Deriving the Schwarzschild solution

About: Deriving the Schwarzschild solution is a research topic. Over the lifetime, 809 publications have been published within this topic receiving 18561 citations.

Event Horizons in Static Vacuum Space-Times

Abstract: The following theorem is established. Among all static, asymptotically flat vacuum space-times with closed simply connected equipotential surfaces ${g}_{00}=\mathrm{constant}$, the Schwarzschild solution is the only one which has a nonsingular infinite-red-shift surface ${g}_{00}=0$. Thus there exists no static asymmetric perturbation of the Schwarzschild manifold due to internal sources (e.g., a quadrupole moment) which will preserve a regular event horizon. Possible implications of this result for asymmetric gravitational collapse are briefly discussed.

Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics

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.

Particlelike solutions of the Einstein-Yang-Mills equations.

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.

Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes

Abstract: A general formalism for understanding the thermodynamics of horizons in spherically symmetric spacetimes is developed. The formalism reproduces known results in the case of black-hole spacetimes and can handle more general situations such as: (i) spacetimes which are not asymptotically flat (such as the de Sitter spacetime) and (ii) spacetimes with multiple horizons having different temperatures (such as the Schwarzschild–de Sitter spacetime) and provide a consistent interpretation for temperature, entropy and energy. I show that it is possible to write Einstein's equations for a spherically symmetric spacetime in the form T dS − dE = P dV near any horizon of radius a with S equal; 1/4(4πa2), |E| = (a/2) and the temperature T determined from the surface gravity at the horizon. The pressure P is provided by the source of Einstein's equations and dV is the change in the volume when the horizon is displaced infinitesimally. The same results can be obtained by evaluating the quantum mechanical partition function without using Einstein's equations or the WKB approximation for the action. Both the classical and quantum analyses provide a simple and consistent interpretation of entropy and energy for de Sitter spacetime as well as for (1 + 2) dimensional gravity. For the Rindler spacetime the entropy per unit transverse area turns out to be 1/4 while the energy is zero. The approach also shows that the de Sitter horizon—like the Schwarzschild horizon—is effectively one dimensional as far as the flow of information is concerned, while the Schwarzschild–de Sitter, Reissner–Nordstrom horizons are not. The implications for spacetimes with multiple horizons are discussed.

The Flatter Regions of Newman, Unti, and Tamburino's Generalized Schwarzschild Space

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.