The cauchy problem and the equations

[shadi_s10]

Hi everyone!
again d'inverno!
to tell the truth I don't really understand what is going on in the cauchy problem!

1)
in section 13.5 "the cauchy problem", it is said that the field equations can be written as the forms in 13.12 to 13.14
can anyone tell me how?

actually I tried to use the riemannian tensor : R_abcd
then I tried to contract indices a and c
but there are some gamma terms which I don not know what to do with.

2)
moreover it is said that if you use the equation that transforms g_ab to g_ab prime (at the end of page 175) , you can get to 13.18

I do not understand how is this one possible:

 

[shadi_s10]

please somebody help me with it!

 

[Altabeh]

Hi everyone!
again d'inverno!
to tell the truth I don't really understand what is going on in the cauchy problem!

1)
in section 13.5 "the cauchy problem", it is said that the field equations can be written as the forms in 13.12 to 13.14
can anyone tell me how?

actually I tried to use the riemannian tensor : R_abcd
then I tried to contract indices a and c
but there are some gamma terms which I don not know what to do with.

These equations can be simply verified. Show us what you've done so far so we could be able to help you out! Please use Latex to typeset equations and if you don't know how to do it, use this picture stuff for a quick understanding.

Your approach is good.

2)
moreover it is said that if you use the equation that transforms g_ab to g_ab prime (at the end of page 175) , you can get to 13.18

I do not understand how is this one possible:

Catching the glimpse of metric indices carefully, plug (13.17) into the transformation formula to get those equations.

AB
AB

 

[shadi_s10]

Hi again!
you are right. the second one was so easy. I derived it correctly.
but about the first one,I mean equations (13.12)-(13.14) here is what I've done:

I know that one can do the rest by using gamma definition which is based on the metric.
but isn't it too long?!
I feel there is something wrong with my solution as first it had derived the rimannian tensors and then the metric and the derivatives. (13.18)
so maybe I should do something else...

 

[Altabeh]

Hi again!
you are right. the second one was so easy. I derived it correctly.
but about the first one,I mean equations (13.12)-(13.14) here is what I've done:

I know that one can do the rest by using gamma definition which is based on the metric.
but isn't it too long?!
I feel there is something wrong with my solution as first it had derived the rimannian tensors and then the metric and the derivatives. (13.18)
so maybe I should do something else...

I think you have serious problems with the understanding of tensor calculus!

First off, the Ricci tensor $$R_{ab}$$ is given by

$$R_{ab}=R^{c}_{abc}=\Gamma_{ab}^{c}_{,c}-\Gamma_{ac}^{c}_{,b}+\Gamma_{ab}^{c}\Gamma_{cd}^{d}-\Gamma_{ad}^{c}\Gamma_{bc}^{d}$$.

Secondly, you can get the Ricci tensor from the following definition:

$$R_{ab}=g^{dc}R_{cabd}$$, (1)

where $$R_{cabd}=g_{ck}R^{k}_{abd}$$ is the covariant Riemann tensor. Do you know anything about the geodesic coordinates? If yes, split $$R_{cabd}$$ into two categories of terms, one containg all terms of second order and the other including in all terms of first order. So spending 10 minutes of your time on the expansion of terms after bringing into consideration the geodesic coordinates (for the sake of convenience, as this is not necessary) you'll end up getting a straightforward expression for $$R_{cabd}$$, only consisting of second derivatives of metric tensor. Then turn to computing the Ricci tensor using (1) which will give you all those equations that you are after by a componential analysis.

AB