Why are the red parts of this expression present $\int_a^b \sqrt{g_{cd}\frac{dx^c}{\color{red}{d\lambda}}\frac{dx^d}{\color{red}{d\lambda}}}d\lambda$?

Question

I am now trying to teach myself about geodesics, and a passage of my notes reads:

In this chapter geodesics have been introduced as generalizations of straight lines through $$\frac{Du^a}{ds}=0$$ Here two alternative characterizations of geodesics are considered.

$(\mathrm{a})$ Consider a curve $C$ joining two fixed points $x_A$ and $x_B$. This can be defined through $x^a=x^a(\lambda)$, where $\lambda \in [a, b]$ is a parameter. The length of the curve can be expressed as an integral $$L=\int_a^b \sqrt{g_{cd}\frac{dx^c}{\color{red}{d\lambda}}\frac{dx^d}{\color{red}{d\lambda}}}d\lambda\tag{?}$$ Geodesics may be defined as curves that minimise (or maximise) $L$ for fixed end points $x_A$ and $x_B$. The integral is awkward to work with due to the parametrisation ambiguity of the integral.

The second method, $(\mathrm{b})$ is no interest to what I have to ask here. The simple question I have is why are there the red parts in denominators of $(\mathrm{?})$. In other words, I think those denominators should be unity, $d\lambda=1$. I say this because from the equation for the differential line segment (squared): $$ds^2=g_{cd} dx^c dx^d\tag{A}$$

Since the action, $$L=\int_{a}^{b} ds = \int_{a}^{b}\sqrt{g_{cd}dx^cdx^d}d\lambda\tag{B}$$ I think the notes are wrong and $(\mathrm{?})$ should actually be $$\int_{a}^{b}\sqrt{g_{cd}dx^cdx^d}d\lambda$$

I thought the above would be enough to convince others that this is all just another typo and I have the right expression, but sadly, I came across these handwritten notes, which I will embed as an image below:

Seeing this only makes me believe that I am the one making the mistake and that $(\mathrm{?})$ truly is correct.

So, how can $(\mathrm{?})$ possibly be correct when it is simply derived from taking the square root of $(\mathrm{A})$ [as in $(\mathrm{B})$]?

Answer 1

Carrying your reasoning forward, since $ds^2=g_{cd}dx^cdx^d$, we'll get $$ \int_a^b ds=\int_a^b \sqrt{g_{cd}dx^c dx^d} =\int_a^b \sqrt{g_{cd}\frac{dx^c}{d\lambda}\frac{dx^d}{d\lambda}}d\lambda. $$

Answer 2

There's a serious problem already in the first half of Equation (B):

$$ L \stackrel?= \int_a^b ds. $$

Strictly speaking, this integral that you wrote in the middle of Equation (B) is a simple single-variable integral that runs from $s=a$ to $s=b$ (that's what your notation means), evaluating to $$ \int_a^b ds = b - a. $$

Surely that is not what you wanted to write. You want an integral where there's a variable $\lambda$ that goes from $\lambda = a$ to $\lambda = b.$ You might write that integral like this: $$ L = \int_a^b \frac{ds}{d\lambda}\, d\lambda. $$

You also say that $ds^2=g_{cd}dx^cdx^d,$ which you later use to justify the substitution $ds = \sqrt{g_{cd}dx^cdx^d}.$ That might be fine and well in the proper context, but in getting from $\int_a^b \frac{ds}{d\lambda}\, d\lambda$ to $\int_a^b \fbox{something else}\, d\lambda$ you need to replace $\frac{ds}{d\lambda},$ not just $ds.$

But if you accept that $ds^2 = g_{cd} dx^c dx^d$ you should also accept that $$ \left(\frac{ds}{d\lambda}\right)^2 = g_{cd} \frac{dx^c}{d\lambda}\frac{dx^d}{d\lambda}. $$

Now take the square root on both sides, substitute for $\frac{ds}{d\lambda}$ in $L = \int_a^b \frac{ds}{d\lambda}\, d\lambda,$ and you have the equation in your notes.