The statement that “a straight line is the shortest distance between two points” is partially true.
We know that light always takes the shortest path between two points, which we generally consider a straight line. However, a straight line is only the shortest distance between two points on a flat surface. We have to go back to the postulates of Euclid of Alexandria, considered by many to be the most important mathematician in history, who affirmed that light traveled in a straight line and demonstrated that a straight line is shorter than two other straight lines that join them. extremes (regardless of curves). But this theorem fails for non-Euclidean geometries, such as spheres and more complex geometries. In fact, everything you learned in school, like parallel lines stay parallel, only refers to Euclidean geometry. In the non-Euclidean universe, parallel lines can actually diverge or converge.
But let’s remember that the Earth is not flat , it is roughly a sphere. If we lived on a flat Earth, we could say that a straight line would be the shortest distance between points A and B. But we have already said that the Earth is shaped like a sphere; to do it through a straight line we would have to cross the Earth) and the shortest distance between two points in this geometric shape is an arc known as “the distance of the great circle”. Thus, on a curved surface, the shortest distance between two points is actually a curve, technically known as a geodesic: it is the trajectories given by the “great circles”. It is the maximum circle that unites them.
Is it a new concept? No way. To better visualize it, let’s think about the shortest route of a plane that flies between London and New York. The route follows a “great circle” path rather than what appears to be a more direct straight line on a flat map. Curved routes at these destinations are not used randomly, but are used because they actually represent the shortest distance between two particular locations on our planet.
Albert Einstein enters the scene
If we combine this concept with Einstein’s equivalence principle, in the theory of general relativity, in which gravity can bend the shape of space and time, then it would appear that light in the presence of gravity follows a curved path or, In other words, gravity bends the path of light, as we have discussed. Thus, it turns out that gravity is nothing more than a curved space or, more specifically, the curvature or deformation of four-dimensional space-time.
Earth’s path as it travels through space is constantly tilting toward the Sun in this way, so much so that the planet traces an almost circular orbit. However, from the perspective of the fourth dimension, it can be seen that the Earth is simply following the shortest path along a geodesic through curved spacetime.
With everything advanced, even large circular distances do not represent the shortest true distance between two given locations . For what reason? Distances are calculated assuming that the Earth is a perfect sphere, but the planet is more of a slightly flattened sphere with different radius values towards the equator and the poles. Therefore, the great circle values have a tolerance of approximately ± 5%.
A bit of history: Euclid’s postulates:
1.- Given two points, a line can be drawn that joins them.
2.- Any segment can be continuously prolonged in an unlimited line in the same direction.
3.- It is possible to draw a center circle at any point and any radius.
4.- All right angles are equal.
5.- If a line, when cutting two others, forms the internal angles of the same side that are smaller than two straight lines, both lines prolonged indefinitely are cut from the side with the angles smaller than two straight lines. Or what is the same: through a point outside a line a single parallel can be drawn.
