Supermassive Black Holes Aren’t Dense


So, the big news of the day is that we have a picture of a black hole event horizon for the first time.  If you haven’t already heard, you should realize that the M87 black hole is really really big.  XKCD provides a nice comparison.  (Though you should also know that the shadow you see is about two and half times bigger than the event horizon itself, so between the orbit of Pluto and Voyager 1.)

You might think that such a massive black hole would be extremely dense.  But you’d be wrong.

Of course, the notion of the “size” of a black hole can be a bit tricky.  All of the mass crashes down into an infinitely small volume at the center — a spacetime singularity (leaving aside any effects of quantum gravity, which we haven’t figured out).  So in that sense a black hole is infinitely dense.  But temporal notions are also vague in this context, so it’s reasonable to say that the black hole hasn’t yet gotten that dense.

When we talk about the size of black hole, we are usually referring to the size of its event horizon, the points of no return surrounding the central singularity.   To get a density, we’d like to divide the mass of the black hole by its volume.  But inside the event horizon it’s hard to even talk about a spatial notion like volume — the spacetime is obviously very non-Euclidean, and that singularity in the middle also messes things up.

But we can pretend that the black hole defined by its event horizon is just a solid sphere, and for most purposes that’s close enough.  So if we take the mass of the M87 black hole and divide it by the volume of a sphere the size of its event horizon what do we find?

We find (if I’ve done my math right) that it has a density of about 50 grams per cubic meter.  Contrast that with the density of air at sea level which is over 1,000 grams per cubic meter.  So the density of the M87 black hole is like the density of air way up in the upper stratosphere on the border of empty space.

This also means that if you had some six billion solar masses of very dilute gas like this, it would form a black hole.  You don’t need high densities.

It’s worth noting that if you were to fall into a black hole the size of M87 you wouldn’t notice any unusual gravitational effects at all as you passed through the event horizon (the burning hot gasses might be annoying though).

With small black holes the tidal effects at the event horizon are really strong.  Your feet would be pulled in a lot faster than your head, so you’d be stretched out like spaghetti.  But the tidal effect at the event horizon of M87 is way less than the effect of the moon on you right now.  So gravitationally speaking, it would be just like falling through empty space.  (Although, if you tried to turn around and escape, you’d fail.)

Indeed, the Earth could be falling into a black hole the size of M87 right now, and we wouldn’t notice any difference.  For several minutes.  Then we’d die.


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