Science tidbits

Landscapes and atmospheric refraction

Sometimes I'm bored and I'm getting involved in discussions with various kinds of pseudoscientists. Such discussions are often a waste of time, but it's possible occasionally to get something out of them - after all, if you want to explain to someone why there are wrong, you need to have a good understanding of the topic yourself. If your knowledge is not enough to counter the opponent's arguments, you need to expand it, and so you are learning. It was the case for me this time.

It all began with two flat-earthers appearing on a certain forum. The exchange started with standard arguments like timezones, seasons, eclipses, the rotation of the sky... what have you. As usual in such cases, those arguments were met with silence or really far-fetched alternative explanations. I'll omit the details, interested people can find standard flat-earth arguments on the web.

Well, you can't sway a person that is completely confident in their beliefs with arguments, so the discussion has become somewhat futile. Both sides stuck to their positions and mulling over the same issues time and time again has started. That is, until one of the flat-earthers started presenting photos which, according to them, proved that the Earth "can't be a ball with a 6371-6378 km radius", with descriptions that can be expressed shortly as "explain THAT!". Alright.

Challenge accepted!

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Events and space-time

The first entry in the series will be quite basic, but I think that some problems will nevertheless be quite interesting. We'll be talking about what is the space-time, events, and we will show where the theory of relativity comes from. So, let's go :)

The notion of space-time is briefly mentioned at school, but usually the profound consequences of combining space and time into a single entity aren't explained too much. To understand this, one must first go a bit deeper into the details of this idea.

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Hyperbolic functions - what sorcery is this?

If you are like me, your first contact with the hyperbolic functions was as "this strange, useless something on the calculator". There were just some weird buttons labeled "sinh" and "cosh". The school finally explained what "sin" and "cos" are, but there was no mention of those variants with the final "h". What is this about? The names suggest some similarity to the trigonometric functions, let's see what happens:

 \begin{array}{ll} \cos (1) = 0.54030230586 & \cosh (1) = 1.54308063482 \\ \cos (10) = -0.83907152907 & \cosh (10) = 11013.2329201 \end{array}

(You will get these results if you have the calculator set to radians - if you use degrees, then the cosine results will be different; it has no influence on the hyperbolic functions and we'll see later why that is.)

Right, these 11 thousand for cosh(10) look very similar to the trigonometric functions. This "h" apparently changes quite a bit, but what exactly...?

If you encountered complex numbers during your later education, you could stumble upon such definitions:

 \begin{array}{ll} \cos x = \frac{e^{ix} + e^{-ix}}{2} & \cosh x = \frac{e^x + e^{-x}}{2} \\ \sin x = \frac{e^{ix} - e^{-ix}}{2i} & \sinh x = \frac{e^x - e^{-x}}{2} \end{array}

Some similarity is visible here, but... Why such a form? What does this have to do with hyperbolas? If you don't know it yet, you will know after reading this article.

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The shape of a black hole's event horizon

Yesterday, while browsing the internet, I stumbled upon a thread which looked like a typical question asked by someone interested in science, and turned out to be a really interesting problem.

The question that has been asked concerned the shape of a black hole. A few people replied that the event horizon (the boundary - or the "surface" in a way - of a black hole) has the shape of a ball (which should be actually described as a sphere, since the horizon is a 2-dimensional surface, and not a 3-dimensional shape). Someone suggested that it's not exactly true, because black holes usually spin, which flattens them. I entered the thread then and said that even when a black hole is spinning, its horizon is still spherical - it's described by an equation like r = const. But is that really so...?

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