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2 d =0 r dr
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(10.30)
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To solve this equation, consider the second derivative of r e 2 (r e 2 ) = e 2 2r = 4 d 2 e dr d dr
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d 2 d2 2r 2 e 2 + 4r e dr dr
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The Schwarzschild Solution
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Let s quickly set this equal to zero to see that this is in fact another way of writing (10.30): 4 d 2 d2 e 2r 2 e 2 + 4r dr dr d dr
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e 2 = 0
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Now divide through by e 2 to get 4 d2 d 2r 2 + 4r dr dr d dr
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Next we divide through by 2r , which gives d2 d 2 dr 2 dr
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2 d =0 r dr
Returning to r e 2 , since this vanishes because it is equivalent to (10.30), we can integrate once r e 2 = constant (10.31)
Now, recall that we had found that r e 2 = e 2 2r d e 2 . dr Let s turn to (10.26). We obtain another of the vacuum equations by setting this equal to zero: R = 1 d 2 1 d 2 1 e 2 e e + + =0 r dr r dr r2
We multiply through by r 2 to get r d 2 d + r e 2 + 1 e 2 = 0 e dr dr
Now we use = to obtain 2r d 2 e + 1 e 2 = 0 dr
The Schwarzschild Solution
Moving 1 to the other side and then multiplying through by 1, we have 2r d 2 e + e 2 = 1 dr
Now the left side is nothing other than r e 2 . And so we have found that r e 2 = 1. Integrating this equation, we nd r e 2 = r 2m where 2m is an unknown constant of integration. We choose this odd designation for the constant of integration because, as we will see, this term is related to the mass. Dividing through by r , we nd that e 2 = 1 2m r
Let s recall the original form of the metric. We had ds 2 = e2 (r ) dt 2 e2 (r ) dr 2 r 2 d 2 + sin2 d 2 Using = , we obtain the coef cients in the metric we have been seeking e2 = 1 2m r and e2 = 1 2m r
(10.32)
The Meaning of the Integration Constant
We nd the constant appearing in the line element by seeking a correspondence between newtonian theory and general relativity. Later, we will discuss the weak eld limit and nd that relativity reduces to newtonian theory if we have (for the moment explicitly showing fundamental constants) gtt 1 + 2 c2
where is the gravitational potential in newtonian theory. For a point mass located at the origin, = G M r
and so
The Schwarzschild Solution
gtt 1 2 Comparing this to (10.32), we set m=
GM c2r
GM c2
where M is the mass of the body in kilograms (or whatever units are being used). Looking at this term you will see that the units of m which appears in the metric has units of length. This constant m is called the geometric mass of the body.
The Schwarzschild Metric
With these results in hand, we can rewrite (10.5) in the familiar form of the Schwarzschild line element: ds 2 = 1 2m r dt 2 dr 2 r 2 d 2 + sin2 d 2 2m 1 r (10.33)
Looking at the metric, we can see that as r gets large, it approaches the form (10.2) of the at space metric.
The Time Coordinate
The correspondence between the Schwarzschild metric in (10.33) and the at space metric tells us that the time coordinate t used here is the time measured by a distant observer far from the origin.
The Schwarzschild Radius
Notice that the metric de ned in (10.33) becomes singular (i.e., blows sky high) when r = 2m. This value is known as the Schwarzschild radius. In terms of the mass of the object that is the source of the gravitational eld, it is
given by
The Schwarzschild Solution
rs =
2GM c2
(10.34)
For ordinary stars, the Schwarzschild radius lies deep in the stellar interior. Therefore we can use the metric given by (10.33) with con dence to describe the region found outside of your average star. As an example, the Schwarzschild radius for the Sun can be used by inserting the solar mass 1.989 1030 kg into (10.34), and we nd that rs 3 km. Remember, we started out by solving the vacuum equations. Since this point lies inside the Sun where matter is present, the solutions we obtained in the previous section cannot apply in that region. It should catch your attention that the metric blows up at a certain value of the radius. However, we need to investigate further to determine whether this is a real physical singularity, which means the curvature of spacetime becomes in nite, or whether this is just an artifact of the coordinate system we are using. While it seems like stating the obvious, it is important to note that the line element is written in terms of coordinates. It may turn out that we can write the metric in a different way by a transformation to some other coordinates. To get a better idea of the behavior of the spacetime, the best course of action is to examine invariant quantities, i.e., scalars. If we can nd a singularity that is present in a scalar quantity, then we know that this singularity is present in all coordinate systems and therefore represents something that is physically real. In the present case, the components of the Ricci tensor vanish, so the Ricci scalar vanishes as well. Instead we construct the following scalar using the Riemann tensor. Rabcd R abcd = 48m 2 r6 (10.35)
This quantity blows up at r = 0. Since this is a scalar, this is true in all coordinate systems, and therefore, we conclude that the point r = 0 is a genuine singularity.
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