Exact gravitational field of the infinitely long rotating hollow cylinder

Abstract

The vacuum line element inside an infinitely long rotating hollow cylinder is the usual flat space line element. It is fitted in a most general way to the general cylindrical vacuum field outside at the singular hypersurface R o = const, representing the infinitely thin hollow cylinder. With the use of the jump conditions at R o = const the surface densities τ x μ , of which the energy-momentum-stress tensor τ^ of the shell consists, are calculated. The physical properties of the cylinder, as derived from the eigenvalues and -vectors of τj, and the generated gravitational field are discussed in full detail.

Rotating cylindrical systems and gravitomagnetism

“…Hence the spacetime is flat inside the shell (R = 0) and the exterior field is static ( B = 0). These conclusions agree with the results of the exact theory [6][7][8], though the general situation is more complicated [6].…”

Hollow cosmic string: The general-relativistic hollow cylinder

“…Let us for convenience translate the radial coordinate r (which is defined by the parametrization (2.1) only up to an additive constant) so that the surface r = 0 corresponds to the cylinder surface. As in Frehland's analysis of the hollow cylinder problem [3], we use the junction method [5] to determine the space-time metric, i. e. the 3 × 3 matrix λ(r), from the knowledge of the energy-momentum tensor T µ ν = κ −1 T µ ν δ(r) concentrated on the surface of the matter cylinder. The metric field λ(r) may be written…”

“…A limiting case which may be of some interest is that of a hollow cosmic string, the energy-momentum tensor being concentrated not on a line but on the surface of an infinitely long cylinder. The gravitational field generated by such a cylinder with given energy density and longitudinal and azimuthal stresses, and rotating about its axis, was in principle determined some time ago by Frehland [3]. One of the results of Frehland's analysis is that, unexpectedly, a rotating hollow cylinder can exist in general relativity only if its energy-momentum is traceless.…”

Dynamics of Rotating Cylindrical Shells in General Relativity

“…It can be shown that the eigenvalue λ = 0 corresponds to the eigenvector ξ µ 1 = η µ , where η µ is the normal vector to the hypersurface r = R(t), and given by Eq. (29). The eigenvalue λ = p z corresponds to the eigenvector ξ µ 2 = z µ , which represents the pressure of the shell in the z-direction.…”