Speed of light and sugary water

TL;DR – The speed of light tells us how many different positions can fit in an interval of time.

I was having a run with a buddy of mine. “The speed of light is fascinating” he panted. “Could we ever go faster?” “If we go faster, do we travel back in time or something?” I frantically answered: “could we talk about it after we lose the mountain lion?”

The fact that the speed of light is the same for all observers is admittedly quite confusing. We are so used to the fact that when we move, everything slows down compared to us. This is convenient when being chased by a mountain lion: as you increase your speed, at some point you’ll be faster than the damn beast (or one of your friends) and you won’t have any problem. But with light is different: no matter how much you run, it outruns you with the same mindboggling speed. It’s a good thing light does not want to maul you.

This has all sort of weird consequences. There’s this bit with Einstein on a train struck by two bolts of lightning, one at the first car and one at the last car. While they are simultaneous for the observer on the platform, they are not for the observer on the train. Which is totally crazy: a train hit by two bolts of lightning at the same time? What are the odds of that?

But why is it that the speed of light is the same for everybody? Is it something truly mystifying that we should contemplate for days and days? Or is it something more obvious? Let’s jump into a moronically insightful thought experiment.

Sugary water

You work at a chocolate factory. It’s an interesting place. The workers are very cheerful even though they seem to never get paid. The owner is a brick short of a load, though that has its advantages. For example, last month he gave you the go ahead on your proposal for a chocolate river with a chocolate waterfall.

This time you are looking into improving sugar distribution. You wanted to make a second smaller river, yet you had the problem that sugar melts only at very high temperature. A visitor fell in and it generated all sorts of issues with the production. These sorts of accidents seem to happen often in this factory, which is very annoying.

At any rate, you decided to dissolve the sugar in water to avoid further boiled-alive visitors. The speed and temperature of the water is set because of union regulations. You have the system setup and you are trying to improve efficiency by moving the sugar as fast as possible.

Your first attempt is able to move 100 grams per second. Which is not that great. You dump more sugar in the water and you are able to improve the flow to a kilogram of sugar per second. Which is good for the production of the new edible marshmallow pillows. With further tuning, you are able to up the speed to two kilograms per second. Which is great for the everlasting gobstopper. But, no matter what you do, you don’t seem to be able to go any further. Seems the most you can get is two kilograms of sugar per second.

You decide to hire a local scientist as a consultant. You describe the factory, your sugar distribution system and show him the designs. He looks troubled and distressed, but very politely tells you the root of the problem: the flow of water is fixed to one liter per second and a liter of water can only dissolve two kilograms of sugar. Given the circumstances, the maximum speed for sugar distribution is two kilograms per seconds.

Unfortunately, you don’t seem to have time to elaborate a new strategy: the very next day the factory is shut down during a surprise inspection by the health department. What unfortunate timing.

Space-time capacity

It seems very logical that one can’t fit an arbitrarily large amount of sugar in a finite amount of water. Maybe it’s not so unreasonable that one can’t fit an arbitrarily large distance in a finite amount of time.

Suppose you have a road that is one kilometer long and suppose you want to park your car there. How many possible spots are there? How many possible states? You may answer: an infinite amount. But that’s problematic. What if the road is just one meter? Wouldn’t you say there are one thousand times fewer possible ways to park, fewer possible states? If you said infinite for the first, you’d have to say infinite for the second. So you would be saying that there is the same number of possible states within a kilometer, a meter or a nanometer. Try parking your car in a nanometer. You can’t do it even if you have one of those small German cars. The actual answer must be: there are one kilometer’s worth of possible states.

Now, suppose you have a second of time and you want to park your car sometime within that second. How many possible spots in time are there? How many possible states? You have the same problem. So the answer must be: a second’s worth of possible states. So, both kilometers and seconds, in a sense, measure the same thing: the number of possible ways you can park your car. That should give you a sense of how important your car is.

But if they are measuring the same thing, shouldn’t we be able to convert one into the other? In the same way that we convert kilometers into meters, or hours into seconds? Shouldn’t there be a constant that tells me how many possible states there are in one second, as measured in kilometers? Indeed there is: 1 second’s worth of initial states corresponds to about 300,000 kilometers worth of initial states.

So, suppose you want to move your car. At every spot in time you’ll want to have a different spot in space. You can’t go through more states in space than you go through in time. So if you move for one second, you can only go through as many as 300,000 kilometers’ worth of initial states: the maximum speed is 300,000 kilometers per seconds.

The fundamental constant we call “the speed of light” is not really a speed: it’s a conversion constant between counting possible initial conditions in space and counting them in time. Maybe we should call it “space-time capacity” or something. It becomes the maximum speed because we can’t go through space without going through time as well. And anything that has no mass, such as light, travels at that speed. So, if you really want to go as fast as the speed of light, you have to shed a few pounds. All of them.