Can black holes have densities comparable to entire galaxies?

[PeterDonis]

It is correct that there are likely no non-rotating BH in existence.

Actually, it's quite possible that there aren't even any slightly rotating BH in existence. The usual assumption is that a BH formed by collapse of a single star would have negligible spin, but I'm not sure that's actually true. We don't have data on the spins of other stars AFAIK, but if we compute the spin parameter of the Sun, we find that it's about 0.5--which is certainly not negligible.

Here's the quick computation: the Sun's mass in geometric units is about $2 \times 10^{30}$ kg, or about 1480 meters in geometric units. The Sun's angular momentum in geometric units is $GL / c^3$, where $L$ is the angular momentum in conventional units, which is about $4.4 \times 10^{41}$. But the geometric parameter $a$ in the Kerr metric is $j / m$, where $j$ is the angular momentum in geometric units and $m$ is the mass in geometric units. So if we take the ratio $L / M$ in conventional units (where $L$ is the angular momentum and $M$ is the mass), and then divide by $c$, we will get $a$. So we get

$$
a = \frac{4.4 \times 10^{41}}{2 \times 10^{30} \times 3 \times 10^8} \approx 740
$$

So $a \approx 0.5 m$. If we assume that a typical star that collapses to a BH has a similar ratio of angular momentum to mass, then we would expect a BH formed by stellar collapse to have a significant spin parameter.

 

[mfb]

Actually, it's quite possible that there aren't even any slightly rotating BH in existence.

You can always have infalling matter (or even a binary merger) that happens to cancel the spin at some point.
They are rare, but I would be careful about "not any" statements.

 

[PeterDonis]

You can always have infalling matter (or even a binary merger) that happens to cancel the spin at some point.

In principle, yes, you could, but I would rate this as extremely unlikely in practice. Basically you would have to have enough mass (either of infalling matter or another hole) that got captured into an orbit with just the right total angular momentum (counting orbital and the spin of the captured mass) to negate a large fraction of the hole's spin. That seems like a highly unlikely coincidence.

 

[mfb]

I didn't argue against unlikely. You can avoid the fine-tuning if infalling matter has angular momentum opposite to the black hole (e.g. from a captured star orbiting in the "wrong" direction), then spin will slowly decrease, reach zero, and the black hole will start spinning in the opposite direction afterwards. We have billions of galaxies in the observable universe - it will have happened somewhere.

 

[EinsteinKreuz]

Actually, it's quite possible that there aren't even any slightly rotating BH in existence. The usual assumption is that a BH formed by collapse of a single star would have negligible spin, but I'm not sure that's actually true. We don't have data on the spins of other stars AFAIK, but if we compute the spin parameter of the Sun, we find that it's about 0.5--which is certainly not negligible.

Here's the quick computation: the Sun's mass in geometric units is about $2 \times 10^{30}$ kg, or about 1480 meters in geometric units. The Sun's angular momentum in geometric units is $GL / c^3$, where $L$ is the angular momentum in conventional units, which is about $4.4 \times 10^{41}$. But the geometric parameter $a$ in the Kerr metric is $j / m$, where $j$ is the angular momentum in geometric units and $m$ is the mass in geometric units. So if we take the ratio $L / M$ in conventional units (where $L$ is the angular momentum and $M$ is the mass), and then divide by $c$, we will get $a$. So we get

$$
a = \frac{4.4 \times 10^{41}}{2 \times 10^{30} \times 3 \times 10^8} \approx 740
$$

So $a \approx 0.5 m$. If we assume that a typical star that collapses to a BH has a similar ratio of angular momentum to mass, then we would expect a BH formed by stellar collapse to have a significant spin parameter.

There is clearly a conflict between theory and observation about rotating black holes. Because there is already observational evidence for black hole in the heart of M87 that shows evidence of rapid rotations by means of its relativistic jet. Seems like GR and quantum gravity need to really catch up and explain how these things exist and hold themselves together.

 

[mfb]

There is clearly a conflict between theory and observation about rotating black holes. Because there is already observational evidence for black hole in the heart of M87 that shows evidence of rapid rotations by means of its relativistic jet. Seems like GR and quantum gravity need to really catch up and explain how these things exist and hold themselves together.

Where is the conflict now? GR works nicely with a < 1.