What the smallness of the metric perturbation means physically is somewhat application-dependent. If you're doing a post-Newtonian expansion, that's an expansion in orders of v, the typical velocities in your gravitational system. If you're dealing with gravitational radiation far away from any...
For some reason I can't edit my post, so:
Just for posterity, my last sentence in the above is of course complete nonsense. A constant cannot be of any order in an expansion parameter other than zero. The real reason is a bit more complicated and has to do with the derivatives of the metric...
Einstein published what I think were the first detailed treatments of the post-Newtonian expansion of GR in 1938, 1940 and 1949. Perhaps not as ground-breaking as some of his quantum work, but those first results are the foundation of an extensive modern PN formalism used in gravitational wave...
I was aware that Weinberg performs the calculations in great detail, but the justification for dropping the g_{i0} to first order wasn't clear to me. Having another look at it, I just realized a mistake I was making, however: I was coupling n-th order terms in the Einstein tensor to n-th order...
Perhaps someone will recognize my problem if I phrase it slightly differently: the original question I had was why, in the PN expansion of the metric, the g_{i0} components are assumed to be zero to first order. See e.g. page 239 of Maggiore. Weinberg does the same thing in his chapter on...
Well, that is exactly the assumption made in post-Newtonian theory. It deals with systems held together by gravitational interaction, thus the virial theorem holds.
Yes, my point is that in the "first order" expansion, i.e. the Newtonian limit, you actually keep a term h_{00,i}, which is second order, not first order at all. To see this, consider the matching to Newtonian gravity, which gives you
h_{00} = \phi = \frac{GM}{r}
where I neglected some...
It does seem like cheating. However, I wouldn't be so worried except for the fact that when going to post-Newtonian order, e.g. Maggiore takes this static Newtonian limit and uses it to assume that there are no first-order metric components in the expansion in v, which doesn't seem to have any...
Yes, that's the argument most books give, but I don't understand it. Once you're done with the calculation of the Newtonian limit, you find that h_{00}, which you kept, is actually second order in v by the virial theorem, so if you were expanding to first order, you should have thrown that out too.
Does anyone have any insight? For example, when I calculate
\Gamma^i_{00}
to second order in v (since you end up with g_{00} to second order, I'm keeping everything to that order), I find not only the standard term, but also a term
g_{i0,0}
which is second order as long as you keep...
After a lot of bouncing around on Google, I found some of the papers in random locations, but am still looking for some of Einstein's:
Einstein A, (1916) Approximate integration of the field equations of gravitation. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften...
Hey all,
I was looking for some older papers by Einstein and De Sitter from around 1916 and my search is coming up blank. Google (Scholar) is not returning anything useful, the journal archives don't go nearly so far back, the particular volumes are not on archive.org, and even my local...