Hypersurface In Spacetime Research Articles

Null hypersurfaces in de Sitter and anti-de Sitter cosmologies

The study of gravitational waves in the presence of a cosmological constant has led to interesting forms of the de Sitter and anti-de Sitter line elements based on families of null hypersurfaces. The forms are interesting because they focus attention on the geometry of null hypersurfaces in spacetimes of constant curvature. Two examples are worked out in some detail. The first originated in the study of collisions of impulsive gravitational waves in which the post-collision spacetime is a solution of Einstein’s field equations with a cosmological constant, and the second originated in the generalization of plane fronted gravitational waves with parallel rays to include a cosmological constant.

Maximal Hypersurfaces in Spacetimes with a Nonvanishing Spacelike Killing Field

We consider four-dimensional vacuum spacetimes which admit a nonvanishing spacelike Killing field. The quotient with respect to the Killing action is a three-dimensional quotient spacetime (M, g). We establish several results regarding maximal hypersurfaces (spacelike hypersurfaces of zero mean curvature) in such quotient spacetimes. First, we show that a complete noncompact maximal hypersurface must either be a cylinder $$S^1 \times {\mathbb {R}}$$ with flat metric or else conformal to the Euclidean plane $${\mathbb {R}}^2$$ . Second, we establish positivity of mass for certain maximal hypersurfaces, referring to a analogue of ADM mass adapted for the quotient setting. Finally, while lapse functions corresponding to the maximal hypersurface gauge are necessarily bounded in the four-dimensional asymptotically Euclidean setting, we show that nontrivial quotient spacetimes admit the maximal hypersurface gauge only with unbounded lapse.

Stability of maximal hypersurfaces in spacetimes: new general conditions and application to relevant spacetimes

We obtain new simple sufficient conditions to ensure the stability and strong stability of maximal hypersurfaces (without boundary) immersed in an arbitrary spacetime. Several applications to maximal hypersurfaces in a spatially open or closed spacetime endowed with an infinitesimal causal symmetry are also given.

Contingency and the order of nature

Many profess faith in the universal rule of deterministic law. I urge remaining agnostic, putting into nature only what we need to account for what we know to be the case: order where, and to the extent that, we see it. Powers and mechanisms can do that job. Embracing contingency and deriving order from powers and mechanisms reduces three kinds of problems: ontological, theological, and epistemological. Ontologically, there is no puzzle about why models from various branches of natural and social science, daily life, and engineering serve us in good stead if all that's happening is physics laws playing themselves out. Also, when universal laws are replaced with a power/mechanism ontology, nothing is set irredeemably by the Big Bang or at some hyper-surface in space-time. What happens can depend on how we arrange things to exploit the powers of their parts. That may be put to significant theological advantage. The epistemological problem comes from philosopher of physics, Erhard Scheibe. Given what we take physics to teach about the universality of interaction, there is just one very large object – the entire universe – to be governed by laws of nature. How then do we ever learn those laws?

GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold [Formula: see text] of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

Conformal smectics and their many metrics

We establish that equally spaced smectic configurations enjoy an infinite-dimensional conformal symmetry and show that there is a natural map between them and null hypersurfaces in maximally symmetric spacetimes. By choosing the appropriate conformal factor it is possible to restore additional symmetries of focal structures only found before for smectics on flat substrates.

Black-hole horizons as probes of black-hole dynamics. II. Geometrical insights

In a companion paper [1], we have presented a cross-correlation approach to near-horizon physics in which bulk dynamics is probed through the correlation of quantities defined at inner and outer spacetime hypersurfaces acting as test screens. More specifically, dynamical horizons provide appropriate inner screens in a 3+1 setting and, in this context, we have shown that an effective-curvature vector measured at the common horizon produced in a head-on collision merger can be correlated with the flux of linear Bondi-momentum at null infinity. In this paper we provide a more sound geometric basis to this picture. First, we show that a rigidity property of dynamical horizons, namely foliation uniqueness, leads to a preferred class of null tetrads and Weyl scalars on these hypersurfaces. Second, we identify a heuristic horizon news-like function, depending only on the geometry of spatial sections of the horizon. Fluxes constructed from this function offer refined geometric quantities to be correlated with Bondi fluxes at infinity, as well as a contact with the discussion of quasi-local 4-momentum on dynamical horizons. Third, we highlight the importance of tracking the internal horizon dual to the apparent horizon in spatial 3-slices when integrating fluxes along the horizon. Finally, we discuss the link between the dissipation of the non-stationary part of the horizon's geometry with the viscous-fluid analogy for black holes, introducing a geometric prescription for a "slowness parameter" in black-hole recoil dynamics.

Electroweak Structure of Composite Hadronic Systems

The electroweak structure of relativistic composite systems defined on the light-front space-time hypersurface starting from field theory and from the Bethe–Salpeter equation is reviewed, with emphasis on the content of the current operator, which acquires two-body terms. The role of different frames in computing elastic form factors is exemplified by discussing results for the pion and nucleon. The projection on to the light-front of the Bethe–Salpeter equation and the associated current operator is sketched. It gives a systematic expansion towards covariance while keeping current conservation in the description of relativistic few-body systems.

Torsion in extra-dimensions

We consider a variant of the 5 dimensional Kaluza-Klein theory within the framework of Einstein-Cartan formalism. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. and the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard K-K theory to the Ricci tensor and the affine connections are completely nullified by the contributions from the torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D space-time from a hypersurface of 5D space-time with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein-Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

KALUZA–KLEIN THEORY WITH TORSION CONFINED TO THE EXTRA DIMENSION

Here we consider a variant of the five-dimensional Kaluza–Klein (KK) theory within the framework of Einstein–Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard KK theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D spacetime from a hypersurface of 5D spacetime with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein–Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

Matching stages of heavy-ion collision models

Heavy ion reactions and other collective dynamical processes are frequently described by different theoretical approaches for the different stages of the process, like initial equilibration stage, intermediate locally equilibrated fluid dynamical stage and final freeze-out stage. For the last stage the best known is the Cooper-Frye description used to generate the phase space distribution of emitted, non-interacting, particles from a fluid dynamical expansion/explosion, assuming a final ideal gas distribution, or (less frequently) an out of equilibrium distribution. In this work we do not want to replace the Cooper-Frye description, rather clarify the ways how to use it and how to choose the parameters of the distribution, eventually how to choose the form of the phase space distribution used in the Cooper-Frye formula. Moreover, the Cooper-Frye formula is used in connection with the freeze-out problem, while the discussion of transition between different stages of the collision is applicable to other transitions also. More recently hadronization and molecular dynamics models are matched to the end of a fluid dynamical stage to describe hadronization and freeze-out. The stages of the model description can be matched to each other on spacetime hypersurfaces (just like through the frequently used freeze-out hypersurface). This work presents a generalized description of how to match the stages of the description of a reaction to each other, extending the methodology used at freeze-out, in simple covariant form which is easily applicable in its simplest version for most applications.

Time-Symmetric Boundary Conditions and Quantum Foundations

Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed hypersurface in spacetime). Such a time-symmetric perspective of BCs, as applied to classical fields, leads to interesting parallels with quantum theory. This paper will map out some of the consequences of this counter-intuitive premise, as applied to covariant classical fields. The most notable result is the contextuality of fields constrained in this manner, naturally bypassing the usual arguments against so-called “realistic” interpretations of quantum phenomena.

Quaternion-Loop Quantum Gravity

It is shown that the Riemannian curvature of the 3-dimensional hypersurfaces in space-time, described by the Wilson loop integral, can be represented by a quaternion quantum operator induced by the SU(2) gauge potential, thus providing a justification for quaternion quantum gravity at the Tev energy scale.

An angular momentum bound at null infinity

We prove an inequality relating the trace of the extrinsic curvature, the total angular momentum, the centre of mass, and the Trautman-Bondi mass for a class of gravitational initial data sets with constant mean curvature extending to null infinity. As an application we obtain non-existence results for the asymptotic Dirichlet problem for CMC hypersurfaces in stationary space-times.

Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized M ¯ n + 1 = I × ϕ F n Robertson–Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given x : M n → M ¯ n + 1 a closed, strongly stable spacelike hypersurface of M ¯ n + 1 with constant mean curvature H, if the warping function ϕ satisfying ϕ ″ ⩾ max { H ϕ ′ , 0 } along M, then M n is either maximal or a spacelike slice M t 0 = { t 0 } × F , for some t 0 ∈ I .

Degree of mapping for general relativistic kinks

The Finkelstein-Misner metrical kinks of general relativity are homotopically nontrivial light cone configurations that can occur on space-time hypersurfaces. The number of kinks corresponds to the winding number of a timelike vector field that that is determined from the metric. This paper uses the usual Euclidean integral formula for degree of mapping as a starting point and so produces a covariant formula that call be applied to counting general relativistic kinks in any dimension. The kink number is calculated for some simple-to-visualize examples in 2 + 1 dimensions. These include hypersurfaces of differing topologies and so have relevance to mechanisms of topology change in semi-classical theories of quantum gravity.

Cancelling Jüttner distributions for space-like freeze-out

We study freeze out process of particles across a three dimensional space-time hypersurface with space-like normal. The problem of negative contribution is discussed with respect to conservation laws, and a simple and practical new one-particle distribution for the post FO side is introduced, the Cancelling Juttner distribution.

Exact global solutions of brane universes and big bounce

Exact global solutions of five-dimensional cosmological models compactified on a S1/Z2 orbifold with two 3-branes are presented, and evolution of a simple model is studied. It is found that on all 4D spacetime hypersurfaces, except a singular one, the expanding universe was started not from a big bang but from a big bounce, and before the bounce the universe was in a deflationary contracting phase. It is also found that whether the energy density on the second brane is positive, zero, or negative depends on the size of the fifth dimension.

Action and Energy of the Gravitational Field

We present a detailed examination of the variational principle for metric general relativity as applied to a quasilocal spacetime region M (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity and thereby assumes a foliation of M into spacelike hypersurfaces Σ. We allow for near complete generality in the choice of foliation. Using a field-theoretic generalization of Hamilton–Jacobi theory, we define the quasilocal stress-energy momentum of the gravitational field by varying the action with respect to the metric on the boundary ∂M. The gravitational stress-energy momentum is defined for a two-surface B spanned by a spacelike hypersurface in spacetime. We examine the behavior of the gravitational stress-energy momentum under boosts of the spanning hypersurface. The boost relations are derived from the geometrical and invariance properties of the gravitational action and Hamiltonian. Finally, we present several new examples of quasilocal energy momentum, including a novel discussion of quasilocal energy momentum in the large-sphere limit toward spatial infinity.

The Hamiltonian Mass and Asymptotically Anti-de Sitter Space-times

We give a Hamiltonian definition of mass for asymptotically hyperboloidal Riemannian manifolds, or for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one.