Inside a black hole



When a star collapses and becomes a black hole, all the information about its geometry, its making, and so on, are lost.
Three parameters are enough to fully describe a black hole : its mass, its electrical charge, and its rotation speed (actually, its angular momentum, but it's quite the same).

We are going to describe the three kinds of black holes :

This approach is very simplified, in particular we won't look at all the details, specially in general relativity and quantum mechanics.


Before diving inside a BH (black hole), we must have a look at three things, which will help us to understand the different phenomena.

Gravitational redshift

One outstanding result of General Relativity is about time. The proper time passes slower when the local gravitational field is stronger.
Consequently, for an outside observer, the apparent time of a remote object, embedded in a gravitational field, is slowing down : for a clock near a black hole, its proper time slows down, and the remote observer sees the delay.

This phenomenon will slow down the frequency of a wave radiated by such an object : in other words, the emission spectrum shifts towards low frequencies.
This shift is called the gravitational redshift, by analogy with the redshift due to the expansion of universe, or Doppler effect.

The change in the wavelength of electromagnetic radiation in a gravitational field is expressed as -,
with G gravitational constant, M the mass of the massive body, c the speed of light, and r the distance between the transmitter and the body.

The energy of a photon is in proportion to its frequency. So we can interpret the gravitational redshift as a loss of energy necessary to escape the gravitational field.

This effect is, of course, very weak for normal gravitational fields, like the Earth's, but it is of considerable significance near a black hole.

The space-time diagram

A space-time diagram is a simple and easy way to describe a space-time continuum, like ours.

Very often, one shows only one space dimension, in order to simplify.

space-time diagram
This is a classical space-time diagram, with only one space dimension.
Red diagonals show the space-time paths of light.
Every object with a mass can only move by following a path (green curve) inside its "light cone".

Red diagonals are called the "light cone". Nothing can move faster than light, so the path of any object is necessarily inside this cone.

As a result of this, the two areas, marked area 1 and 2 are causally independent : no event from one of them can act upon a point of the other one ; to do such a thing, one would have to travel faster than light.

The Penrose space-time diagram

This diagram comes from the English physicist Roger Penrose. It's a space-time diagram which has been closed (in the mathematical meaning) by bringing the infinities back to lines.
Such a diagram is, by no way, an exact description of the universe. Its aim is only to show causal relationships.

Here is such a diagram, always limited to one space dimension.

Penrose diagram
Like in a standard space-time diagram, light travels on the diagonals (purple lines), and other objects can only follow pathes like the blue one.

This diagram is the result of a compactification of space, by means of an appropriate change of coordinates. As it is drawn here, it shows an infinite space-time universe, without beginning nor end.

With this knowledge, we will be able to see what happens when we move near a black hole.


The Schwarzschild black hole

This is the most simple, idealized, model. Certainly, it doesn't exist in the actual universe, but it allows us to start on the main concepts, in the easiest way.
Its name comes from the German astronomer Schwarzschild, who was the first person to succeed in solving the General Relativity equations near a massive object, in an empty space.

With these conditions, the metric of the space-time can be expressed as -
where -are the polar coordinates, and - the Schwarzschild radius.
When r is equal at -, the metric is no more defined, but a change of coordinates (Kruskal coordinates) show that this is not an actual singularity.
When r tends towards infinity, ie far away from the mass, we find again the metric of a flat Minkowski space-time.

Let's consider an observer moving towards a black hole. What will he notice?


We are now drawing the space around a BH, with the help of a Penrose space-time diagram. We shall use the coordinates system of Kruskal.

Penrose diagram of a static BH
This diagram is the description of a universe, with only one everlasting black hole, situated at a distance r =0.

The thick diagonals show the horizon of the black hole. If something crosses it, following the blue path, it can't go back, and has no other choice but to strike the singularity.

On the Penrose diagram, we can notice two things :

Is this sketch an actual view of the universe ?
In fact, there is an hypothesis : the BH is everlasting. This is not true when the BH comes from the collapse of a star.

non-eternal black hole
In this case, the black hole is no longer eternal, it only appears when the star collapses.

There is no white hole, no parallel universe in this case.


Using a space-time diagram is another way to depict a BH. On this diagram (with only two space dimensions), we shall draw the light cones of some points around the BH.
Let's remember that around a BH, space-time itself is curved, and light cannot travel in a straight line.

Light cones around a static black hole

Space-time around a BH is curved, so the light cones are directed towards the "inside". At a characteristic distance from the singularity, these cones are so tilted that their "outer side" becomes vertical in the diagram. These "sides" form a surface (it's the red cylinder).

This surface is called the event horizon.

Source : Penrose (Scientific American)

On this surface, light is motionless in relation to the outer space. But the light speed is the same in all of the systems of reference, so it is the horizon itself which is moving at the light speed in the curved space-time of the black hole.

For an outside observer, no information can come from the horizon of the black hole : the time, in the vicinity of the black hole, has stopped.