GR in Newtonian Limit: Understanding Weak Fields & Inequalities

[Silviu]

Hello! I am reading A first course in General Relativity by Schutz and at a point he proves that for a weak gravitational field and assuming $\Lambda = 0$ we have $\Box \bar{h}^{\mu \nu} = -16\pi T^{\mu \nu}$. Leaving the notations aside, he says that for a weak gravitational field (and non-relativistic speeds) we have $|T^{00|}>>|T^{0i}|>>|T^{ij}|$ and this implies $|\bar{h}^{00}|>>|\bar{h}^{0i}|>>|\bar{h}^{ij}|$. Can someone explain to me why do we have this last inequality? This is Chapter 8.3 in the second edition. Like for example, taking $T^{0i}=0$, we get $\Box\bar{h}^{0i}=0$. Why does this implies in any way that $\bar{h}^{0i}$ is very small? Thank you!

 

[Orodruin]

The right-hand side is the source term. Without a source, your solution is identically $\bar h^{0i} = 0$ (assuming homogeneous boundary conditions).

 

[Silviu]

The right-hand side is the source term. Without a source, your solution is identically $\bar h^{0i} = 0$ (assuming homogeneous boundary conditions).

Sorry I am a bit confused, if $\bar{h}^{0i}$ is a constant, doesn't the equality still holds?

 

[Orodruin]

If it is a constant it does not satisfy homogenous boundary conditions. Also, you could just absorb it into the zeroth order metric.