I’m following the lecture here, and this question is with respect to the content on the whiteboard at the timestamp provided.

Consider Newtonian spacetime $(mathcal{M}, mathcal{O}, mathcal{A}, nabla, t)$ where $(mathcal{M}, mathcal{O}, mathcal{A})$ is a smooth 4-dimensional manifold where all charts $(mathcal{U}, x)$ are of the form

$$x^0 : mathcal{U} to mathbb{R}\

x^1 : mathcal{U} to mathbb{R}\

vdots\

x^3 : mathcal{U} to mathbb{R}

$$

where $x^0 = t|_mathcal{U}$ is the restriction of the absolute time function $t$ to the chart domain $mathcal{U}$ and $nabla$ is a prescribed covariant derivative operator with given connection coeffitient functions.

I’m trying to reevaluate the claim on the right, namely

$$ 0 = nabla dt.$$

As I understand it, one does so by picking any direction $frac{partial}{partial x^a}$ from a chart-induced basis for $a=0,1, ldots, 3$, then act on this vectorfield with the prescribed covariant derivative operator and apply the result to the $(0,1)$-tensor field $dt$, which is the gradient of the 0th component function: $d(x^0)$. The action of the covariant derivative along a vectorfield on a $(p, q)$-tensor field yields again a $(p, q)$-tensor field, hence we can look at the resulting $(0,1)$-tensor-field, i.e. a vector field, component-wise for components $b=0, ldots, 3$. Formally, we apply the rules for covariant derivatives

$$

left(nabla_{frac{partial}{partial x^a}} d(x^0)right)_b = frac{partial}{partial x^a}left(d(x^0)_bright) – Gamma_{b m}^n d(x^0)_n left(frac{partial}{partial x^a}right)^m

$$

According to the notes on the blackboard, the first term should vanish entirely, while the second one should reduce to $Gamma_{ba}^0$. I’m havin trouble to see this.

Let’s start with the first term: $d(x^0)$ is a covector, so $d(x^0)_b$ is the 0th component function. $left(frac{partial}{partial x^a}right)$ is a vector field, which can be applied to a fucntion to yield another function. Alright. Still I can’t see how I need to proceed there.

Secondly, I’m stuck with the rightmost term. Somehow, $left(frac{partial}{partial x^a}right)^m$ must reduce to $delta_a^m$ and $d(x^0)_n$ must reduce to $delta_n^0$, however I don’t see how.

Basically, as far as I understand, the whole expression needs to be the zero function in the end.

Can anyone explain where I seem to go wrong? Or help me understand the step missing?