GEODESIC
Geodesic, curves that are the shortest paths between near points, are re-made tools for many Riemannian Geometry tasks. [1]The importance of geodesics is similar to straight lines in Euclidean space and great circles on a sphere.
Before illustrating the Geodesic, we should understand metric[2]. The metric is a mathematical tool which at any point gives us a grid target to the object along which we can measure length and angles. Imagining points A and B are on the surface of a sphere, if we ask an ant to walk from A to B, since it remains on the surface of the sphere, it must have a whole walk(it must have walked through whole path/trajectory). To measure the length of the whole path travelled by the end, we would add the length of each of step. At each of the steps, metric gives a grid within which we can measure the small distance that the ant has just covered. By repeating this process for each step, we can get the total length it has walked. But now we can’t know which path has shortest distance. For finding the true distance between A and B, we can imagine hundreds of ants walking between A and B on the sphere. With the metric of all of the paths, so comparing with all the path , the shortest path from A to B we can find ,determines the distance become two points, this type of trajectory is called a geodesic.
By combining the metric of all paths and comparing them, we can find the shortest path between A and B which determines the distance between the two points. The trajectory between those two points is called a geodesic.
We can understand it as generalization of a straight line . On the surface of a plane geodesics are simplified straight lines so we can measure it with a ruler.
The surface of a sphere, the geodesics, is known as great circles which are the cover of the greatest possible circumference. For example ,as the earth lines of longitude are geodesics on earth, on the other hand apart from equator lines of latitude are not geodesics ,because they are not the great circle.
When looking at the more complex surface, geodesics will take many interesting shapes like the universe. Our universe is a hyper surface with four dimensions, three dimensions of space and one of time ，we call it spaced time, all objects in the universe evolve through space -time. At this time the metrics within measure geodesics are more complex. In the universe all objects tend to follow geodesics.
In special relativity, the spaces of universe are empty, space-time is almost flat, so the geodesics is a straight line because bodies in the universe don’t experience any force. But in general relativity, close to a massive body space time gets bent, for bodies that are spherical, isolated and stable over time, the geodesics correspond to deflected trajectories or orbits. If the moon moves around the earth, it follows the geodesic as a straight line but in a curved geometry. At this time geodesic is not only in the universe but also describes the trajectory through time in space.
