We consider diagonal cylindrically symmetric metrics, with an interior representing a general nonrotating fluid with anisotropic pressures. An exterior vacuum Einstein-Rosen spacetime is matched to this using Darmois matching conditions. We show that the matching conditions can be explicitly solved for the boundary values of metric components and their derivatives, either for the interior or exterior. Specializing to shearfree interiors, a static exterior can only be matched to a static interior, and the evolution in the nonstatic case is found to be given in general by an elliptic function of time. For a collapsing shearfree isotropic fluid, only a Robertson-Walker dust interior is possible, and we show that all such cases were included in Cocke's discussion. For these metrics, Nolan and Nolan have shown that the matching breaks down before collapse is complete, and Tod and Mena have shown that the spacetime is not asymptotically flat in the sense of Berger, Chrusciel, and Moncrief. The issues about energy that then arise are revisited, and it is shown that the exterior is not in an intrinsic gravitational or superenergy radiative state at the boundary.

Shearfree cylindrical gravitational collapse

The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (Levi--Civita) space--time. A match to a cylinder source is achieved for $-\frac{1}{2} \frac{1}{2}$, this case being without a Newtonian limit; the positive (negative) values of $\si$ correspond to a positive (negative) fluid density. The range of $\si$ for which a source has previously been matched to the Levi--Civita metric is $0\leq\si<\frac{1}{2}$ for a cylinder source.

Perfect-fluid cylinders and walls - sources for the Levi-Civita space-time

The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (Levi-Civita) spacetime. A match to a cylinder source is achieved for , where is the mass per unit length in the Newtonian limit , and a match to a wall source is possible for , this case being without a Newtonian limit; the positive (negative) values of correspond to a positive (negative) fluid density. The range of for which a source has previously been matched to the Levi-Civita metric is for a cylinder source.

Perfect-fluid cylinders and walls - sources for the Levi-Civita spacetime

Two exact solutions of Einstein’s field equations corresponding to a cylinder of dust with net zero angular momentum are considered. In one of the cases, the dust distribution is homogeneous, whereas in the other, the angular velocity of dust particles is constant [Nuovo Cimento B 21, 64 (1974)]. For both solutions the junction conditions to the exterior static vacuum Levi–Civita space–time were studied. From this study we find an upper limit for the energy density per unit length σ of the source equals 1/4 for both cases. Thus the homogeneous cluster provides another example [Class. Quant. Grav. 8, 727 (1991); J. Math. Phys. 27, 152 (1980)] where the limit of σ is 1/4. It was also found that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density ρ0.

The Levi–Civita space–time

We consider two exact solutions of Einstein's field equations corresponding to a cylinder of dust with net zero angular momentum. In one of the cases, the dust distribution is homogeneous, whereas in the other, the angular velocity of dust particles is constant [1]. For both solutions we studied the junction conditions to the exterior static vacuum Levi-Civita spacetime. From this study we find an upper limit for the energy density per unit length $\sigma$ of the source equal ${1\over 2}$ for the first case and ${1\over 4}$ for the second one. Thus the homogeneous cluster provides another example [2] where the range of $\sigma$ is extended beyond the limit value ${1\over 4}$ previously found in the literature [3,4]. Using the Cartan Scalars technics we show that the Levi-Civita spacetime gets an extra symmetry for $\sigma={1\over 2}$ or ${1\over 4}$. We also find that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density $\rho_0$.

The Levi-Civita spacetime

The case of a cylindrically symmetric cluster of particles moving in circles perpendicular to the axis of symmetry is discussed on the basis of the equations of general relativity. A number of special solutions are obtained, and two interesting results on the relation between the gravitational mass and the sum of the free particle masses for these clusters are deduced.

H. Bondi

Papers

Stationary Cylindrically Symmetric Clusters of Particles in General Relativity