Articles

Geodesics - from intuition to equations

The simplest kind of geometry, taught in schools, is the so called Euclidean geometry - named after an ancient Greek mathematician, Euclid, who described its basics in the 4th century BC in his "Elements". It is based on the notions of points, straight lines and planes and it seems to correspond perfectly to our everyday experiences with various shapes. However, we can notice problems for which Euclidean geometry is insufficient even in our immediate surroundings.

Let's imagine, for example, that we are airline pilots and our task is to fly as quickly as possible from Warsaw, Poland to San Francisco. We take a world map and knowing from Euclidean geometry that a straight line is the shortest path between two points, we draw such a line from Warsaw to San Francisco. We're getting ready to depart and fly along the course we plotted... but fortunately, our navigator friend tells us that we fell into a trap.

The trap is that the surface of the Earth isn't flat! The map we used to plot our straight line course is just a projection of a surface that is close to spherical in reality. Because of that, the red line on the map below is not the shortest path - the purple line is:

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How science helps us understand the world

I've been stumbling upon posts expressing various kinds of doubts against science on the Internet recently. Either something isn't proved enough, or scientific theories are too abstract, or they are even absurd. All these posts seem to have one thing in common - a fundamental misunderstanding regarding the way science works, or the way it should work. Because of this, I decided to attempt to explain the issue here - what science does, what it doesn't do, and why. Enjoy!

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A continuation of flat Earth debunking

In the previous entry, I described the story of a discussion with some flat-Earthers and how I created a refraction caculator in order to have stronger arguments. Today I'm going to write a bit about how this situation developed further (with the calculator, not with the flat-Earthers - I don't think anybody expects that I managed to convince a pseudoscientist? ;) ).

Let me just recap quickly on what the discussion concerned. It is that one of the flat-Earthers insists that some landscapes look the way they should on a flat Earth, and not how they should on a spherical Earth. He supports his claims by showing some photos he took and calculating some proportions of distances between characteristic points or sizes of some visible objects. It's actually a very reasonable approach - provided that one does everything earnestly, ie. calculates what proportions one should get on a flat Earth, and what they should be on spherical Earth. As it turns out - which is what the previous entry was about - that a fully correct analysis must even take atmospheric refraction into account, and it is negligible for most purposes.

The refraction calculator I created on this occasion had one major drawback - it allowed only for tracing a single light ray at a time. Because of this, for every photo you had to choose some specific points and calculate e.g. ratios of some angles. This actually still enables getting some interesting results, but isn't very attractive visually - it's just comparing numbers. So I came up with an idea of using computers to improve the situation a bit: what if I could create software that would simulate multiple rays at once, instead of just one, check where they hit the Earth's surface and generate a whole panorama based on that...?

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Special Relativity without assumptions about the speed of light

Introduction

As the Special Theory of Relativity seems to contradict the common sense, it remains a somewhat magical topic for the regular people. The consequences of this theory seem to be so far removed from everyday life, that it's quite hard to admit them as the correct description of the surrounding reality.

Most people have their first contact with SR at school and its introduction there looks somewhat like this: near the end of the 19th century people discovered the electromagnetic waves. The equations describing these waves imply a specific speed of their propagation, denoted $c$ and equal to about 300 000 km/s. It was quite interesting, since nothing seemed to imply any frame of reference for this speed. Since all known waves required a medium to propagate, it was assumed that the electromagnetic waves are no different and travel in something called the aether, and that the speed arising from the equations is relative to the aether.

Once people decided that aether should exist, the logical next step was to try and detect it. One of the ideas was to measure the speed of the Earth relative to the aether. Some attempts were made, but the results were unexpected - it seemed that the Earth is not moving in the aether. It was strange, especially considering that the Earth changes its velocity in its motion around the Sun, so even if it did stand still in the aether at one point, it shouldn't at another one - but the measured speed was always 0. People then tried to modify the concept of the aether to explain the results and started performing more sensistive experiments. One of these was the famous Michelson-Morley experiment, which, just like the earlier attempts, failed to detect the motion of the Earth, too.

The scientists were rather confused with these results. It seemed that the speed of light was constant regardless of the motion of the observer, which was quite extraordinary. To better illustrate what is so strange about this situation, let us imagine that we are in a car standing at an intersection, and that there is another car in front of us. Once the traffic light turns green, the car in front of us starts moving and accelerates to 15 m/s, so its distance from us starts to grow by 15 meters every second. We start moving shortly afterwards. Once we are moving at 5 m/s, we expect the car ahead to be leaving us behind by 10 m every second, but once we check that, we are surprised to discover that the distance is still growing by 15 m/s. We accelerate to 10 m/s - and the distance is still growing by 15 m/s. We accelerate more and more, but we can't seem to start catching up to the car in front, even though our friend, a policeman, was standing with a radar near the road and told us that the speed of that car was always just 15 m/s. Light seemed to behave just like such a weird car.

The 20th century came and various people were proposing different explanations - among them were Lorentz, Poincare, and eventually Einstein. In 1905, Einstein presented a theory known today as Special Relativity, which was based on 3 assumptions:

1. The space(time) is homogeneous and isotropic, ie. there are no special points or directions in the Universe.
2. There are no special inertial frames of reference, the laws of physics are the same in all of them - this is the so called Galilean relativity principle.
3. The speed of light is the same in all frames of reference - this was a conclusion from the Michelson-Morley experiment.

Thus the aether became unnecessary - from that moment on, $c$ was just a universal speed, independent of who is measuring it. Coincidentally, this also has some unusual consequences, such as time passing slower for moving observers, or contraction of moving objects.

There is still a loophole, though. One could argue - and some people do - that the third assumption is not adequately proven. The Michelson-Morley experiment could have been not sensitive enough, or it could give a null result under some specific circumstances, even though the speed of light is not really constant. Thus, SR can be (and, according to some, just is) wrong.

This is all true, but not many people are aware that this third assumption isn't actually needed to obtain SR. I'm going to show here how this is possible.

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Lorentz transformations, light cones

In the previous article:

• What are events and spacetime?
• What are world lines?
• Simple spacetime diagrams
• How does the inseparability of space and time influence their perception by observers?

Most of the illustrations in the last article used rotations, but it turned out eventually that rotations aren't the correct transformations that would let us look at the spacetime from the point of view of different observers. Now we will take a look at transformations that actually describe reality - the Lorentz transformations.

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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.

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.

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:

(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:

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.

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...?

Part 3 - the metric

We already mentioned the notion of the magnitude of a vector, but we said nothing about what it actually is. On a plane it's easy - when we move by $v_x$ in the $x$ axis and by $v_y$ in the $y$ axis, the distance between the starting and the ending point is $\sqrt{v_x^2 + v_y^2}$ (which can be seen by drawing a right triangle and using the Pythagorean theorem - see the picture). It doesn't have to be always like that, though, and here is where the metric comes into play.

The metric is a way of generalizing the Pythagorean theorem. The coordinates don't always correspond to distances along perpendicular axes, and it is even not always possible to introduce such coordinates (but let's not get ahead of ourselves). We want then to have a way of calculating the distance between points $\Delta x^\mu$ apart, where $x^\mu$ are some unspecified coordinates.