In "Tensors" by A. Das on p166 the author derives the first integral (5.202) of the geodesic equation. To achieve this, he uses the chain rule

$$\frac{dg_{jk}}{d\tau} = \frac{\partial g_{jk}}{\partial x^i}\frac{dx^i}{d\tau},$$

where $g$ is the metric tensor, $x^i$ are the coordinates and $\tau$ is the affine parameter. But $x^i$ are also $x^i=x^i(\tau)$, the parametric equations of the geodesic curve. This is confusing.

Does $g$ actually depend on $\tau$? One would think it depends on the spatial coordinates $x^i$ only. One would also think different notation should be used for spatial coordinates to distinguish them from the parametric curve functions. Is the above chain rule correct?

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    $\begingroup$ I am not sure if I understood your question correctly. But when you are working with total derivative, you should note that It doesn't matter whether your function depends on your affine parameter explicitly or not. In other words, metric depends on spatial coordinates and spatial coordinates depends on proper time, so it's meaningful to take total derivative with respect to proper time. Partial derivative is different story though. (en.wikipedia.org/wiki/Total_derivative) and (en.wikipedia.org/wiki/Partial_derivative) $\endgroup$ – Paradoxy Jul 11 '19 at 22:30

The components of the metric tensor are functions which eat four spacetime coordinates and spit out real numbers:

$$g_{ij} : \mathbb R^4 \rightarrow \mathbb R$$ $$ {\mathbf{x}} \mapsto g_{ij}({\mathbf{x}})$$

A parametric curve $\gamma$ is a function which eats a parameter value and spits out four spacetime coordinates:

$$\gamma : \mathbb R \rightarrow \mathbb R^4$$ $$\tau \mapsto {\mathbf{x}}(\tau)$$

If you compose them, you get a map which eats a parameter value and spits out a real number:

$$g_{ij}\circ \gamma : \mathbb R \rightarrow \mathbb R$$ $$ \tau \mapsto (g_{ij}\circ \gamma)(\tau) = g_{ij}\big({\mathbf{x}}(\tau)\big)$$

The chain rule then gives us that

$$\frac{d}{d\tau} \big(g_{ij}\circ \gamma\big)(\tau) = \frac{\partial g_{ij}}{\partial x^k} \frac{dx^k}{d\tau}$$

The notation in your book is mildly ambiguous but standard. Any time the idea of a "total derivative" comes up, just recognize that you're differentiating the composition of a function with some parameterized curve.


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