Nobody Expected This SILLY Problem with Relativity - Coordinate Singularity

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Relativity is known for being a challenging theory to understand. And when it was being developed, it faced many boss-level challenges - conceptual, mathematical, and political. But in this video, I want to look at what I think is one of the funnier ones.

After Einstein published his field equations, the most fundamental equations of general relativity, there was a race to find solutions to them. These solutions would correspond to different spacetime geometries that we could observe, such as gravitational waves travelling through space, or the warping of spacetime around a black hole.

One of the first non-trivial solutions to be published was by Karl Schwarzschild. It shows the geometry of spacetime around a spherical body, such as a planet, star, or black hole. It describes what is known as the “interval” between two events in the spacetime that we’re studying.

For this video, we’re not super interested in the exact mathematical form of the solution. What we’re interested in instead is what happens at two specific coordinate values. The first, r = 0, represents the centre of the black hole. We see that at this point there is a divide-by-zero-error, meaning we have a zero in the denominator. This means the solution suggests an infinitely warped spacetime at this point. This is known as a singularity, and does seem to exist in reality.

The second is r = r_S, which is where the event horizon of the black hole exists. In Schwarzschild's formulation, this also appears to blow up to infinity due to a zero in the denominator. However, we see how changing coordinates eliminates this singularity. This means the singularity at r = r_S does NOT exist in reality, and is just a problem with our choice of coordinates.

We also look at an intuitive way to understand coordinate singularities by comparing Cartesian and Plane Polar coordinates on a simple piece of paper.

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Chapters:
0:00 - The FUN challenge with relativity
1:05 - Karl Schwarzschild's genius solution
2:25 - The problem with Karl Schwarzschild's genius solution
5:10 - Coordinates matter! Choose wisely

Videos in Cards:
2:23 - youtube.com/watch?v=ci06S-jNn8U&t=3s