Spacelike Foliations Research Articles

The Potential of a Thick Present through Undefined Causality and Non-Locality.

This paper elaborates on the interpretation of time and entanglement, offering insights into the possible ontological nature of information in the emergence of spacetime, towards a quantum description of gravity. We first investigate different perspectives on time and identify in the idea of a “thick present” the only element of reality needed to describe evolution, differences, and relations. The thick present is connected to a spacetime information “sampling rate”, and it is intended as a time symmetric potential bounded between a causal past of irreversible events and a still open future. From this potential, spacetime emerges in each instant as a space-like foliation (in a description based on imaginary paths). In the second part, we analyze undefined causal orders to understand how their potential could persist along the thick present instants. Thanks to a C-NOT logic and the concept of an imaginary time, we derive a description of entanglement as the potential of a logically consistent open choice among imaginary paths. We then conceptually map the imaginary paths identified in the entanglement of the undefined orders to Closed Time-like Curves (CTC) in the thick present. Considering a universe described through information, CTC are interpreted as “memory loops”, elementary structures encoding the information potential related to the entanglement in both time and space, manifested as undefined causality and non-locality in the emerging foliation. We conclude by suggesting a possible extension of the introduced concepts in a holographic perspective.

Codimension Two Spacelike Submanifolds Through a Null Hypersurface in a Lorentzian Manifold

Most important examples of null hypersurfaces in a Lorentzian manifold admit an integrable screen distribution, which determines a spacelike foliation of the null hypersurface. In this paper, we obtain conditions for a codimension two spacelike submanifold contained in a null hypersurface to be a leaf of the (integrable) screen distribution. For this, we use the rigging technique to endow the null hypersurface with a Riemannian metric, which allows us to apply the classical Eschenburg maximum principle. We apply the obtained results to classical examples as generalized Robertson–Walker spaces and Kruskal space.

Foliations by Spacelike Hypersurfaces on Lorentz Manifolds

In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We find an equation that relates the foliation with the ambient manifold and applies it to investigate conditions for the leaves being totally umbilical or geodesic. Using the Maximum principle with the mentioned equation we obtain an obstruction for the existence of totally geodesic spacelike foliations in a spacetime with positive Ricci curvature on the direction N.

The Euler current and relativistic parity odd transport

For a spacetime of odd dimensions endowed with a unit vector field, we introduce a new topological current that is identically conserved and whose charge is equal to the Euler character of the even dimensional spacelike foliations. The existence of this current allows us to introduce new Chern-Simons-type terms in the effective field theories describing relativistic quantum Hall states and (2+1) dimensional superfluids. Using effective field theory, we calculate various correlation functions and identify transport coefficients. In the quantum Hall case, this current provides the natural relativistic generalization of the Wen-Zee term, required to characterize the shift and Hall viscosity in quantum Hall systems. For the superfluid case this term is required to have nonzero Hall viscosity and to describe superfluids with non s-wave pairing.

Schmidt boundaries of foliated space-times

For every -dimensional foliated Lorentzian manifold , where is a codimension q space-like foliation, we build its Q-completion and Q-boundary . These are analogs, within transverse Lorentzian geometry of foliated manifolds, to the b-completion and b-boundary (due to (Schmidt 1971 Gen. Relativ. Gravit. 1 269–80)). The bundle morphism (mapping the -component of the Levi–Civita connection 1-form of into the unique torsion-free adapted connection on the bundle of Lorentzian transverse orthonormal frames) is shown to induce a surjective continuous map of the adapted boundary () of onto its Q-boundary. Map is used to characterize as the set of end points , in the topology of , of all Q-incomplete curves . As an application we determine a class of b-boundary points, where , g is Schwartzschildʼs metric, and is the codimension two foliation tangent to the Killing vector fields and .

Spacetimes with singularities

Abstract We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed.

Relative entropy as a measure of inhomogeneity in general relativity

We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.

Binary black hole coalescence in the large-mass-ratio limit: The hyperboloidal layer method and waveforms at null infinity

We compute and analyze the gravitational waveform emitted to future null infinity by a system of two black holes in the large mass ratio limit. We consider the transition from the quasi-adiabatic inspiral to plunge, merger, and ringdown. The relative dynamics is driven by a leading order in the mass ratio, 5PN-resummed, effective-one-body (EOB), analytic radiation reaction. To compute the waveforms we solve the Regge-Wheeler-Zerilli equations in the time-domain on a spacelike foliation which coincides with the standard Schwarzschild foliation in the region including the motion of the small black hole, and is globally hyperboloidal, allowing us to include future null infinity in the computational domain by compactification. This method is called the hyperboloidal layer method, and is discussed here for the first time in a study of the gravitational radiation emitted by black hole binaries. We consider binaries characterized by five mass ratios, $\nu=10^{-2,-3,-4,-5,-6}$, that are primary targets of space-based or third-generation gravitational wave detectors. We show significative phase differences between finite-radius and null-infinity waveforms. We test, in our context, the reliability of the extrapolation procedure routinely applied to numerical relativity waveforms. We present an updated calculation of the gravitational recoil imparted to the merger remnant by the gravitational wave emission. As a self consistency test of the method, we show an excellent fractional agreement (even during the plunge) between the 5PN EOB-resummed mechanical angular momentum loss and the gravitational wave angular momentum flux computed at null infinity. New results concerning the radiation emitted from unstable circular orbits are also presented.

Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field

A new technique to study spacelike hypersurfaces of constant mean curvature in a spacetime which admits a timelike gradient conformal vector field is introduced. As an application, the leaves of the natural spacelike foliation of such spacetimes are characterized in some relevant cases. The global structure of this class of spacetimes is analyzed and the relation with its well-known subfamily of generalized Robertson–Walker spacetimes is exposed in detail. Moreover, some known uniqueness results for compact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes are widely extended. Finally, and as a consequence, several Calabi–Bernstein problems are solved obtaining all the entire solutions on a compact Riemannian manifold to the constant mean curvature spacelike hypersurface equation, under natural geometric assumptions.

N+1formalism in Einstein-Gauss-Bonnet gravity

Towards the investigation of the full dynamics in a higher-dimensional and/or a stringy gravitational model, we present the basic equations of the Einstein-Gauss-Bonnet gravity theory. We show the $(N+1)$-dimensional version of the Arnowitt-Deser-Misner decomposition including Gauss-Bonnet terms, which shall be the standard approach to treat the space-time as a Cauchy problem. Because of the quasilinear property of the Gauss-Bonnet gravity, we find that the evolution equations can be in a treatable form in numerics. We also show the conformally transformed constraint equations for constructing the initial data. We discuss how the constraints can be simplified by tuning the powers of conformal factors. Our equations can be used both for timelike and spacelike foliations.

On the radius of injectivity of null hypersurfaces

We investigate the regularity of past (future) boundaries of points in regular, Einstein vacuum spacetimes. We provide conditions, expressed in terms of a space-like foliation and which imply, in particular, uniform L 2 L^2 bounds for the curvature tensor, sufficient to ensure the local nondegeneracy of these boundaries. More precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries N ± ( p ) \mathcal {N}^{\pm }(p) of the causal past (future) sets J ± ( p ) \mathcal {J}^{\pm }(p) . Such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the Einstein equations. They are particularly important in construction of an effective Kirchoff-Sobolev type parametrix for solutions of wave equations on M \mathbf {M} . Such parametrices are used by the authors in a forthcoming paper to prove a large data break-down criterion for solutions of the Einstein vacuum equations.

FOLIATION OF THE SCHWARZSCHILD AND REISSNER–NORDSTRÖM SPACE–TIMES BY FLAT SPACE-LIKE HYPERSURFACES

It is known that spherically symmetric space–times admit flat space-like foliations. In this paper a simple procedure for the complete foliation of the Schwarzschild space–time by flat space-like hypersurfaces is developed, using the fact that unforced geodesics are orthogonal to such hypersurfaces. The method is then extended to obtain a complete foliation of the Reissner–Nordström space–time by such hypersurfaces. In this case, as there is a barrier beyond which the geodesics do not go, a complete foliation is obtained by analytically continuing the hypersurfaces beyond the barrier. A duality between imaginary time-like geodesics and space-like hypersurfaces is noted, which also provides complete foliation — but the hypersurfaces are not flat. The features of our foliation are discussed.

Local Existence of Dynamical and Trapping Horizons

Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, we prove that under a suitable stability condition S is contained in "horizon" i.e., a smooth 3-surface foliated by marginally outer trapped slices which lie in the leaves of the given foliation. We also show that under rather weak energy conditions this horizon must be either achronal or spacelike everywhere. Furthermore, we discuss the relation between "bounding" and "stability" properties of marginally outer trapped surfaces.

General relativity histories theory

The canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. After presenting the developments in the History Projection Operator histories theory in the last seven years - giving special emphasis on the novel temporal structure of the formalism - we show how this problem can be solved in the histories formulation of general relativity. We implement the 3+1 decomposition using metric-dependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction we have a coexistence of the spacetime diffeomorphisms group Diff(M) and the Dirac algebra of constraints.

Histories approach to general relativity: I. The spacetime character of the canonical description

The problem of time in canonical quantum gravity is related to the fact that the canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. We show how this problem can be solved in the histories formulation of general relativity. We implement the 3 + 1 decomposition using metric-dependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction, we also have the coexistence of the spacetime diffeomorphisms group, Diff(M), and the Dirac algebra of constraints.

Local prescribed mean curvature foliations in cosmological spacetimes

A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance inherent in general foliations of spacetimes. Moreover, in contrast to the situation of the more special constant mean curvature foliations, which play an important role in the global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature.

Note on flat foliations of spherically symmetric spacetimes

It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings.

Spacelike metric foliations

In this paper, we investigate spacelike metric foliations in lightlike complete spacetimes. When such a foliation satisfies the strong energy condition Ric V ( e) ≥ 0 for timelike vectors e, it must be totally geodesic, and the metric is of higher rank, in the sense that each point of the spacetime is contained inside a flat, totally geodesic, timelike rectangle. If in addition Ric V( e) = 0, then the metric is (at least locally) a product metric, with the leaves of the foliation tangent to one of the factors.

Quantum fields at any time

The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schr\"odinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schr\"odinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac constraint quantization of parametrized field theory. In particular, it is shown that the Schr\"odinger picture physical states satisfy a functional Schr\"odinger equation which includes a slice-dependent $c$-number quantum correction, in accord with a proposal of Kucha\ifmmode \check{r}\else \v{r}\fi{}. The spatial diffeomorphism invariance of the Schr\"odinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes.

Cauchy-characteristic evolution of Einstein-Klein-Gordon systems.

A Cauchy-characteristic initial value problem for the Einstein-Klein-Gordon system with spherical symmetry is presented. Initial data are specified on the union of a space-like and null hypersurface. The development of the data is obtained with the combination of a constrained Cauchy evolution in the interior domain and a characteristic evolution in the exterior, asymptotically flat region. The matching interface between the space-like and characteristic foliations is constructed by imposing continuity conditions on metric, extrinsic curvature and scalar field variables, ensuring smoothness across the matching surface. The accuracy of the method is established for all ranges of $M/R$, most notably, with a detailed comparison of invariant observables against reference solutions obtained with a calibrated, global, null algorithm.