Static Schwarzschild Research Articles

Group theoretical quantization of Schwarzschild and Taub-NUT

Stationary spherically symmetric gravity is equivalent to a nonlinear coset sigma model on SL(2, R )/SO(2) coupled to a gravitational remnant. Classically there are stationary solutions besides the static Schwarzschild metric labeled by the Schwarzschild mass m and the Taub-NUT charge l. Imposing the SL(2, R ) symmetry at the quantum level the Wheeler-DeWitt equation becomes related to the Casimir operator on the coset, which makes the system amenable to exact quantization.

Critical behavior near the singularity in a scalar field collapse.

The critical behavior of a field near the singularity in the spherical symmetric scalar field collapse is studied. We consider the supercritical case in which a black hole singularity is formed by the strong imploding scalar field at some advanced time $v$. We find that the field evolution near the singularity can be divided into the following two stages. The spacetime structure near the onset of the first singularity is shown to be well approximated by a self-similar solution. In this self-similar stage the horizon mass linearly increases with $v$. After the self-similar stage ends, the logarithmic behavior becomes remarkable and the system evolves toward the Schwarzschild spacetime with the advanced time. According to this evolution, the strength of the curvature singularity decreases as $I\ensuremath{\simeq}{\ensuremath{\chi}}^{\ensuremath{-}\frac{2(f+2)}{(1+f)}}$, where $\ensuremath{\chi}$ is the circumference radius and $f$ runs from 0 to 1 with the advanced time $v$. In the final stage of gravitational collapse ($v\ensuremath{\rightarrow}\ensuremath{\infty}$), the scalar field dies away as $\mathrm{exp}(\ensuremath{-}kv)$ inside the apparent horizon and the system smoothly approaches the static Schwarzschild space-time. We find that the power-law behavior of the black hole mass is crucially related to the logarithmic behavior of the field. We also propose our main idea that the critical exponent $\ensuremath{\beta}$ of the mass power law is a decreasing function of $v$, which is due to the area law of the apparent horizon.

Canonical quantization of spherically symmetric gravity in Ashtekar's self-dual representation

We show that the quantization of spherically symmetric pure gravity can be carried out completely in the framework of Ashtekar's self-dual representation. Consistent operator orderings can be given for the constraint functionals yielding two kinds of solutions for the constraint equations, corresponding classically to globally nondegenerate or degenerate metrics. The physical state functionals can be determined by quadratures and the reduced hamiltonian system possesses two degrees of freedom, one of them corresponding to the classical Schwarzschild mass squared and the canonically conjugate one representing a measure for the deviation of the nonstatic field configurations from the static Schwarzschild one. There is a natural choice for the scalar product making the two fundamental observables self-adjoint. Finally, a unitary transformation is performed in order to calculate the triad-representation of the physical state functionals and to provide for a solution of the appropriately regularized Wheeler-DeWitt equation.

The radial force on a charged particle in superimposed magnetic fields on Schwarzschild spacetime

Following the approach of optical reference geometry we derive the expression for the total force in the radial direction acting on a charged particle in magnetic fields superimposed on the static Schwarzschild background and show the possible existence of bound orbits for particles in the field of ultra compact objects at distancesr⩽3m wherein the Lorentz force counterbalances both the gravitational and centrifugal forces.

Spherically symmetric spacetimes admitting inheriting conformal Killing vector fields

Perfect fluid spherically symmetric spacetimes admitting a proper inheriting conformal Killing vector (CKV) are studied. It is found that, other than Friedmann-Robertson-Walker (FRW) spacetimes, the only known examples of proper CKV perfect fluid spherically symmetric spacetimes in which the CKV is inheriting are conformal FRW spacetimes, static Schwarzschild interior spacetimes or generalized Gutman-Be'spalko-Wesson spacetimes and all of these spacetimes are either conformally flat or have a CKV which is either parallel to or orthogonal to the fluid 4-velocity. A general theorem is proven in which the (restricted) form of the CKV (and conformal factor) is given should a perfect fluid spherically symmetric spacetime admit a proper inheriting CKV. Various results on the non-existence of perfect fluid spherically symmetric spacetimes admitting a proper inheriting CKV are then derived, thereby providing further validity of a general non-existence conjecture (at least in the perfect fluid case).

Approximate stress-energy tensor for evaporating black holes.

We approximate the Israel-Hartle-Hawking and Unruh stress-energy tensors of quantum fields in the neighborhood of a static Schwarzschild black hole. The approximate tensors are constructed from the conservation equations in the ultrastatic optical metric of the black hole and the expected regularity properties at the horizon, in the respective vacua. Our scheme reproduces the approximation of Page, Brown, and Ottewill for the Israel-Hartle-Hawking vacuum and yields a tensor for the Unruh state. The latter exhibits the correct behavior at infinity and predicts luminosities in good agreement with numerical calculations.

Superstring collisions at planckian energies

By evaluating and resumming the large-s behaviour of string loops to all orders we are able to obtain an explicitly unitary operator for the light (closed) superstring S-matrix above planckian energies: it is dominated by graviton exchange at large impact parameters and by absorption at small impact parameters. In an intermediate, eikonal region a semiclassical description emerges as if, for small deflection angles at least, each string was moving in a static Schwarzschild metric.

Gravitational lens

A plastic lens is described which will bend light rays parallel to the lens axis in the same way as a static (Schwarzschild) gravitational field. This allows one to observe the optical distortions caused by a (nonrotating) black hole directly. The double-imaging property is discussed with particular application to the recently announced double quasar which is suspected to be a gravitational double image of a single quasar. A detailed cosmological model calculation is performed in order to place limits on the mass of the galaxy acting as the lens. It is seen that if the double quasar is a double image and if the lens is a massive galaxy then we will have established that the quasar redshift is cosmological rather than gravitational. The gravitational lens thus occupies a key role in resolving this question.

Physical interpretation of N.U.T. metric

AbstractA metric is constructed, which represents the gravitational field of two spherical static bodies of masses +b and −b, together with a massless source of angular momentum between them. When the separation of the two masses is allowed to tend to infinity the N.U.T. metric can be obtained from the limiting process. This supports the view that the N.U.T. solution represents a static Schwarzschild body together with a rotating semi-infinite spike of pure angular momentum.

Approximate solutions of the general relativity field equations with a scalar meson field

It was shown by Birkhoff ((1), p. 253) that every spherically symmetric solution of the field equations of general relativity for empty space,may be reduced, by suitable coordinate transformations, to the static Schwarzschild form:where m is a constant. This result is known as Birkhoff's theorem and excludes the possibility of spherically symmetric gravitational radiation. Different proofs of the theorem have been given by Eiesland(2), Tolman(3), and Bonnor ((4), p. 167).