Non-Eclidean Geometry and General Relativity

(I studied these two subjects independently at different times, then realized that they complemented each other. So I wrote this analysis.)

Euclid lived around 300 B.C. and wrote a work called the Elements. In it he described the geometry that we all learned in high school. Euclid began by giving 5 axioms or propositions which were considered to be self-evident. From these 5 axioms he deduced many theorems. The result was Euclidean geometry, which was considered to be a true and accurate description of the world in which we live.

However, a problem continually arose for mathematicians concerning Euclid’s geometry. The 5th axiom did not seem self- evident. In simple terms the 5th axiom states that parallel lines will remain equidistant from each other and never meet or diverge from one another. Mathematicians questioned how the properties of something that extended to infinity could be self-evident.

In the 19th century non-euclidean geometry emerged when some mathematicians tried assuming that parallel lines do meet or diverge from one another, which actually is true on curved surfaces. For example, lines that are parallel at the equator converge at the poles on the surface of the Earth. They found that coherent geometries could be created with a change in the fifth axiom. In these geometries, triangles, for instance, have more than or less than 180 degrees. Whereas Euclidean geometry says that all triangles have exactly 180 degrees. These new geometries were considered an irrelevant oddity of mathematics.

In the 20th century Albert Einstein published his Theory of General Relativity, which stated that the presence of massive bodies such as a planet or star causes the space around it to be curved. Einstein's theory was first proved experimentally by scientists who observed, during a solar eclipse, the emergence of a star from behind the sun sooner than possible unless the light was curving around the sun. This bending of light is now known as the gravitational lensing effect. Since gravity was known to only affect objects with mass, this was proof that space itself was curved because light has no mass. According to Einstein, the moon revolves around the Earth, not because an invisible force pulls on the moon, but because the mass of the Earth curves the space around it. The moon is moving in a straight line at constant speed in accordance with Newtonian physics, but the space it is moving in is curved. This can be demonstrated by drawing a straight line on a flat piece of paper (representing the moon’s movement through space), then bending the paper into a tube until the ends of the line meet. The straight line has now become a circle on the curved paper.

The advent of Einstein’s Relativity reveals that non-euclidean geometries are actually the most accurate description of our universe because space, itself, is curved. Euclidean geometry is still suitable for small scale applications like constructing buildings and surveying, but on a cosmic scale, it is not valid.