The mystery of black holes



In 1916, the German astronomer Schwarzschild, with the support of Einstein's works, calculated the size of a star whose escape velocity would be equal to the speed of light .
This speed is an absolute limit, and according to relativity, nothing can go faster.

What's more, we know that the size of a neutron star decreases when its mass increases, because gravity becomes stronger than the degeneracy pressure.
If the mass of such a star increases, there comes a time, called the Oppenheimer-Volkoff limit, when the escape velocity becomes equal to the speed of light, and nothing will be able to escape the star.

Let us notice that the cohesion of a neutron star is also depending of the strong nuclear interaction. As the behaviour of this interaction is poorly understood under a high gravity, the Oppenheimer-Volkoff limit is not precisely known. It is contained between 1.5 and 3 solar masses.

mass-radius relationship
On this graph, you can see that the size of a black hole horizon (the Schwarzschild radius) is equal to a neutron star size when its mass is about 2.5 Solar masses (this mass is called the Oppenheimer-Volkoff limit).
Notice that, for a white dwarf or a neutron star, the size decreases when the mass increases !

If nothing, not even light, can't escape, this star becomes invisible : Such an object is called a black hole.
The black hole has no material surface ; the original matter of the star is shrunk to an infinitely dense point, called a singularity.
The "surface" of the black hole is called the horizon, its size is called 'Schwarzschild radius'.

If G is the gravitational constant and M the mass of a body whose radius is R, the escape velocity is -.
In the case of a black hole, this velocity is equal to c the speed of light,
hence the Schwarzschild radius of a black hole is expressed as -

Everything which could happen beyond the horizon is trapped, and can only increase the mass of the black hole.

Contraty to Hollywood movies, a black hole is not a "cosmic vacuum cleaner" : it can only catch objects which come very near. If we replaced the Sun with a black hole, we couldn't notice the difference (at least in terms of gravity, we would miss the heat !)


We can view two scenarios to explain the creation of a black hole :

  It must be said that some people, with supporting arguments, think that a black hole is physically impossible.
All the theory about black holes must be considered with the utmost care, by keeping in mind the fact that it's only a mathematical theory, at the moment, but whose physical reality is becoming more and more obvious.

Effects of a black hole

Einstein's general relativity describes gravity as a curvature of the space-time continuum. The more concentrated the mass, the more curvature you obtain.
If we draw the framework of space-time as a plane (actually there are 4 dimensions : 3 for the space, and one for time), we can visualize this curvature, in an illustrative way.

space-time wrapping around a massive body

In the case of a black hole, the curvature may have no end : there would be a tear in the fabric of space-time . We must use the conditional here, because we are entering a field where there is no absolute certainty...

In this space-time, light travels along the shortest path. If space is flat, i.e. non-curved, the path is of course a straight line.
Near a mass, this is no longer true : the mass can act towards the light like an optical lens.

deflection of light
gravitational lens

The effect of a gravitational lens appears here.

Warning : this picture is very simplistic. It is the whole of space-time which is curved. This means that, not only space, but time itself is modified by the central mass.

The more concentrated the mass, the larger the effect. This is a way to detect a black hole if it lies between a star and us.


By analysing the same effect of the bending of light rays, we can try to guess the appearance of a black hole with an accretion disk.

black hole

black hole

Because of the bending of light, a black hole would appear like this hat shape.
(source J-A. Marck/J-P. Luminet).


Kerr's black hole

The idea of the black hole is the result of calculations from general relativity, due to Schwarzschild. He calculated the size of the horizon of a static black hole. Kerr improved the calculations when the black hole is rotating.
In this case, the curvature of space-time looks different, and the singularity is no longer concentrated into a point, but into a ring inside the horizon.

déformation de l'espace-temps au voisinage d'un trou noir en rotation
In this case, space-time is not only curved like a funnel, but it's twisted to follow the rotation of the black hole.
A light ray, directed at the black hole, would follow the red path.

(Source : Sky and Telescope, J. Bergeron)


References :
Schwarzschild's Spacetime (R. Salgado)
Black Holes : A General Introduction (J-P. Luminet)