How would you use Mathematica to extract a metric tensor from a metric formula? [duplicate]

Question

I'm trying to follow an explanation for extracting a metric tensor from a metric formula and having trouble following the mathematical explanation. I'm sure I would understand it better if written in a language I understand. In Mathematica, how would you create the metric tensor from this formula?$$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$

Answer 1

Its very simple, its the coefficient matrix of second derivatives with respect to the list of differentials

  (1/2 Outer[(D[d\[Theta]^2 + Sin[\[Theta]] d\[Phi]^2, #1, #2] &),#,#]&)[
       {d\[Theta],d\[Phi]}]

   {{1,0},{0,Sin[\[Theta]]}}

Answer 2

The metric in the form you've written computes the distance between two points separated by infinitesimal distances. In your case, you've only specified two parameters so you're working in a 2-d case. This particular metric describes the surface of a sphere. The basis for your metric is comprised from two "one-forms" - in this case the differentials, ${d\theta,d\phi}$. The choice of basis depends on your choice of coordinate system.

The expression is written $ds^2=g_{\mu\nu} dx^\mu dx^\nu$, with repeated indices summed over if one if upper and the other lower, and with $dx^{i}=d\theta$ or $d\phi$, so all the metric components are zero except for $g_{\theta\theta} = 1$ and $g_{\phi\phi}= sin(\theta)^2$. In Mathematica code, you could write this as

EDIT: This is a better way to code it; use two lists of differentials instead of one with squared terms.

In[2]:= {Dt[th], Dt[phi]} . DiagonalMatrix[{1, Sin[th]^2}] . {Dt[th], Dt[phi]}
(*Out[2]= Dt[th]^2 + Dt[phi]^2 Sin[th]^2 *)

The differentials represent distances between two points you're concerned with in the coordinate system you've selected. For example, if P1 has coordinates $(\theta_1,\phi_1)$ and P2 has coordinates $(\theta_1+d\theta, \phi_1+d\phi)$, your metric gives you the square of the interval/distance between points P1 and P2. For finite separations, perform an integral over the infinitesimals.