## Sunday, July 14, 2019

The Schwarzschild radius of a nonrotating black hole grows linearly with mass with a constant of proportionality of about 3 km per solar mass.  (This linear relationship is surprising, since sphere volume grows cubically with radius.  This results in supermassive black holes having surprisingly low average density.)  This proportionality constant per solar mass can be calculated to high precision from the standard gravitational parameter, using the "units" program:

2 gauss_k^2 au^3 day^-2 c^-2 = 2.953250077 km = 2953250077 micrometer (to 1 micrometer precision)

Diameter is 5.9065002 km per solar mass.

However, we typically won't know the mass of a black hole in solar mass units to high precision.

We do know the mass of the earth in solar mass units to high precision.  If a supervillain were to stop earth's rotation then crush the earth into a black hole, our event horizon would have a radius of

2 gauss_k^2 au^3 day^-2 solarmass^-1 earthmass c^-2 = 8.8700560 mm, accurate to 8 significant digits (0.1 nm).  The diameter would be twice that: 1.7740112 cm.

These high precision results are interestingly possible despite not knowing G, the gravitational constant, to high precision.

Inspiration was LIGO's discovery of two 30-solar-mass black holes merging.  Just how large were these objects that whipped around each other so fast that their orbital frequency corresponded to a sound audible to humans?  Answer: each had radius about 90 km.

The peak frequency was 250 Hz, and we get 2 peaks (one per black hole) for each orbit, so they were orbiting at 125 Hz or 7500 RPM.  (Does general relativity make this calculation completely wrong?  Time goes slower in strong gravity.)

Both the size and frequency are are human-comprehensible numbers.  We can imagine an object spinning at 7500 RPM -- we often build such things.  We can sort of imagine an object, maybe a drum or spinning arm, with diameter or length 2*90 km = 110 miles.  Next, just spin that thing at 7500 RPM.

What is the largest object we can make out of conventional materials that can spin at 7500 RPM without breaking apart?  It's a rather dangerous experiment.

Smaller merging black holes can get closer so would probably have higher peak frequency.