The media often portray black holes as high-density ‘cosmic vacuum cleaners’. But are they really always very dense?

Let’s do the maths:

Let’s consider the interior of an event horizon with the Schwarzschild radius RS as a black hole, because nothing can escape from it. Since no direction is preferred, we have a sphere with radius RS:

RS =

2 G M

c2

Obviously, we can assign a black hole radius to any mass.

Task: Calculate your own Schwarzschild radius by inserting your own mass into the formula.

Good news: As far as we know, there is no mechanism that could turn you into a black hole.

So far, only supermassive black holes with a radius larger than our solar system have been observed in the universe.

Exercise: Calculate the mass of a black hole whose Schwarzschild radius is approximately 40 AU (600 million km) (this is roughly the semi-major axis of the dwarf planet Pluto).

Task: Now calculate the mass density in this sphere, assuming (unrealistically) that the mass in this sphere is distributed homogeneously!

Mass Density ρ =

M

V

     with a volume of the sphere: V =

4

3

 π R3

Compare the mass density calculated here with typical material densities that we know on Earth: which materials have similar densities?

The black hole at the centre of our Milky Way is much larger than the solar system. What is its mass density and what material densities is it comparable to?

nota bene: By definition, we have no chance of ever looking inside this RS sphere. Therefore, we can assume whatever we want a priori – our physics no longer applies within this radius because it is a singularity of space-time.

What is a singularity? … something similar to a pole (Wikipedia link) in 4D.

Solution:

Schwarzschild radius of a human being: If we use the mass of a very large teenager, we get a Schwarzschild radius that is smaller than the radius of an electron.

(Carried out with a course in ASL 2005,

the largest course participant was a 1.98 m tall, strong young man,

so let’s say: around 100 kg mass)

Density of a supermassive black hole: Using the size of the SoSy, we get a black hole density comparable to chlorine or basalt. Even larger black holes would therefore also reach the density of air. Conclusion: Black holes can have all densities and do not necessarily have to be particularly dense.

(carried out with a physics class at a Berlin grammar school in 1999,

as well as numerous courses in the ASLs)

Why is that?

In general relativity, the equations operate with the quotient of mass M/ radius R.

In contrast, in the mathematical equation for the mass density, the radius R grows much fast than M, namely to the power of 3 (R³).

Simple Verbal Explanations

When someone asks me during my public observatory tours, ‘What are black holes?’, I always refer to the simplest, historically first solution of the theory of relativity, the so-called Schwarzschild solution (named after Karl Schwarzschild (Wikipedia)). According to this, a black hole is:

‘a sphere in space from which light cannot escape.’

Incidentally, ‘everything can enter this sphere because it is not bounded by outer space, i.e. it does not have a hard surface like the Earth. However, once something is inside, it can never escape because not even light can escape.’

Furthermore, it is easy to see

why light cannot escape.

Well, because the escape velocity is greater than the speed of light. Everyone knows that a rocket needs a certain escape velocity to launch into space from Earth. If it does not reach escape velocity, it will not reach space, but will only rise as high as a thrown ball or an aeroplane, for example, and then fall back down at some point (ballistic trajectory). For Earth, the escape velocity is approximately 11.2 m/s, for the Sun it is greater, for the Moon it is smaller (which is why astronauts always bounce around on TV pictures from the 1960s).

If the escape velocity of an object is greater than the speed of light, any light emitted by someone there must ‘fall back’ and cannot escape. Since light is the fastest thing in the universe (see the theory of relativity), and everything else is slower, nothing else can escape from this sphere from which light cannot escape.

Do black holes always have high density? – No, that’s popular science nonsense. You can calculate with school maths that there are also black holes with the density of air.

Do they attract everything? – No! If you are far enough away, you orbit around them and eventually you don’t feel it anymore. Just as the Earth attracts us, but TV satellites don’t fall down, they orbit around the Earth, and on Jupiter you hardly notice the Earth’s gravitational pull at all.

It is important to keep in mind that

The Schwarzschild solution is not yet the complete physics of black holes; it is only the simplest solution. There are many other types of black holes that have more properties than just curving space (e.g. angular momentum, magnetic fields, etc.), but this does not make the solution described above wrong, only more comprehensive.

In summary:

A black hole is a sphere from which light cannot escape

and which affects the rest of the universe in the same way that masses do (it curves space) and may also have angular momentum and a magnetic field – but no other properties.

Types of Black Holes

Although it is possible to calculate a Schwarzschild radius for any mass, whether this is physically realistic is another matter entirely. Physics currently postulates approximately three sizes of black holes:

Supermassive black holes

are observed in the centres of galaxies and quasars, but we do not know where they come from, how they are formed, etc.

(as of 1999)

Stellar black holes

Black holes with the mass of a star. This is a mathematical concept in the theory of star formation, i.e. it is conceivable that such a thing could form under certain conditions. However, to my knowledge, none have been definitively observed – but there are a few candidates.

(as of 1999)

Mini black holes by Steven Hawking

(purely hypothetical) they are supposed to be so tiny that they can whizz through us every day without any health consequences for us.

(as of 1999)

Historical Ideas

With the modern idea of black holes as objects from which light cannot escape, we can look at historical books and assess whether this idea (albeit probably under a different name) also existed in historical precursors. In the ‘treasure trove of human thought,’ it is unlikely that someone today has an idea for the very first time.


In fact, we find what we are looking for: for example, in Pierre Simon Laplace. In his book on celestial mechanics, he actually thinks of bodies with a higher escape velocity than the speed of light. Of course, in 1796, Laplace did not yet have the theory of relativity, which states that the speed of light in a vacuum is the upper limit for travel speeds in the universe. And, of course, Laplace did not anticipate Schwarzschild’s solution to general relativity and therefore had a completely different picture of his ‘corps obscure’ than we have today of black holes. But one of the basic ideas can be found there.


We probably don’t need to look much further back than the early modern period for such ideas, since before Newton’s law of gravitation was formulated, a completely different picture of physics and the mechanisms that hold the world together at its core prevailed. At the very least, Newton’s theory of gravity is a necessary prerequisite for these ideas, because Einstein’s theory of gravity (ART) can be regarded as a refinement or extension of Newton’s. If one were to find a statement in Aristotle or al-Haytham that suggests something similar to black holes, I would consider it implausible because their physics does not include dynamics (the study of forces).