How is the "size" of a black hole defined?

“Size” doesn’t seem to have meaning in a black hole, and yet I hear people referring to “bigger” and “smaller” black holes.
My rudimentary understanding of the concept of size is that it’s a description of distance which is a description of the time it takes for a particle moving at a known rate to cross an area. But a photon, for instance, takes an infinite amount of time to cross a black hole. Holes are, as I understand it, ruptures in space, and thus can’t be defined in terms of space and size. So why do people say that a black hole is “bigger?” My hunch is that this is just shorthand to say that they have more sucking power.

The size of a black hole is defined as that radius at which the surface escape velocity is equal to the speed of light in a vacuum. It’s called the Schwartzchild Radius. It equals twice the Newtonian constant of gravitation multiplied by the mass of the star and divided by the square of the velocity of light in a vacuum.

A star of about 2 solar masses is thought to be the smallest that can form a black hole. For that mass the Schwartzchild Radius equals about 6 km. The physical star itself can be a lot smaller than this, but the black hole remains at that radius.

You can see that the size of the black hole rises linearly with the mass of the star that formed it.

You have to distinguish between the size to an external observer and to one inside the black hole. The latter will always see it as infinitely large since no matter how fast he–or a photon–moves, it will never get out. But the external observer sees an event horizon and can measure its diameter.

Incidentally, to an observer falling into a black hole, especially a very massive one, nothing special happens at the event horizon. No bells ring nor lights flash and, if it is big enough, there is not even an unusual tidal force.

I once read that technically one has to speak of the circumference of an event horizon rather than it’s radius. Is this so?

The formula for the radius may be found here. It simplifies to r[sub]s[/sub] = 1.48×10[sup]−27[/sup] m/kg.
The density of matter required to form a black hole gets surprisingly low, surprisingly fast.
For example, you can calculate how large a body of air, at 25 °C and 1 Atm pressure, could get before it turns into a black hole.
The density of air being 1.168 kg/m[sup]3[/sup], the mass of a sphere of such air will equal 4.89 r^3 kg. Setting that equal to the Schwarzschild radius of 1.48e-27 X m gives 7.24e-27 r^3 = r[sub]s[/sub].
Solving for r gives a radius of 11.7 billion kilometers.
That number is in the same range as Pluto’s distance from the sun, 7.4 billion kilometers at aphelion.

That’s not a bad way of thinking about it, actually. The notion of “radius” sort of implies a measuring stick stretched from the center of the black hole to the event horizon. However, time & space get rather mixed up inside a black hole, so much so that the “radial” coordinate one would naively used to measure distances outside the black hole starts measuring time intervals inside the black hole instead. In a very real sense, there’s no “center” of the black hole that you can measure spatial distances to. However, you can meaningfully talk about the circumference of a circle going around the outside of the event horizon, and the circumference of this circle is indeed 2*pi times the Schwarzchild radius.

Even more useful than the circumference of the event horizon is its surface area. Among other interesting properties, the surface area of a black hole is analogous to entropy in thermodynamics (or, depending on how you look at it, it’s just another kind of energy). No matter what you do to a black hole or set of black holes, you won’t be able to get the total area of event horizon to decrease. If, for instance, you collide two black holes together to form a single, larger black hole, the total mass of the new hole might be less than the sum of the masses of the two smaller ones (a lot of energy can be released, mostly as gravitational waves), but the area of the event horizon of the new hole will be at least equal to the sums of the areas of the old ones (this puts a hard limit on how much energy can be released in such a collision).

That’s still a hell of a lot of mass to collect in one locale. The average density of the galactic core is estimated to me even much lower than that, on the order of a few grams per cubic meter. You can also have fun with timelike curves in space around the ergosphere of a rotating black hole, but unfortunately you can’t get off the ride until you come back to or pass your starting point, so as a time machine isn’t not terribly useful. You could spin a black hole until it becomes a Kerr ring singularity and is “naked”, but most right thinking [sup]*[/sup] people believe that there is some physical law that would prevent this from happening, as it would discombobulate the whole cause-and-effect assumption that underlies our willingness to invest in the commodities market.

Another thing to note about black holes is that they have only three fundamental properties: mass, electric charge, and spin. From these qualities you can derive all other properties. All black holes have “no hair”; that is, two black holes with the same values for these properties are as indistinguishable as two electrons, and in fact some people (most promenently John Wheeler, Christopher Isham, and Brian Greene) have suggested that electrons and other fundamental particles may in fact be something like very tiny black holes that are somehow both bound from decaying and fit into a precise, unalterable combination of mass, charge, and spin. Wheeler developed his theory of geometrodynamics as a precursor to a theory of quantum gravity and unified relativistic quantum field theory i.e. essentially a Theory of Everything (TOE) that would unite General Relativity and quantum mechanics. Wheeler couldn’t get it to work out and in fact was his own biggest critic, but his work still underpins modern developments in QG.

Stranger

[sup]* “…who are largely recognizable as being right thinking people by the mere fact that they hold this view.” – Douglas Adams[/sup]

What? It’s only 3.9 billion suns worth, less than 0.7% of the mass of the Milky way. :wink:

So you can’t measure the radius of the black hole, but, as Hari Seldon pointed out, you can measure its diameter which will be twice the Schwartzchild radius. From outside, then, can’t you postulate a hypothetical center?

From inside you can’t do it but who cares about those inside with whom all communication is impossible for us? :wink:

The last sentence actually highlights the problem: inside the event horizon of a black hole, nothing whatsoever “exists”, insofar as an observer on the outside is concerned. A black hole is just a massive particle with mass, electric charge, and angular momentum; hence it has “no hair”, i.e. no other inherent properties. Without this, it can’t properly have a center, though we can infer what would be its geodesic center from its mass and angular momentum. This may seem like mere semantics, but when you’re dealing with a region that can’t be described by existing physical laws it’s probably a mistake to give it terms that sound like you know what you’re on about.

As previously noted, the surface area of the event horizon can’t get smaller; this alone probably prevents the formation of a natural Kerr ring singularity–it will either radiate away energy as gravity waves or virtual particles while contracting, or will hit some kind of limit where you can’t give it any more angular momentum or (in the case of a Kerr-Newman construct) electical charge without the event horizon becoming larger, preventing the formation of a naked singularity. Even though the field equations can detail what properties a naked Kerr singularity would have, it seems unlikely (at least, from my limited comprehension of gravitation and GR), such a condition probably won’t arise in nature. Whether you could create one artificially (by holding conditions just so and forming a singularity without an event horizon) is unknowned, but it would probably be unstable.

Stranger

How would a black hole manifest having an electric charge? By an otherwise inexplicable reluctance of particles with the same charge to approach it? But from the outside anyway a black hole is simply a rather bizarre region of space. How the heck does space have an electric charge?

A black hole is a bizarre region of space[time] deformed by the presence of mass. It’s likely (in fact, almost certain) that some of this mass has to have an electric charge, and unless all of these charges exactly cancel out, a black hole will have a net charge. Mathematically, this complicates things slightly, and a hole with a significant charge will have enough repulsion to overcome gravity and push other charged particles away, the electromagnetic force being significantly stronger (at short range) than gravity, so there’s a limit to just how much charge a black hole of a specific mass can have.

Stranger

It’s impossible to spin a BH hole faster than it can survive, thus a naked singularity is impossible. For example if you tried to increase the spin of a maximally spinning hole by throwing spinning matter into it, centrifugal forces will prevent the matter from reaching the horizon.

See: (Israel, W. 1986,“Third Law of Black Hole Dynamics - A Formulation and Proof,” Phys. Rev.Lett. 57, 397.)

I thought that all forces are conveyed by particles. If nothing can come back over the event horizon of a black hole, how can the effects of a black hole’s charge be felt outside the event horizon?

Charge interactions are mediated by virtual photons and virtual particles aren’t constrained by the speed of light.

In fact, a black hole will typically only fluctuate by the charge of a few electrons above and below neutral, so great is the tendancy for a net charge to be neutralized.

Nitpick: The ratio of the electromagnetic force to the gravitational force for a given pair of particles is the same everywhere, since they both fall off as r[sup]-2[/sup]. So it’s not really accurate to say that the electromagnetic force is less “at short range”.

Just as an aside Richard Feynman in his Feynman Lectures and Six Easy Pieces has an interesting (but impossible) thought experiment that pretty clearly drives home the incredible forces involved in electromagnetism.

He says something like “if you had two grains of sand thirty meters apart and one were made of all positive charge and the other all negative charge, then the force between them would be three million tons.

Three million tons!! - that’s a lot of tons. No wonder protons and electrons tend to form neutral systems.

Yes and no… Electric charges tend to pair up with opposite electric charges, forming various sorts of multipoles, which do in fact fall off quicker than 1/r[sup]2[/sup]. But gravity doesn’t have this option, since there are no negative “gravitational charges”. So in practice, gravity is stronger at long ranges.