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The Schwarzschild chart is a spacetime diagram that shows the world lines of light rays moving radially towards and away from a black hole. Recall that spacetime diagrams are drawn with time (ct) against distance (x), so that both axes have the same units (meters, feet or whatever). In flat spacetime, light's spacetime path (or world line) has a slope of + or - 1, depending on which direction is plotted. Figure 1 shows how light's world lines deviate from the unity slopes near and inside a black hole.
Figure 1 
The vertical cyan line at the left of the diagram represents the central singularity of the black hole and the vertical red line the event horizon. The yellow lines represent radially infalling photons and the orange lines represent photons moving (or attempting to move) radially outward. It is clear that the slopes of the in- and outgoing photon paths become asymptotically vertical near the event horizon. This means that the infalling photons slow down to a crawl near the event horizon and never quite reach it. Huh?
This characteristic was a big puzzle for many years after the Schwarzschild vacuum's discovery in 1916. A French mathematician, Paul Painlevé, eventually overcame it in 1921 by introducing a set of 'free-fall' coordinates. Where the Schwarzschild chart is plotted using the time as measured by a distant observer, the Painlevé chart plots the time as measured by an observer that is falling freely towards and into the black hole at (negative) escape velocity. This is equivalent to thinking that the observer is motionless in a coordinate system that is falling into the black hole at negative escape velocity. At the event horizon, this coordinate system falls in at the speed of light; hence, not even light can move outward there.
Figure 2 shows how the world lines of photons have changed. Gone is the problem of "asymptotically approaching the vertical". Ingoing photons appear to be moving 'faster than light' and smoothly sail over the event horizon, to fall into the central singularity. This is obviously due to the fact that the coordinate system beneath them is also 'falling inward'. Outgoing photons generally progress outward 'slower than light', because the coordinate system that they move on drags them somewhat inward. Locally, relative to the coordinate system around them, all photons still move at their normal speed c.
Figure 2 
Inside the event horizon, things are looking normal, except that photons attempting to go outward from just inside the event horizon are dragged into the singularity by the 'faster than light infalling' coordinate system. Exactly at the event horizon, an outgoing photon can be loosely said to 'move' outward forever without going anywhere. Technically such a photon is red-shifted an infinite amount, has zero frequency and hence zero energy - it simply don't exist!
The curved green lines are the world line of the free falling observer, riding "stationary" on the in-falling coordinate system. Note how the green world lines all reach a slope of -1 at the event horizon - the speed of light in a static coordinate system. In case you are curios as to what the relationship between the distant observers time and the free-fall observer's time is:
t' = t + 2 r1/2 + ln|(r1/2 - 1)/(r1/2 + 1)|,
in highly normalized form, where t' is the free-fall time, t the distant stationary observer's time and r is expressed in multiples of the event horizon radius (re=2GM/c2), e.g., r = 2 means twice the event horizon radius (r = 4GM/c2) where M is the mass of the hole in SI units. [paragraph edited for clarity]
Recall that the clocks of observers closer to a black hole runs slower than the clocks of distant observers. Add to that the fact that the free-fall observer's speed causes time dilation and it starts to make sense. However awkward it may appear, the Painlevé chart makes life much easier in grasping and calculating effects around and inside black holes than what the Schwarzschild chart does.
In the next post, we will look at an even more useful chart of the Schwarzschild vacuum and 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 credit: From this Andrew Hamilton page:
(http://casa.colorado.edu/~ajsh/schwp.html)
Jorrie
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