Deriving the Schwarzschild solution

Abstract: We are concerned with the pulse of gravitational radiation given off when a star falls into a "black hole" near the center of our galaxy. We look at the problem of a small particle falling in a Schwarzschild background ("black hole") and examine its spectrum in the high-frequency limit. In formulating the problem it is essential to pose the correct boundary condition: gravitational radiation not only escaping to infinity but also disappearing down the hole. We have examined the problem in the approximation of linear perturbations from a Schwarzschild background geometry, utilizing the decomposition into the tensor spherical harmonics given by Regge and Wheeler (1957) and by Mathews (1962). The falling particle contributes a $\ensuremath{\delta}$-function source term (geodesic motion in the background Schwarzschild geometry) which is also decomposed into tensor harmonics, each of which "drives" the corresponding perturbation harmonic. The power spectrum radiated in infinity is given in the high-frequency approximation in terms of the traceless transverse tensor harmonics called "electric" and "magnetic" by Mathews.

Abstract: We study the static spherically symmetric Einstein-Yang-Mills equations with SU(2) gauge group and find numerical solutions which are nonsingular and asymptotically flat. These solutions have a high-density interior region with sharp boundary, a near-field region where the metric is approximately Reissner-N\o{}rdstrom with Dirac monopole curvature source, and a far-field region where the metric is approximately Schwarzschild.

Abstract: THE discovery of pulsars and the general conviction that they are neutron stars resulting from gravitational collapse have strengthened the belief in the possible presence of Schwarzschild black-holes—or Schwarzschild horizons—in nature, the latter being the ultimate stage in the progressive spherical collapse of a massive star. The stability of these objects, which has been discussed in a recent report1, ensures their continued existence after formation. Inasmuch as the infinite redshift associated with it and its behaviour as a one-way membrane make the Schwarzschild horizon at once elusive and intriguing, it is important to explore theoretically all possible modes in which the presence of such a black-hole manifests itself. In what follows, we present a partial summary of some results obtained from an investigation of the scattering of gravitational waves by a Schwarzschild horizon.

Abstract: The ``generalized Schwarzschild'' metric discovered by Newman, Unti, and Tamburino, which is stationary and spherically symmetric, is investigated. We find that the orbit of a point under the group of time translations is a circle, rather than a line as in the Schwarzschild case. The time‐like hypersurfaces r = const which are left invariant by the group of motions are topologically three‐spheres S 3, in contrast to the topologyS 2 × R (or S 2 × S 1) for the r = const surfaces in the Schwarzschild case. In the Schwarzschild case, the intersection of a spacelike surface t = const and an r = const surface is a sphere S 2. If σ is any spacelike hypersurface in the generalized metric, then its (two‐dimensional) intersection with an r = const surface is not any closed two‐dimensional manifold, that is, the generalized metric admits no reasonable spacelike surfaces. Thus, even though all curvature invariants vanish as r → ∞, in fact R μναβ = O(1/r 3) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which g μν − ημν = O(1/r). An apparent singularity in the metric at small values of r, which appears to be similar to the spurious Schwarzschild singularity at r = 2m, has not been studied. If this singularity should again be spurious, then the ``generalized Schwarzschild'' space would represent a terminal phase in the evolution of an entirely nonsingular cosmological model which, in other phases, contains closed spacelike hypersurfaces but no matter.