Lorentzian Spacetime Research Articles

A classification of spherically symmetric spacetimes

A complete classification of locally spherically symmetric four-dimensional Lorentzian spacetimes is given in terms of their local conformal symmetries. The general solution is given in terms of canonical metric types and the associated conformal Lie algebras. The analysis is based upon the local conformal decomposition into 2+2 reducible spacetimes and the Petrov type. A variety of physically meaningful example spacetimes are discussed.

Quantum astrometric observables: Time delay in classical and quantum gravity

A class of diffeomorphism invariant, physical observables, so-called astrometric observables, is introduced. A particularly simple example, the time delay, which expresses the difference between two initially synchronized proper time clocks in relative inertial motion, is analyzed in detail. It is found to satisfy some interesting inequalities related to the causal structure of classical Lorentzian spacetimes. Thus it can serve as a probe of causal structure and in particular of violations of causality. A quantum model of this observable as well as the calculation of its variance due to vacuum fluctuations in quantum linearized gravity are sketched. The question of whether the causal inequalities are still satisfied by quantized gravity, which is pertinent to the nature of causality in quantum gravity, is raised, but it is shown that perturbative calculations cannot provide a definite answer. Some potential applications of astrometric observables in quantum gravity are discussed.

Spinors: A Mathematica package for doing spinor calculus in General Relativity

The Spinors software is a Mathematica package which implements 2-component spinor calculus as devised by Penrose for General Relativity in dimension 3+1. The Spinors software is part of the xAct system, which is a collection of Mathematica packages to do tensor analysis by computer. In this paper we give a thorough description of Spinors and present practical examples of use.Program summaryProgram title: SpinorsCatalogue identifier: AEMQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEMQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 117039No. of bytes in distributed program, including test data, etc.: 300404Distribution format: tar.gzProgramming language: Mathematica.Computer: Any computer running Mathematica 7.0 or higher.Operating system: Any operating system compatible with Mathematica 7.0 or higher.RAM: 94Mb in Mathematica 8.0.Classification: 1.5.External routines: Mathematica packages xCore, xPerm and xTensor which are part of the xAct system. These can be obtained at http://www.xact.es.Nature of problem: Manipulation and simplification of spinor expressions in General Relativity.Solution method: Adaptation of the tensor functionality of the xAct system for the specific situation of spinor calculus in four dimensional Lorentzian geometry.Restrictions: The software only works on 4-dimensional Lorentzian space-times with metric of signature (1, −1, −1, −1). There is no direct support for Dirac spinors.Unusual features: Easy rules to transform tensor expressions into spinor ones and back. Seamless integration of abstract index manipulation of spinor expressions with component computations.Running time: Under one second to handle and canonicalize standard spinorial expressions with a few dozen indices. (These expressions arise naturally in the transformation of a spinor expression into a tensor one or vice versa.)

Boundary conditions and unitarity: the Maxwell-Chern-Simons system in AdS3/CFT2

Abstract We consider the holography of the Abelian Maxwell-Chern-Simons (MCS) system in Lorentzian three-dimensional asymptotically-AdS spacetimes, and discuss a broad class of boundary conditions consistent with conservation of the symplectic structure. As is well-known, the MCS theory contains a massive sector dual to a vector operator in the boundary theory, and a topological sector consisting of flat connections dual to U(1) chiral currents; the boundary conditions we examine include double-trace deformations in these two sectors, as well as a class of boundary conditions that mix the vector operators with the chiral currents. We carefully study the symplectic product of bulk modes and show that almost all such boundary conditions induce instabilities and/or ghost excitations, consistent with violations of unitarity bounds in the dual theory.

Pseudo-Riemannian VSI spaces II

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). Using an algebraic classification of pseudo-Riemannian spaces in terms of the boost-weight decomposition, we first show more generally that a space which is not characterized by its invariants must possess the SG1-property. As a corollary, we then show that a VSI space must possess the NG-property (these results are the analogues of the alignment theorem, including corollaries, for Lorentzian spacetimes). As an application we classify all 4D neutral VSI spaces and show that these belong to one of two classes: (1) those that possess a geodesic, expansion-free, shear-free, and twist-free null congruence (Kundt metrics), or (2) those that possess an invariant null plane (Walker metrics). By explicit construction we show that the latter class contains a set of VSI metrics which have not previously been considered in the literature.

Higher rank Killing tensors and Calogero model

(n+2)-dimensional Lorentzian spacetime which admits irreducible Killing tensors of rank up to n is constructed by applying the Eisenhart lift to the Calogero model.

The Nature of Gravitational Field and its Legitimate Energy–Momentum Tensor

In this paper we show how a gravitational field generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel and nonnull nonmetricity spacetimes) or that we even can dispense all those geometrical structures and simply represent the gravitational field as a field, in the Faraday sense, living in Minkowski spacetime . The explicit Lagrangian density for this theory is given and the field equations (which are a set of four Maxwell's-like equations) are shown to be equivalent to Einstein's equations . We also analyze whether the teleparallel formulation can give a mathematical meaning to “Einstein's most happy thought”, i.e. the equivalence principle. Moreover we discuss the Hamiltonian formalism for our theory and its relation to one of the possible concepts for energy of the gravitational field which emerges from it and the concept of ADM energy. One of the main results of the paper is the identification in our theory of a legitimate energy-momentum tensor for the gravitational field expressible through a really nice formula.

On neutrino Oscillations and Predicting the 125 GEV Two Photon Emission State from p-p Collisions Based on the 5D Homogeneous Space-Time Projection Model

Previously the 5D homogeneous space-time metric was introduced with explicitly given projection operators in matrix form which map the 5D space-time manifold into a Lorentzian space-time. Based on this projection model, vector field and spinor solutions are found to be expressible in terms of SU(2)xL and SU(3)xL, where L is the 4D Lorentz space-time group. The spinor solutions give the SU(2) leptonic states arising from space-time projection, whereas the SU(3) representation arises from conformal projection and gives the quarks, and due to gauge requirement leads to mesons and baryons. This process of mapping the 5D space-time manifold into the 4D space-time is at the basis of an analysis of the recent CERN experimental results, the presence of neutrino oscillations and the observed 125 GeV resonance in the p-p collisions, respectively. In fact, it is found that the spinor solution contains an oscillating phase, and the 125 GeV resonance is shown to be predictable, thereby 1) eliminating the need to introduce a Higgs vacuum, and 2) can be shown possibly to be an indicator for a missing heavy baryon octet.

Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua

The elliptic Einstein–DeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. The Ricci–DeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the Einstein–DeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, Kaluza–Klein, locally AdS or have extremal horizons. Using a maximum principle, we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that the Ricci–DeTurck flow preserves these classes of manifolds. As an example, we simulate the Ricci–DeTurck flow for a manifold with asymptotics relevant for AdS5/CFT4. Our maximum principle dictates that there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N2c) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.

ASYMPTOTICALLY VANISHING COSMOLOGICAL CONSTANT IN THE MULTIVERSE

We study the problem of the cosmological constant in the context of the multiverse in Lorentzian space–time, and show that the cosmological constant will vanish in the future. This sort of argument was started by Sidney Coleman in 1989, and he argued that the Euclidean wormholes make the multiverse partition function a superposition of various values of the cosmological constant Λ, which has a sharp peak at Λ = 0. However, the implication of the Euclidean analysis to our Lorentzian space–time is unclear. With this motivation, we analyze the quantum state of the multiverse in Lorentzian space–time by the WKB method, and calculate the density matrix of our universe by tracing out the other universes. Our result predicts vanishing cosmological constant. While Coleman obtained the enhancement at Λ = 0 through the action itself, in our Lorentzian analysis the similar enhancement arises from the front factor of eiS in the universe wave function, which is in the next leading order in the WKB approximation.

Some spacetimes with higher rank Killing–Stäckel tensors

By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four- and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson–Schouten–Nijenhuis algebra. We discuss the extension to the quantum regime.

Optical metrics and projective equivalence

Trajectories of light rays in a static spacetime are described by unparametrized geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the Universe as static may determine the geometry of the light rays differently. More specifically, we classify Lorentzian metrics admitting more than one hyper-surface orthogonal timelike Killing vector and analyze the projective equivalence of the resulting optical metrics. These metrics are shown to be projectively equivalent up to diffeomorphism if the static Killing vectors generate a group $SL(2,\mathbb{R})$, but not projectively equivalent in general. We also consider the cosmological $C$ metrics in Einstein-Maxwell theory and demonstrate that optical metrics corresponding to different values of the cosmological constant are projectively equivalent.

Discriminating the Weyl type in higher dimensions using scalar curvature invariants

We consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics. It is possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions using alignment theory. In the case of the Weyl tensor, and using bivector theory, the associated Weyl curvature operator will have a restricted eigenvector structure for algebraic types II and D, which leads to necessary conditions on the discriminants of the associated characteristic equation which can be manifestly expressed in terms of polynomial scalar curvature invariants. The use of such necessary conditions in terms of scalar curvature invariants will be of great utility in the study and classification of black hole solutions in more than four dimensions.

The random discrete action for two-dimensional spacetime

A one-parameter family of random variables, called the Discrete Action, is defined for a two-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this discrete action is calculated for various regions of 2D Minkowski spacetime, . When a causally convex region of is divided into subregions using null lines the mean of the discrete action is equal to the alternating sum of the numbers of vertices, edges and faces of the null tiling, up to corrections that tend to 0 as the discreteness scale is taken to 0. This result is used to predict that the mean of the discrete action of the flat Lorentzian cylinder is zero up to corrections, which is verified. The ‘topological’ character of the discrete action breaks down for causally convex regions of the flat trousers spacetime that contain the singularity and for non-causally convex rectangles.

Supersonic flows and their interaction with propagating acoustic signals: Acoustic black holes in the laboratory.

Work in particle physics and general relativity (GR) has established that deep connections exist between acoustics and GR. Most remarkable is the fact that acoustic wave propagation in fluids is governed by an effective Lorentzian spacetime geometry: Acoustic waves follow the geodesics of a (curved) acoustic metric. This provides an entirely new way of looking at acoustic propagation, and in principle provides valuable theoretical tools since much of the machinery developed by the GR community over the past several decades can be directly applied to acoustic systems expressed in this framework. Notably, supersonic liquid flows are predicted to have completely analogous properties to spacetime regions near (gravitational) black holes. We present the status of a developing research program at NRL, designed to begin exploring these connections via laboratory experiments, numerical simulations, and theoretical development. [Work supported by the Office of Naval Research.]

Motion4D-library extended

Motion4D-library extended

GeodesicViewer – A tool for exploring geodesics in the theory of relativity

The GeodesicViewer realizes exocentric two- and three-dimensional illustrations of lightlike and timelike geodesics in the general theory of relativity. By means of an intuitive graphical user interface, all parameters of a spacetime as well as the initial conditions of the geodesics can be modified interactively. New version program summary Program title: GeodesicViewer Catalogue identifier: AEFP_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFP_v2_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 76 202 No. of bytes in distributed program, including test data, etc.: 1 722 290 Distribution format: tar.gz Programming language: C++, OpenGL Computer: All platforms with a C++ compiler, Qt, OpenGL Operating system: Linux, Mac OS X, Windows RAM: 24 MBytes Classification: 1.5 External routines: • Motion4D (included in the package) • Gnu Scientific Library (GSL) ( http://www.gnu.org/software/gsl/) • Qt ( http://qt.nokia.com/downloads) • OpenGL ( http://www.opengl.org/) Catalogue identifier of previous version: AEFP_v1_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 413 Does the new version supersede the previous version?: Yes Nature of problem: Illustrate geodesics in four-dimensional Lorentzian spacetimes. Solution method: Integration of ordinary differential equations. 3D-Rendering via OpenGL. Reasons for new version: The main reason for the new version was to visualize the parallel transport of the Sachs legs and to show the influence of curved spacetime on a bundle of light rays as is realized in the new version of the Motion4D library ( http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_0.html). Summary of revisions: • By choosing the new geodesic type “lightlike_sachs”, the parallel transport of the Sachs basis and the integration of the Jacobi equation can be visualized. • The 2D representation via Qwt was replaced by an OpenGL 2D implementation to speed up the visualization. • Viewing parameters can now be stored in a configuration file (.cfg). • Several new objects can be used in 3D and 2D representation. • Several predefined local tetrads can be choosen. • There are some minor modifications: new mouse control (rotate on sphere); line smoothing; current last point in coordinates is shown; mutual-coordinate representation extended; current cursor position in 2D; colors for 2D view. Running time: Interactive. The examples given take milliseconds.

Open problems and results in the group theoretic approach to quantum gravity via the BMS group and its generalizations

The Bondi-Metzner-Sachs group B is the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of General Relativity. However, in quantum gravity, complexified or euclidean versions of General Relativity are frequently considered. McCarthy has shown that there are forty-two generalizations of B for these versions of the theory and a variety of further ones, either real in any signature, or complex. A firm foundation for quantum gravity can be laid by following through the analogue of Wigner's programme for special relativity with B replacing the Poincare group P. Here the main results which have been obtained so far in this research programme are reported and the more important open problems are stated.

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-dimensional Minkowski space for datas that are invariant under the action of a co-compact Fuchsian group.

Time dilation in relativistic two-particle interactions

We study the orbits of two interacting particles described by a fully relativistic classical mechanical Hamiltonian. We use two sets of initial conditions. In the first set (dynamics 1) the system's center of mass is at rest. In the second set (dynamics 2) the center of mass evolves with velocity $V$. If dynamics 1 is observed from a reference frame moving with velocity $\ensuremath{-}V$, the principle of relativity requires that all observables must be identical to those of dynamics 2 seen from the laboratory frame. Our numerical simulations demonstrate that kinematic Lorentz space-time transformations fail to transform particle observables between the two frames. This is explained as a result of the inevitable interaction dependence of the boost generator in the instant form of relativistic dynamics. Despite general inaccuracies of the Lorentz formulas, the orbital periods are correctly predicted by the Einstein's time dilation factor for all interaction strengths.