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

[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.

 

[topsquark]

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.
\(\displaystyle T = \left ( \frac{2GM}{r} - 1 \right ) ^{1/2} e^{r/(4GM)} ~ sinh \left ( \frac{t}{4GM} \right )\)

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

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

\(\displaystyle X = \left ( 1 - \frac{2GM}{r} \right ) ^{1/2} e^{r/(4GM)} ~ sinh \left ( \frac{t}{4GM} \right )\)
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

 

[MoreCoffee]

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.