Black Holes in .NET

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A black hole is a region of spacetime where gravity is so strong that not even light can escape. In nature, it is believed that black holes form at the end of a stars lifetime, when a massive star runs out of fuel and ends its life in collapse. We shall begin our study of black holes by taking a closer look at the Schwarzschild solution. As we will see, according to classical general relativity, a black hole is completely characterized by just three parameters. These are
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This characterization results in three general classes of black holes that are studied:
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Static black holes with no charge, described by the Schwarzschild solution Black holes with electrical charge, described by the Reissner-Nordstr m solution Rotating black holes, described by the Kerr solution
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In this chapter we will consider two cases: the Schwarzschild and Kerr black holes. To begin, let s review the problem of coordinate singularities and see how to remove the singularity from the Schwarzschild metric.
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When studying black holes you will often see an in nite redshift being discussed. Let s take a moment to see what happens to light as it is emitted upward in a gravitational eld, that is from an observer located at some inner radius ri to an observer positioned at some outer radius ro . The key to seeing what happens to the light is to see how time passes for each observer. In other words we are interested in the period of the light wave as seen by each observer. Recall that the proper time is the time a given observer measures on his or her own clock. For the Schwarzschild metric, for a stationary observer the proper time relates to the time as measured by a distant observer via the relationship 2m dt r
d =
It is a simple matter to calculate a redshift factor by comparing the proper time for observers located at two different values of r . This is best illustrated by an example. EXAMPLE 11-1 Consider two xed observers located nearby a Schwarzschild black hole. One observer, located at r1 = 3m, emits a pulse of ultraviolet light to a second observer located at r2 = 8m. Show that the second observer nds that the light has been redshifted to orange. SOLUTION 11-1 To nd the redshift factor, we simply calculate the ratio d 2 /d 1 where
d i =
2m dt ri
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Let s denote the redshift factor by . Then we have d 2 = = d 1 1 1 1 1
2m r2 2m r1 2m 8m 2m 3m
dt dt =
1 1
1 4 2 3
2m r2 2m r1 3 4 1 3
1 1
9 3 = 4 2
We can take the wavelength of ultraviolet light to be 1 400 nm. The wavelength that the second observer sees is then 2 = 1 = 1.5 400 nm = 600 nm This is in the orange region of the spectrum, which roughly runs from 542 nm to 620 nm.
Coordinate Singularities
Let s step back for a moment and review the distinction between coordinate and curvature singularities. First recall the Schwarzschild metric given in (10.33): ds 2 = 1 2m r dt 2 dr 2 r 2 (d 2 + sin2 d 2 ) (1 2m/r )
It s pretty clear that the Schwarzschild metric exhibits unusual behavior at r = 2m. For r > 2m, gtt > 0 and grr < 0. However, notice that for r < 2m, the signs of these components of the metric reverse. This means that a world line along the t-axis has ds 2 < 0 and so describes a spacelike curve. Meanwhile, a world line along the r -axis has ds 2 > 0 and so describes a timelike curve. The time and space character of the coordinates has reversed. This indicates that a massive particle inside the Schwarzschild radius could not remain stationary at a constant value of r. Now let s take a direct look at r = 2m. Considering gtt rst, we see that at r = 2m gtt = 1 2m =1 1=0 2m
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