γrθr=0\begin{align*} 22)\quad \Gamma_{12}^1 &= \frac{g^{11}}{2} \left[ g_{11,2}+ \cancel{g_{21,1}}^0 – \cancel{g_{12,1}}^0 \right] \\ &= \frac{g^{11}}{2} \left[ g_{11,2} \right]\\ &= \quad \frac{g^{11}}{2} \frac{d}{d\theta} \left(1-\frac{2M}{r}\right)^{-1}\\ \Gamma_{r\theta}^{r} &= 0 \end{align*}
23)Γ122=g222[g14/2+g22,1−g12,2]=g222[g22,1]=12r2[ddr(r2)]=12r2Γrθθ=1r\begin{align*} 23)\quad \Gamma_{12}^2 &= \frac{g^{22}}{2} \left[ g_{14/2} + g_{22,1} – g_{12,2} \right] \\ &= \frac{g^{22}}{2} \left[ g_{22,1} \right] \\ &= \frac{1}{2r^{2}} \left[ \frac{d}{dr} \left( r^{2} \right) \right] \\ &= \frac{1}{2r^{2}} \\ \Gamma_{r\theta}^\theta&=\frac{1}{r} \end{align*}
24)Γ123=g33[g131,2+g231,1−g12,3]=12r2sin2⁡θ[0]=0\begin{align*} 24)\quad \Gamma_{12}^{3} &= {g^{33}} \left[ g_{131,2} + g_{231,1} – g_{12,3} \right] \\ &= \frac{1}{2r^{2}\sin^{2}\theta} \left[ 0 \right]\\ &= 0 \end{align*}
25)Γ130=g002[g10,3+g30,1−g13,0]=0\begin{align*} 25)\quad \Gamma_{13}^0 &= \frac{g^{00}}{2} \left[ g_{10,3}+ g_{30,1}- g_{13,0}\right]\\ &= 0 \end{align*}
26)Γ131=g112[g11,3+g31,10−g13,10]=12(1−2Mr)ddϕ(1−2Mr)−1=0\begin{align*} 26)\quad \Gamma_{13}^1 &= \frac{g^{11}}{2}\left[g_{11,3} + \cancel{g_{31,1}}^0 – \cancel{g_{13,1}}^0\right]\\ &=\frac{1}{2}\left(\frac{1-2M}{r}\right)\frac{d}{d\phi}\left(1-\frac{2M}{r}\right)^{-1}\\ &=0 \end{align*}
27)Γ132=g222[g1213+g3211−g1312]=0\begin{align*} 27)\quad\Gamma_{13}^{2}&=\frac{g^{22}}{2}[g_{1213}+g_{3211}-g_{1312}]\\ &=0 \end{align*}
28)Γ133=g332[g1313+g3311−g1313]=g332[g3311]=12r2sin2⁡θd(r2sin2⁡θ)dr=12r2sin2⁡θ2rsin2⁡θΓrϕϕ=1r\begin{align*} 28)\quad\Gamma_{13}^{3}&=\frac{g^{33}}{2}[g_{1313}+g_{3311}-g_{1313}]\\&=\frac{g^{33}}{2}[g_{3311}]\\&=\frac{1}{2r^{2}\sin^{2}\theta}\frac{d(r^{2}\sin^{2}\theta)}{dr}\\&=\frac{1}{2r^{2}\sin^{2}\theta}\frac{2r\sin^{2}\theta}{}\\ \Gamma_{r\phi}^{\phi } &=\frac{1}{r} \end{align*}
29)Γ220=g002[g20,2+g20,2−g22,0]=g002(−dr2dt)=0\begin{align*} 29)\quad{\Gamma}_{22}^{0} &= \frac{g^{00}}{2} \left[ g_{20,2} + g_{20,2} – g_{22,0} \right] \\ &= \frac{g^{00}}{2} \left(-\frac{dr^2}{dt}\right)\\ &=0 \end{align*}
30) Γ221=g112[g21,2+g21,2−g22,1]=12(1−2Mr)[−ddr(r2)]=12(1−2Mr)(−2r)=−r(1−2Mr)\begin{align*} 30)\ \Gamma_{22}^{1}&=\frac{g^{11}}{2}\left[g_{21,2}+g_{21,2}-g_{22,1}\right]\\ &=\frac{1}{2}(1-\frac{2M}{r})\left[-\frac{d}{dr}(r^{2})\right]\\ &=\frac{1}{2}(1-\frac{2M}{r})(-2r)\\ &=-r(1-\frac{2M}{r}) \end{align*}
31)Γ)222=g222[g22,2+g22,2−g22,2]=12r2[ddθ(r2)]=0\begin{align*} 31) \quad \Gamma){22}^2 &= \frac{g^{22}}{2} \left[ g_{22,2} + g_{22,2} – g_{22,2} \right] \\ &= \frac{1}{2r^{2}} \left[ \frac{d}{d\theta} (r^{2}) \right]\\ &= 0 \end{align*}
32)Γ223=g332[g23,2+g23,2−g22,3]=12r2sin2⁡θ[−ddϕ(r2)]=0\begin{align*} 32)\quad\Gamma_{22}^{3} &= \frac{g^{33}}{2} \left[g_{23,2} + g_{23,2} -g_{22,3} \right]\\ &=\frac{1}{2r^{2}\sin^{2}\theta}\left[-\frac{d}{d\phi}\left(r^{2}\right)\right] \\ &=0 \end{align*}
33)Γ230=g002[g20,3+g30,2−g23,0]=0\begin{align*} 33)\quad \Gamma_{23}^{0} &= \frac{g^{00}}{2} \left[ g_{20,3} + g_{30,2} – g_{23,0} \right] \\ &= 0 \end{align*}
34)Γ231=g112[g21,3+g31,2−g23,1]=0\begin{align*} 34)\quad\Gamma_{23}^{1} &=\frac{g^{11}}{2} [g_{21,3}+g_{31,2}-g_{23,1}]\\ &=0 \end{align*}
35)Γ232=g222[g22,3+g32,2−g23,2]=g222(g22,3)=12r2(ddϕr2)=0\begin{align*} 35)\quad \Gamma_{23}^2&= \frac{g^{22}}{2}\left[g_{22,3}+g_{32,2}-g_{23,2}\right] \\ &=\frac{g^{22}}{2}(g_{22,3})\\ &=\frac{1}{2r^{2}}\left(\frac{d}{d\phi}r^{2}\right)\\ &=0\end{align*}
36)Γ233=g332[g23+g33−g23]=g332[g33]=12r2sin2⁡θddθ(r2sin2⁡θ)Γθϕϕ=12r2sin2⁡θφ2sin⁡θcos⁡θΓθϕϕ=cos⁡θsin⁡θ=cot⁡θ\begin{align*} 36)\quad\Gamma_{23}^{3}& = \frac{g^{33}}{2} \left[ g_{23} + g_{33} – g_{23} \right] \\ &= \frac{g^{33}}{2} \left[ g_{33} \right] \\ &=\frac{1}{2r^{2}\sin^{2}\theta}\frac{d}{d\theta}(r^{2}\sin^{2}\theta)\\ \Gamma_{\theta \phi}^{\phi}&=\frac{1}{2r^{2}\sin^{2}\theta}\quad \varphi^{2} \sin\theta \cos\theta \\ \Gamma_{\theta \phi}^{\phi}&=\frac{\cos\theta}{\sin\theta}=\cot\theta \end{align*}
37)Γ331=g002[g30,3+g30,3−g33,0]=0\begin{align*} 37)\quad \Gamma_{33}^{1} &= \frac{g^{00}}{2} \left[ g_{30,3} + g_{30,3} – g_{33,0} \right] \\ &= 0 \end{align*}
38)Γ331=g112[g31,3+g31,3−g33,1]=12(1−2Mr)(−ddr(r2sin2⁡θ))=−12(1−2Mr)2rsin2⁡θΓϕϕr=−r(1−2Mr)sin2⁡θ\begin{align*} 38)\quad\Gamma_{33}^1&=\frac{g^{11}}{2}\left[g_{31,3}+g_{31,3}-g_{33,1}\right]\\ &=\frac{1}{2}\left(1-\frac{2M}{r}\right)\left(-\frac{d}{dr}\left(r^{2}\sin^{2}\theta\right)\right)\\ &=-\frac{1}{2}\left(1-\frac{2M}{r}\right)2r\sin^{2}\theta\\ \Gamma_{\phi \phi}^r&=-r(1-\frac{2M}{r})\sin^{2}\theta \end{align*}
39)Γ332=g222[g32,3+g32,3−g33,2]=12r2(−ddθ(r2sin2⁡θ))=−12r22r2sin⁡θcos⁡θ=−sin⁡θcos⁡θ\begin{align*} 39)\quad\Gamma_{33}^2&=\frac{g^{22}}{2}\left[g_{32,3}+g_{32,3}-g_{33,2}\right]\\ &=\frac{1}{2r^{2}}\left(-\frac{d}{d\theta}\left(r^{2}\sin^{2}\theta\right)\right)\\ &=-\frac{1}{2r^{2}}2r^2\sin\theta\cos\theta\\ &=-\sin{\theta}\cos{\theta} \end{align*}
40)Γ333=g332[g33,3+g33,3−g33,3]=g332ddϕ(r2sin2⁡θ)=0\begin{align*} 40)\quad \Gamma_{33}^{3} &= \frac{g^{33}}{2} \left[ g_{33,3} + g_{33,3} – g_{33,3} \right] \\ &= \frac{g^{33}}{2} \frac{d}{d\phi}\left(r^2 \sin^2{\theta}\right)\\ &=0 \end{align*}

The non-vanishing Christoffel symbols of Schwarzschild metric are:

1)Γrtt=Mr2(1−2Mr)−12)Γrrr=−Mr2(1−2Mr)−13)Γttr=Mr2(1−2Mr)4)Γθθr=−r(1−2Mr)5)Γϕϕr=−rsin2⁡θ(1−2Mr)6)Γθrθ=1r7)Γϕrϕ=1r8)Γθϕϕ=cot⁡θ9)Γϕϕθ=−sin⁡θcos⁡θ\begin{align*} 1)\quad\Gamma_{rt}^t&=\frac{M}{r^2}\left(1 – \frac{2M}{r}\right)^{-1}\\ 2)\quad\Gamma_{rr}^r&=\frac{-M}{r^2}\left(1 – \frac{2M}{r}\right)^{-1}\\ 3)\quad\Gamma_{tt}^r&=\frac{M}{r^2}\left(1 – \frac{2M}{r}\right)\\ 4)\quad\Gamma_{\theta \theta}^r&=-r\left(1 – \frac{2M}{r}\right)\\ 5)\quad\Gamma_{\phi \phi}^r&=-r\sin^2{\theta}\left(1 – \frac{2M}{r}\right)\\ 6)\quad\Gamma_{\theta r}^\theta&=\frac{1}{r}\\ 7)\quad\Gamma_{\phi r}^\phi&=\frac{1}{r}\\ 8)\quad\Gamma_{\theta \phi}^\phi&=\cot{\theta}\\ 9)\quad\Gamma_{\phi \phi}^\theta &=-\sin{\theta}\cos{\theta} \end{align*}

References:

  1. Schutz, B. F. (1985). A first course in general relativity (Vol. 2). Cambridge university press.

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