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While this is well behaved mathematically, the fact that gtt vanishes means that the surface r = 2m is a surface of in nite redshift. Something unusual is obviously going on, and we will examine this behavior again later. But for now let s consider the other components of the metric. While nothing unusual happens to g and g , we see that grr behaves very badly. In fact, this term blows up: grr = 1 1 2m r as r 2m
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When a mathematical expression goes to in nity at some point, we call that point a singularity. However, in geometry and in physics and hence in general relativity, the presence of a singularity must be explored carefully. The rst question to ask is whether or not the singularity is physically real or whether it is due to the choice of coordinates we have made. In this case, we will nd that while the surface r = 2m has some unusual properties, the singularity is due to the choice of coordinates, and so is a coordinate singularity. Simply put, by using a different set of coordinates we can write the metric in such a way that the singularity at r = 2m is removed. However, we will see that the point r = 0 is due to in nite curvature and cannot be removed by a change in coordinates. We have already seen a way to investigate this question: construct invariant quantities invariant quantities will not depend on our particular choice of coordinates. In 10, we found that Rabcd R abcd = 48m 2 r6
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This invariant (it s a scalar) tells us that the curvature tensor does blow up at r = 0, but that at r = 2m, nothing unusual happens. This tells us that we can remove the singularity at r = 2m by changing to an appropriate coordinate system.
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We can study these problems further by examining the behavior of light cones near r = 2m. Consider paths along radial lines, which means we can set
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d = d = 0. In this case, the Schwarzschild metric simpli es to ds 2 = 1 2m r dt 2 dr 2 1 2m r
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To study the paths of light rays, we set ds 2 = 0. This leads to the following relationship, which gives the slope of a light cone: 2m dt = 1 dr r
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(11.1)
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The rst thing to notice that far from r = 2m, that is as r , this equation becomes dt = 1 dr Therefore in this limit we recover the motion of light rays in at space (integration gives t = r modulo a constant, just what we would expect for light cones in Minkowski space). Now let s examine the behavior as we approach smaller r , speci cally approaching r = 2m. It will be helpful to examine the positive sign, which corresponds to outgoing radial null curves. Then we can write (11.1) as dt r = dr r 2m Notice that as r 2m, dt/dr . This tells us that the light cones are becoming more narrow as we approach r = 2m (at r = 2m, the lines become vertical). This effect is shown in Fig. 11-1. We can nd the key to getting rid of the singularity by integrating (11.1) to get time as a function of r . Once again, if we take the positive sign, which applies for outgoing radial null curves, then integration gives t = r + 2m ln |r 2m| (we are ignoring the integration constant). The form of t (r ) suggests a coordinate transformation that we can use. We now consider the tortoise coordinate, which will allow us to write down the metric in a new way that shows only the curvature singularity at the origin.