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r r=2m
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Fig. 11-2. Using Eddington-Finkelstein coordinates (v, r ) removes the coordinate singularity at r = 2m. As r gets smaller, light cones tip over. For r < 2m, all geodesics directed toward the future head toward r = 0.
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Kruskal Coordinates
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Kruskal-Szekeres coordinates allow us to extend the Schwarzschild geometry into the region r < 2m. Two new coordinates which we label u and v are introduced. They are related to the Schwarzschild coordinates t, r in the following ways, depending on whether r < 2m or r > 2m: r > 2m: u = er/4m v = er/4m t r 1 cosh 2m 4m t r 1 sinh 2m 4m (11.5)
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r < 2m: u = er/4m 1 v = er/4m 1 r t sinh 2m 4m t r cosh 2m 4m (11.6)
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The Kruskal form of the metric is given by ds 2 = 32m 3 r/2m e du 2 dv 2 + r 2 d 2 + sin2 d 2 r (11.7)
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These coordinates are illustrated in Fig. 11-3. These coordinates exhibit the following features:
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The outside world is labeled by O is the region r 2m, which corresponds to u |v|. The line u = v corresponds to the Schwarzschild coordinate t while u = v corresponds to t . The region inside the event horizon, which is r < 2m, corresponds to v > |u|. In Kruskal coordinates, light cones are 45 .
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We also have the following relationships: u2 v 2 = r 1 er/2m 2m and t v = tanh u 4m (11.8)
The coordinate singularity at r = 2m corresponds to u 2 v 2 = 0. The real, curvature singularity r = 0 is a hyperbola that maps to v 2 u2 = 1 Once again we can examine the paths of light rays by studying ds 2 = 0. For the Kruskal metric, we have ds 2 = 0 = This immediately leads to du dv
32m 3 r/2m e du 2 dv 2 r
In Kruskal coordinates massive bodies move inside light cones and have slope dv du
r=0 v I O' A r = 2m, t = + r = const O u
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I' r = 2m, t = r=0
Fig. 11-3. An illustration of Kruskal coordinates. Regions O and O are outside the event
horizon and so correspond to r > 2m. Regions I and I correspond to regions r < 2m. The hyperbola r = const is some constant radius outside of r = 2m; it could, for example, represent the surface of a star.
which tells us that the velocity of light is 1 everywhere. Therefore there is no boundary to light propagation in these coordinates. Furthermore,
u serves as a global radial marker. v serves as a global time marker. The metric is equivalent to the Schwarzschild solution but does not correspond to at spherical coordinates at large distances. There is no coordinate singularity at r = 2m. It still has the real singularity at r = 0.
In Fig. 11-3, the dashed line indicated by A represents a light ray traveling radially inward. The slope is 1 and in Kruskal coordinates it must hit the singularity at r = 0.
The Kerr Black Hole
Observation of astronomical objects like the Earth, Sun, or a neutron star reveals one fact: most (if not all) of them rotate. While the Schwarzschild solution still works as a description of the spacetime around a slowly rotating object, to accurately describe a spinning black hole we need a new solution. Such a solution is given by the Kerr metric. The Kerr metric reveals some interesting new phenomena that are wholly unexpected. For example, we will nd that an object that is placed near a spinning black hole cannot avoid rotating along with the black hole no matter what state of motion we give to the object. Put a rocket ship there. Fire the most powerful
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engines that can be constructed so that the rocket ship will move in a direction opposite to that in which the black hole is rotating. But the engines cannot help no matter what we do the rocket ship will be carried along in the direction of rotation. In fact, we will see below the rotation even effects light! We re also going to see that a rotating black hole has two event horizons. In between these event horizons is a region called the ergosphere, and we will see that it is there where the effects of rotation are felt. It is also possible to extract energy from a Kerr black hole using a method known as the Penrose process. Let s get started examining the Kerr black hole by making some de nitions. As you know a spinning object is characterized by its angular momentum. When describing a black hole, physicists and astronomers give the angular momentum the label J and are usually concerned with angular momentum per unit mass. This is given by a = J/M, and if we are using the gravitational mass for M then the units of a are given in meters. With the Kerr metric, the effects of spinning on the spacetime around the black hole will be seen by the presence of angular momentum in the metric along with mixed or cross terms. These are terms of the form dt d that indicate a change in angle with time a rotation. The Kerr metric is a bit complicated, and so we are simply going to state what it is. To simplify notation, we make the following de nitions: = r 2 2mr + a 2 = r 2 + a 2 cos2 where, as de ned above, a is the angular momentum per unit mass. With these de nitions the metric describing the spacetime around a rotating black hole is ds = 1
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