In part 1 of the Special Relativity series, two points were brought up. Nothing can travel faster than light, and time slows and objects contract when they travel very fast. In part 2, we covered the second point, finding out that time dilation and length contraction are simply logical consequences of light traveling at c in a vacuum relative to every observer. But what about the first point? We briefly covered that it doesn’t make much sense. If we have a rocket with an unlimited amount of fuel, we can always make it go faster than it was going before. In principle, there is no limit to how fast it can go! But there actually is! The cosmic speed limit is simply c, the speed of light in a vacuum. And we can prove it. Remember the light-clock thought experiment we did in part 2? Time slows down when the object (in this case, you) is in motion. The light-clock can help us see how much time slows.
When traveling at subluminal speeds, the answer is just a bit of geometry and algebra away. But let’s try and break the rules. You’ve been told you’re entire life that you can’t travel faster than light. But this is a thought experiment! Let’s try it and see what happens. You are on a bus traveling faster than light, and your friend is left behind at the bus stop. You set up a light clock next yourself. The light clock is prepared to function properly, by having the light beam set vertically from your perspective so that it never escapes. This is possible, because no matter what speed the bus is going, you are in an inertial frame of reference. Now, what does your friend see? Umm… Something’s wrong here. Let’s back up for a bit. From your perspective, the light beam is set up in the clock in such a way that it never escapes. You can do this because, with respect to you, everything behaves like you’re not moving. So your friend should see it stay in the clock too. But if it did, it would need to travel faster than light just to keep up with you, because you’re traveling faster than light! But that goes against the fact your friend is supposed the see the light traveling at c, which is the very point we started this entire process with! This is an impossible situation.
The very notion of going faster than light in a vacuum is incompatible with the rules we’ve come to know, so something ends up breaking. Going faster than c inevitably causes an inherent contradiction. Okay, that makes sense, but it still seems a bit iffy. What if we try to approach light speed and see what happens? Busses have kinda gotten old at this point, so let’s stick you on a rocket. Specifically, that rocket with the unlimited amount of fuel that continues to make itself go faster than before. From your perspective, you start at rest, and experience acceleration due to the burst of fuel.
You are now going at a new constant velocity, and therefore, you are in a new inertial frame of reference. This new situation is exactly the same as the original, so there’s nothing stopping you from accelerating by the same amount yet again. From your perspective, the same thing can go on indefinitely, always accelerating you by the same amount as before. Hey! But I thought there was supposed to be a cosmic speed limit. Doesn’t this situation mean that you can increase your speed by the same amount as always forever? Well, yes and no. Let’s take a look at what this situation looks like from your friend’s perspective. Let’s assume that the rocket starts at rest relative to your friend. He can observe the rocket accelerate, increasing its speed. And then the rocket accelerates again, gaining even more speed, but not by the same amount. As we learned last time, time dilation and length contraction occur when traveling at high speeds. And if shorter distances and extended times mean anything, it’s that speeds that are lower than expected. As the rocket gets faster, its speed gets dampened by the effects of relativity.
This happens in such a way that you can always increase your speed, but the dampening always keeps you from reaching the speed of light in a vacuum. Now of course, all your friend would notice is that it’s getting harder and harder for the rocket to accelerate, like it has more inertia than before. It looks, just like space and time, as though mass is relative. As long as an object has mass, this will happen to it. How do we know for sure? Particles in particle accelerators have been brought to speeds over 99% c, but no matter how hard you push, you cannot get these particles to surpass the cosmic speed limit. The only thing that all of this doesn’t apply to is light itself, since light, by definition, does travel at c in a vacuum.
What does everything look like from the light’s perspective? If we place a light clock next to the light beam itself, we find out that light experiences no time at all. A photon would experience zero time traveling from one end of the observable universe to the other. But if you asked the light how it could travel such a far distance in literally no time at all, it would tell you the answer was simple. For the light, the entire length of the universe has contracted to zero. It would tell you that its entire life lasts a marvelous instant, being everywhere it will ever be at once, because everywhere it will ever be is right here. Wow, what an ego!.
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