Christoffel Symbols in Terms of $\sqrt{|g|}$

The notation $g$ usually means the determinant of the metric tensor $g_{\mu\nu}$:

$$g = \det(g_{\mu\nu}).$$

A standard identity relates the contracted Christoffel symbols to the metric volume factor:

$$\boxed{\Gamma^\mu_{\mu\nu}=\partial_\nu\ln\sqrt{|g|} =\frac{1}{\sqrt{|g|}}\partial_\nu\sqrt{|g|}}$$

For a positive-definite Riemannian metric, $g>0$, so the absolute-value signs can be omitted. For a Lorentzian metric, $g$ is typically negative, and the invariant expression uses $\sqrt{|g|}$.

This is a formula for the contraction over one upper and one lower index. It is not a formula for every individual $\Gamma^\rho_{\mu\nu}$.

Derivation from the Christoffel formula

For the Levi-Civita connection,

$$\Gamma^\rho_{\mu\nu} =\frac{1}{2}g^{\rho\sigma} \left(\partial_\mu g_{\sigma\nu} +\partial_\nu g_{\sigma\mu} -\partial_\sigma g_{\mu\nu}\right).$$

Set $\rho=\mu$ and sum over $\mu$:

$$\Gamma^\mu_{\mu\nu} =\frac{1}{2}g^{\mu\sigma} \left(\partial_\mu g_{\sigma\nu} +\partial_\nu g_{\sigma\mu} -\partial_\sigma g_{\mu\nu}\right).$$

The first and third terms cancel after relabeling the dummy indices $\mu$ and $\sigma$. Therefore,

$$\Gamma^\mu_{\mu\nu} =\frac{1}{2}g^{\mu\sigma}\partial_\nu g_{\mu\sigma}.$$

Now use Jacobi’s determinant identity:

$$\partial_\nu g = g\,g^{\mu\sigma}\partial_\nu g_{\sigma\mu}.$$

Consequently,

$$\Gamma^\mu_{\mu\nu} =\frac{1}{2g}\partial_\nu g =\partial_\nu\ln\sqrt{|g|}.$$

The last equality follows because, away from a degenerate metric,

$$\partial_\nu\ln\sqrt{|g|} =\frac{1}{2}\frac{\partial_\nu g}{g}.$$

For a Levi-Civita connection, the lower indices are symmetric:

$$\Gamma^\rho_{\mu\nu}=\Gamma^\rho_{\nu\mu}.$$

Thus the equivalent contraction

$$\Gamma^\mu_{\nu\mu}$$

has the same value.

Worked example: polar coordinates

The Euclidean plane has metric

$$ds^2=dr^2+r^2d\theta^2.$$

In coordinates $(r,\theta)$, the metric matrix is

$$(g_{\mu\nu})= \begin{pmatrix} 1&0\\ 0&r^2 \end{pmatrix}, \qquad g=\det(g_{\mu\nu})=r^2.$$

For $r>0$,

$$\sqrt{g}=r.$$

The determinant formula gives

$$\Gamma^\mu_{\mu r} =\frac{1}{\sqrt{g}}\frac{\partial}{\partial r}\sqrt{g} =\frac{1}{r}\frac{\partial r}{\partial r} =\frac{1}{r},$$

and

$$\Gamma^\mu_{\mu\theta} =\frac{1}{\sqrt{g}}\frac{\partial}{\partial\theta}\sqrt{g}=0.$$

Direct calculation gives the nonzero Christoffel symbols

$$\Gamma^r_{\theta\theta}=-r, \qquad \Gamma^\theta_{r\theta} =\Gamma^\theta_{\theta r}=\frac{1}{r}.$$

Therefore,

$$\Gamma^\mu_{\mu r} =\Gamma^r_{rr}+\Gamma^\theta_{\theta r} =0+\frac{1}{r},$$

in agreement with the determinant formula.

Relation to divergence

The identity is especially useful when writing the divergence of a vector field $V^\mu$. By definition,

$$\nabla_\mu V^\mu =\partial_\mu V^\mu+\Gamma^\mu_{\mu\nu}V^\nu.$$

Substitute the determinant expression:

$$\nabla_\mu V^\mu =\partial_\mu V^\mu +\frac{1}{\sqrt{|g|}} \left(\partial_\nu\sqrt{|g|}\right)V^\nu.$$

Combining the terms gives the coordinate-invariant volume-element form

$$\boxed{\nabla_\mu V^\mu =\frac{1}{\sqrt{|g|}} \partial_\mu\left(\sqrt{|g|}V^\mu\right)}.$$

The corresponding volume element is

$$\mathrm{d}V=\sqrt{|g|}\,\mathrm{d}x^1\cdots\mathrm{d}x^n.$$

This explains geometrically why $\sqrt{|g|}$ appears: it converts coordinate volume into metric volume.

What the formula does not tell you

Knowing $g=\det(g_{\mu\nu})$ alone is generally insufficient to reconstruct all Christoffel symbols. The determinant provides only one scalar function, whereas the metric contains many independent components. Different metrics can have the same determinant but different connections.

For example, in two dimensions, the metrics

$$\begin{pmatrix}e^{2f(x)}&0\\0&e^{-2f(x)}\end{pmatrix} \quad\text{and}\quad \begin{pmatrix}1&0\\0&1\end{pmatrix}$$

both have determinant $1$, but the first generally has nonzero Christoffel symbols when $f$ is nonconstant, while the second has zero Christoffel symbols in Cartesian coordinates.

To calculate an individual symbol, use

$$\Gamma^\rho_{\mu\nu} =\frac{1}{2}g^{\rho\sigma} \left(\partial_\mu g_{\sigma\nu} +\partial_\nu g_{\sigma\mu} -\partial_\sigma g_{\mu\nu}\right).$$

The compact $\sqrt{|g|}$ formula is therefore best understood as a formula for the contracted connection and as a convenient route to divergence and volume-element identities.