Constant Time Slices Research Articles

Gravitational energy in stationary spacetimes

Static observers remain on Killing-vector worldlines and measure the rest-mass + kinetic energies of particles moving past them, and the flux of that mechanical energy through space and time. The total mechanical energy is the total flux through a spacelike cut at one time. The difference between the total mass energy and the total mechanical energy is the total gravitational energy, which we prove to be negative for certain classes of systems. For spherical systems, Misner, Thorne and Wheeler define the total gravitational energy in this way. To obtain the gravitational energy density analogous to that of electromagnetism we first use Einstein's equations with integrations by parts to remove second-order derivatives. Next we apply a conformal transformation to reexpress the scalar 3-curvature of the 3-space. The resulting density is non-local. We repeat the argument for mechanical energies as measured by stationary observers moving orthogonally to constant time slices like the ‘zero angular momentum’ observers of Bardeen who exist even within ergospheres.

Vacuum energy density for massless scalar fields in flat homogeneous spacetime manifolds with nontrivial topology

Although the observed universe appears to be geometrically flat, it could have one of 18 global topologies. A constant-time slice of the spacetime manifold could be a torus, M\obius strip, Klein bottle, or others. This global topology of the universe imposes boundary conditions on quantum fields and affects the vacuum energy density via the Casimir effect. In a spacetime with such a nontrivial topology, the vacuum energy density is shifted from its value in a simply connected spacetime. In this paper, the vacuum expectation value of the stress-energy tensor for a massless scalar field is calculated in all 17 multiply connected, flat, and homogeneous spacetimes with different global topologies. It is found that the vacuum energy density is lowered relative to the Minkowski vacuum level in all spacetimes and that the stress-energy tensor becomes position-dependent in spacetimes that involve reflections and rotations.

Reconstructing the Universe

We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.

HAWKING RADIATION VIA TUNNELING FROM KERR BLACK HOLES

In this paper, Parikh's recent work, which treats Hawking radiation as a tunneling process in a spherically symmetric spacetime, is extended to the stationary axisymmetric Kerr spacetime. A generalized Painleve coordinate system (Painleve–Kerr coordinates) is introduced. The new coordinates are well-behaved at the event horizon. Constant-time slices are just flat Euclidean space in radial and satisfy Landau's condition of clock synchronization. The tunneling process across the Kerr black hole event horizon is investigated and the correctd radiation spectrum is obtained. The result is consistent with an underlying unitary theory.

New coordinates for Kerr–Newman black hole radiation

A simple coordinate system of Kerr–Newman space–time is introduced. The new coordinates have several attractive properties: the time direction is a Killing vector, the metric is smooth at the horizons, constant-time slices are just flat Euclidean space in radial and satisfy Landau's condition of clock synchronization. The usefulness of the coordinates is demonstrated by calculating the rate at which uncharged particles tunnel across the event horizon. When self-gravitation is taken into account, the radiation deviates from precisely thermal spectrum. The corrected spectrum is consistent with an underlying unitary theory.

On the roots of the Poincaré structure of asymptotically flat spacetimes

The analysis of canonical vacuum general relativity by Beig and Ó Murchadha (1987 Ann. Phys., NY 174 463–498) is extended in numerous ways. The weakest possible power-type fall-off conditions for the energy–momentum tensor of the matter fields, the metric, the extrinsic curvature, the lapse and the shift are determined which, together with the parity conditions, are preserved by the energy–momentum conservation law Tab;b = 0 and the evolution equations for the geometry. The algebra of the asymptotic Killing vectors, defined with respect to a foliation of the spacetime, is shown to be the Lorentz Lie algebra for slow fall-off of the metric, but it is the Poincaré algebra for 1/r or faster fall-off.It is shown that the applicability of the symplectic formalism already requires the 1/r (or faster) fall-off of the metric. The connection between the Poisson algebra of the Beig–Ó Murchadha Hamiltonians (and, in particular, the constraint algebra) and the asymptotic Killing vectors is clarified. Their Hamiltonian H[Ka] is shown to be constant in time modulo constraints for those asymptotic Killing vectors Ka that are defined with respect to the foliation by the constant time slices.The energy–momentum and angular momentum are defined by the boundary term Q[Ka] in H[Ka] even in the presence of matter. Although the energy–momentum is well defined even for slightly faster than r−1/2 fall-off, we show that the angular momentum and centre-of-mass are finite only if the metric falls off as 1/r or faster. Q[Ka] is constant in time for those Ka that are asymptotic Killing vectors with respect to the foliation by the constant time slices. If the foliation corresponds to proper time evolution (i.e., its lapse tends to 1 at infinity), then Q[Ka] reproduces the ADM energy, the spatial momentum and spatial angular momentum, but the centre-of-mass deviates from that of Beig and Ó Murchadha by the spatial momentum times the coordinate time. The spatial angular momentum and the new centre-of-mass form an anti-symmetric Lorentz tensor, which transforms in the expected way under Poincaré transformations.

New coordinates for de Sitter space and de Sitter radiation

We introduce a simple coordinate system covering half of de Sitter space. The new coordinates have several attractive properties: the time direction is a Killing vector, the metric is smooth at the horizon, and constant-time slices are just flat Euclidean space. We demonstrate the usefulness of the coordinates by calculating the rate at which particles tunnel across the horizon. When self-gravitation is taken into account, the resulting tunneling rate is only approximately thermal. The effective temperature decreases through the emission of radiation.

Structure formation in the Lemaître-Tolman model

Structure formation within the Lema\^{\i}tre-Tolman model is investigated in a general manner. We seek models such that the initial density perturbation within a homogeneous background has a smaller mass than the structure into which it will develop, and the perturbation then accretes more mass during evolution. This is a generalization of the approach taken by Bonnor in 1956. It is proved that any two spherically symmetric density profiles specified on any two constant time slices can be joined by a Lema\^{\i}tre-Tolman evolution, and exact implicit formulas for the arbitrary functions that determine the resulting LT model are obtained. Examples of the process are investigated numerically.

Nonexistence of conformally flat slices of the Kerr spacetime

Initial data for black hole collisions are commonly generated using the Bowen-York approach based on conformally flat 3-geometries. The standard (constant Boyer-Lindquist time) spatial slices of the Kerr spacetime are not conformally flat, so that use of the Bowen-York approach is limited in dealing with rotating holes. We investigate here whether there exist foliations of the Kerr spacetime that are conformally flat. We limit our considerations to foliations that are axisymmetric and that smoothly reduce in the Schwarzschild limit to slices of constant Schwarzschild time. With these restrictions, we show that no conformally flat slices can exist.

Initial data for superposed rotating black holes

The standard approach to initial data for both analytic and numerical computations of black hole collisions has been to use a conformally flat initial geometry. Among other advantages, this choice allows the simple superposition of holes with arbitrary mass, location and spin. The conformally flat restriction, however, is inappropriate to the study of Kerr holes, for which the standard constant-time slice is not conformally flat. Here we point out that for axisymmetric arrangements of rotating holes, a nonconformally flat form of the 3-geometry can be chosen which allows fairly simple superposition of Kerr holes with arbitrary mass and spin. We present initial data sets representing locally Kerr holes at large separation, and representing rotating holes close enough so that outside a common horizon the spacetime geometry is a perturbation of a single Kerr hole.

Renormalization group approach in Newtonian cosmology

We apply the renormalization group (RG) method to examine the observable scaling properties in Newtonian cosmology. The original scaling properties of the equations of motion in our model are modified for averaged observables on constant time slices. In the RG flow diagram, we find three robust fixed points: Einstein-de Sitter, Milne and Quiescent fixed points. Their stability (or instability) property does not change under the effect of fluctuations. Inspired by the inflationary scenario in the early Universe, we set the Einstein-de Sitter fixed point with small fluctuations, as the boundary condition at the horizon scale. Solving the RG equations under this boundary condition toward the smaller scales, we find generic behavior of observables such that the density parameter $\Omega$ decreases, while the Hubble parameter $H$ increases for smaller averaging volume. The quantitative scaling properties are analyzed by calculating the characteristic exponents around each fixed point. Finally we argue the possible fractal structure of the Universe beyond the horizon scale.

Evolving the Bowen-York initial data for spinning black holes

The Bowen-York initial value data typically used in numerical relativity to represent spinning black hole are not those of a constant-time slice of the Kerr spacetime. If Bowen-York initial data are used for each black hole in a collision, the emitted radiation will be partially due to the ``relaxation'' of the individual holes to Kerr form. We compute this radiation by treating the geometry for a single hole as a perturbation of a Schwarzschild black hole, and by using second order perturbation theory. We discuss the extent to which Bowen-York data can be expected accurately to represent Kerr holes.

Ω 0 < 1 from inflation

An inflationary scenario that leads to $\Omega _0<1$ today is presented. An epoch of `old' inflation during which the smoothness and horizon problems are solved is followed by a shortened epoch of `new' inflation. Old inflation exits through the nucleation of a single bubble, leading to negative spatial curvature on slices of constant cosmic time. We calculate the spectrum of density perturbations in such a scenario.

An adaptive optimal-kernel time-frequency representation

Time-frequency representations with fixed windows or kernels figure prominently in many applications, but perform well only for limited classes of signals. Representations with signal-dependent kernels can overcome this limitation. However, while they often perform well, most existing schemes are block-oriented techniques unsuitable for on-line implementation or for tracking signal components with characteristics that change with time. The time-frequency representation developed in the present paper, based on a signal-dependent radially Gaussian kernel that adapts over time, surmounts these difficulties. The method employs a short-time ambiguity function both for kernel optimization and as an intermediate step in computing constant-time slices of the representation. Careful algorithm design provides reasonably efficient computation and allows on-line implementation. Certain enhancements, such as cone-kernel constraints and approximate retention of marginals, are easily incorporated with little additional computation. While somewhat more expensive than fixed kernel representations, this new technique often provides much better performance. Several examples illustrate its behavior on synthetic and real-world signals.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Quasilocal energy for a Kerr black hole.

The quasilocal energy associated with a constant stationary time slice of the Kerr spacetime is presented. The calculations are based on a recent proposal in which quasilocal energy is derived from the Hamiltonian of spatially bounded gravitational systems. Three different classes of boundary surfaces for the Kerr slice are considered (constant radius surfaces, round spheres, and the ergosurface). Their embeddings in both the Kerr slice and flat three-dimensional space (required as a normalization of the energy) are analyzed. The energy contained within each surface is explicitly calculated in the slow rotation regime and its properties are discussed in detail. The energy is a positive, monotonically decreasing function of the boundary surface radius. It approaches the Arnowitt-Deser-Misner (ADM) mass at spatial infinity and reduces to (twice) the irreducible mass at the horizon of the Kerr black hole. The expressions possess the correct static limit and include negative contributions due to gravitational binding. The energy at the ergosurface is compared with the energies at other surfaces. Finally, the difficulties involved in an estimation of the energy in the fast rotation regime are discussed.

Rolling radii and a time-dependent dilation

Three families of solutions of the background field equations for the bosonic string are presented. These solutions describe a D-dimensional space-time for which the constant time slices are tori with D −1 “rolling radii” - that is, radii which are allowed to vary with time, along with a time-dependent dilation field. Each of the three families (which correspond to D = 26, D > 26, or D < 26) includes solutions where d radii increase as the other D− d−1 radii decrease, where 0⩽ d⩽ D − 1. It is shown that these solutions can be extended to order α' in nonlinear sigma model perturbation theory, and it is conjectured that this continues to any given order in α'.