Raising/Lowering Metric Indices: Explained

[WWCY]

TL;DR Summary

What are the rules?

If I have a metric of the form $g_{\mu \nu} = f_{\mu \nu} + h_{\mu \nu}$ where $f_{\mu \nu}$ is the background metric and $h_{\mu \nu}$ the perturbation, how do I raise and lower indices of tensors?

For instance, I was told that $G_{ \ \nu}^{\mu} = f^{\mu \nu '} G_{\nu ' \nu }$. But shouldn't $G_{ \ \nu}^{\mu} = g^{\mu \nu '} G_{\nu ' \nu }$ be true as well? What exactly are the "rules" behind these operations?

PS If the answer is related to differential geometry, could I be also be pointed to a (introductory) source that explains it?

Cheers.

 

[Ibix]

You raise and lower indices using the metric $g$ with upper or lower indices as appropriate. However, if you are perturbing the metric then the components of $h$ are small compared to those of $f$, and components of $(h)^2$ and derivatives thereof are negligible.

I would imagine that if you expand out both tensors in $g^{\mu\nu'}G_{\nu'\nu}$ in terms of $f$ and $h$ you will find that it is equal to $f^{\mu\nu'}G_{\nu'\nu}$ plus terms like $h^{\mu\nu'}h_{\nu'\nu}$ and derivatives thereof, which you would neglect.

 

[robphy]

First realize that "raising" and "lowering" of indices are really just shortcuts:
Given $v^a$, then $v_b \equiv v^a g_{ab}$.
So, the real object is $v^a g_{ab}$, but it gets tiring to write that all of the time.
So, a convention is adopted to define the shortcut $v_b$.

So, $v^a g_{ab}=v^a f_{ab}+v^a h_{ab}$.
Which term do want to declare a shortcut for?The following references suggest using the "background" unperturbed metric (in your notation, $f_{ab}$).From the following reference, a scan of Geroch's 1972 General Relativity notes [maybe posted by Ashtekar?]
http://www.gravity.psu.edu/links/general_relativity_notes.pdfSee Section 39. Linearization (page 143 [pdf page 145]).
On page 144 [pdf page 146] between equations (156) and (157), Geroch declares

We shall hereafter raise and lower indices with $g_{ab}$, the unperturbed metric".

which makes sense because your unperturbed metric likely corresponds to an exact solution [not necessarily Minkowski] that you know about, rather than some perturbed-metric.

However, the notation in those notes might be a little confusing since
in the first paragraph of that section,
he started by writing $g_{ab}=\eta_{ab}+\gamma_{ab}$ as an example when perturbing from Minkowski spacetime.
But between equations (153) and (154), he essentially does the following:

Using a family of metrics $g_{ab}(\lambda)$, where $\lambda \in[0,1)$
the perturbed metric is [in general]
$\begin{align*} g_{ab}(\lambda) &= \left(g_{ab}(\lambda)\right|_{\lambda=0}+\lambda\ \frac{d}{d\lambda}\left(g_{ab}(\lambda)\right|_{\lambda=0}\\ &= g_{ab}(0)+\lambda\ \frac{d}{d\lambda}\left(g_{ab}(\lambda)\right|_{\lambda=0} \end{align*}$
in terms of the unperturbed metric $g_{ab}(0)$ (which he now calls "$g_{ab}"$ ... between (153) and (154).)

For another reference, look at Wald's text [which is influenced by Geroch, as noted in the preface]

• Section 4.4 Linearized Gravity: The Newtonian Limit and Gravitational Radiation,
  

  $g_{ab} =\eta_{ab} + \gamma_{ab}$
  ...
  In order not to have $\gamma_{ab}$ hidden in a raised or lowered index,
  it is convenient to raise and lower tensor indices with $\eta_{ab}$ and $\eta^{ab}$ rather than $g_{ab}$ and $g^{ab}$ . We will adopt this notational convention for the remainder of this section with one exception :
  The tensor $g^{ab}$ itself will still denote the inverse metric, not $\eta^{ac}\eta^{bd}g_{cd}$. It should be noted that in the linear
  approximation we have
  $g^{ab}= \eta^{ab} -\gamma^{ab}$ (4 .4 .2)
  since the composition of the right-hand sides of equations (4 .4.1) and (4 .4 .2) differs
  from the identity operator only by terms quadratic in $\gamma_{ab}$.

• Section 7.5 on Perturbations,
  

  $g_{ab} = {}^0 g_{ab}+\gamma_{ab}$
  ...
  (between Eq 7.5.14 and 7.5.15)
  Since all quantities aside from $\gamma_{ab}$, now refer only to the background metric ${}^0 g_{ab}$, we
  shall in the following drop the superscript zero on the background metric and its
  derivative operator, and we shall use the background metric $g_{ab}$ and its inverse $g^{ab}$ to raise and lower indices.