Space-time Splitting Research Articles

On the canonical form of the electromagnetic field

Using the decomposition of an antisymmetric 2-tensor as a sum of two orthogonal bivectors, the various canonical forms of the electromagnetic tensor field are analyzed, recovering known results. However, introducing 1+3 spacetime splitting techniques, the canonical forms are associated to special frames and observers and this helps to clarify the role played by “measurable” quantities (electric and magnetic fields, Poynting vector) in the classification problem itself. Keywords: Electromagnetic field, Canonical form Mathematics Subject Classification (2000): 83A05, 78A25

Brane Worlds in Collision

We obtain an exact solution of the supergravity equations of motion in which the four-dimensional observed Universe is one of a number of colliding D3 branes in a Calabi-Yau background. The collision results in the ten-dimensional spacetime splitting into disconnected regions, bounded by curvature singularities. However, near the D3 branes the metric remains static during and after the collision. We also obtain a general class of solutions representing p-brane collisions in arbitrary dimensions, including one in which the universe ends with the mutual annihilation of a positive-tension and a negative-tension 3 brane.

Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory

We study nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary. We establish the basic regularity and compactness properties (assuming L p -bounds on the curvature for p>2) as well as the Fredholm theory in a compact model case. The motivation for studying this boundary value problem lies in the construction of an instanton Floer homology for 3-manifolds with boundary. The present paper is part of a program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres.

BANACH SPACE VALUED CAUCHY–RIEMANN EQUATIONS WITH TOTALLY REAL BOUNDARY CONDITIONS

The main purpose of this paper is to give a general regularity result for Cauchy–Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic Lp-regularity results hold true under one crucial assumption: The Banach space is isomorphic to a closed subspace of an Lp-space. (Equivalently, the totally real submanifold is modelled on a closed subspace of an Lp-space.)Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah–Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy–Riemann equation with totally real boundary condition.

The general form of supersymmetric solutions of N = (1, 0) U(1) and SU(2) gauged supergravities in six dimensions

We obtain necessary and sufficient conditions for a supersymmetric field configuration in the N = (1, 0) U(1) or SU(2) gauged supergravities in six dimensions, and impose the field equations on this general ansatz. It is found that any supersymmetric solution is associated with an structure. The structure is characterized by a null Killing vector which induces a natural 2 + 4 split of the six-dimensional spacetime. A suitable combination of the field equations implies that the scalar curvature of the four-dimensional Riemannian part, referred to as the base, obeys a second-order differential equation; surprisingly, for a large class of solutions the equation in the SU(2) theory requires the vanishing of the Weyl anomaly of N = 4 SYM on the base. Bosonic fluxes introduce torsion terms that deform the structure away from a covariantly constant one. The most general structure can be classified into terms of its intrinsic torsion. For a large class of solutions the gauge field strengths admit a simple geometrical interpretation: in the U(1) theory the base is Kähler, and the gauge field strength is the Ricci form; in the SU(2) theory, the gauge field strengths are identified with the curvatures of the left-hand spin bundle of the base. We employ our general ansatz to construct new solutions; we show that the U(1) theory admits a symmetric Cahen–Wallach4 × S2 solution together with a compactifying pp-wave. The SU(2) theory admits a black string, whose near horizon limit is AdS3 × S3, which is supported by a self-dual 3-form flux and a meron on the S3. In the limit of the zero 3-form flux we obtain the Yang–Mills analogue of the Salam–Sezgin solution of the U(1) theory, namely R1,2 × S3. Finally we obtain the additional constraints implied by enhanced supersymmetry, and discuss Penrose limits in the theories.

PETROV TYPES AND SPECIAL REFERENCE FRAMES

The Petrov classification is studied with the language of spacetime splitting with respect to a generic timelike unit vector field. Canonical forms of the Weyl tensor for each Petrov type are introduced in terms of preferred reference frames. As an application, the canonical forms of the Bel–Robinson superenergy density and flux are constructed and applications to Bel's definition of gravitational radiation are considered.

Quasi-local conservation equations in general relativity

A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza–Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner–Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne et al. We discuss the remaining quasi-local conservation equations briefly.

Causal differencing of flux-conservative equations applied to black hole spacetimes

We give a general scheme for finite-differencing partial differential equations in flux-conservative form to second order, with a stencil that can be arbitrarily tilted with respect to the numerical grid, parametrized by a `tilt' vector field . This can be used to centre the numerical stencil on the physical lightcone, by setting , where is the usual shift vector in the 3 + 1 split of spacetime, but other choices of the tilt may also be useful. We apply this `causal differencing' algorithm to the Bona-Massó equations, a hyperbolic and flux-conservative form of the Einstein equations, and demonstrate long-term stable causally correct evolutions of single black hole systems in spherical symmetry.

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.

The Intrinsic Derivative and Centrifugal Forces in General Relativity: I.

Everyday experience with centrifugal forces has always guided thinking on the close relationship between gravitational forces and accelerated systems of reference. Once spatial gravitational forces and accelerations are introduced into general relativity through a splitting of spacetime into space-plus-time associated with a family of test observers, one may further split the local rest space of those observers with respect to the direction of relative motion of a test particle world line in order to define longitudinal and transverse accelerations as well. The intrinsic covariant derivative (induced connection) along such a world line is the appropriate mathematical tool to analyze this problem, and by modifying this operator to correspond to the observer measurements, one understands more clearly the work of Abramowicz et al. who derine an "optical centrifugal force" in static axisymmetric spacetimes and attempt to generalize it and other inertial forces to arbitrary spacetimes. In a companion article the application of this framework to some familiar stationary axisymmetric spacetimes helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of de Felice and others on circular orbits in black hole spacetimes into a more general context.

Averaging of a locally inhomogeneous realistic universe.

We present an averaging scheme in general relativity which allows us to study the effect of local inhomogeneity on the global behavior of the universe. The scheme uses 3+1 splitting of spacetime and introduces Isaacson averaging on the spatial hypersurface to get the averaged geometry. As a result of the averaging, the Friedmann-Robertson-Walker (FWR) geometry is derived in the first-order approximation for a wide class of inhomogeneous nonlinear matter distribution. The deviation from the FRW expansion is derived to the next order in terms of the anisotropic distribution of an effective stress-energy tensor. Using a simple model of inhomogeneity we show that the average effect of the inhomogeneity behaves like a negative spatial curvature term and thus has a tendency to extend the age of the universe. \textcopyright{} 1996 The American Physical Society.

Space-Time Spectral Element Methods for One-Dimensional Nonlinear Advection-Diffusion Problems

The following space-time Galerkin spectral element methods are developed and applied to solve the Burgers equation with small viscosity: (a) coupled methods, consisting of an explicit method for hyperbolic dominated equations and an implicit method for parabolic dominated equations; (b) two splitting methods which solve the hyperbolic substep explicitly and the parabolic one implicitly (one uses spectral elements in the explicit part and the other uses the Adams-Bashforth multistep method). A subcycling technique, in which several convective steps are taken for each implicit viscous step was also investigated for the two splitting methods. A stability analysis of the four methods is performed and subsequent results are debated. A convergence study and a comparison of computer execution time for the four methods is made and the results are discussed. Comparative study leads to the conclusion that the space-time spectral element splitting method with subcycling is superior to the other methods presented in terms of robustness and computer execution time. The number of subcycles should be kept low (2-3) in order to avoid significant loss of accuracy. The coupled explicit method is also applied to the solution of the one-dimensional coupled continuity, momentum, and energy equations for non-isothermal flow of an ideal gas with temperature dependent properties in a cylindrical duct of variable radius.

The gravitomagnetic dynamo effect in accretion disks of rotating black holes

view Abstract Citations (23) References (20) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Gravitomagnetic Dynamo Effect in Accretion Disks of Rotating Black Holes Khanna, Ramon ; Camenzind, Max Abstract We present a previously unknown dynamo effect that arises from the coupling of the gravitomagnetic field of a Kerr black hole with electromagnetic fields. The axisymmetric dynamo equations are derived in the 3 + 1 split of Kerr spacetime. They are formally identical to their equivalents in flat space, augmented by a general relativistic source term for the poloidal magnetic field provided by the gravitomagnetic field. There is no need for any particular small-scale plasma motions. The effect does, however, require finite conductivity and is enhanced by anomalous or turbulent magnetic diffusivity. The results of our time-dependent numerical simulations of the mean field dyanmo in a turbulent accretion disk prove that the gravitomagnetic dynamo has growing modes in the vicinity of a rotating black hole. Publication: The Astrophysical Journal Pub Date: November 1994 DOI: 10.1086/187611 Bibcode: 1994ApJ...435L.129K Keywords: Accretion Disks; Black Holes (Astronomy); Dynamo Theory; Gravitational Effects; Magnetohydrodynamics; Stellar Magnetic Fields; Stellar Rotation; Computerized Simulation; Magnetic Effects; Numerical Analysis; Plasma Oscillations; Poloidal Flux; Relativity; Astrophysics; ACCRETION; ACCRETION DISKS; BLACK HOLE PHYSICS; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS: MHD; RELATIVITY full text sources ADS |

Generalised splitting of spacetime

Normally, when a spacetime splitting is considered the ADM 3+1 split is brought to mind. In this paper, the idea of spacetime splitting is extended to include anm + n splitting of spacetime. The global spacetime has dimension (m + n) and the foliating spaces have dimensionm. There aren independent normals to each of these foliating spaces, thus givingn different extrinsic curvatures. The generalised Gauss-Weingarten and the generalised Gauss-Codazzi equations associated with this splitting are derived. These generalised equations reduce to the familar ADM equations when a 3+1 split is considered. The generalised equations are found to have a particularly elegant form when an orthogonal splitting of spacetime is examined.

Nonlinear wave solutions to the planar vacuum Einstein equations.

We use the 3+1 split of spacetime and the York splitting into free and constrained variables to set up initial data and solve the vacuum Einstein equations in plane symmetry. We present numerical solutions for free data in the form of traveling waves in both the linear and nonlinear regimes. No evidence of nonlinear wave propagation is found and we demonstrate that for our class of metrics the nonlinearity lies in the Coulomb'' or nonradiative components of the Riemann curvature tensor.

On the variability of physical constants and conformal transformations

The problem of fundamental physical constants variability is discussed on the basis of extended conformal symmetry principles including the concept of local mass variability. It is shown, in particular, that the canonical determination of conformal weights can lead, in general, to non-invariance of the known elementary charge-to-mass ratio. The introduction of active time-like and space-like conformal transformations leads also to possibility of simultaneous variations of dimensional constants without any observable changes in the evolution of the Univrse. The generally covariant method of space-time splitting for conformal renormalization of the metric is considered in terms of Lie derivatives.

Anisotropic cosmological models in ten-dimensional supergravity

A new class of ten-dimensional supergravity cosmologies is presented. The solutions given are of the anisotropic Bianchi-Kantowski-Sachs types I–IX with space-time splitting M 10 = R × M 3 × T 6 .

Electrovac compactification of gauged d = 7 supergravity

Solutions of the classical field equations of gauged N = 2, d = 7 supergravity are found, even if the scalar field potential has no critical points. This leads to the splitting of the seven-dimensional spacetime to a product (AdS) 2 × M 2 × M 3, where M 2 and M 3 may be compoct or noncompact. For a wide range of parameter values, compactification on S 3 from d = 7 to d = 4 occurs. For none of these solutions supersymmetry is preserved.

Existence of CMC Cauchy surfaces and spacetime splitting

In this paper, we review results on the existence (and nonexistence) of constant mean curvature spacelike hypersurfaces in the cosmological setting, and discuss the connection to the spacetime splittng problem. It is a pleasure to dedicate this paper to Robert Bartnik, who has made fundamental contributions to this area.