Curvedness of a line

I found a very interesting anecdote expressing the marriage between Geometry, Physics and real life. Please enjoy and feel free to share your experiences in this regard.

Suppose you are a 16^th century explorer and you want to travel the world while making note

of where you have been, what you have seen, and all the while keeping track of where it was

that you saw certain things, i.e. `Here there be monsters... but over there the islanders love

to party.' Being a kind soul, you want to make sure that other explorers are not forced into

the same hardships you have endured throughout your gallant forays into the unknown. So

you make a chart of this region of the world and that region of the world, carefully choosing the coordinates and the scaling factor.

But Wait!!!

What about the intersection going from one chart to an adjacent one? How are we going to manage that?

Surely, you must be saying to yourself, one could just continue on in a straight line into the next chart and close your eyes while amidst the fluff created by needing Mathematics here. Well that sounds like a great idea; I really like that idea. Just one problem “What is a straight line? You and I both know what we mean by a straight line, but the confusion comes in when we try to compare my straight lines to yours. Consider the example of a boat sailing on the ocean with a perfectly north heading from a point on the equator at longitude 14 degrees. Consider a fellow traveller in a similarly oriented boat sailing with the same heading yet he is located at 18 degrees longitude. They both continue in what appears to be a straight line. Now what if you are watching this from the international space station, if such a station exists? (depending on what century we are imagining at the moment) Your perspective grants you the ability to notice that they are not at all going in straight lines, they are going along an arc of a circle concentric with the earth and sharing the same radius. This is simply anecdotal; but it does give the flavor that we need to be precise by what we mean by going straight.

In Euclidean space, going straight means just that. The problem with our example is this whole curvature of the surface of the earth. There is a similar problem with the universe as a whole. There exists curvature in the fabric of space-time. This curvature gives rise to one of the most geometric of physical theories, Einstein's General Relativity. In Einstein's formulation of the governing equations, there is a magical object called the “stress-energy tensor" which tells space-time how to curve. Space-time, being a jovial conversational partner, in turn tells particles how to move. This lovely colloquium of fields and motions is

a very common construct in physical theories and each one of those lends itself to a very geometrical expression.

Adapted from a paper submitted by Douglas Ortego Department of Mathematics Colorado State University