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r =2m
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Fig. 11-1. Using Schwarzschild coordinates, light cones close up as they approach
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The Path of a Radially Infalling Particle
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In the Schwarzschild geometry a radially infalling particle falling from in nity with vanishing initial velocity moves on a path described by 2m 1 r dt =1 d dr d
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From these equations, we nd that dr dr/d 2m = = dt/d dt r Integration yields 2 3/2 t t0 = r 3/2 r0 + 6m r 6m r0 3 2m r + 2m r0 2m + 2m ln r0 + 2m r 2m 1 2m r
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In the limit that r becomes close to 2m, this becomes r 2m = 8m e (t t0 )/2m This indicates that if we choose to use t as the time coordinate, the surface r = 2m is approached but never passed. We recall that t corresponds to the proper time of a distant observer. Therefore for an outside observer far from the black hole, a falling test body will never reach r = 2m. Revisiting the path of a radially infalling particle using the particle s proper time, we have dr 2m = d r We assume the particle starts from r = r0 at proper time = 0 . Cross multiplying terms and integrating, we have 1 2m
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r0 r
r dr =
where the primes denote dummy integration variables. Integrating both sides gives 2 3/2 r0 r 3/2 = 0 3 2m Looking at this equation, the mysterious surface r = 2m makes no appearance. The body falls continuously to r = 0 in nite proper time, in contrast to the result seen by an outside observer. In fact, one can say that the entire evolution of the physical universe has occurred by the time the body passes the surface r = 2m. To study the spacetime inside r = 2m, we need to remove the coordinate singularity. We take this up in the next section.
Eddington-Finkelstein Coordinates
The rst attempt to get around the problem of the coordinate singularity was made with Eddington-Finkelstein coordinates. First we introduce a new coordinate r called the tortoise coordinate
r = r + 2m ln along with two null coordinates u = t r From (11.2) we nd dr = dr + = = 2m (r/2m 1) 1 2m dr = dr + and v = t + r r 1 2m
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(11.2)
(11.3)
dr (r/2m 1)
(r/2m 1) dr r dr dr + = (r/2m 1) (r/2m 1) 2m (r/2m 1) dr 1 2m/r
Now we use (11.3) to write dt = dv dr = dv
dr 1 2m r
dt 2 = dv 2 2
dv dr dr 2 + 1 2m r 1 2m r
Substitution of this result into the Schwarzschild metric gives the EddingtonFinkelstein form of the metric ds 2 = 1 2m r dv 2 2 dv dr r 2 d 2 + sin2 d 2 (11.4)
While the curvature singularity at r = 0 is clearly evident, in these new coordinates the metric is no longer singular at r = 2m. Once again, let s consider the radial paths of light rays by setting d = d = 0 and ds 2 = 0. This time we nd 1 2m r dv 2 2 dv dr = 0
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Dividing both sides by dv 2 , we obtain 1 2m r 2 dr =0 dv
If we set r = 2m, then we have dr/dv = 0; that is, the radial coordinate velocity of light has vanished. We can integrate to nd that r (v) = const, which describes light rays that stay right where they are, neither outgoing nor ingoing. Rearranging terms gives 2 dv = dr 1 2m r Integrating, we nd that v (r ) = 2 (r + 2m ln |r 2m|) + const This equation gives us the paths that radial light rays will follow using (v, r ) coordinates. If r > 2m, then as r increases, v increases. This describes the behavior we would expect for radial light rays that are outgoing. On the other hand, if r < 2m, as r decreases, v increases. So the light rays are ingoing. In these coordinates, light cones no longer become increasingly narrow and they make it past the line r = 2m; however, the fact that the time and radial coordinates reverse their character inside r = 2m means that light cones tilt over in this region (see Fig. 11-2). In summary, we have found the following:
The surface de ned by r = 2m is a coordinate singularity. We can nd a suitable change of coordinates to remove the singularity. However, the surface r = 2m de nes a one-way membrane called the event horizon. It is possible for future-directed lightlike and timelike curves to cross from r > 2m to r < 2m, but the reverse is not possible. Events inside the event horizon are hidden from external observers. Moving in the direction of smaller r , light cones begin to tip over. At r = 2m, outward traveling photons remain stationary. For r < 2m, future-directed lightlike and timelike curves are directed toward r = 0. The Schwarzschild coordinates are well suited for describing the geometry over the region 2m < r < and < t < ; however, another coordinate system must be used to describe the point r = 2m and the interior region.
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