Stressless Schwarzschild

This self-contained pedagogical simple explicit 6-step derivation of the Schwarzschild solution, in “ $3 + 1$ ” formulation and conformal spatial gauge, (almost) avoids all affinity, curvature and index gymnastics.     

Three-dimensional Schwarzschild spacetimes

In the following we find all of the three-dimensional flat spacetimes which are static and rotationally symmetric.     

Schwarzschild mass uncertainty

In this work we research on the thermodynamical properties of the Bardeen black holes. We compute the series of new thermodynamic quantities such as local temperature, heat capacity, free energy of this kind of black hole in detail. We further analyze the thermodynamical characteristics of the Bardeen black hole by varying its charge $q$ to check the existence and stability of the black hole.     

Rindler modified Schwarzschild geodesics

The mysterious attractive constant radial force acted in the past on Pioneer spacecrafts—the so-called Pioneer anomaly—is considered within the context of Rindler acceleration. As an idea this is tempting since it is reminiscent of the cosmological constant. Fortunately the anomalous force acts radially toward the sun so that it differs from the mission of a cosmological constant. Without resorting to the physical source responsible for such a term we investigate the modified Schwarzschild geodesics. The Rindler acceleration naturally affects all massive / massless particle orbits. Stable orbits may turn unstable and vice versa with a finely-tuned acceleration parameter. The overall role of the extra term, given its attractive feature is to provide confinement in the radial geodesics.     

The Schwarzschild H space

The usual construction of anH space from asymptotic data onI ⁺ uses only the radiative component of the asymptotic gravitational field and ignores the longitudinal part of the field [2]. While Penrose and Sparling [1] have given the construction of a deformed twistor space encoding the Schwarzschild structure by exponentiating the linearized spin 2 monopole field, one would like to have a directH space construction as a step toward constructingH spaces containing all the asymptotic degrees of freedom. The first step of giving a directH space construction for the Schwarzschild asymptotic data is contained in the following. Several interesting new structures arise in this process.     

Schwarzschild Metrics and Quasi-Universes

The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows us to obtain some results that otherwise are less apparent or even hidden in other coordinate systems.     

An irradiated Schwarzschild object

A generalization of the Schwarzschild space-time, a Petrov type-II electrovacuum solution, is obtained. The metric is stationary but the Maxwell tensor contains an arbitrary function of time. The curvature invariants and singularities are investigated usingSU(2) spin coefficient techniques.Dedicated to Achille Papapetrou on the occasion of his retirement.On leave from the Central Research Institute for Physics, Budapest.     

Radiation Fields on Schwarzschild Spacetime

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.     

Schwarzschild geometry,once more

Schwarzschild geometry exhibits interesting features when the field equations are decomposed with respect to a system of freely falling observers. We only use quantities behaving like tensors under a restricted group of transformations of the reference system. Moreover Lorentz transformations with non-constant velocities drastically change the physical picture of the theory.     

Finslerian Extension of Schwarzschild Metric

Continuing [1], the concept of the static and spherically symmetric gravitational field is examined in the context of the fibered Finslerian approach. The approximate analysis of the associated field equations is elaborated in a systematic manner, expanding the equations with respect to the lowvelocity parameters and in the post-Newtonian way. An exact solution which has the meaning of direct extension of the Schwarzschild metric for the Finslerian case has been found assuming that fibres are of constant curvatures.     

Schwarzschild Solution on the Brane

In this paper it is shown that Schwarzschild solution is possible in brane world for some specific choices of brane matter and the non-local effects from the bulk. A conformally flat bulk space time with fine-tuned vacuum energy (brane tension) shows that Schwarzschild solution may also be the vacuum solution for brane world scenario.     

Isotropic coordinates and Schwarzschild metric

Written in terms of isotropic coordinatesr, t, the Schwarzschild metric as usually given is static, i.e., admits a timelike Killing vector for all values ofr andt. Therefore the region within the event horizon cannot be accounted for. This deficiency is remedied here, by finding the general spherically symmetric vacuum metric in isotropic coordinates.     

Schwarzschild and Synge once again

We complete the historical overview about the geometry of a Schwarzschild black hole at its horizon by emphasizing the contribution made by Synge in [6] to its clarification.     

Entropy Quantization of Schwarzschild Black Hole

The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.     

Stability of Scalar Modified Schwarzschild Solution

It is shown that the Schwarzschild solution in general relativity, reconsidered adding to the vacuum a massless scalar field, is stable to perturbations from radiation fields of spin s = 0, ±1/2, ±1, ±2.     

Interior Schwarzschild Problem and Its Integration

The interior Schwarzschild metric for a static,spherically symmetric perfect fluid can be parametrizedwith two independent functions of the radial coordinate.These functions are easily expressed in terms of (radial) integrals involving the fluidenergy density and pressure. The pressure is, however,not independent, but is determined in terms of thedensity by one of Einstein's equations, theOppenheimer–Volkov (OV) equation. An approximate integral to theOV equation is presented which is accurate for slowlyvarying, realistic, densities, and exact in theconstant-density limit. It makes it possible to findcompletely integrated accurate solutions to the interiorSchwarzschild metric in terms of the density only. Somepost-Newtonian consequences of the solution are given aswell as the resulting general relativistic pressure for an energy densityr^-1/2.     

Analytical time-like geodesics in Schwarzschild space-time

Time-like orbits in Schwarzschild space-time are presented and classified in a very transparent and straightforward way into four types. The analytical solutions to orbit, time, and proper time equations are given for all orbit types in the form r = r(λ), t = t(χ), and τ= τ (χ), where λ is the true anomaly and χ is a parameter along the orbit. A very simple relation between λ and χ is also shown. These solutions are very useful for modelling temporal evolution of transient phenomena near black holes since they are expressed with Jacobi elliptic functions and elliptic integrals, which can be calculated very efficiently and accurately.