# Black Holes and Dinosaurs

So the last topic was on the idea of infinity (remember that we have to be careful about infinity – it is not a number, just a concept of a really big number).

But once you get into these abstract ideas, you might be tempted to ask: what’s the use? Well, Mathematics is often useful: you can mathematically describe a beautiful butterfly of chaos or come up with the probability of detecting extraterrestrial civilizations within our own Galaxy. But what about infinity? Does physical infinity actually exist? Let’s drop down some ideas which were previously thought to be infinite but later shown not.

#### The size of the universe.

The size of the universe is finite but unbounded. In other words, the universe does have an edge, although the edges are getting farther away from each other every day. The fate of the universe is a very well studied topic, but yet to have any conclusion. [1] If the universe has finite size, then everything inside has to have finite size as well.

#### The speed of light, $\displaystyle c$.

When you switch on the light, you instantly see the effect. Light can travel around the Earth about 7.5 times in a second; in our daily life, light seems like it has infinite speed. But it actually takes 8 minutes for the light from the Sun to travel to the Earth. You may have heard the media going crazy when a group of scientists announced that they observed neutrinos appearing to have travelled faster than $\displaystyle c$. The fuss was because if this had been true, Einstein’s Special Relativity would be violated, since the theory was based on the assumption that nothing travels faster than $\displaystyle c$. If this had been true, we would therefore need to come up with a brand new theory – a revolutionary prospect, seeing as Einstein’s theory of relativity is the basis of almost all of modern physics. But for now, the speed limit of nature is still $\displaystyle c$.

So, there is no infinite size or infinite speed. What else could possibly be infinite?

#### Dense Talk

This may not be the first unit that would come up in your mind but let’s take a moment and talk about density. What would infinite density look like? Density is mass divided by volume, $\displaystyle \rho=m/V$. We just talked about no mass being infinite, so let’s keep the mass constant. In this case, making the volume infinitely small will get us closer to infinite density. Now, a smaller volume means smaller radius, since $\displaystyle V = \frac{4}{3} \pi (radius)^3$ (in three dimensions) or $\displaystyle V = \pi (radius)^2$ (in two dimensions). [2]

Now let’s turn our attention to the concept of escape velocity, $\displaystyle v_e = \sqrt{2GM/r}$. This is the minimal speed needed to get out of an object’s gravitational field, a bit like the minimal speed that little dinosaurs needed to achieve in order to escape being eaten by big dinosaurs. (The relevance of the dinosaurs will become clear in a moment – bear with me.) In the above equation, $\displaystyle G$ is just a constant and $\displaystyle M$ is the mass that we decided to keep constant. Then $\displaystyle r$ (radius) is the only thing that is varying and we’re making it smaller and smaller. So the escape velocity will keep getting bigger and bigger. But let’s remind ourselves: the speed limit of nature is $\displaystyle c$. What happens when we have high enough density – which means small enough radius – such that the escape velocity is faster than $\displaystyle c$?

We have a black hole.

Nothing travels faster than the $\displaystyle c$, so no light is travelling to us from the black holes. Hence, it is literally just a “black hole in space”.

So, using only very simple steps and with a little help from the idea of infinity, Mathematics allowed us predict the existence of black holes. This is, in fact, what happened in the 1960s when mathematicians first came up with the idea of black holes. It wasn’t until much later that black holes were actually observed by astronomers. [3]

Beautiful, isn’t it?

But how did astronomers observe black holes if they’re black?

Let’s continue with our analogy of the dinosaurs and let’s imagine a small island (say, Island A) surrounded by water. A batch of small dinosaur eggs were laid in the center of the island. When they hatch, the small dinosaurs are programmed to crawl towards the water. To do that, however, they must pass through the forest where big dinosaurs live; since the small dinosaurs cannot crawl faster than the big dinosaurs, no small dinosaurs ever get to the water. Consequently, no one from outside of Island A will ever know that those small dinosaurs ever existed. This is much like the case of black holes, where small dinosaurs are photons (light) and big dinosaurs are gravity.

Let’s say that on another island (say, Island B), the same batch of eggs are laid, but the small dinosaurs are actually able to get to the water this time – the forest is only inhabited by medium dinosaurs who cannot catch all the small dinosaurs. When the small dinosaurs get to the water, they travel straight unless it is to avoid danger. Obviously, Island A is something that they must avoid. Hence the small dinosaurs change trajectory when they close to Island A. Because you know that small dinosaurs have a property which predicts that they will travel straight unless they are faced with danger, you know that if the trajectory is changed, there was a dangerous island in the middle. To bring this back to black holes, light normally travels straight, but will change its trajectory when it gets close to a black hole. When this distortion is observed, astronomers can indirectly infer the existence of the black hole. This is called gravitational lensing.

Is knowing a little more about black holes going change your day tomorrow? Probably not. Would dinosaurs have survived if they had known about black holes? Doubt it. But who doesn’t want to know a little more about black holes?

 Notes: [1] Physicists are currently studying to see whether the universe will expand forever or stop expanding at one point. [2] We’re making the assumption that we’re dealing with circles or spheres (which isn’t such a bad assumption). [3] The above prediction of the existence of black holes used Newtonian mechanics which is not the best approximation when looking at large scales such as stars and planets. Mathematicians used General Relativity and this is one of the solutions that the theory gives.