Riemann volume forms: Absolute value of the determinant of the metric, or not? When using differential geometry in the context of General Relativity, you are usually taught that the general invariant scalar volume element reads
$$ \mathrm{d}V=\sqrt{|g|}\mathrm{d}x^n $$
for an n-dimensional Riemannian manifold. With $g$ being the determinant of the metric tensor. Cf. for example [1,2] and also the Wikipedia article on volume forms [3].
Now, my question is: Is it just a convention that we use the $\textit{absolute value}$ of the determinant when defining the volume form?
In the Wikipedia article [3] it is mentioned that volume forms are in general not unique. Does this imply that we could equally say that $\mathrm{d}V=\sqrt{g}\mathrm{d}x^n$ is the volume form of choice?
PS: Note that this would imply for the Minkowski metric ($g=\mathrm{diag}(-1,1,1,1)$), that in one incident that volume form would be real and in the other imaginary. See for example this article [4] and the consequences for the complex structure of Quantum Mechanics.
[1] http://www.blau.itp.unibe.ch/newlecturesGR.pdf Eq. 4.47 and 4.51
[2] https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll2.html Eq. 2.48
[3] https://en.wikipedia.org/wiki/Volume_form#Riemannian_volume_form
[4] Lindgren, Liukkon 2019: https://www.nature.com/articles/s41598-019-56357-3
 A: The absolute value is "just convention" in some sense, but it is better motivated than your choice of definition might suggest.
In $\mathbb{R}^n$, one typically defines a notion of "volume" such that the volume of the box is given by the product of its side lengths.
$$
\text{volume}\left([a_1,b_1]\times\cdots\times[a_n,b_n]\right)=\prod_{i=1}^n|b_i-a_i|
$$
This (eventually) leads to the Lebesgue measure on $\mathbb{R}^n$. One could just as easily define the volume of such a box to be some other (negative, or even complex) number, but this would be a rather unusual choice, generally speaking. If one were to make such a choice, it would probably be best to not call the result "volume" to avoid confusion.
On an pseudo-Riemannian $n$-manifold $(M,g)$, there is an analogue of the Labesgue measure which has essentially all the same properties. In the oriented case*, it is most easily described by a differential $n$-form $\omega_g$ known as the Riemann volume form. The analogue of the "volume of a box" condition is an "infinitesimal cube" condition
$$\tag{1}
\omega_g(e_1,\cdots,e_n)=1 \\
e_1,\cdots,e_n\in T_pM\ \ \ \text{any oriented orthonormal basis}
$$
This condition uniquely determines $\omega_g$, and implies the coordinate definition: in oriented coordinates $x^1,\cdots,x^n$, we have $\omega_g=\sqrt{|g|}dx^1\wedge\cdots\wedge dx^n$.
One could just as easily use some other (complex) number in place of $1$ in $(1)$, but the result wouldn't be the Riemann volume form (as it is commonly understood), but some scalar multiple of it.
*There is actually no need for an orientation, provided one uses an $n$-density in place of an $n$-form.
