B How Far Is It To The Center Of The Milky Way/Tootsie Pop?

Please don't tag your threads with the "A" prefix unless you are prepared for an answer appropriate for a physics PhD candidate - in the relativity forum that would mean that you have studied and understand something at the level of http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2010.pdf.

We have corrected the thread tag for you.

 

As you seemed to have intuitively understood, you cannot just lay a measuring rod to measure that distance. And because of the curvature of spacetime it would not mean the same thing anyway.

Instead, what is measured is the area of the sphere around the singularity. The unmeasurable radial coordinate is related to that measurable area by ##A=4\pi r^2##

 

The quick answer is that you don't - the "distance" from the central singularity to anywhere else is undefined, as is the "diameter" passing through the singularity.

The longer answer: although the distance to the singularity is undefined, and therefore there is no such thing as a radius or diameter, there is a quantity that we can use outside the event horizon. Consider two imaginary spheres centered on the black hole. The first one has surface area ##A_1=4\pi{r_1}^2## and the second has surface area ##A_2=4\pi{r_2}^2## - we can measure these surface areas and calculate ##r_1## and ##r_2## from them. These values ##r_1## and ##r_2## are not distances from the center, and the difference between them is not the distance between the surface of the smaller sphere and the larger sphere - but we can use them to label points around the black hole.

If you have a spherical galaxy with a black hole in the center, and someone says that it is ##X## lightyears across... They are really saying that the surface area of the galaxy is ##X\pi## square lightyears. If the black hole weren't there then the galaxy would be ##X## lightyears across... But it is there, so the distance across is undefined.