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Schwarzschild geometry and black holes |
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It is clear that as ε → 0 the integral of this goes like In ε, which diverges. We would also |
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1, because the divergence comes from the [1 |
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doesn’t contain E. Therefore a particle reaches the surface r 2M only after an infinite coordinate time has elapsed. Since the proper time is finite, the coordinate time must be behaving badly.
Inside r = 2M
To see just how badly it behaves, let us ask what happens to a particle after it reaches r = 2M. It must clearly pass to smaller r unless it is destroyed. This might happen if at r = 2M there were a ‘curvature singularity’, where the gravitational forces grew strong enough to tear anything apart. But calculation of the components Rα βμν of Riemannian tensor in the local inertial frame of the infalling particle shows them to be perfectly finite: Exer. 20, § 11.7. So we must conclude that the particle will just keep going. If we look at the geometry inside but near r = 2M, by introducing ε := 2M − r, then the line element is
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Since ε > 0 inside r = 2M, we see that a line on which t, θ , φ are constant has ds2 < 0: it is timelike. Therefore ε (and hence r) is a timelike coordnate, while t has become spacelike: even more evidence for the funniness of t and r! Since the infalling particle must follow a timelike world line, it must constantly change r, and of course this means decrease r. So a particle inside r = 2M will inevitably reach r = 0, and there a true curvature singulaity awaits it: sure destruction by infinite forces (Exer. 20, § 11.7). But what happens if the particle inside r = 2M tries to send out a photon to someone outside r = 2M in order to describe his impending doom? This photon, no matter how directed, must also go forward in ‘time’ as seen locally by the particle, and this means to decreasing r. So the photon will not get out either. Everything inside r = 2M is trapped and, moreover, doomed to encounter the singularity at r = 0, since r = 0 is in the future of every timelike and null world line inside r = 2M. Once a particle crosses the surface r = 2M, it cannot be seen by an external observer, since to be seen means to send out a photon which reaches the external observer. This surface is therefore called a horizon, since a horizon on Earth has the same effect (for different reasons!). We shall henceforth refer to r = 2M as the Schwarzschild horizon.
Coordinate systems
So far, our approach has been purely algebraic – we have no ‘picture’ of the geometry. To develop a picture we will first draw a coordinate diagram in Schwarzschild coordinates, and on it we will draw the light cones, or at least the paths of the radially ingoing and outgoing null lines emanating from certain events (Fig. 11.10). These light cones may be calculated by solving ds2 = 0 for θ and φ constant:
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11.2 Nature of the surface r = 2M |
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Much needs to be said about this. First, two light cones are drawn for illustration. Any |
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45◦ line is a radial null line. Second, only u and υ are plotted: θ and φ are suppressed; |
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therefore each point is really a two sphere of events. Third, lines of constant r are hyper- |
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bolae, as is clear from Eq. (11.68). For r > 2M, these hyperbolae run roughly vertically, |
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being asymptotic to the 45◦ line from the origin u = υ = 0. For r < 2M, the hyperbolae |
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run roughly horizontally, with the same asymptotes. This means that for r < 2M, a timelike |
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line (confined within the light cone) cannot remain at constant r. This is the result we had |
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before. The hyperbola r = 0 is the end of the spacetime, since a true singularity is there. |
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Note that although r = 0 is a ‘point’ in ordinary space, it is a whole hyperbola here. How- |
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ever, not too much can be made of this, since it is a singularity of the geometry: we should |
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not glibly speak of it as a part of spacetime with a well-defined dimensionality. |
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Our fourth remark is that lines of constant t, being orthogonal to lines of constant r, |
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are straight lines in this diagram, radiating outwards from the origin u = υ = 0. (They are |
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orthogonal to the hyperbolae r = const. in the spacetime sense of orthogonality; recall |
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our diagrams in § 1.7 of invariant hyperbolae in SR, which had the same property of |
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being orthogonal to lines radiating out from the origin.) In the limit as t → ∞, these lines |
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approach the 45◦ line from the origin. Since all the lines t = const. pass through the ori- |
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gin, the origin would be expanded into a whole line in a (t, r) coordinate diagram like |
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Fig. 11.10, which is what we guessed after discussing that diagram. A world line cross- |
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ing this t = ∞ line in Fig. 11.11 enters the region in which r is a time coordinate, and so |
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cannot get out again. The true horizon, then, is this line r = 2M, t = +∞. |
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Fifth, since for a distant observer t really does measure proper time, and an object that |
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falls to the horizon crosses all the lines t = const. up to t = ∞, a distant observer would |
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conclude that it takes an infinite time for the infalling object to reach the horizon. We have |
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already drawn this conclusion before, but here we see it displayed clearly in the diagram. |
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There is nothing ‘wrong’ in this statement: the distant observer does wait an infinite time |
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to get the information that the object has crossed the horizon. But the object reaches the |
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horizon in a finite time on its own clock. If the infalling object sends out radio pulses each |
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time its clock ticks, then it will emit only a finite number before reaching the horizon, so |
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the distant observer can receive only a finite number of pulses. Since these are stretched |
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out over a very large amount of the distant observer’s time, the observer concludes that |
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time on the infalling clock is slowing down and eventually stopping. If the infalling ‘clock’ |
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is a photon, the observer will conclude that the photon experiences an infinite gravitational |
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redshift. This will also happen if the infalling ‘object’ is a gravitational wave of short |
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wavelength compared to the horizon size. |
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Sixth, this horizon is itself a null line. This must be the case, since the horizon is the |
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boundary between null rays that cannot get out and those that can. It is therefore the path |
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of the ‘marginal’ null ray. Seventh, the 45◦ lines from the origin divide spacetime up into |
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four regions, labeled I, II, III, IV. Region I is clearly the ‘exterior’, r > 2M, and region II |
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is the interior of the horizon. But what about III and IV? To discuss them is beyond the |
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scope of this publication (see Misner et al. 1973, Box 33.2G and Ch. 34; and Hawking and |
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Ellis 1973), but one remark must be made. Consider the dashed line in Fig. (11.11), which |
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could be the path of an infalling particle. If this black hole were formed by the collapse of |
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a star, then we know that outside the star the geometry is the Schwarzschild geometry, but |
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