In previous posts, I discussed the concept of a line element. It is a description in terms of differentials of a distance equation.
The nice thing about these line elements is that they exist for every surface and easily generalize to higher dimensions (which for cylindrical and spherical coordinates in three dimensions).
So, at this point, we know how to find the arc length of a curve on a surface by integrating ds over that curve.
Now, a new concept: a geodesic is a curve between two points that has the smallest possible arc length for curves between those two points.
As an easy example, a straight line is a geodesic in the plane and a great circle is a geodesic on a sphere.
So, how do we find the equations for a geodesic?
This will involve finding the minimum value of an integral over all possible curves between two points. Following the intuition from calculus, we want to take a derivative *with respect to the curves* and set that derivative equal to zero.
The procedure is called the calculus of variations and is a very general and very powerful way of finding *functions* that minimize certain integrals. Just like we find ordinary algebraic or trigonometric equations when minimizing functions in calculus, in the calculus of variations, we get a *differential* equation whose solution is the function we seek. This differential equation is called the Euler-Lagrange equation for the integral we are trying to minimize.
Next, we can determine that our surface (or, better, higher dimensional manifold) is *curved* if two geodesics that start out parallel start to accelerate either towards or away from each other. Imagine the great circles on a sphere that we have two of them through close points that are parallel (in the sense of having the same direction). Those two great circles will inevitably get closer together as we move away from our initial points. This is because a sphere is curved.
What this ultimately says is that the line element (now also called a metric) tells us a great deal about the geometry of the surface (or manifold) we are investigating. It tells us which curves are geodesics and those tell us about the curvature of the manifold.
next step: General Relativity.
The nice thing about these line elements is that they exist for every surface and easily generalize to higher dimensions (which for cylindrical and spherical coordinates in three dimensions).
So, at this point, we know how to find the arc length of a curve on a surface by integrating ds over that curve.
Now, a new concept: a geodesic is a curve between two points that has the smallest possible arc length for curves between those two points.
As an easy example, a straight line is a geodesic in the plane and a great circle is a geodesic on a sphere.
So, how do we find the equations for a geodesic?
This will involve finding the minimum value of an integral over all possible curves between two points. Following the intuition from calculus, we want to take a derivative *with respect to the curves* and set that derivative equal to zero.
The procedure is called the calculus of variations and is a very general and very powerful way of finding *functions* that minimize certain integrals. Just like we find ordinary algebraic or trigonometric equations when minimizing functions in calculus, in the calculus of variations, we get a *differential* equation whose solution is the function we seek. This differential equation is called the Euler-Lagrange equation for the integral we are trying to minimize.
Next, we can determine that our surface (or, better, higher dimensional manifold) is *curved* if two geodesics that start out parallel start to accelerate either towards or away from each other. Imagine the great circles on a sphere that we have two of them through close points that are parallel (in the sense of having the same direction). Those two great circles will inevitably get closer together as we move away from our initial points. This is because a sphere is curved.
What this ultimately says is that the line element (now also called a metric) tells us a great deal about the geometry of the surface (or manifold) we are investigating. It tells us which curves are geodesics and those tell us about the curvature of the manifold.
next step: General Relativity.
