Schwarzschild Space Research Articles

THE QUASI BOUND STATES OF DIRAC FERMION IN SCHWARZSCHILD METRIC

In this paper, the two-component Dirac equation in Schwarzschild background space time is reduced to a one component second order equation. The effective potential and the L-S coupling term are given. Near the horizon, there is a potential wall. When K2>3η2, there is another potential well by the wall. So the particle resonating in the well will be absorbed by the horizon after penetrating through the wall. With WKB approximation, the resonance spectrum and its width are obtained. For the case of Re-issner-Nordstrom background space time, the same procedure is also carried out, and the results are discussed briefly.

Quantum tachyons in Schwarzschild space–time

The wave equation of a spinless tachyon is studied in Schwarzschild space–time. In contrast to earlier approaches to the problem, it is shown that tachyonic static solutions satisfy a simple second-order linear differential equation regardless of the mass of the black hole and the mass parameter of the tachyon. Physical implication of the present approach is discussed. Using Langer modification of the WKB (Wentzel–Kramers–Brillouin) boundary condition an expression similar to the Bohr–Sommerfeld quantization condition is derived.

Space-time topology and a new class of Yang-Mills instanton

Different families of Yang-Mills instantons exist in space-time of different topology. Two (self-dual) examples are provided in Schwarzschild space having SU(2) Pontryagin numbers ±1 and ±2 n 2 ( n = integer). The latter solution describes a dyon. The Reissner-Nordstrom geometry also admits of non-self-dual solutions.

Solutions of the time-dependent Klein-Gordon equation in a Schwarzschild background space

Time-dependent solutions of the Klein-Gordon equation for a massive scalar meson field in a Schwarzschild background space are obtained by the use of asymptotic methods. The solutions are found for all angular momentum states and are valid over all space exterior in a Schwarzschild radius provided the black hole is large, and the energy is very much less than the rest-mass energy of the scalar pi meson.

The massive scalar meson field in a Schwarzschild background space

The radial equation of a static massive scalar meson field in a Schwarzschild background space is solved by asymptotic methods leading to solutions over the whole range. This approach is used to obtain the form of the Yukawa potential in the presence of a large Schwarzschild black hole, and to show that the meson field fades away to zero as the event horizon is reached, in agreement with the 'no hair' conjecture.

Path-integral derivation of black-hole radiance

The Feynman path-integral method is applied to the quantum mechanics of a scalar particle moving in the background geometry of a Schwarzschild black hole. The amplitude for the black hole to emit a scalar particle in a particular mode is expressed as a sum over paths connecting the future singularity and infinity. By analytic continuation in the complexified Schwarzschild space this amplitude is related to that for a particle to propagate from the past singularity to infinity and hence by time reversal to the amplitude for the black hole to absorb a particle in the same mode. The form of the connection between the emission and absorption probabilities shows that a Schwarzschild black hole will emit scalar particles with a thermal spectrum characterized by a temperature which is related to its mass, $M$, by $T=\frac{\ensuremath{\hbar}{c}^{3}}{8\ensuremath{\pi}}\mathrm{GMk}$. Thereby a conceptually simple derivation of black-hole radiance is obtained. The extension of this result to other spin fields and other black-hole geometries is discussed.

Quantum field theory in Schwarzschild and Rindler spaces

The problem of defining a scalar quantum field in the space-times described by the Schwarzschild and Rindler metrics is discussed. The matrix elements of the field operators are found by calculating the Green's functions for the fields. The requirement of positive frequencies for asymptotic timelike separations combined with a careful analysis of the continuity conditions at the event horizons yields a unique prescription for the Green's function. This is turn defines the vacuum state. In the Schwarzschild space the vacuum is shown to be stable and the lowest-energy state. In the Rindler space the quantization procedure yields the same results as quantization in Minkowski coordinates.

Dipole electromagnetic radiation in a Schwarzschild space: The wavetail to order m/r

The author considers Maxwell's equations in a Schwarzschild space and determine the retarded solutions to order m2/r2 of an electric dipole located at the centre of symmetry. The dipole moment p(t) is assumed to be variable in the interval t1 t2) to order m/r are discussed in some detail.

Gravitationskollaps und Lichtgeschwindigkeit im Gravitationsfeld

AbstractAn elementary criterion of the stability of a matter sphere against gravitational collapse is given by the circular velocity condition of POINCARÉ: In a space with a spherically symmetric gravitation potential ϕ (r) and with a spherically symmetric metric gik (e.g., a SCHWARZSCHILD space time) the circular velocity V* of a particle on the surface r = R of the matter‐sphere must be (This condition is a consequence of the virial theorem and of the POINCARÉ theorem.) ‐ However, EINSTEIN's axiom of causality implies that this velocity V* must be smaller than the local velocity of light v: V*2 < v2. And this local velocity v is a function of the gravitation potential ϕ, too: v = v [ϕ].In the case of NEWTON's or EINSTEIN's theory the spherically symmetric gravitation potential is given by the NEWTONian function ϕ = fM/r. In the special theory of relativity, we would have v = c (c = EINSTEIN's fundamental velocity) and grr = 1. Therefore, the specialrelativistic stability condition is R > fMc−2. ‐ But in the NEWTONian theory v is depending of the gravitation potential and depends of the boundary condition for the light propagation, also. According to the ansatz of LAPLACE (1799) we have: (emanation‐theory of light). But, according to SOLDNER (1801), we have Therefore, we are finding in the case of LAPLACE the same condition R > fMc−2 as in the SRT. But, in the case of SOLDER's ansatz non condition for stability is resulting. ‐ In the general relativistic theories the local velocity of light is given by EINSTEIN 's expression According to EINSTEIN's theory of “static gravitation” (1911/12) we have grr = 1 and therefore the formula and according to the GRT (with ‐ gω = grr−1) we have the formula Therefore, the Hilbert‐Laue condition r= R > 3fMc−2 results as stability condition.From the gravo‐optical point of view, in GRT and for the classical ansatz of LAPLACE “black‐holes” with bounding states of light result for R ≤ 2fM−2. But, no “black‐holes” are existing according to SOLDNER's ansatz. However, in GRT each black‐hole must be a “collapsar”. But according to the classical theory of LAPLACE we have uncollapsed “black‐ holes” for the domain .

Maximal foliations of extended Schwarzschild space

Smooth families of spherically symmetric maximal surfaces which are spacelike except at r = 2m are explicitly constructed in Schwarzschild space. Such surfaces should be useful in the study of initial value problems.

Non-lorentz transformations of reference systems

The structure of transformations is investigated which describe the transition from an inertial reference system to a noninertial one and conversely. As a result of such transition the curvature of the space of events is modified, and one is thus able to discuss its observational character. This leads to a new interpretation of the general relativity theory in which the space curvature is now related only to the noninertial motion of the observer. Transition from a noninertial reference system corresponding to the Schwarzschild space to an inertial reference system (plane space) is considered as an example.

Point Charge in the Vicinity of a Schwarzschild Black Hole

The electrostatic field of a point charge at rest in Schwarzschild space is derived. The solution is used to study the problem of a point charge slowly lowered into a nonrotating black hole. We find that the electric field of the charge remains well behaved as the charge is lowered and that all the multipole moments except the monopole fade away. We conclude that a Reissner-Nordstrom black hole is produced.

Tidal Gravitational Accelerations near an Arbitrary Timelike Geodesic in Schwarzschild Space

It is suggested that a ``physical'' definition of a singularity in a space-time manifold might be that it is a point where the relative accelerations of nearby timelike geodesics become infinite. Along an arbitrary timelike geodesic in Schwarzschild space, we construct an orthonormal tetrad of 4-vectors which are used to define ``elevator coordinates'' in a neighborhood of the geodesic. We use these coordinates to determine the tidal gravitational accelerations near the geodesic, and we point out that these accelerations are finite (and continuous) at r = 2m, the ``Schwarzschild surface,'' although they are unbounded as r approaches 0.

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

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 S3, in contrast to the topology S2 × R (or S2 × S1) 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 S2. 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 Rμναβ = O(1/r3) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which gμν − 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.