It's all undefined!

  

Singularity

This is our first blog. So, let's start with some basics. Because, why not? So, here we go...

Mathematics defines a singularity as a point where a given mathematical object such as a function is not defined. The function reaches an undefined value such as infinity or is non-differentiable at the point.

Singularity

We can see this from an example of the function, 1/x. As x approaches zero the value of the function increases significantly. At zero, the value is not defined or we say that it reaches infinity. In Mathematics, we escape such situations by not defining function at these points. But what happens when we try to apply these equations in our real, practical life? What do these results imply?

There might be one of these two possibilities: Either mathematics is not describing reality if we look too closely or, we may claim the reality in actuality has infinities!

Let us try to understand the probability of the existence of these singularities from some examples and physical equations.

Let us take the case of a vortex flow, the velocity of fluid particles spinning in the whirlpool is inversely proportional to the distance of fluid particles from the centre of the vortex. As the fluid reaches the centre of the vortex (or its radius approaches zero) its velocity approaches infinity. At a radius equal to zero or at the centre of the vortex an infinite velocity is expected. But in actual this phenomenon is restricted by many factors.  Like the size of water molecules (2.75 angstroms), repulsion between atoms, electrons etc. Thus, the infinite velocity is not reached and singularity (at r = 0) does not exist in this example, as proposed by the mathematical equation.

A vortex

Let us take another example.

Suppose, we are walking on the equator at such a speed that we can cross all the time zones in a 24-hour span. We would complete a full circle of the earth in 24 hours. Now, if we move northwards, say near the north pole, at 60 degrees latitude, and move with the same speed, we would cross all the time zones twice in the same 24-hour span. At a latitude of 89.4 degrees this count increases to 100. As we get closer to the north pole, the number of times we cross the time zones at the same speed and same time interval increases steeply. At the north pole, as we all know all the time zones intersect at a point, the no. of times we cross the time zones reaches infinity, whatever that may mean.

But if instead of seeing these time zones on a globe, we use a map; we will see these time zones as parallel non-intersecting lines with no singularity. Thus, singularity is removed by a mere change of coordinates. This type of singularity is called co-ordinate singularity, i.e., a mere change in co-ordinates can bring or remove a singularity.

An artist's hilarious depiction of co-ordinate singularity

Let's take the example of something all of us are most interested in - The case of Gravitational Singularity!

If we take Newton’s laws of gravitation, we see that the gravitational force between two bodies is inversely proportional to the square of the distance between them. When the two bodies approach each other, the gravitational force between them increases. For a finite body, the whole mass can be supposed to be concentrated at the centre of gravity. When we reach that point, we are most probably inside the body and all the mass is over us. But, what if the whole mass was concentrated on one point?

The point would have infinite density and a point near to that would experience infinite gravitational pull. This infinite force would lead to infinite acceleration, thus, breaking the laws of physics. This point is the gravitational singularity!

Artist's rendering of a black hole sucking matter from a blue giant companion star.

Thus, we can define gravitational singularity as a point where infinite density develops as infinite space-time approaches it. This was the case with Newton’s law of Gravitation! But, as we know, Newton’s law of Gravitation is actually not that universal and it gives wrong answers where the gravitational pull is too strong, like near a star or a black hole. But what happens when we take into consideration the General Theory of Relativity?

Actually, it gives even more singularities!

For understanding this we need some high-level mathematics which is outside the scope of this blog. We use Schwarzschild Metric. This is obtained when we solve Einstein field equations for the simple case of a spherically symmetric mass in an empty universe. When we simplify it to allow movements towards or away from a massive object, this equation reduces to

This allows us to compare two points or events in space-time around a massive object from the perspective of different observers. 

Here, ∆s is the space-time interval, r_s is the Schwarzschild radius, ∆t is the time interval and ∆r is the change in position. 

As it can be seen, we can obtain about two singularities from this equation. One when r = 0, and one when r = r_s. The first case is similar to as given by Newton’s law of Gravitation. However, the second case is that of the event horizon (Event horizon is a boundary in space-time where the gravitational pull becomes so great, that escape is impossible, even for light. We will talk about it more in further articles). This is the case of a co-ordinate singularity and not an actual singularity and can be resolved by Eddington- Finkelstein tortoise co-ordinates (Big name? I know). Once, the event horizon is crossed, we still have a central singularity to deal with. Unfortunately, this cannot be done away with a simple change in coordinates. So, this gives rise to the most obvious question. Does this point of singularity actually exist? Actually, Einstein’s theory and the Schwarzschild solution derived from it, suggests it must exist. The apparent inevitability of this singularity may be evidence that general relativity is incomplete.

The two singularities predicted by the equation

So, I would leave this topic here only, for now. Hope it was useful and you liked and enjoyed reading it. Will see you soon. Till then, bye.

Danke!