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The Schwarzschild vacuum refers to the spacetime geometry around an isolated, spherically symmetric, homogeneous, non-rotating and uncharged mass. Wow, that's a mouthful of restrictions! It simply means that we are looking at an ideal case in order to make a complex situation as simple as possible. Karl Schwarzschild derived this exact solution to the field equations only months after Einstein published it in 1916.
The solution can be partially pictured as shown in fig.1 below. It represents a spatial view of the Schwarzschild vacuum. Space far from an isolated mass becomes exponentially flat. Nearer to the mass, it can be pictured as having a changing slope, as shown. In certain cases, the slope may get infinitely large and end up in a singularity.
Normal masses, like stars and planets do not have a singularity at the center. However, if the mass is concentrated into a small enough volume, it contracts unstoppable into a singularity, as shown below, forming a black hole with an event horizon. This can happen to massive stars that run out of fuel.
Figure 1: 
Figure 1 shows only two of the three spatial dimensions, i.e., two dimensions of space (the white circles) that curve into a hypothetical third dimension, sometimes called 'hyper-space'. The temporal (or time) dimension of spacetime is suppressed for clarity.
The red circle is the event horizon of the black hole. Anything reaching that circle (in 3-d it's a virtual spherical "membrane") cannot ever escape again. The reason for this is that gravity is so strong there that the force required to move outward from the event horizon approaches infinity. So this is a "one-way membrane" - things (even light) can only go inward.
One of the consequences of gravity is that, from a distant observers point of view, all processes appear to be slowed down near a gravitating mass. This apparent slowdown gets more acute the closer the process is to the mass. All processes at the event horizon apparently stop - that is from the point of view of any stationary observer outside of the event horizon.
The result is that when a source of (say) yellow light is placed at some distance from the mass, then an observer that is farther from the mass will observe the light as red shifted in wavelength and an observer closer to the mass will observe the light as blue shifted. This is depicted in Fig. 2 below. The effect is normally called 'gravitational redshift', despite the fact that gravitational blueshift is also possible.
Figure 2: 
The yellow waves in fig. 2 represents light that travels between observers at the same distance from the hole - no shift in wavelength. The yellow/red wave is an outgoing (red shifted) light ray and the green/blue waves are ingoing (blue shifted) light rays.
The white circles are drawn at equally spaced coordinate distance from the hole, i.e., if they are projected onto a flat surface, say at the bottom of the picture, they will form equally spaced concentric circles. The varying slope of the curve makes it clear that the 'proper radial distance' between the circles increases as you go closer to the black hole. Proper radial distance is what a stationary observer near the black hole would measure in the radial direction, using a local ruler, following the curvature of space.
It is obviously not possible for any stationary observer to directly measure the proper distance to the center of the black hole. Rulers and observers would all be swallowed by the hole and radar signals would never return an echo. It is however possible, in principle at least, for a slow moving observer to measure the circumference of the white circle by means of a local ruler. This circumference corresponds to the usual C = 2Пr of Euclidean geometry, hence the radial distance parameter r is usually referred to as 'circumferential radius'.
The reason why this is an issue, is the following. Let a local observer (say Pam) determine that the circumference of a white circle (2Пr) at her position is 6.282 million miles. This means that her circumferential radial distance is 1 million miles (the yellow radius, r1 in fig. 3 below). Let Pam move closer to the black hole by a radial distance of 100 000 miles (or 10% of her original circumferential radius) as measured by her own yardstick. Let Pam then measure the circumference at her new location in order to determine the new circumferential radius (r2).
Figure 3: 
One would naively expect the circumference and radius to have decreased by 10%. This is not what Pam would find, because, depending on the mass of the black hole, it would have decreased by somewhat less, as shown in Fig. 3.
Pam's twin brother Jim, playing 'distant stationary observer', would not have agreed with Pam on the amount that she has moved in the radial direction. The difference obviously lurks in the sloped nature of Pam's radial space relative to Jim's flat radial space (recall that the space far from the hole is exponentially approaching flatness, i.e., the horizontal).
Relativists tell us (e.g., Thorne in "Black Holes and Time Warps", paraphrased) that we can forget about the "curving into hyper-space", provided that we apply the correct transformation rules between the values measured by different observers. We will return to that in a later post. In the next post we will continue with this discussion of black holes, specifically looking at the temporal (time) picture in the Schwarzschild vacuum, or the so-called Schwarzschild chart.
More details are available on this web page from Relativity 4 Engineers and its PDF download. Credit: figures were taken from http://casa.colorado.edu/~ajsh/schwp.html, and modified where required for explanatory purposes. There are some very interesting animations and even 'morphing' on the casa.colorado web page, although the discussions are somewhat cryptic.
Jorrie
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