Physics

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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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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