Schwarzschild holes

The Schwarzschild black hole is named after Karl Schwarzschild, who in 1916 derived the relativistic solution for the gravitational field surrounding a nonrotating, electrically neutral sphere. To describe the space-time outside such a sphere one needs to develop the metric that tells us how the space-time is curved. Since we are dealing with spherical symmetry, it is convenient to use spherical coordinates $(r,\theta ,\phi)$. The Schwarzschild metric is given by the following expression:

$$ds^2=-\left(1-\frac{r_g}{r}\right)c^2 dt^2+\left(1-\frac{r_g}{r}\right)^{-1}dr^2 +r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2,$$ (25)

where $r_g$ is the Schwarzschild radius,

$$r_g \equiv \frac{2GM}{c^2},$$ (26)

$M$ is the mass of the black hole, $c$ is the speed of light.

At large distances, space-time is essentially flat. However, the behavior of space and time at $r=r_g$ is remarkable. For a photon traveling radially inwards, $d\theta =d\phi =0$, and according to equation (25), $dr/dt=c(1-r_g/r)$. Here it is taken into account that $ds^2=0$ for any photon. At $r\gg r_g$, $dr/dt\simeq c$ as expected in a flat space-time. However, at $r=r_g$, $dr/dt=0$. Time has slowed to a complete stop, as measured from our vantage point at rest a large distance away. From our viewpoint, nothing ever happens at the Schwarzschild radius, because light is frozen in time at the Schwarzschild radius. The spherical surface at $r_g$ acts as a barrier and prevents us from receiving any information from within. For this reason, a star that has collapsed down within the Schwarzschild radius is called a black hole. It is enclosed by the event horizon, the spherical surface at $r_g$. The Schwarzschild coordinates (25) behaves badly near $r=r_g$; there $g_{tt}=-(1-r_g/r)$ becomes zero, and $g_{rr}=(1-r_g/r)^{-1}$ becomes infinite. But this is due to a strange behavior of the coordinate system, rather than of the space-time geometry itself. The real singularity is at the center, a point of zero volume and infinite density where all of the black hole's mass is located. Space-time is infinitely curved at the singularity.

The Schwarzschild metric is a vacuum solution of Einstein's field equations, it is valid only in the empty space outside the object. The mathematical form of the metric is different in the object's interior, and will not be discussed here. Note that the event horizon is a mathematical surface and need not coincide with any physical surface. Although the interior of a black hole, inside the event horizon, is a region that is forever hidden from us on the outside, its properties may still be calculated.

To investigate the properties of the event horizon we let a very brave, and suicidal explorer fall in from far away towards the black hole, and we let her fall freely and radially into the event horizon. As she approaches the event horizon she notices no slowing of time, instead her proper time continues normally, and she encounters no frozen stellar matter since it has fallen through long ago. At $r=r_g$ there is a reversal of the roles of $t$ and $r$ as time-like and space-like coordinates. This means that the explorer has the option to turn on her jets and escape from the black hole whenever $r>r_g$, but inside the event horizon, $r<r_g$, she cannot escape. There the further decrease in $r$ represents the passage of time, and therefore it is impossible to remain at rest. Within the event horizon of a nonrotating black hole, all worldlines converge at the singularity. Even photons are pulled in towards the center. This means that the explorer never has an opportunity to glimpse the singularity because no photons can reach her from there. She can, however, see the light that falls in behind her from events in the outside universe.