# Restrictions on 1st order perturbation theory

• I
Hello guys,
I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory.
I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski ##\eta_{\mu\nu}## + perturbation ## h_{\mu\nu} = \epsilon \gamma_{\mu\nu} ## ignoring terms of order ##\epsilon^2## in subsequent calculations.

1. Why can we do so if Schwarzschild's (exterior) solution is supposed to work even in the vecinity of big black holes where one could think the metric wouldn't be just Minkowski + a small perturbation.

2. If I'm working in coordinates that force the spacetime to be conformally-flat (like in FLRW(##k=0##) using cartesian coordinates and conformal time) is it possible to decompose ##g_{\mu\nu}## as ## \eta_{\mu\nu} + h_{\mu\nu}## ?, even more, given that ## g_{\mu\nu} ## is conformally flat, ie: ## g_{\mu\nu}=\Omega\eta_{\mu\nu} ##, then ## h_{\mu\nu} = (\Omega-1)\eta_{\mu\nu} ## ?

Thanks in advance for any idea you can comment below.

Orodruin
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I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric gμνgμνg_{\mu\nu} as Minkowski ημνημν\eta_{\mu\nu} + perturbation hμν=ϵγμνhμν=ϵγμν h_{\mu\nu} = \epsilon \gamma_{\mu\nu} ignoring terms of order ϵ2ϵ2\epsilon^2 in subsequent calculations.
You might be able to argue for the form of the Schwarzschild metric in the linear regime, but to derive the Schwarzschild solution what you assume is typically spherical symmetry, not the linear regime.

You might be able to argue for the form of the Schwarzschild metric in the linear regime, but to derive the Schwarzschild solution what you assume is typically spherical symmetry, not the linear regime.

I know a way to deduce the solution in terms of one factor A(r) that has to be (##1-2GM/r##).
In order to complete the solution and find the equivalence between both terms you have to use perturbation theory to make that factor consistent with special relativity, ie: if I do so at first order, ##h_{00}## gives me a relation between that factor and the classical newtonian potential, but I'm not sure if I can do that. (I mean, I get the line element that I want but I'm not sure if the supposition of neglecting ##\epsilon^2## in the computations is correct).

PeterDonis
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I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski ##\eta_{\mu\nu}## + perturbation ##h_{\mu\nu}## = ##\epsilon \gamma_{\mu\nu}## ignoring terms of order ##\epsilon^2## in subsequent calculations.

Can you give more details, and/or a reference? This is not something I'm familiar with.

Why can we do so if Schwarzschild's (exterior) solution is supposed to work even in the vecinity of big black holes where one could think the metric wouldn't be just Minkowski + a small perturbation.

Indeed, which is why I'm skeptical that this can actually be done. Hence my request for a reference.

If I'm working in coordinates that force the spacetime to be conformally-flat

Whether or not a spacetime is conformally flat is a geometric fact about the spacetime, independent of any choice of coordinates.

Can you give more details, and/or a reference? This is not something I'm familiar with.

Sadly, I'd have to write my lecture notes in PDF in order to give you more details about it but I think that would take a long time, more than desired. I'm going to wait a bit more to see what comments arrive now and... maybe, some days later I can upload my notes and then tag you to them.

Well, it's even worse than that, haha.
(I have to fix some steps in the calculations that were made in my class - I think there are better ways to develop them in a more formal fashion and I can improve the steps before uploading them, but, again, it takes time).

PeterDonis
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I'd have to write my lecture notes in PDF

Your profile says you are an undergrad. Are these lecture notes from a class you're taking? What class and what textbook is being used?

Your profile says you are an undergrad. Are these lecture notes from a class you're taking? What class and what textbook is being used?

Just teacher and student discussions. Our classes follow my teacher's hand-written lecture notes and new demonstrations he prepares for each specific class to exhibit an interesting fact of the topic under exposition. There's no book, he just writes down his demonstrations and we discuss it with him within the class.