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Just like the Special Theory of relativity, the General Theory of Relativity has amazing results. Nowdays there are still projects to test general relativity, and all the tests so far have shown that Einstein was right after all. The Universe as a Not-Euclidean Any of you who has taken geometry in school, Euclidean geometry that is, knows that the arc of a circle divided by its diameter is p, you know that the angles of a triangle add up to 180°, that two parallel lines never intersect and many other Euclidean laws of geometry. As a surprising result of the general theory of relativity, these laws were proved not to hold true in reality. This doesn't mean that Euclidean laws are wrong or false, on the contrary, they are true when they are dealt with as a mathematical abstraction. This is because Mathematics is a science which has certain laws that hold only when dealing with them as mathematical abstractions and not in reality. For example 1+1=2, mathematically that's true, but if we add up a litter of water and a litter of milk we will still have one litter of liquid in that case 1+1=1. These things about Mathematics were known way before Einstein was even born, however it was thought that Euclidean Geometry holds true in reality and we will see it doesn't. Let's suppose we have a galleon frame of reference, in it we have a frame of reference K which is a big disk rotating uniformly with respect to the galilean frame. We have two observers, one on K and the other in the galilean frame. The observer in K observes that as he approaches de edge of the circle a gravitational field appears and its intensity increases as he approaches the edge and that this field disappears when he is on the center of the circle. This gravitational field in classical mechanics is known as centrifugal force and it is an outward-pulling force which can be observed while in rotational motion. The observer in the galilean frame measures the diameter and the arc of the circle using a ruler, divides and he obtains 3.14... which is equal to p. The observer in the disk then measures the diameter of the disc and its arc with the same ruler. As he approaches the edge, relative to the observer in the galilean frame, the ruler must contract (we stated in special relativity that moving bodies contract). So, when he can not possible have the same measures for arc and diameter than the observer in the galilean frame, his measures of arc will be smaller (because of the contraction) and so will be the ones for diameter but in a smaller scale. So, when the observer divides the results he obtained, he will not obtain p, but a larger number. This example shows that the universe is not necessarily Euclidean, it can't be considered using only straight lines (like in Classical Mechanics and in Special Relativity). This doesn't mean we must disregard special relativity, since when there are no gravitational fields present and the frame of reference is at a constant motion this theory holds true.
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