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DoctorKokktor t1_j6np6h4 wrote

Reply to comment by NIRPL in ELI5 why time slows down as you go faster by -cool--beans-

Good question, sorry for not clarifying in my original post. An axiom is a statement that we take to be self-evident. I.e. it's a statement that we don't need to prove because we just ASSUME that it's true. Then, based on that assumption (i.e. axiom), we try to deduce certain conclusions.

This is also how we do things in geometry (and indeed every single branch of math)! You may have heard of Euclid's five postulates (another word for axiom). All of Euclidean geometry can be derived by using the five axioms that Euclid laid out some 2000 years ago.

However, it turns out that if you refuse to accept the fifth axiom of Euclid, then you can deduce/derive other kinds of geometries. These are called non-euclidean geometries.

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NIRPL t1_j6nxhmx wrote

Thank you for explaining!

Regarding the non-euclidean geometries link, the images describing elliptical Euclidean and hyperbolic lines, why is the identified 90° angle significant? Or is that the point? Regardless of bend, we can assume a 90° angle will exist?

If I'm completely off-base with my question and wasting your time, feel free to say so lol

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DoctorKokktor t1_j6nynl9 wrote

Another good question :)

So basically, Euclidean geometry is founded on five posulates, the last of which is called the "parallel postulate". It turns out that the statement that describes this parallel postulate is ambiguous, and so you can have multiple descriptions/variations of this postulate (you can even omit the 5th postulate altogether!), all of which allow you to derive entirely new geometries.

Those 90 degree angles in different shapes are a consequence of the different statements of the parallel postulate. Those figures shown in the article are all examples of parallel lines, which at first seem preposterous but are a natural consequence of accepting a different version of Euclid's parallel postulate.

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NIRPL t1_j6oe0q2 wrote

I sincerely hope you are an educator. You have a talent for explanations. Thank you for taking the time to respond and teach me something new today.

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DoctorKokktor t1_j6om43l wrote

Thank you for your kind words :)

Incidentally, Einstein's theory of special and general relativity are also a different kind of geometry! Just like how Euclidean geometry takes place in Euclidean space, special relativity takes place on what's known as "Minkowski" space or spacetime, whereas general relativity takes place on what's known as a pseudo-Riemannian manifold.

These words sound complex but just remember that they are merely names for some kind of foreign geometry that are derived from a completely different set of axioms than the normal Euclidean geometry that we studied in high school. This article is very nice at introducing the history and context behind non-euclidean geometries!

But you don't even have to think about something as abstract as relativity to realize that non-euclidean geometries are everywhere. Even the surface of the earth has a non-euclidean geometry (the geometry that takes place on the surface of the earth is called "spherical geometry"). In this sort of geometry, you can have triangles which can have two right angles!

The reason I am introducing all these different types of geometries is because in the end, relativity theory is a geometric theory of spacetime. If you can understand the context behind some of the different kinds of geometries, then you can understand the context behind relativity theory as well! In this way, relativity won't seem as mysterious anymore. It's still very counter-intuitive, but at least you can understand that relativity is just a consequence of choosing some set of axioms, and drawing conclusions from there, just like any other sort of non-euclidean geometry.

Sorry for all these other links haha. This topic is extremely interesting to me and I wanted to share some resources to hopefully get you excited about it too! :)

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