A Question about Weinberg's GR book

[LordShadow_05]

TL;DR Summary

Why does Weinberg discard a term of the ricci tensor he should be taking into account in the post-newtonian approximation of fourth order?

To anyone who has studied Weinberg's book. Does anyone know why Weinberg discards the fourth order term of the purely spatial components of the ricci tensor? It's the chapter 9 (post-newtonian approximation) of his GR book. It doesn't make sense to me because he includes the R_{00} term of fourth order but not R_{ij}.

 

[Alien101]

It seems that Weinberg's treatment reflects the physical hierarchy in gravitational systems: time components (energy) drive the dynamics while spatial components (stress) provide corrections. Including fourth-order R₀₀ but not R_ij maintains the appropriate balance between accuracy and computational tractability in the post-Newtonian framework.This is a standard and well-justified approach in post-Newtonian theory, used successfully in gravitational wave astronomy and precision tests of General Relativity.

 

[LordShadow_05]

It seems that Weinberg's treatment reflects the physical hierarchy in gravitational systems: time components (energy) drive the dynamics while spatial components (stress) provide corrections. Including fourth-order R₀₀ but not R_ij maintains the appropriate balance between accuracy and computational tractability in the post-Newtonian framework.This is a standard and well-justified approach in post-Newtonian theory, used successfully in gravitational wave astronomy and precision tests of General Relativity.

Ok, now I understand. Thank you very much!

 

[TSny]

The reason that $^4_{R_{i j}}$ is not included in the list (9.1.26) through (9.1.29) is because this list is meant to include only the Ricci terms that can be calculated from the list of affine connection terms given in (9.1.16) through (9.1.22) on page 215. Note how all the connection terms that appear on the right sides of (9.1.26) through (9.1.29) are in the list (9.1.16) through (9.1.22). At the top of page 216, Weinberg refers to the connection terms in (9.1.16) through (9.1.22) as the “known” affine connection terms.

You can show that $^4_{R_{i j}}$ depends on affine connection terms not included in (9.1.16) through (9.1.22), such as $^4_{\Gamma^0_{0i}}$ and $^3_{\Gamma^0_{ij}}$. So, $^4_{R_{i j}}$ is not included in (9.1.26) through (9.1.29).

If you proceed through Weinberg’s analysis on pages 216-220, you will see that he needs only the Ricci terms listed in (9.1.26) through (9.1.29).

 

[LordShadow_05]

The reason that $^4_{R_{i j}}$ is not included in the list (9.1.26) through (9.1.29) is because this list is meant to include only the Ricci terms that can be calculated from the list of affine connection terms given in (9.1.16) through (9.1.22) on page 215. Note how all the connection terms that appear on the right sides of (9.1.26) through (9.1.29) are in the list (9.1.16) through (9.1.22). At the top of page 216, Weinberg refers to the connection terms in (9.1.16) through (9.1.22) as the “known” affine connection terms.

You can show that $^4_{R_{i j}}$ depends on affine connection terms not included in (9.1.16) through (9.1.22), such as $^4_{\Gamma^0_{0i}}$ and $^3_{\Gamma^0_{ij}}$. So, $^4_{R_{i j}}$ is not included in (9.1.26) through (9.1.29).

If you proceed through Weinberg’s analysis on pages 216-220, you will see that he needs only the Ricci terms listed in (9.1.26) through (9.1.29).

Oh I see. So the thing is that even if $^4_{R_{i j}}$ is of fourth order it doesn't contribute because we see that it is constructed with components of Christoffel's symbol that don't appear in the geodesic equation, so those components aren't measurable, right? I have calculated this component of the Ricci tensor including the "unknown" affine connection terms and I obtained the following:
$$^4_{R_{i j}}=\partial_j ^4_{\Gamma^0_{i0}}-\partial_0 ^3_{\Gamma^0_{ij}}+\partial_j ^4_{\Gamma^k_{ik}}-\partial_k ^4_{\Gamma^k_{ij}}+ ^2_{\Gamma^0_{i0}} ^2_{\Gamma^0_{j0}}+ ^2_{\Gamma^m_{ik}} ^2_{\Gamma^k_{jm}} - ^2_{\Gamma^k_{ij}} ^2_{\Gamma^0_{k0}} - ^2_{\Gamma^m_{ij}} ^2_{\Gamma^k_{mk}}$$
I find that this component of the Ricci tensor isn't only constructed with "unknown" affine connection terms but also with "known" ones. Then, why don't we take into account those terms which contain "known" affine connection terms?
Also, at the end of that same page (216) he uses the harmonic gauge $\Gamma^{\lambda}=0$. For the component $\lambda=0$ he obtains the equation 9.1.35. For obtaining that equation he made use of the "unknown" affine connection term $^3_{\Gamma^0_{ij}}$. So why can he use the "unknown" terms here but not in the Ricci tensor?
Thank you very much for your comment.

Edit: my formula doesn't show up and I don't know why. I have attached an image.

 

[TSny]

Tλλ

Oh I see. So the thing is that even if $^4_{R_{i j}}$ is of fourth order it doesn't contribute because we see that it is constructed with components of Christoffel's symbol that don't appear in the geodesic equation, so those components aren't measurable, right? I have calculated this component of the Ricci tensor including the "unknown" affine connection terms and I obtained the following:
$$^4_{R_{i j}}=\partial_j ^4_{\Gamma^0_{i0}}-\partial_0 ^3_{\Gamma^0_{ij}}+\partial_j ^4_{\Gamma^k_{ik}}-\partial_k ^4_{\Gamma^k_{ij}}+ ^2_{\Gamma^0_{i0}} ^2_{\Gamma^0_{j0}}+ ^2_{\Gamma^m_{ik}} ^2_{\Gamma^k_{jm}} - ^2_{\Gamma^k_{ij}} ^2_{\Gamma^0_{k0}} - ^2_{\Gamma^m_{ij}} ^2_{\Gamma^k_{mk}}$$
I find that this component of the Ricci tensor isn't only constructed with "unknown" affine connection terms but also with "known" ones. Then, why don't we take into account those terms which contain "known" affine connection terms?

The objective is to determine the free-fall equations of motion to order $\frac {\overline{v}^4}{\overline r}.$ This requires knowing the specific $\overset{\small N}{g}_{\mu \nu}$’s on the right-hand sides of (9.1.16) – (9.1.22). Specifically, we need to find these: $\overset{\small 2}{g}_{00}$, $\overset{\small 4}{g}_{00}$, $\overset{\small 3}{g}_{i0}$, and $\overset{\small2}{g}_{ij}$.

These $\overset{\small N}{g}_{\mu \nu}$’s are determined from the Einstein field equations $\overset{\small 2}{R}_{00} = -8\pi G \overset{\small 0}{S}_{00}$, $\overset{\small 4}{R}_{00} = -8\pi G\overset{\small 2}{S}_{00}$, $\overset{\small 3}{R}_{i0} = -8\pi G \overset{\small 1}{S}_{i0}$, and $\overset{\small 2}{R}_{ij} = -8\pi G \overset{\small 0}{S}_{ij}$, where $S_{\mu \nu} = T_{\mu \nu} - \frac 1 2 g_{\mu \nu} {T^{\lambda}}_{\lambda}$.

Setting up $\overset{\small 4}{R}_{ij} = -8\pi G \overset{\small 2}{S}_{ij}$ would not help. It would result in an equation involving unknown $\overset{\small N}{g}_{\mu \nu}$’s that we do not need for the equations of motion. So, $\overset{\small 4}{R}_{ij} = -8\pi G \overset{\small 2}{S}_{ij}$ is not going to be helpful in determining the $\overset{\small N}{g}_{\mu \nu}$’s that we need.

Also, at the end of that same page (216) he uses the harmonic gauge $\Gamma^{\lambda}=0$. For the component $\lambda=0$ he obtains the equation 9.1.35. For obtaining that equation he made use of the "unknown" affine connection term $^3_{\Gamma^0_{ij}}$. So why can he use the "unknown" terms here but not in the Ricci tensor?

$\overset{\small 3}{\Gamma^0}_{ij}$ does not appear in the list (9.1.16)-(9.1.22). But this term is nevertheless needed in deriving equation (9.1.35), which can be used to help simplify (9.1.30)-(9.1.33). I don’t see any problem here.