Special Relativity

Ever since my high school physics class, special relativity has always bugged me. I thought I understood the concept, and I was able to work through the thought experiments presented by my teacher, but I always had one in my head that bugged me. Recently, I decided to work out the problem on paper, and resolved my issue! Yay! I will try to explain my findings here. My explanation may be completely off-base, but the concepts and results I talk about are definitely correct (take your disagreements up with Einstein!).

In the text below, I will try to introduce the problem, but that is going to require some background. I will attempt to give it; if it isn't coherent, you'll have to read a real book about it. I promise I'll do my best, but even really clever physicists can't really explain this stuff in a short post, so my task seems a little daunting.

To understand special relativity, you have to understand that the speed of light (conveniently denominated c)is constant. The speed of light is the speed of light, no matter the reference frame. That's a little peculiar, because in traditional physics, the reference frame is quite important. If I throw my luggage down from my moving train to my sister on the platform, she perceives the velocity of the luggage to be the speed of my throw plus the speed of the train.
Is this true for light? In other words, if I kindly decide to shine a flashlight at my sister instead, does she measure the photons moving at c plus the speed of the train? Before Einstein, physicists believed that she would. In truth, she does not.

If you find this intuitively wrong and creepy, then at least you've understood what I'm saying. To make sure you understand the enormity of the situation, imagine that the train is traveling at .99c. I can measure the photons coming out of my flashlight at c. My sister measures the photons coming at her at c. What the heck is going on here?

It turns out that this result can be explained via time and length dilation. But what are time and length dilation, you ask? Let's focus on time dilation, and leave length dilation to a more complete text.

One of the implications of special relativity is that the faster you move, the slower time moves. The best way to understand why this happens is to understand that time is a dimension, just like the three we're used to (x, y and z, i.e., length, width, and height). In other words, as I'm sitting here typing, I'm stationary in the usual three dimensions, but moving through the time dimension. Einstein suggested that everything moves through the four dimensions at the speed of light. Thus, I am typically traveling through the time dimension at the speed of light, while staying constant in the usual three dimensions. Due to conservation of momentum, if I want to move in some direction, I have to be moving through the time dimension slower, in order to keep moving through all four at the speed of light.

If you accept that as true, then the dilation of time through speed makes perfect sense. If I'm on a train moving at .99c, then I can't be moving through the time dimension as fast as if I'm stationary. You'll have to accept my word that at .99c, both time and length are dilated by a factor of 10. Or, you can look up the Lorentz equations for time and length dilation, and calculate the result yourself. Either way, let's move on and figure out the implications!

At this point, my explanation has left a lot of room for confusion, so let me try to specify what I'm talking about. When I say that time is dilated by a factor of 10 at .99c, I don't mean that on the train, I perceive my life to be going by at 1/10th normal speed. Since the train is moving at a constant speed, I can rationally argue that I'm not moving at all. Time is moving at the speed I expect. To my sister on the platform, however, I'm moving at .99c, and my time is moving at 1/10th of the speed of hers. How can both of these experienced realities be true?

In fact, the confusion goes further. As I already established, to me and everyone else on the train, time is moving normally, because we are at rest. When I look down at the platform, I see it moving by me at .99c. Thus, I must see my sister moving through time at 1/10th the normal rate. While this result seems unbelievable, it actually resolves the question of how light always travels at a constant speed.

Imagine that I'm on the train moving .99c, and I shoot one photon in the same direction. Einstein tells me that I must measure the photon moving at c, because the speed of light is always constant. My sister on the platform sees me go by on the train, and measures the photon moving at c. How can this possibly be?

First, let's imagine that I measure how far away the photon is from me, one second after I shoot it. Since the speed of light is always c, I measure the photon as being 186,000 miles away. What does my sister measure? Since the photon always moves at the speed of light, after one second, the photon moves 186,000 miles away from her. But the train is also moving at .99c, so in that time, it has moved .99*186,000 miles away from her. Effectively, she should see the light as 1,860 miles away from the train. So which is it- 186,000 or 1,860?

The key is that my sister times and measures in her reference frame. When my sister measures the photon as 1860 miles away from my train, since my reference frame is length dilated, those 1,860 miles correspond to 18,600 miles in my frame. Furthermore, since she measured after 1 of her seconds, she sees only 1/10th of a second has elapsed on the train. After 1/10th of a second at the speed of light, the photon should be 18,600 miles away.

Don't get me wrong, I'm happy that I haven't come up with a contradiction to one of the pillars of modern physics. But this result is still pretty weird, at least to me. To my sister, my time frame is slowed by a factor of 10. To me, time is moving normally, and since my sister is the one moving at .99c, her time frame is slowed by a factor of 10. So if we reverse the whole observation frame of this thought experiment, when I measure my photon after 1 second, I see that only 1/10th of a second has elapsed in my sister's time frame. But before, we were saying that when she measures the photon, it looks like 1/10th of a second has elapsed in the train reference frame. This brain-melting result just illustrates that time is relative. There is no absolutely correct notion of when events occur: time depends on frames of reference. Since everybody not experiencing acceleration has an equal claim to be in motion, everybody has a right to talk about how fast or slow time is moving.

This is merely one of the many strange implications of special relativity. I hope it made a bit of sense.