Proving the Existence of Locally Inertial Coordinates

This question is regarding the proof of the existence of locally inertial coordinates, outlined in Sean Carroll’s Spacetime and Geometry book (Chapter 2, page 74). In particular, I believe that extra factors of $2$ or $frac{1}{2}$ should be present in the Taylor expansions the proof uses – these are later flagged in red. I’ve checked the errata here.

Given a manifold $M$, at any point $p in M$, we want to show that we can move from a coordinate system $x^{mu}$ to $x^{hat{mu}}$ such that:

1. $g_{hat{mu} hat{nu}}(p) = eta_{hat{mu} hat{nu}}$
2. $partial_{hat{sigma}} g_{hat{mu} hat{nu}}(p) = 0$

For simplicity, choose the origin of both sets of coordinates to coincide with $p$. We also know the metric components transform like:

$$g_{hat{mu} hat{nu}} = frac{partial x^{mu}}{partial x^{hat{mu}}}frac{partial x^{nu}}{partial x^{hat{nu}}} g_{mu nu} tag{2.48}$$

Carroll now Taylor expands both sides of this equation to show that there are enough degrees of freedom to satisfy constraints 1 and 2. However, I have obtained extra factors of 2 when trying to perform the expansion myself.

The expansion of the old coordinates $x^{mu}$ looks like
$$x^{mu} = left(frac{partial x^{mu}}{partial x^{hat{mu}}}right)_p x^{hat{mu}} + frac{1}{2} left(frac{partial^2 x^{mu}}{partial x^{hat{mu}_1} partial x^{hat{mu}_2}}right)_p x^{hat{mu}_1} x^{hat{mu}_2} + dots$$
[…] Then using some extremely schematic notation, the expansion of $(2.48)$ to second order is
$$(hat{g})_p + (hat{partial} hat{g})_p hat{x} + (hat{partial} hat{partial} hat{g})_p hat{x}hat{x} = left(frac{partial x}{partial hat{x}} frac{partial x}{partial hat{x}} gright)_p + left(frac{partial x}{partial hat{x}} frac{partial^2 x}{partial hat{x} partial hat{x}}g + frac{partial x}{partial hat{x}} frac{partial x}{partial hat{x}} hat{partial}g right)_p hat{x} + left(frac{partial x}{partial hat{x}} frac{partial^3 x}{partial hat{x} partial hat{x} partial hat{x}}g + frac{partial^2 x}{partial hat{x} partial hat{x}} frac{partial^2 x}{partial hat{x} partial hat{x}}g + frac{partial x}{partial hat{x}} frac{partial^2 x}{partial hat{x} partial hat{x}} hat{partial}g + frac{partial x}{partial hat{x}} frac{partial x}{partial hat{x}} hat{partial} hat{partial}g right)_p hat{x} hat{x}$$

I agree with the expansion of $x^{mu}$, but disagree with the expansions of the left and right hand side of $(2.48)$. I’m introducing a book-keeping parameter $epsilon$ to help track the order of particular terms to make them easier to group together.

Looking at the LHS:

$$g_{hat{mu} hat{nu}} (x^{hat{alpha}}) = g_{hat{mu} hat{nu}} (0) + epsilon partial_{hat{sigma}} g_{hat{mu} hat{nu}} (0) x^{hat{sigma}} + epsilon^2 frac{1}{2} partial_{hat{sigma}} partial_{hat{rho}} g_{hat{mu} hat{nu}} (0)x^{hat{sigma}}x^{hat{rho}} tag{A} $$

Therefore, I would have actually expected the LHS to be $$(hat{g})_p + (hat{partial} hat{g})_p hat{x} + color{red}{frac{1}{2}} (hat{partial} hat{partial} hat{g})_p hat{x}hat{x}$$

Looking at the RHS:

The expansion of $g_{mu nu}$ follows (A), but with $g_{hat{mu} hat{nu}}$ replaced with $g_{mu nu}$.

$$frac{partial x^{mu}}{partial x^{hat{mu}}} = left(frac{partial x^{mu}}{partial x^{hat{mu}}}right)_p + epsilon left(frac{partial^2 x^{mu}}{partial x^{hat{mu}} partial x^{hat{sigma}}}right)_p x^{hat{sigma}} + frac{1}{2} epsilon^2 left(frac{partial^3 x^{mu}}{partial x^{hat{mu}} partial x^{hat{sigma}} partial x^{hat{rho}}}right)_p x^{hat{sigma}} x^{hat{rho}} $$

We can now substitute both these expansions into $(2.48)$.

• $epsilon^0$ terms: the only way to get such a term is by multiplying by the $epsilon^0$ parts of the expansion in each of the 3 terms that are being multiplied on the RHS of $(2.48)$. This corresponds to $g_{mu nu}(p) left(frac{partial x^{mu}}{partial x^{hat{mu}}}right)_p left(frac{partial x^{nu}}{partial x^{hat{nu}}}right)_p$ in agreement with Carroll.
• $epsilon$ terms: we need to get an order $epsilon$ term from one of the multiplicative factors on the RHS of $(2.48)$ and multiply it by the $epsilon^0$ terms from the expansion of the other factors. I believe we should have:

$$left(frac{partial x^{mu}}{partial x^{hat{mu}}}right)_p left(frac{partial^2 x^{nu}}{partial x^{hat{nu}} partial x^{hat{sigma}}}right)_p g_{mu nu}(p) x^{hat{sigma}} + left(frac{partial x^{nu}}{partial x^{hat{nu}}}right)_p left(frac{partial^2 x^{mu}}{partial x^{hat{mu}} partial x^{hat{sigma}}}right)_p g_{mu nu}(p) x^{hat{sigma}} + left(frac{partial x^{mu}}{partial x^{hat{mu}}}right)_p left(frac{partial x^{nu}}{partial x^{hat{nu}}}right)_p partial_{hat{sigma}} g_{mu nu}(p) x^{hat{sigma}}$$

Since $g_{mu nu} = g_{nu mu}$, this corresponds to:

$$left(color{red}{2} frac{partial x}{partial hat{x}} frac{partial^2 x}{partial hat{x} partial hat{x}}g + frac{partial x}{partial hat{x}} frac{partial x}{partial hat{x}} hat{partial}g right)_p hat{x}$$

in disagreement with Carroll.

• $epsilon^2$ terms: using similar reasoning to above, I believe that Carroll’s $hat{x} hat{x}$ coefficient should actually be

$$left(color{red}{2} frac{partial x}{partial hat{x}} frac{partial^3 x}{partial hat{x} partial hat{x} partial hat{x}}g + frac{partial^2 x}{partial hat{x} partial hat{x}} frac{partial^2 x}{partial hat{x} partial hat{x}}g + color{red}{2}frac{partial x}{partial hat{x}} frac{partial^2 x}{partial hat{x} partial hat{x}} hat{partial}g + frac{partial x}{partial hat{x}} frac{partial x}{partial hat{x}} hat{partial} hat{partial}g right)_p$$

Summary of Question

I was wondering if these extra factors in the expansion of $(2.48)$ that I’ve highlighted in red are actually correct; it doesn’t affect the overall argument (which counts the number of free parameters we have), but I’m still quite confused why these identified factors aren’t present.