# 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. 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. 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?

• first of all, he will have to cope with tidal forces : these forces are created by the difference in the gravitational field between two points, which are at different distances from the massive object which generates the field. On Earth, these forces are negligible, but in this case, they can take huge values when we approach the horizon.

• if he looks at an object which is falling into a BH, he will notice that the light, radiated by this object, is shifted towards longer wavelengths, due to the gravitational redshift.
This shift tends to infinity near the BH. As the observed time slows down, the observer sees the object falling towards the horizon, but never reaching it, and the frequency of its radiated light becomes weaker and weaker. Warning : for the falling object, time does not have the same value, this is one of the aspects of Relativity. Inside such an object, the travel time is finite, and quite short, because of the acceleration due to the intense gravitational field.
On the right diagram, the path of the particle is shown in blue within its proper time, and in yellow for a distant observer.

• by going nearer to the horizon, he will cross the photon sphere.
Around a massive body, any object can be put into orbit, if its speed is the right one in relation with its altitude. The lower it is, the faster it must travel.
Around a BH, gravity is so intense that there is a height where the orbiting speed is the speed of light : this is the photon sphere, so called because only photons can travel at the speed of light, and hence, orbit around the black hole.
This sphere is only ethereal, the orbits are very unstable. We can see here three interesting things :
• the photon sphere : 1.5 times bigger than the Schwarzschild radius.
• the event horizon : the "border" of the black hole. Its distance from the singularity is called the Schwarzschild radius.
• the singularity : a point where space and time have an infinite curvature. Physically, we must say that the meaning of this curvature is less than obvious.

• next, he will cross the event horizon. At this stage, he can't go back and escape the BH. When he crosses the horizon, time and space are swapped : what was in front of him becomes his future. In a simple sense, this means that he can't stay still, and he is forced to fall into the singularity. This one is called a spatial singularity.

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. 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 :

• on the left side, there appears an area, symmetrical to our universe, that we can call a parallel universe.
These two universes can't have any contact, except beyond the horizon of the black hole.

• The existence of another singularity is the second interesting thing. This is a white hole, sometimes called a white fountain, where nothing can enter.
The opposite of a black hole, one is only able to go out of it, due to the fact that we can't travel back in time.

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. 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. 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.