einstein

* Here’s how it works. If you are at some point in curved space and want to know the distance to a neighboring point—infinitesimally close—
then things can be complicated if you have just the Pythagorean theorem and some general geometry to use. The distance to a nearby point
to the north may need to be computed differently from the distance to one to the east or to one in the up direction. You need something
comparable to a little scorecard at each point of space to tell you the distance to each of these points. In four-dimensional spacetime your
scorecard will require ten numbers for you to be able to deal with all the questions pertaining to spacetime distances to nearby points. You
need such a scorecard for every point in the spacetime. But once you have those scorecards, you can figure out the distance along any
curve: just add up the distances along each infinitesimal bit using the scorecards as you pass them. These scorecards form the metric
tensor, which is a field in spacetime. In other words, it is something defined at every point, but that can have differing values at every point. I
am grateful to Professor John D. Norton for helping with this section.