ADM connections
I always wondered what the connections, Riemann, and Ricci of an arbitrary metric represented in ADM form was.
Granted an easy way around the Riemann and Ricci is via the Gauss-Codazzi-Ricci.
My curiousity wasn't satisfied, but halfway through this I got bored.
If you want to see it in more detail, check out the automatic generation on my symmath page
here.
Another source: 2008 Alcubierre "Introduction to 3+1 Numerical Relativity" Appendix B.
metric:
$g_{uv} = \left[\matrix{
-\alpha^2 + \beta^k \beta^l \gamma_{kl} & \beta_j \\
\beta_i & \gamma_{ij}
}\right]$
inverse:
$g^{uv} = \left[\matrix{
-1/\alpha^2 & \beta^j / \alpha^2 \\
\beta^i / \alpha^2 & \gamma^{ij} - \beta^i \beta^j / \alpha^2
}\right]$
metric partials:
$g_{tt,\mu} = -2 \alpha \alpha_{,\mu} + 2 {\beta^k}_{,\mu} \beta_k$
$g_{ti,\mu} = \beta_{i,\mu}$
$g_{ij,\mu} = \gamma_{ij,\mu}$
4-metric connection coefficients:
$\Gamma_{uvw} = \frac{1}{2}(g_{uv,w} + g_{uw,v} - g_{vw,u})$
individually:
$\Gamma_{ttt}
= \frac{1}{2} g_{tt,t}
= -\alpha \alpha_{,t} + \beta^k \beta_{k,t}$
$\Gamma_{tti} = \Gamma_{tit} = \frac{1}{2} g_{tt,i} = -\alpha \alpha_{,i} + \beta^k \beta_{k,i}$
$\Gamma_{itt} = g_{it,t} - \frac{1}{2} g_{tt,i} = \beta_{i,t} + \alpha \alpha_{,i} - \beta^k \beta_{k,i}$
$\Gamma_{tij} = \frac{1}{2}(g_{ti,j} + g_{tj,i} - g_{ij,t}) = \frac{1}{2}(\beta_{i,j} + \beta_{j,i} - \gamma_{ij,t})$
$\Gamma_{itj} = \Gamma_{ijt} = \frac{1}{2}(g_{ij,t} + g_{it,j} - g_{tj,i}) = \frac{1}{2}(\gamma_{ij,t} + \beta_{i,j} - \beta_{j,i})$
$\Gamma_{ijk} = \frac{1}{2}(g_{ij,k} + g_{ik,j} - g_{jk,i}) = \frac{1}{2}(\gamma_{ij,k} + \gamma_{ik,j} - \gamma_{jk,i}) = {^3\Gamma}_{ijk}$
Connections / Christoffel of 2nd kind:
${\Gamma^u}_{vw} = g^{ur} \Gamma_{rvw}$
individually:
${\Gamma^t}_{tt} = g^{tt} \Gamma_{ttt} + g^{tk} \Gamma_{ktt}$
$ = -1/\alpha^2 (
-\alpha \alpha_{,t} + \beta^j {\beta^k}_{,t} \gamma_{jk} + \frac{1}{2} \beta^j \beta^k \gamma_{jk,t}
) + \beta^k / \alpha^2 (
{\beta^j}_{,t} \gamma_{kj} + \beta^j \gamma_{kj,t} + \alpha \alpha_{,k} - \beta^j {\beta^l}_{,k} \gamma_{jl} - \frac{1}{2} \beta^j \beta^l \gamma_{jl,k}
)$
$ = \frac{1}{\alpha^2}(
\alpha \alpha_{,t}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,t}
+ \alpha \alpha_{,k} \beta^k
- \beta^k \beta^j {\beta^l}_{,k} \gamma_{jl}
- \frac{1}{2} \beta^k \beta^j \beta^l \gamma_{jl,k}
)$
${\Gamma^t}_{ti} = {\Gamma^t}_{it} = g^{tt} \Gamma_{tti} + g^{tj} \Gamma_{jti} $
$ = -1 / \alpha^2 ( -\alpha \alpha_{,i} + \beta^j {\beta^k}_{,i} \gamma_{jk} + \frac{1}{2} \beta^j \beta^k \gamma_{jk,i} )
+ \frac{1}{2} \beta^j / \alpha^2 (\gamma_{ij,t} + {\beta^k}_{,i} \gamma_{jk} + \beta^k \gamma_{jk,i} - {\beta^k}_{,j} \gamma_{ik} - \beta^k \gamma_{ik,j})
$
$ = 1 / \alpha^2 (
\alpha \alpha_{,i}
- \frac{1}{2} \beta^j {\beta^k}_{,i} \gamma_{jk}
+ \frac{1}{2} \beta^j \gamma_{ij,t}
- \frac{1}{2} \beta^j {\beta^k}_{,j} \gamma_{ik}
- \frac{1}{2} \beta^j \beta^k \gamma_{ik,j}
)$
${\Gamma^i}_{tt} = g^{it} \Gamma_{ttt} + g^{ij} \Gamma_{jtt}$
$ = \beta^i / \alpha^2 (-\alpha \alpha_{,t} + \beta^j {\beta^k}_{,t} \gamma_{jk} + \frac{1}{2} \beta^j \beta^k \gamma_{jk,t})
+ (\gamma^{ij} - \beta^i \beta^j / \alpha^2) (
{\beta^k}_{,t} \gamma_{jk} + \beta^k \gamma_{jk,t} + \alpha \alpha_{,j} - \beta^k {\beta^l}_{,j} \gamma_{kl} - \frac{1}{2} \beta^k \beta^l \gamma_{kl,j})$
$ = \frac{1}{\alpha^2} (
- \beta^i \alpha \alpha_{,t}
- \beta^i \beta^j \alpha \alpha_{,j}
+ \gamma^{ij} \alpha \alpha_{,j}
+ {\beta^i}_{,t}
- \frac{1}{2} \beta^i \beta^j \beta^k \gamma_{jk,t}
+ \gamma^{ij} \beta^k \gamma_{jk,t}
+ \beta^i \beta^j \beta^k {\beta^l}_{,j} \gamma_{kl}
+ \frac{1}{2} \beta^i \beta^j \beta^k \beta^l \gamma_{kl,j}
- \beta^k {\beta^l}_{,j} \gamma^{ij} \gamma_{kl}
- \frac{1}{2} \gamma^{ij} \beta^k \beta^l \gamma_{kl,j}
)
$
${\Gamma^t}_{ij} = g^{tt} \Gamma_{tij} + g^{tk} \Gamma_{kij}$
$= -\frac{1}{2} 1/\alpha^2 ({\beta^k}_{,j} \gamma_{ki} + \beta^k \gamma_{ki,j} + {\beta^k}_{,i} \gamma_{kj} + \beta^k \gamma_{kj,i} - \gamma_{ij,t})
+ \frac{1}{2} \beta^k / \alpha^2 (\gamma_{ij,k} + \gamma_{ik,j} - \gamma_{jk,i})$
$= \frac{1}{2} 1/\alpha^2 (
- {\beta^k}_{,i} \gamma_{kj}
- {\beta^k}_{,j} \gamma_{ki}
+ \gamma_{ij,t}
- \beta^k \gamma_{kj,i}
- \beta^k \gamma_{jk,i}
+ \beta^k \gamma_{ij,k}
)$
$= \frac{1}{2} 1/\alpha^2 (
- {\beta^k}_{,i} \gamma_{kj}
- {\beta^k}_{,j} \gamma_{ki}
+ \gamma_{ij,t}
- \beta^k ({^3\Gamma})_{kij}
)$
${\Gamma^i}_{tj} = {\Gamma^i}_{jt} = g^{it} \Gamma_{ttj} + g^{ik} \Gamma_{ktj}$
$ = \beta^i / \alpha^2 (-\alpha \alpha_{,j} + \beta^k {\beta^l}_{,j} \gamma_{kl} + \frac{1}{2} \beta^k \beta^l \gamma_{kl,j})
+ \frac{1}{2} (\gamma^{ik} - \beta^i \beta^k / \alpha^2) (\gamma_{kj,t} + {\beta^l}_{,j} \gamma_{lk} + \beta^l \gamma_{lk,j} - {\beta^l}_{,k} \gamma_{lj} - \beta^l \gamma_{lj,k})$
$ = 1 / \alpha^2 (
- \beta^i \alpha \alpha_{,j}
+ \frac{1}{2} \beta^i \beta^k {\beta^l}_{,j} \gamma_{lk}
+ \frac{1}{2} \beta^i \beta^k {\beta^l}_{,k} \gamma_{lj}
- \frac{1}{2} \beta^i \beta^k \gamma_{kj,t}
+ \frac{1}{2} \beta^i \beta^k \beta^l \gamma_{lj,k}
)
+ \frac{1}{2} \gamma^{ik} \gamma_{kj,t}
+ \frac{1}{2} {\beta^i}_{,j}
- \frac{1}{2} \gamma_{lj} {\beta^l}_{,k} \gamma^{ik}
+ \frac{1}{2} \gamma^{ik} \beta^l \gamma_{lk,j}
- \frac{1}{2} \gamma^{ik} \beta^l \gamma_{lj,k}
$
${\Gamma^i}_{jk} = g^{it} \Gamma_{tjk} + g^{il} \Gamma_{ljk}$
$= \frac{1}{2} 1 / \alpha^2 (
\beta^i {\beta^l}_{,k} \gamma_{lj}
+ \beta^i {\beta^l}_{,j} \gamma_{lk}
+ \beta^i \beta^l \gamma_{jk,l}
- \beta^i \gamma_{jk,t}
)
+ \frac{1}{2} (
\gamma^{il} \gamma_{lj,k}
+ \gamma^{il} \gamma_{lk,j}
- \gamma^{il} \gamma_{jk,l}
)$
$= \frac{1}{2} 1 / \alpha^2 (
\beta^i {\beta^l}_{,k} \gamma_{lj}
+ \beta^i {\beta^l}_{,j} \gamma_{lk}
+ \beta^i \beta^l \gamma_{jk,l}
- \beta^i \gamma_{jk,t}
)
+ {^3\Gamma^i}_{jk}$
Connection contraction:
$\Gamma^u = {\Gamma^u}_{vw} g^{vw}$
$\Gamma^t = {\Gamma^t}_{ab} g^{ab}
= {\Gamma^t}_{tt} g^{tt} + 2 {\Gamma^t}_{ti} g^{ti} + {\Gamma^t}_{ij} g^{ij}$
$= -1/\alpha^4 (
\alpha \alpha_{,t}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,t}
+ \alpha \alpha_{,k} \beta^k
- \beta^k \beta^j {\beta^l}_{,k} \gamma_{jl}
- \frac{1}{2} \beta^k \beta^j \beta^l \gamma_{jl,k}
)
+ 2 \beta^i / \alpha^2 (
-1 / \alpha^2 (
-\alpha \alpha_{,i}
+ \beta^j {\beta^k}_{,i} \gamma_{jk}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,i}
)
+ \frac{1}{2} \beta^j / \alpha^2 (
\gamma_{ij,t}
+ {\beta^k}_{,i} \gamma_{jk}
+ \beta^k \gamma_{jk,i}
- {\beta^k}_{,j} \gamma_{ik}
- \beta^k \gamma_{ik,j}
)
)
+ \frac{1}{2} (\gamma^{ij} - \beta^i \beta^j / \alpha^2) 1/\alpha^2 (
- {\beta^k}_{,i} \gamma_{kj}
- {\beta^k}_{,j} \gamma_{ki}
+ \gamma_{ij,t}
- \beta^k \gamma_{kj,i}
- \beta^k \gamma_{jk,i}
+ \beta^k \gamma_{ij,k}
)$
$= - 1 / \alpha^3 \alpha_{,t}
+ 1 / \alpha^3 \alpha_{,k} \beta^k
- 1 / \alpha^2 \gamma^{ij} {\beta^k}_{,j} \gamma_{ik}
- 1 / \alpha^2 \gamma^{ij} \beta^k \gamma_{jk,i}
+ \frac{1}{2} 1/\alpha^2 \gamma^{ij} \gamma_{ij,t}
+ \frac{1}{2} 1/\alpha^2 \gamma^{ij} \beta^k \gamma_{ij,k}
$
2008 Alcubierre says:
$\Gamma^t = -\frac{1}{\alpha^3} ( \alpha_{,t} - \beta^m \alpha_{,m}) - \frac{1}{\alpha} K$
$\Gamma^k = {\Gamma^k}_{ab} g^{ab}
= {\Gamma^k}_{tt} g^{tt} + 2 {\Gamma^k}_{ti} g^{ti} + {\Gamma^k}_{ij} g^{ij}$
$= -1/\alpha^2 \frac{1}{\alpha^2} (
- \beta^k \alpha \alpha_{,t}
- \beta^k \beta^j \alpha \alpha_{,j}
+ \gamma^{kj} \alpha \alpha_{,j}
+ {\beta^k}_{,t}
- \frac{1}{2} \beta^k \beta^j \beta^i \gamma_{ji,t}
+ \gamma^{kj} \beta^i \gamma_{ji,t}
+ \beta^k \beta^j \beta^i {\beta^l}_{,j} \gamma_{il}
+ \frac{1}{2} \beta^k \beta^j \beta^i \beta^l \gamma_{il,j}
- \beta^i {\beta^l}_{,j} \gamma^{kj} \gamma_{il}
- \frac{1}{2} \gamma^{kj} \beta^i \beta^l \gamma_{il,j}
)
+ 2 \beta^i / \alpha^2 (
1 / \alpha^2 (
- \beta^k \alpha \alpha_{,i}
+ \frac{1}{2} \beta^k \beta^m {\beta^l}_{,i} \gamma_{lm}
+ \frac{1}{2} \beta^k \beta^m {\beta^l}_{,m} \gamma_{li}
- \frac{1}{2} \beta^k \beta^m \gamma_{mi,t}
+ \frac{1}{2} \beta^k \beta^m \beta^l \gamma_{li,m}
)
+ \frac{1}{2} \gamma^{km} \gamma_{mi,t}
+ \frac{1}{2} {\beta^k}_{,i}
- \frac{1}{2} \gamma_{li} {\beta^l}_{,m} \gamma^{km}
+ \frac{1}{2} \gamma^{km} \beta^l \gamma_{lm,i}
- \frac{1}{2} \gamma^{km} \beta^l \gamma_{li,m}
)
+ (\gamma^{ij} - \beta^i \beta^j / \alpha^2) (
\frac{1}{2} 1 / \alpha^2 (
\beta^k {\beta^l}_{,i} \gamma_{lj}
+ \beta^k {\beta^l}_{,j} \gamma_{li}
+ \beta^k \beta^l \gamma_{ji,l}
- \beta^k \gamma_{ji,t}
)
+ {^3\Gamma^k}_{ij}
)
$
$=
+ 1 / \alpha^3 \beta^k \alpha_{,t}
- 1 / \alpha^3 \beta^k \beta^j \alpha_{,j}
- 1 / \alpha^3 \gamma^{kj} \alpha_{,j}
- 1 / \alpha^4 {\beta^k}_{,t}
+ 1 / \alpha^2 \gamma^{kj} \beta^i \gamma_{ij,t}
- 1 / \alpha^4 \gamma^{kj} \beta^i \gamma_{ij,t}
+ 1 / \alpha^4 \beta^i \beta^j \beta^k {\beta^l}_{,i} \gamma_{jl}
+ 1 / \alpha^4 \beta^i {\beta^l}_{,j} \gamma^{kj} \gamma_{il}
+ 1 / \alpha^2 \beta^i {\beta^k}_{,i}
- 1 / \alpha^2 \beta^i \gamma_{li} {\beta^l}_{,j} \gamma^{kj}
+ 1 / \alpha^2 \gamma^{ij} \beta^k {\beta^l}_{,i} \gamma_{lj}
- 1 / \alpha^4 \beta^i \beta^j \beta^k {\beta^l}_{,i} \gamma_{lj}
+ \frac{1}{2} 1 / \alpha^4 \gamma^{jk} \beta^i \beta^l \gamma_{il,j}
+ \frac{1}{2} 1 / \alpha^4 \beta^i \beta^j \beta^k \beta^l \gamma_{il,j}
+ 1 / \alpha^2 \gamma^{kj} \beta^i \beta^l \gamma_{lj,i}
- 1 / \alpha^2 \gamma^{kj} \beta^i \beta^l \gamma_{li,j}
+ \frac{1}{2} 1 / \alpha^2 \gamma^{ij} \beta^k \beta^l \gamma_{ij,l}
- \frac{1}{2} / \alpha^4 \beta^i \beta^j \beta^k \beta^l \gamma_{ij,l}
- \frac{1}{2} / \alpha^4 \beta^i \beta^j \beta^k \gamma_{ij,t}
- \frac{1}{2} / \alpha^2 \gamma^{ij} \beta^k \gamma_{ij,t}
+ \frac{1}{2} / \alpha^4 \beta^i \beta^j \beta^k \gamma_{ij,t}
+ {^3\Gamma^k}_{ij} \gamma^{ij}
- 1 / \alpha^2 \beta^i \beta^j {^3\Gamma^k}_{ij}
$
2008 Alcubierre says:
$\Gamma^i = {}^3 \Gamma^i + \beta^i (\frac{1}{\alpha^3} (\alpha_{,t} - \beta^m \alpha_{,m}) + \frac{1}{\alpha} K) - \frac{1}{\alpha^2} ({\beta^i}_{,t} - {\beta^i}_{,m} \beta^m) - \frac{1}{\alpha} \alpha_{,j} \gamma^{ij}$
Connection derivatives used for Riemann tensor:
${\Gamma^t}_{tt,i}$
$ = (\frac{1}{\alpha^2}(
\alpha \alpha_{,t}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,t}
+ \alpha \alpha_{,k} \beta^k
- \beta^k \beta^j {\beta^l}_{,k} \gamma_{jl}
- \frac{1}{2} \beta^k \beta^j \beta^l \gamma_{jl,k}
))_{,i}$
$ = -2 \frac{\alpha_{,i}}{\alpha^3} (
\alpha \alpha_{,t}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,t}
+ \alpha \alpha_{,k} \beta^k
- \beta^k \beta^j {\beta^l}_{,k} \gamma_{jl}
- \frac{1}{2} \beta^k \beta^j \beta^l \gamma_{jl,k}
) + \frac{1}{\alpha^2} (
\alpha_{,i} \alpha_{,t}
+ \alpha \alpha_{,ti}
+ {\beta^j}_{,i} \beta^k \gamma_{jk,t}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,ti}
+ \alpha_{,i} \alpha_{,k} \beta^k
+ \alpha \alpha_{,ki} \beta^k
+ \alpha \alpha_{,k} {\beta^k}_{,i}
- {\beta^k}_{,i} \beta^j {\beta^l}_{,k} \gamma_{jl}
- \beta^k {\beta^j}_{,i} {\beta^l}_{,k} \gamma_{jl}
- \beta^k \beta^j {\beta^l}_{,ki} \gamma_{jl}
- \beta^k \beta^j {\beta^l}_{,k} \gamma_{jl,i}
- \frac{1}{2} {\beta^k}_{,i} \beta^j \beta^l \gamma_{jl,k}
- \beta^k {\beta^j}_{,i} \beta^l \gamma_{jl,k}
- \frac{1}{2} \beta^k \beta^j \beta^l \gamma_{jl,ki}
)$
$ = \frac{1}{\alpha^2} (
-2 \alpha_{,t} \alpha_{,i}
+ \alpha_{,i} \alpha_{,t}
- \alpha_{,i} \alpha_{,j} \beta^j
+ \alpha \alpha_{,j} {\beta^j}_{,i}
+ \alpha \alpha_{,ti}
+ \alpha \alpha_{,ij} \beta^j
- \alpha^{-1} \alpha_{,i} \beta^j \beta^k \gamma_{jk,t}
+ \alpha^{-1} \alpha_{,i} \beta^j \beta^k \gamma_{jk,l} \beta^l
+2 \alpha^{-1} \alpha_{,i} \beta^j \gamma_{jk} {\beta^k}_{,l} \beta^l
+ {\beta^j}_{,i} \beta^k \gamma_{jk,t}
- {\beta^j}_{,i} {\beta^k}_{,l} \beta^l \gamma_{jk}
- \beta^j {\beta^k}_{,l} {\beta^l}_{,i} \gamma_{jk}
- \beta^j \gamma_{jk} {\beta^k}_{,li} \beta^l
- \frac{1}{2} \beta^j \beta^k {\beta^l}_{,i} \gamma_{jk,l}
- \beta^j {\beta^k}_{,l} \beta^l \gamma_{jk,i}
- {\beta^j}_{,i} \beta^l \beta^k \gamma_{jl,k}
+ \frac{1}{2} \beta^j \beta^k \gamma_{jk,ti}
- \frac{1}{2} \beta^k \beta^j \beta^l \gamma_{jl,ki}
)$
${\Gamma^t}_{ti,t}$
${\Gamma^t}_{ti,j}$
${\Gamma^t}_{ij,\mu}
= \frac{1}{2} (1/\alpha^2 (
- {\beta^k}_{,i} \gamma_{kj}
- {\beta^k}_{,j} \gamma_{ki}
+ \gamma_{ij,t}
- \beta^k ({^3\Gamma})_{kij}
))_{,\mu}$
${\Gamma^i}_{tt,i}$
${\Gamma^i}_{ti,t}$
${\Gamma^j}_{ti,j}$
${\Gamma^j}_{tj,i}$
${\Gamma^k}_{ij,k} = {^3\Gamma^k}_{ij,k}$
${\Gamma^k}_{ki,j} = {^3\Gamma^k}_{ki,j}$
Extrinsic curvature (of spatial indexes):
(From ADM worksheet)
$K_{ij} = -\alpha {\Gamma^i}_{ij}$
Riemann tensor representations used for the Ricci tensor:
${R^i}_{tit} = {\Gamma^i}_{tt,i} - {\Gamma^i}_{ti,t} + {\Gamma^i}_{ti} {\Gamma^t}_{tt} + {\Gamma^i}_{ji} {\Gamma^j}_{tt} - {\Gamma^i}_{tt} {\Gamma^t}_{ti} - {\Gamma^i}_{jt} {\Gamma^j}_{ti}$
${R^t}_{tti} = {\Gamma^t}_{ti,t} - {\Gamma^t}_{tt,i} + {\Gamma^t}_{tt} {\Gamma^t}_{ti} + {\Gamma^t}_{jt} {\Gamma^j}_{ti} - {\Gamma^t}_{ti} {\Gamma^t}_{tt} - {\Gamma^t}_{ji} {\Gamma^j}_{tt}$
$= {\Gamma^t}_{ti,t} - {\Gamma^t}_{tt,i} + {\Gamma^t}_{jt} {\Gamma^j}_{ti} - {\Gamma^t}_{ji} {\Gamma^j}_{tt}$
${R^j}_{tji} = {\Gamma^j}_{ti,j} - {\Gamma^j}_{tj,i} + {\Gamma^j}_{tj} {\Gamma^t}_{ti} + {\Gamma^j}_{kj} {\Gamma^k}_{ti} - {\Gamma^j}_{ti} {\Gamma^t}_{tj} - {\Gamma^j}_{ki} {\Gamma^k}_{tj}$
${R^t}_{itj} = {\Gamma^t}_{ij,t} - {\Gamma^t}_{ti,j} + {\Gamma^t}_{tt} {\Gamma^t}_{ij} + {\Gamma^t}_{kt} {\Gamma^k}_{ij} - {\Gamma^t}_{tj} {\Gamma^t}_{it} - {\Gamma^t}_{kj} {\Gamma^k}_{it}$
${R^t}_{ijk} = {\Gamma^t}_{ik,j} - {\Gamma^t}_{ij,k} + {\Gamma^t}_{lj} {\Gamma^l}_{ik} - {\Gamma^t}_{lk} {\Gamma^l}_{ij}$
$= {\Gamma^t}_{ik,j} - {\Gamma^t}_{ij,k} + {\Gamma^t}_{lj} {\Gamma^l}_{ik} - {\Gamma^t}_{lk} {\Gamma^l}_{ij}$
${R^k}_{ikj} = {\Gamma^k}_{ij,k} - {\Gamma^k}_{ki,j} + {\Gamma^k}_{tk} {\Gamma^t}_{ij} + {\Gamma^k}_{lk} {\Gamma^l}_{ij} - {\Gamma^k}_{tj} {\Gamma^t}_{ik} - {\Gamma^k}_{lj} {\Gamma^l}_{ik}$
All unique representations of fully covariant Riemann metric tensor:
$R_{titj} = R_{itjt} = -R_{tijt} = -R_{ittj}$
$= g_{tt} {R^t}_{itj} + g_{tk} {R^k}_{itj}$
$= (-\alpha^2 + \beta^k \beta^l \gamma_{kl}) {R^t}_{itj} + \beta^l \gamma_{lk} {R^k}_{itj}$
$R_{tijk} = -R_{itjk} = R_{jkti} = -R_{jkit} = g_{tt} {R^t}_{ijk} + g_{tl} {R^l}_{ijk}$
$ = (-\alpha^2 + \beta^l \beta^m \gamma_{lm}) {R^t}_{ijk} + \beta^m \gamma_{lm} {R^l}_{ijk}$
$ = (-\alpha^2 + \beta^l \beta^m \gamma_{lm}) + \beta^m \gamma_{lm} {R^l}_{ijk}$
$R_{ijkl} = g_{it} {R^t}_{jkl} + g_{im} {R^m}_{jkl}$
Ricci tensor:
$R_{tt} = {R^t}_{ttt} + {R^i}_{tit} = {R^i}_{tit}$
$R_{ti} = R_{it} = {R^t}_{tti} + {R^j}_{tji}$
$R_{ij} = {R^t}_{itj} + {R^k}_{ikj}$
Gaussian curvature:
$R = g^{tt} R_{tt} + 2 g^{ti} R_{ti} + g^{ij} R_{ij}$
Einstein tensor:
$G_{tt} = R_{tt} - \frac{1}{2} g_{tt} R$
$G_{ti} = G_{it} = R_{ti} - \frac{1}{2} g_{ti} R$
$G_{ij} = R_{ij} - \frac{1}{2} g_{ij} R$