Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 23; Problem 23.6.
Another example of a solution to the Einstein equation is a simple model of a cosmic string. Cosmic strings are postulated objects that are left over from the big bang. They are virtually one-dimensional structures with a radius much smaller than an atomic nucleus, but with lengths of hundreds of thousands of light years. As a model of a cosmic string, suppose we have an infinite, straight string stretching along the axis, and that the string is axially symmetric, that is, that its structure depends only on the radial coordinate measured from the axis. The metric describing the string is a generalization of the cylindrical coordinate system:
For an ordinary cylindrical system in flat space, .
[The interpretation of the coordinate is qualitatively different from the Schwarzschild metric, where we assumed spherical symmetry and used this to write down the angular components of the metric as those that apply in flat space, that is . This choice results in the radial coordinate being a circumferential coordinate, in that the circumference of a circle with radial coordinate is , but the distance from the origin to a point on the circle is not . In the cylindrical metric here, is not a circumferential coordinate because the metric component , so the circumference of a circle of radius is (as you can verify by setting and integrating over from 0 to for a fixed ). However, because , the coordinate here does represent the actual distance from the axis to a point on a circle with coordinate .]
We take the stress-energy tensor to be
[Moore doesn’t explain where these come from, but we’ll just accept this for now.] From the definition of the stress-energy tensor is the energy density.
The scalar is
The non-zero components of can be found from 2 by lowering the first index:
From 3 we therefore have
All off-diagonal components of are zero since both and are diagonal. Thus
Using the Ricci tensor worksheet we can work out in terms of . The only non-zero terms are those involving a derivative of with respect to on its own (that is, not multiplied by some other derivative), or in terms of the notation of the worksheet, those terms involving either or on their own. We have (where a subscript 1 indicates a derivative with respect to ):
The only components of involving these two derivatives on their own are and :
Moore now says that we require the metric to be non-singular at (actually he says ‘non-singular at the origin’ although I assume he means ‘non-singular at all points on the axis, since there’s nothing special about here). It’s not entirely clear to me why we would require this since the Schwarzschild metric is singular at . He also says that this requirement leads to the metric reducing to the flat space metric as . Again, this isn’t exactly obvious; there are lots of metrics that are finite at so why choose flat space? Anyway, let’s plow onwards…
If we require as to give us the flat, cylindrical metric near the axis, then as . We can then integrate 24 to give, for points outside the string’s radius :
where is a constant of integration, is the radius of the string and
is the string’s energy density (energy per unit length). [Recall from above that the circumference of a circle of radius is and is the energy density (per unit volume), so the energy in a cylindrical shell of radius , thickness and unit length is .] In the limiting case of no string at all, and the metric reduces to flat space where so and we have for :
Since is a constant for , we can integrate this directly to get
where is a constant of integration. For very small we should have and since is very small, we’d expect to be small, so would be close to zero.
The resulting metric is
We can redefine the angular coordinate (in a way similar to the redefinition of the time coordinate used in deriving Birkhoff’s theorem) by defining
to get what appears to be a flat space metric:
However, remember that the radial coordinate is not the same as that used in flat space, since the circumference of a circle of radius is given by , so is actually slightly smaller than . Also, the new axial coordinate covers for a complete circle.