Determining the density of a singularity

[Perpendicular]

I have a question about singularities. Determining the density of a singularity involves the mathematically absurd and undefined function of dividing by zero.

What I don't get is this : How can a mathematically absurd entity exist in reality ?

Also, if you multiply the density of a singularity by its volume to get its mass, you get the mass of the singularity as 0. Is that not absurd ?

Ultimately, what I want to know is : How can a real object be defined in absurd and undefined mathematics? That seems to be a revokement of everything I've ever learnt.

 

[nicksauce]

First point: You can't multiply the density of a singularity by its volume and get 0 as its mass. The density is infinite, so this operation is not defined.

Second point: Most people don't expect singularities to exist physically, but rather as a sign that GR breaks down and that a quantum theory of gravity is needed to properly explain what is going on on those scales.

 

[Matterwave]

Either that or we just don't have the right math to work with it...but I think the more popular idea is that GR breaks down.

 

[nearc]

dividing by zero is not a mathematical absurdity, to be sloppy a/0 = infinity, however to be more presice its the limit as we approach zero

 

[Matterwave]

Dividing by zero is undefined. It was never defined as the limit as we approach zero. The limit is a limit, the division is a division.

Even for the limit, the 2-sided limit does not exist, and the 1-sided limit diverges which means that the limit does not exist within the real numbers.

 

[JesseM]

Dividing by zero is undefined. It was never defined as the limit as we approach zero. The limit is a limit, the division is a division.

Even for the limit, the 2-sided limit does not exist, and the 1-sided limit diverges which means that the limit does not exist within the real numbers.

You can have extensions of the real numbers like the Riemann sphere which include an infinite number which allows division of a nonzero number by zero to have a well-defined answer. In general there's nothing intrinsically wrong with having certain quantities go to infinity in a physical theory, for example the idea of point particles with finite mass in Newtonian gravity (or point particles with finite charge in classical electromagnetism) needn't lead to any breakdowns in your ability to predict later states from earlier ones in a deterministic manner despite the presence of infinite densities. The reason GR is thought to break down in the neighborhood of singularities is not just because infinities are bad, but because specific aspects of quantum theory seem to conflict with classical GR once you get to the Planck scale. If not for this type of conflict physicists would probably be happy to accept the idea that singularities might be real physical objects.