What does this correspond to geometrically?

Question

I recently was playing around with some maths and pondered the following:

Let us define $$ \Delta \vec s_{12} = \vec s_1 - \vec s_2 $$

Squaring both sides:

$$ |\Delta s_{12} |^2 = |\vec s_1|^2 + |\vec s_2|^2 + 2 s_1 \cdot s_2 $$

Choosing both $|s_1|=|s_2|$ and dividing by $|s_1|^2$:

$$ \frac{|\Delta s_{12} |^2}{|s_1|^2} = 2(1 + \cos(z_{12})) $$

Where $z_{12}$ is the angle. Multiplying both sides with $ 1 - \cos(z_{12})$ we get:

$$ \frac{|\Delta s_{12} |^2}{|s_1|^2} \cos^2(\frac{z_{12}}{2})= \sin^2(z_{12}) $$

Taking $\Delta s \to ds $, $z_{12} \to \epsilon$ and multiplying $ d \epsilon$ both sides and integrating around in a loop.

$$\oint \frac{|d s |^2}{|s|^2} \cos^2(\frac{\epsilon}{2}) d \epsilon = \oint \sin^2(\epsilon) d \epsilon $$

I was originally thinking of this in Hyperbolic or Euclidean geometry in $2$ dimensions. Is this correct? Can this be extended to $4$ dimensions? What does this correspond to geometrically?

Answer 1

$$|\Delta s_{12} |^2 = |\vec s_1|^2 + |\vec s_2|^2 + 2 s_1 \cdot s_2$$

That should be

$$|\Delta s_{12} |^2 = |\vec s_1|^2 + |\vec s_2|^2 - 2 s_1 \cdot s_2$$

Choosing both $|s_1| = |s_2|$

This means that your calculations only apply to a circle around the origin. It may seem like infinitesimally you can still apply it to other loops, but you are only picking up the contribution in the direction of $d\theta$, and dropping that in the direction of $dr$. So, no, it only holds for the circle.

Multiplying both sides with $1 - \cos(z_{12})$

Because of the earlier sign change, this would be $1 + \cos(z_{12})$. However, the next line has another error that cancels out the previous (perhaps the sign error was only when you copied it into here?)$$\cos^2 \frac \theta 2 = \frac{1 + \cos \theta}2\ne \frac{1 - \cos \theta}2$$ so the next formula is correct again.

Taking $\Delta s \to ds, z_{12} \to \epsilon$

The cosine of $z_{12}$ is the length of $\Delta s$ divided by the radius. If $\Delta s$ becomes infinitesimal, then $z_{12}\to \pi/2$ I.e. $\epsilon = \pi/2, d\epsilon = 0$, and your integrations are over a constant instead of a variable.

$$\oint \frac{|d s |^2}{|s|^2} \cos^2(\frac{\epsilon}{2}) d \epsilon = \oint \sin^2(\epsilon) d \epsilon$$

Even without the issue with $\epsilon$ being constant, you should notice that the interior of the LHS contains an infinitesimal you are not integrating over, while the RHS does not. That alone is enough to tell you you've made some errors.