The wondrous Weyl vacuums
The Weyl vacuums comprise the family of all static axisymmetric vacuum solutions of the EFE. This family includes such familiar examples as the Schwarzschild vacuum solution, as well as the exact solutions corresponding to other isolated objects, such as a uniform density disk, ring, or rod. (These terms require, as it turns out, considerable qualification!) They were found by Weyl in about 1918, and can be expressed in the following memorable form:
<br />
ds^2 = -\exp(2 \, u) \, dt^2 + \exp(-2 \, u) \; <br />
\left( \exp(2 \, v) \; (dz^2 + dr^2) + r^2 \, d\phi^2 \right),
-\infty < t, \, z < \infty, \; \; 0 < r < \infty, \; \; -\pi < \phi < \pi
where u, \, v are axisymmetric (functions of z,r only) and satisfy the vacuum field equations, which reduce to the following simple "triangularized" system:
<br />
u_{zz} + u_{rr} + \frac{u_r}{r} = 0, \; \;<br />
v_r = r \, \left( u_r^2 - u_z^2 \right), \; \; v_z = 2 \, r \, u_z \, u_r<br />
Here, the first equation involves only u and it is the Laplace equation, written in cylindrical coordinates and applied to an axisymmetric function.
That is, to obtain a Weyl vacuum solution, you can choose an arbitrary axisymmetric harmonic function and then obtain v by quadrature (partial integration). Needless to say, this is delightful because it means we can rather easily obtain a great variety of examples of exact axisymmetric static vacuum solutions.
Even better, if we expand everything to first order in u, we find the weak-field Weyl vacuums are controlled by axisymmetric harmonic functions, which correspond to the Newtonian gravitostatic potential:
<br />
ds^2 = -(1+2\, u) \, dt^2 + (1-2 \, u) \; <br />
\left( dz^2 + dr^2 + r^2 \, d\phi^2 \right),
-\infty < t, \, z < \infty, \; \; 0 < r < \infty, \; \; -\pi < \phi < \pi
where
<br />
u_{zz} + u_{rr} + \frac{u_r}{r} = 0<br />
Thus, we have a clear and immediate interpretation of the master metric function u in the weak-field approximation.
However, things are not as simple as this summary would have you believe! To begin to see the difficulties, note that in the Weyl canonical chart (the cylindrical-like chart written above), the radial coordinate r does not possesses the "Schwarzschild property", according to which the area of the sphere t=t_0, \, r=r_0 should be 4 \, \pi \, r_0^2. Nor does it have the "arc length property", according to which r_2-r_1 should be the length of a radial segment with \phi=\phi_0, \; t=t_0, \; z=z_0, \; r_1 < r < r_2. Thus, it is more challenging to give a geometric interpretation of these coordinates than it is to interpret the Schwarzschild coordinates. That is, we want to give a coordinate-free characterization of the radial coordinate r we are using, as we know how to do for the Schwarzschild chart for the Schwarzschild vacuum, but it is not so obvious how to do that here.
Why should anyone care about such quibbles? One reason to see why this issue is important is to ask how to express the Schwarzschild vacuum using the Weyl canonical chart. Since the Schwarzschild vacuum is the unique spherically symmetric static vacuum, and since we seemingly obtained above a simple correspondence between Newtonian potentials and Weyl vacuum solutions, we naturally expect that this should be given by starting with the axisymmetric harmonic function u(z,r) = -m/\sqrt{z^2+r^2}, which is of course the spherically symmetric Newtonian potential (independent of time). But in fact this gives a vacuum solution called the Chazy-Curzon vacuum, which is not in fact a spherically symmetric spacetime at all! It turns out that to obtain the Schwarzschild vacuum, we need to start with the Newtonian potential of a finite thin rod with just the right "effective mass density"! In a similar vein, I note that the Newtonian potential of a thin uniform density ray (i.e., limiting case of a rod with one endpoint drawn off to infinity) gives a Weyl vacuum which is isometric to the Rindler wedge, i.e. is locally flat.
There is much, much more to say, but I'll only hint at a few more points. First, the Weyl vacuums can be generalized to the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions of the EFE, which includes the Kerr vacuum, and which involves a system of coupled second order PDEs in two variables (the simplest formulation does not involve the metric functions directly but certain derived variables), plus a quadrature for a third metric function. Second, the Ernst and Weyl families are closely related to several other families of exact solutions. Perhaps the most interesting of these are the colliding plane wave models (CPW), and then close formal analogies led Chandrasekhar to the discovery that the Kerr and Schwarzschild vacuums can be realized (locally!) as CPW models. Third, if we compute the point symmetry group of the Weyl or Ernst system, we can find explicit solutions obeying symmetry conditions (since the Ernst system, unlike the Weyl system, is a nonlinear system of coupled PDEs, the general solution is not easy to write down!), but we also find that "mass parameters" as in the Schwarzschild vacuum are not so straightforward as Newtonian intuition suggests. Ultimately, this is because the very notion of "parameterization of solutions" is fraught with difficulties when we try to apply this notion to the solution space of a nonlinear PDE!
A good place to start reading about any topic involving exact solutions is the monograph on exact solutions of the EFE, by Kramer et al., which is cited in full at http://www.math.ucr.edu/home/baez/RelWWW/HTML/reading.html#advanced
However, for this topic an even better source is the superb review article by Bonnor, "Physical interpretation of vacuum solutions", Gen. Rel. Grav. 24 (1992): 551-573. For the connection with CPW models, try part II of this paper plus the monograph by Griffiths, Colliding Plane Waves in General Relativity, Oxford University Press, 1991. Various arXiv eprints discuss specific Weyl vacuum solutions, some of are "named", such as the Bach-Weyl dumbell, the Chazy-Curzon asymmetric particle, and so on. There is in fact a substantial literature on the Weyl vacuums; this is one of the best studied classes of vacuum solutions, yet in many respects it remains curiously misunderstood.
