Greetings.

- Ricci tensor perturbations involve sum over spacetime components (in FLRW ->{t, r, \theta \varphi} ). How to expand spacetime sum (greek indices) into time and purely space components (latin indices)?
- For each Gamma and \delta{\Gamma} there are subrules (see related image):
*Gamma^{t}_{t t}= ...*Gamma^{t}_ {t i}=... How to setup and substitute this relations in master equations?

The core of programm is

```
import cdb.core.manip as mp
import cdb.core.component as cmp
{t,r, \theta, \varphi}::Coordinate.
a::Depends(t).
{\mu, \nu, \rho, \sigma, \alpha, \beta, \gamma, \lambda, \tau, \chi, \iota, \upsilon, \omega}::Indices(curved,position=fixed,values={t,r, \theta, \varphi}).
{a,b,c,d,e,f,g,i,j,k,l,m,n,p,s}::Indices(position=fixed,values={r, \theta, \varphi})
gb{#}::LaTeXForm("\bar{g}").
gb_{\mu \nu}::Symmetric.
gb::Diagonal;
{gb^{\mu}_{\nu}, gb_{\mu}^{\nu}}::KroneckerDelta.
back := gb\_\{t t\}=\-1\, gb\_\{r r\}=a**2/(1-k*r**2), gb_{\theta \theta}=a**2*r**2, gb_{\varphi \varphi}=a**2*r**2*\sin{\theta}**2, gb^{t t}=-1, gb^{r r}=(1-k*r**2)/(a**2), gb^{\theta \theta}=1/(r**2_a**2), gb^{\varphi \varphi}=1/(a**2_r\*\*2\*\sin\{\theta\}\*\*2\)\, gb\_\{t r\}=0\, gb\_\{t \theta\}=0\, gb\_\{t \varphi\}=0\, gb\_\{r \theta\}=0\, gb\_\{r \varphi\}=0\, gb\_\{\theta \varphi\}=0\, gb\_\{r t\}=0\, gb\_\{\theta t\}=0\, gb\_\{\varphi t\}=0\, gb\_\{\theta r\}=0\, gb\_\{\varphi r\}=0\, gb\_\{ \varphi \theta\}=0;
```

In the part we define components of ricci perturbations:

```
gs{#}::LaTeXForm("\overline{\gamma}")
dGamma{#}::LaTeXForm("\delta\Gamma");
dRic{#}::LaTeXForm("\delta R").
perric := \delta{R_{\mu \nu}} = \partial_{\sigma}{dGamma^{\sigma}_{\mu \nu}} - \partial_{\nu}{dGamma^{\sigma}_{\mu \sigma}} + dGamma^{\rho}_{\mu \nu} \Gamma^{\sigma}_{\rho \sigma} + \Gamma^{\rho}_{\mu \nu} dGamma^{\sigma}_{\rho \sigma} - dGamma^{\rho}_{\mu \sigma} \Gamma^{\sigma}_{\nu \rho} - \Gamma^{\rho}_{\mu \sigma} dGamma^{\sigma}_{\nu \rho};
dr1 = cmp.get_component(perric, $t,t$);
dr2 = cmp.get_component(perric, $t,i$);
dr3 = cmp.get_component(perric, $j,k$);
```

How to expand sum?

We try to define relation for non perturbed Gamma!:

`eq1 := {\Gamma}^{t}_{i j}=(a \partial_{t}{a} {gs}_{i j});`

But error appear:
`RuntimeError: Free indices on lhs and rhs do not match.`

Thank you in advance for your help.