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I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be utilized? I have searched online, but all the literature I have encountered assumes you accept it as the volume element.

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Towards the end of Chapter 2 of Sean Carroll's lecture notes: preposterousuniverse.com/grnotes –  DJBunk May 9 '13 at 12:34
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Doesn't the wikipedia article explain it sufficiently? It seems to have a nice simple example too. –  twistor59 May 9 '13 at 12:45

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I) Pragmatically speaking, the most important property of $\sqrt{-g}$ for model building purposes, is not per se the fact that $\sqrt{-g}d^{4}x$ measures the volume element of a 4-dimensional Parallelepiped with infinitesimal edges $dx^0, \ldots, dx^3$.

II) A more important property is that $\sqrt{-g}d^{4}x$ transforms as a scalar (i.e. is invariant) under general coordinate transformations $$x^{\prime \mu} ~=~ f^{\mu}(x^0 , \ldots, x^3).$$ In more detail, the factor $\sqrt{-g}$ transforms as a density under general coordinate transformations, cf. also this Phys.SE answer.

III) Example: The Einstein-Hilbert action

$$S_{EH}[g]~:=~\int R \sqrt{-g}d^{4}x$$

is invariant under change of coordinates, because the scalar curvature $R$ and volume element $\sqrt{-g}d^{4}x$ are both invariant under general coordinate transformations. Thus the action $S_{EH}[g]$ is a geometric quantity that doesn't depend on the choice or coordinates.

IV) The factor $\sqrt{-g}$ is the so-called canonical density on a Lorentzian manifold. If the Lorentzian manifold is endowed with another density $\rho$, one could in principle use that to build actions and get interesting theories.

V) Finally let us mention that on a symplectic manifold $(M,\omega)$, the canonical density is instead given by the Pfaffian

$$ {\rm pf}(\omega_{ij}) $$

of the symplectic matrix $\omega_{ij}$.

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Another thing one could mention is that given an (ordered) basis of orthonormal 1-forms $\theta_i$ this volume form can be written as $\theta_1 \wedge \theta_2 \wedge \dots \wedge \theta_n$ up to sign (which depends on choice of orientation and the signature of the metric) –  alexarvanitakis May 9 '13 at 20:28

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