So on special relativity - Galileo and the first relativity

I was trying to do this next week, but I thought it would be better if I just went into this bit, while I can still freshly remember things. So last time, we were seeing reference frames about how the whole concept of frames of reference is developed in physics and how things may change when we change perspective. There were some concepts introduced before, like observer, origin and coordinates so we will talking with those terms in this post. If you want to understand a bit more about this, I recommend you to check the last post I made.

Now we already know what are reference frames and that is cool and all, but the initial question still remained. Why do we even care about those frames in the first place.

A good example I am going to use was given to me by a graduate that used this for the students he was tutoring. Let's use the train example again, but this time let's use three people instead of two. And the names for those people are... Let's say Ed, Murray and Richard. And let's say Ed is standing still on the platform watching the train, that is moving with a constant speed v to the right. Murray is in the train moving to the left also with speed v. And Richard is outside in the platform alongside Ed, but it is moving to the left at a speed that may or may not be the same as v.

Actually here is a visual example of what this situation could be like. I made it in MS Paint so it won't be impressive, but I believe it is simple enough for you to understand what is going on.

I picked these images from google and mixed them up to make a similar scenario that I was describing, so feel free to use it if you are willing to.

Now that we have that figured out, here is a question: Which way is Murray moving?

Now some of you may say that it is obvious he is moving to the left, because that is what the example shows. However, you may be in disagreement with Ed. Since he is standing outside, he sees Murray as not moving, since the movement he makes to the left is cancelled by the movement of the train to the right, and overall to Ed, Murray seems to be in the same place. And if we were to ask Richard he would be saying that Murray is moving to the right, since he is moving to the left outside of the train, it seems to him that the train and everything inside of it is moving away from him to the right, including Murray. And let's not forget Murray himself, which thinks he isn't moving at all and instead that the back of the train is moving towards him (Since he is the observer, he doesn't move in its own referential). And which of them is correct? According to their perspective, to their frames, all of them. And here we see why frames of reference are so important to physics. The way that Murray is moving depends largely on the observer, and on its referential. And depending on our referential, things move in different directions or in different ways and shapes. The greatest example of this "relative" motion is the apparent movement of the Sun around Earth. We all know that the opposite is true, but our daily experience shows the Sun moving from East to West everyday. And that apparent movement of the Sun is because our referential is on Earth, and to all effects we are not moving (at least the Earth isn't) in our referential, and that ultimately would lead us to falsely believing that everything in the skies revolved around the Earth, and it took the works of astronomers like Copernicus, Keppler and Galileo to change this mentality into what closely resembles modern day description of the Solar System (at the time, Newton wasn't born yet, so they didn't have the full picture yet). We will come back to Galileo...

So, as we have seen, the motion of a body changes if you change referentials, and so there what we call in physics relative motion. What does that mean? It means, for instance in our early example, that Murray is not moving relatively to Ed, that he is moving to the right relatively to Richard. So things are in different kinds of motion relatively to other things. But if that is what is happening, how can we predict the motion of something accurately, since depending on the observer we would get totally different results? The person who found a way to solve this problem was Galileo (I told you he was coming back).

He was with a question similar to our own, but instead of a train, he was dealing with a boat. In his book, Dialogue Concerning the Two Chief World Systems (or if you are the kind who likes original titles Dialogo supra i due massimi sistemi del mondo, in italian), he was trying to disprove an idea people had at the time that the Earth could not be rotating, or else we would feel the speed on the surface. And he presented an argument that was quite ingenious. Let's say you are in a ship, moving at a couple of mph in smooth, calm water. If you were below the decks, you couldn't be able to know if the ship was docked or if it was moving, therefore the idea that the Earth was moving because we didn't notice the motion was not valid. This was an idea that not only helped to push forward the idea of heliocentrism, but was also the first example of relative motion. And Galileo himself went on to develop a mathematical framework that could allow us describe relative motion on different frames, what would later be described in Newtonian Mechanics as Galilean Relativity, or also called Galilean Invariance. (What is interesting is that Einstein used a similar analogy to that of Galileo's boat, to establish his Equivalence principle, which is the fundamental principle behind General Relativity). 

Ups, before I am done, I would like to talk a bit about inertial frames, since they are the frames usually considered in Galilean and Special Relativity. The name inertial comes from inertia, which in physics is resistance to motion, and pretty much Newton's first law is all about how inertia plays a role in the motion of bodies. A body in inertia is a body that is not moving, or if it is moving, moves at a constant velocity (velocity is better than speed at this point, since we are talking about a vector), unless forces are applied to it. What this implies is that to change a body's motion, you need to change its velocity (either its speed or direction) and that can only happen with the existence of a force. So, with no force, the bodies are either not moving, or they move in a straight in constant speed. In the same way, an inertial frame would be one where the frame is not moving or moving at a constant velocity. This does not mean that every object in the frame must also not change velocity, it just means that the frame of reference itself should not move, or it should move at constant velocity. Let's use Ed and Richard from the early example, and let's say that Murray's velocity changes over time (he would move faster, for example). As long as they keep moving at constant velocity or remain stopped, Ed and Richard can be frames of reference to study the motion of the train and of Murray. Now Murray himself can't be an inertial frame anymore, because he is not in inertial motion. So as long as our frames of reference are inertial, we can use Galilean (and later, Special) Relativity. 

I believe this is a good way to give a historical perspective onto this concept. Knowing a bit about how things developed in history allows us to appreciate and enjoy more the results we obtain. This way, you will know that the math behind Special Relativity didn't appear out of nowhere, but it came from initial developments and discoveries from the past. Next time, I will go a bit more into the mathematical description of Galilean Relativity and hopefully show where it fails. I am so excited, we are getting into the fun part...