The Phantom Singularity | Space Time

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Matt O'Dowd (0:58)

The Singularity, the point of infinite density at the core of a black hole. But also so much more. In mathematics, singularities come in wild and wonderful varieties.

Isaac Newton's Universal Law of Gravitation was an incredible insight when he figured it out in the late 1600s. In fact, we still use it to fly spacecraft around the solar system today. However, it has its problems. Let's look at the math.

Newton's equation gives you the gravitational force exerted between two masses, m1 and m2, that are distance r apart. Straightforward enough, that r squared in the denominator spells trouble. It means the force gets larger the closer the masses are to each other. That makes sense, but what about when r gets really close to zero? Then the result of the equation, the force, becomes extremely large, and is infinite when r becomes equal to zero.

That doesn't really make a lot of sense. Infinite force means infinite acceleration, which means, well, physics breaks. According to Newton's law, in order to fill that infinite gravitational acceleration, you need to get zero distance from an object's center of mass. That means all of that object's mass would need to be concentrated at that center.

A single point of zero size, which means infinite density, and that, of course, would make it a black hole. We often use the word singularity to describe the hypothetically infinitely dense core of a black hole. But in math, the meaning of this word is much more general. You know what? Instead of me trying to explain mathematical singularities, how about we get a real mathematician to do this properly? Guys, meet Kelsey Houston Edwards of the new PBS show Infinite Series. Hey, Kelsey. Hey, Matt.

Kelsey Houston Edwards (3:03)

Thanks for having me on.

Matt O'Dowd (3:05)

Kelsey, the math for black holes goes to infinity for different properties and in different locations. What does this mathematical weirdness tell us?

Kelsey Houston Edwards (3:15)

Well, mathematicians use the word singularity pretty broadly. It's really just any point that causes problems. Commonly, these problematic points are where quantities become bigger and bigger, approaching infinity, as they do near a black hole.

Some singularities come about from your choice of reference frame or coordinate system. An example of a frame-dependent singularity that might be familiar to Space Time viewers is the event horizon of the black hole. I'll leave that to you to explain.

Here on Earth, the north and south pole are examples of coordinate singularities. It's possible to pass through time zones infinitely quickly, but only because of your choice of spherical coordinates.

Matt O'Dowd (3:57)

Alright, that makes sense. But the gravitational singularity at the center of a black hole is a so-called real singularity, right? I mean, the curvature and the density are infinite from any frame of reference.

Kelsey Houston Edwards (4:10)

Right. And there's no way to avoid a horrible crushing death just by switching coordinate systems. But the reality of the black hole singularity may give reason to doubt the theory that predicts such a thing. In fact, it's happened many times before.

From models of the movement of water to human population growth, mathematics predicts a physical singularity and we've been forced to reject the corresponding theory.

Matt O'Dowd (4:35)

So you're saying Einstein is wrong? Blasphemy?

Actually, Einstein himself agreed on this point. Guys, you should check out Kelsey's show Infinite Series, where she goes into much more depth on the nature of singularities. It's a math show, by the way, so it's sometimes about real stuff.

Kelsey Houston Edwards (4:55)

Mathematicians are lucky. Being limited by reality is so boring.

Matt O'Dowd (5:00)

So does the fact that it includes a singularity mean there's something fundamentally wrong with Newton's law of gravitation? Well, we already know the law isn't really so universal. When the gravitational field is too strong, say near a star or a black hole, Newton's law gives the wrong answers, and we need Einstein's general theory of relativity, which is the far more complete theory of gravity.