|
The Schwarzschild vacuum of the isolated static black hole was shown on two charts so far: the Schwarzschild chart and the Painlevé chart. Both have their place and utility, but one of the easiest charts to work with is the so-called ingoing Eddington-Finkelstein chart. (There is also an outgoing Eddington-Finkelstein chart, but we will leave it alone for now).
Eddington discovered this technique way back in the 1920s, but it took decades before Finkelstein rediscovered it and realized that it has a very special utility. We will discuss this later, but first, what does the Ingoing Eddington-Finkelstein chart of fig. 1 attempt to tell us?
Figure 1 
Unlike the Schwarzschild and the Painlevé charts, which have curved spacetime trajectories (or world lines) for infalling photons, the Ingoing Eddington-Finkelstein chart sports straight world lines for infalling photons (the yellow lines). This is achieved by fiddling with the time coordinate scale to achieve what is called Finkelstein time (tF), expressed as a function of Schwarzschild time (t): and the Schwarzschild coordinate distance (r).
tF = t + ln|(r - 1)|,
where r is in multiples of the event horizon radius (re = 2GM/c2), e.g., r = 2 means twice the event horizon radius. This looks just like flat spacetime, at least for infalling photons. It may appear like 'cheating', but it's no different from plotting power law relationships on a log scale, giving straight lines.
So what is the significance of Finkelstein time? It comes from ideas of how black holes may form. When a very massive star runs out of nuclear fuel, it must collapse under its own gravity. Part of its mass is blown off in the resulting supernova, but if the remaining core has enough mass, it contracts all the way to a singularity.
However, a distant static observer will observe the star to contract slower and slower as its size approaches the event horizon asymptotically, but never quite reaching it. This is similar to a photon approaching the horizon on the Schwarzschild chart. This led to the idea of "frozen stars", i.e., the star freezes in time at the event horizon and never contracts further.
David Finkelstein rediscovered the Eddington chart and showed that the so-called Schwarzschild singularity at the event horizon is just a result of the coordinate chart chosen. If that final contraction is pictured according to Finkelstein time, or, better stated, the time that would be experienced by an observer riding on the surface of the collapsing star, the Schwarzschild singularity disappears.
The star smoothly contracts through the event horizon and all the way to a singularity, while time seems to flow quite normal to the observer riding on its surface. Other things, like tidal gravity might not seem so normal, however!
Figure 2 illustrates the collapse of a uniform, pressureless spherical mass to a black hole on an Ingoing Eddington-Finkelstein chart.
Figure 2 
The world line of the surface of the collapsing sphere is shown in white. Initially (bottom of graph, at say time zero) there was no event horizon. As soon as the sphere shrinks to r = 2 (recall that this means twice the future event horizon radius), an event horizon (thick red line) starts to form and expand. When the collapsing sphere reaches radius r = 1, the event horizon is fully formed and stays at constant radius from there.
In the mean time, the sphere is still collapsing towards the singularity stage, which happens when the white line reaches r = 0, meaning that the star is now contracted to a dimensionless point. The piece of vertical cyan line at the top left represents the newly formed central singularity.
Note that the Ingoing Eddington-Finkelstein chart is not the same as the Eddington chart or the Finkelstein chart - in the latter two the space coordinates are also modified, not just the time. In Eddington-Finkelstein charts the Schwarzschild radial coordinate is retained, so that the circumference of a circle remains 2Πr.
In the next post, we will perform some calculations using all of these charts. The Schwarzschild vacuum is also discussed to some depth in the download from this page in Relativity 4 Engineers.
Graphics from these excellent Andrew Hamilton pages: http://casa.colorado.edu/~ajsh/schwp.html
http://casa.colorado.edu/~ajsh/collapse.html
| 
