Black Holes

In the Schwarzschild solution there is always a natural lower limit on the radius coordinate for a star. For a normal star this lower limit would be . For a star that has undergone gravitational collapse , the Schwarzschild Radius, is the lower bound (-remember it's  in normal units). After that value the radius coordinate becomes a timelike variable. If we want to investigate the properties of the collapsed object, called the Black Hole, we need to get a better set of coordinates than the Schwarzschild coordinates. A better set is found by adopting the following new time coordinate.

(22.1)

The line element, written in terms of proper time , can then be calculated to be

(22.2)

These coordinates are called the Eddington-Finkelstein Coordinates.

In these coordinates none of the metric elements becomes infinite at the Schwarzschild Radius.

We should be able to cross over the  boundary without problems.

Now consider what light does as it falls radially inwards on the Black Hole. Using the above line element we use the radial constraint condition  to arrive at the following differential bifurcation equation for null geodesics (  ):

(22.3)

The bifurcation logic implies that we have

(22.4)

with solution

(22.5)

OR

(22.6)

with solution

(22.7) We will now figure out what these solutions mean physically.

From the equation (22.1) we have the derivative relation

(22.8)

The first bifurcation solution then implies the following

(22.9)

Note that this means

(22.10)

Hence the first bifurcation equation solution must correspond to ingoing null geodesics.

(22.11)

What about the 2nd bifurcation equation? From equation (22.8) we see that

(22.12)

Note that this means

(22.13)

Hence the second bifurcation equation solution must correspond to outgoing null geodesics.

The diagram below show a v versus r diagram with representative null geodesics going in towards the black hole and going away from the black hole. The v-r axes are drawn as oblique axes. The ingoing null geodesics, , are parallel to the r axis, a line drawn  to the vertical. They extend down to the mass source, situated at . The outgoing null geodesics asymptotically coincide with the value  when .

A photon starting on the outside of the black hole will travel through the Schwarzschild Radius. A photon that starts on the inside of the Schwarzschild Radius, on the other hand, never has a path that leads outside of the black hole. It is trapped. If we had a dense collection of mass with a physical surface boundary less than its Schwarzschild Radius, light from the surface could not escape from it to the outside . An outside observer could detect the object's existence through its gravitational field but that observer would not be able to see it. This is the reason why such objects came to be known as black holes. The sphere defined by  shouts off the outside world from observing what's inside. For this reason the region where  is said to constitute the Event Horizon.

The large 3million solar mass black hole called Sagittarius A, situated at the center of the Milky Way Galaxy, is confined to a region no larger than our solar system. It would be quite possible to pass the Schwarzschild Radius boundary of this black hole without anything untoward happening. However, once you were within this boundary, no signals could ever be sent to the outside world. All paths would point to the singularity at the center.

Another set of coordinates that turn out to be very useful in describing the black hole, and actually do a more complete job of doing this than any other set of known coordinates, are called Kruskal Coordinates. In Kruskal coordinates both the r and the t coordinates of the Schwarzschild solution are transformed in a transcendental way as follows.

(22.14)

(22.15)

In terms of proper time, this leads to the line element

(22.16)

Note that this line element has no problems with the Schwarzschild Radius. The variable r in these equations has no status as a coordinate anymore but is defined implicitly by the relation

(22.17)

With these coordinates the radial null geodesics are given by straight lines, not just for the ingoing geodesics, but also for the outgoing geodesics. Their equations are

(22.18)

which are lines that are at  to the vertical and horizontal in a u-v spacetime diagram. This is then very similar to the spacetime diagrams in Special Relativity.

 The Kruskal solution is called the Maximal Extension solution since it takes in regions of the manifold that are not covered by other coordinate systems. This extended manifold is represented in the Kruskal diagram shown above. Note that this solution implies the existence of a parallel universe on the left hand side of the diagram. There is no causal contact with this universe since all paths towards it must travel inside the black hole and all inside paths crash into the singularity at .