How to obtain components of the metric tensor?

[Matter_Matters]

In coordinates given by $x^\mu = (ct,x,y,z)$ the line element is given
$$(ds)^2 = g_{00} (cdt)^2 + 2g_{oi}(cdt\;dx^i) + g_{ij}dx^idx^j,$$
where the $g_{\mu\nu}$ are the components of the metric tensor and latin indices run from $1-3$. In the first post-Newtonian approximation the space time metric is completely determined by two potentials $w$ and $w^i$. The Newtonian potential is contained within $w$ and the relativistic potential is contained with $w^i$. What I don't understand is:

Often in the literature of the first post Newtonian approximation it is just quoted that the components of the metric tensor are given by:
$$\begin{split} g_{00} &= -exp(-2w/c^2), \\ g_{0i} &= -4w^i/c^3, \\ g_{ij} &= - \delta_{ij}\left( 1 +2w/c^2 \right). \end{split}$$

How are these metric components derived?

 

[davidge]

I wonder why Van Hees liked the question but did not give a answer to it.

Would answering that question be too hard?

 

[dextercioby]

"Often in the literature" means nothing. What is the DEFINITION of the w functions?

 

[Matter_Matters]

"Often in the literature" means nothing. What is the DEFINITION of the w functions?

Did you read the question?

 

[Matter_Matters]

I wonder why Van Hees liked the question but did not give a answer to it.

Would answering that question be too hard?

As far as I can tell the answer is actually quite non-trivial. Which is a shame because the coefficients looks like a generalisation of the isotopic and harmonic Schwarzschild line element just incorporating two potentials.

 

[WWGD]

Did you read the question?

It seems you made reference to w but did not define it.

 

[George Jones]

You might try one of Norbert Straumann's books,

 

[pervect]

"Often in the literature" means nothing. What is the DEFINITION of the w functions?

I'd assume the OP is talking about, for instance , the IAU 2000 recommendation B1.3, for instance, <<link>> or <<link>> Though there are some relevant updates to these resolutions - in 2006, I think?

From the first link, which is better formatted.

$$w^i(t,x) = G \int d^3 x' \frac{\sigma^i(t, x')}{\left|x - x' \right|}$$

There's also a defintiion for w, which is a bit more complicated, it has a second time derivative term but is otherwise rather similar to $w^0$. I won't redo it here, this should be enough to locate it in the above links. I'm not really sure where this second time derivative term came from, exactly, by the way.

The above link tells us that s and $s^i$ are the gravitational mass and current densities, but I suspect they meant $\sigma$ and $\sigma^i$ were the gravitational mass and current densities :(. The second link is more official and doesn't seem to have this problem, but it's rather poorly formatted.

I regard these equations (and the reference presents them) as serving as a definition of the coordinate system, rather than explaining where they came from. The OP wants, I think, an explanation of where they came from, not just a statement of what they are. The IAU recommendations would be suitable for the former purpose (definition) but not the later purpose (understanding).

Misner, Thorne, Wheeler (henceforth MTW) has some discussion of where similar-looking (but perhaps not exactly identical) expressions are derived in their text "Gravitation". I'm not sure if it would be the best place to learn from - it's what I have, but I don't think the OP has it, and it's a bit old.

I'll just try to give some basic insight.

We start with $g_{ab} = \eta_{ab} + h_{ab}$ where $\eta_{ab}$ represents a flat-space Minkowskii metric, and $h_{ab}$ is "small" pertubation. Then we come up with the idea that we can expand the pertubation as a power series in terms of a small parameter $\epsilon = M/r$, where M is the mass of some perturbing body, and the location of said perturbing body is treated as if it were in the flat space-time given by the metric $\eta_{ab}$. I've written this as a single perturbing mass M, but we generalize from this to multiple perturbing bodies of mass $m_i$, and then to perturbing mass densities - and mass currents, in the IAU approach, which seems to include more terms than MTW does. Applying the field equations to this and keeping "significant" terms, where "singificant" is defined by the order of the approximation, we have the direct approach of generating the PPN metric coefficients. One can be more clever about this and observe what the form of the metric coefficients might be, but I'm not going to try to be more clever in this short post. MTW talks a bit about how to be more clever, though.