MHB Did you watch the PBS Space-time Math video on Singularities?

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The discussion centers around a PBS Space-time Math video that explores the concept of the Schwarzschild radius in black holes. It highlights that the singularity at this radius is not an actual singularity but rather an artifact of the coordinate system used. The Kruskal-Szekeres coordinate system is introduced as an alternative that treats the Schwarzschild radius as an ordinary point, eliminating the perceived singularity. The discussion acknowledges the complexity of this coordinate system, noting that while it resolves the issue at the Schwarzschild radius, a true singularity still exists at the center of the black hole. The mathematical intricacies involved in these concepts are recognized, emphasizing the respect for physicists who navigate such challenging topics.
MoreCoffee
I watched this earlier today. I hope this is the right place to mention it. It's a PBS Space-time Math video clip. I hope some of you like it.

 
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I watched the video and there is another that is listed in the end notes. The video contain a link to "The Phantom Singularity." In it the host describes what happens at the Schwarzschild radius of a black hole. The show mentions that the singularity at the Schwarzschild radius is not a singularity at all, but an artifact of our coordinate system. In case anyone is wondering about a coordinate system that doesn't blow up, here is an example of a coordinate system where the Schwarzschild radius is just an ordinary point. They are called the Kruskal-Szekeres coordinate system in one dimension of space (labeled "r") and the time dimension (labeled "t"). We change the coordinate system to have variables X and T.
[math]T = \left ( \frac{2GM}{r} - 1 \right ) ^{1/2} e^{r/(4GM)} ~ sinh \left ( \frac{t}{4GM} \right )[/math]

[math]X = \left ( \frac{2GM}{r} - 1 \right ) ^{1/2} e^{r/(4GM)} ~ cosh \left ( \frac{t}{4GM} \right )[/math]
for the outside of the black hole (r > 2GM), and

[math]T = \left ( 1 - \frac{2GM}{r} \right ) ^{1/2} e^{r/(4GM)} ~ cosh \left ( \frac{t}{4GM} \right )[/math]

[math]X = \left ( 1 - \frac{2GM}{r} \right ) ^{1/2} e^{r/(4GM)} ~ sinh \left ( \frac{t}{4GM} \right )[/math]
for the inside of the black hole (0 < r < 2GM).

There is no question that this new coordinate system is a rather tough one to work with (and is not intuitive at all like Cartesian and Spherical coordinate systems.) But there is no discontinuity at the Schwarzschild radius. The lack of a singularity in these new coordinates means there really isn't a singularity there despite what other coordinate systems are used.

There is still a singularity at the center of the mass however. No coordinate system gets you out of that one.

-Dan
 
Co-ordinate system singularities applied to the event horizon of a black-hole is interesting and informative. The mathematics is fairly complicated too. It gives one a feeling of respect for Physicists.
 
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