# Sunrises on the compass

by chroot
Tags: compass, sunrises
 P: 243 I'm basically a novice when it comes to differential geometry, so I don't think that I can help, but I will try. The metric of a 2-sphere is given by: $$ds^2=R^2(d\theta^2+\sin^2 \theta d\phi^2)$$ Denote the tangent space at point A with $V_a$ and the tangent vector of the equator curve with $A^\mu$. Similary denote the tangent space at B with $V_b$ and the parallely transported $A^\mu$ with $B^\mu$. There exists a scalar field covering the entire area of the sphere through which we can define functions (with two variables - $\theta$ and $\phi$). These will be important as through them we can parametarise curves. The equator curve can parametarized by the function $a(\theta,\phi)$ by setting $\theta=0$. The terminator curve can also be parametarised this way. Since I don't know what that is I don't know how to parametarise it. Anyway, let's denote it's parametar by $c(\theta,\phi)$. $A^\mu$ can be defined as $$A^\mu=\frac{dx^\mu}{da}$$ Now we wish to parallely transport $A$ to point B. Introduce it's dual $A_\mu$. $$B^\mu B_{\mu} = \int_0^{\theta_B}\int_{\phi_A}^{\phi_B} \frac{d(A^\mu g_{\mu\mu} A^\mu)}{dc} d\theta d\phi$$ This defines $B^\mu$. The vector pointing to the sun $S$, expressed in the coordinate basis of A, $X^A_\mu$ will be equivalent to $A$. $$S =\sum_{\mu=1}^2 A^\mu X^A_\mu$$ Since it doesn't change direction in point B, we don't parallely transport it, we only express it in the coordinate basis of B using the vector transformation law. $$S^{B \nu} =\sum_{\nu=1}^2 A^\mu \frac{\partial x^{B \nu}}{\partial x^{A \mu}}$$ Since we know that this particular 2-sphere is embedded in Euclidian three-dimensional space we can find the coordinates of the basis of any point, in three-dimensional space, by imposing that they be orthonomal to the position vector, $\vec{r}$, which has it's origin at the center of the Earth. The angle between $S$ and $B$ at $V_b$ can be found by simple methods of vector algebra once we define it's coordinate basis through the above procedure. Hope I helped.