Real Kasner and related complex “windmill” vacuum spacetime metrics

The Kasner family of vacuum solutions of Einstein's field equations admits a simply-transitiveH ₄, a four-parameter local homothetic group of motions which has an AbelianG ₃ subgroup. It is shown that a complex transformation of coordinates and constants exists which maps this family from the normal Kasner form into a form of vacuum metrics whose Weyl tensors are each Petrov type I and which were published in 1932 by Lewis. These metrics also admit a similarH ₄; however for one particular metric (for one parameter value) theH ₄ becomes aG ₄ and the resultant metric is one which was rediscovered by Petrov in 1962. These Lewis metrics are thus shown to be Kasner metrics over complex fields. Here they are calledwindmill metrics because of the rotating relationship between the coordinates and the Killing vector fields admitted. The principal null directions of thereal Kasner and the windmill metrics are discussed; the two families then provide illustrations of two degenerate classes of spacetime metrics whose Weyl tensors are of Petrov type I, as discussed elsewhere by Arianrhod and McIntosh. An extension of the windmill-type generation of metrics to some other families of metrics is also discussed.     

TypeN twisting vacuum gravitational fields

For vacuum, typeN, twisting gravitational fields the Einstein field equations reduce to partial differential equations for two functions, one real, the other complex, which may be regarded as initial data on a localJ ⁺. If it is assumed that in some frame these initial data take on a certain product form, one factor involving only a spatial variable, the other only a retarded time variable, then these equations become relatively tractable and reduce further to two ordinary differential equations. Rejecting all solutions which lead to Minkowski space or to zero twist leaves just two possibilities. One corresponds to the only explicitly known spacetime of the kind, namely that of Hauser. The other leads to new typeN twisting metrics. However, these metrics can be constructed explicitly only once a single nonlinear third-order ordinary differential equation has been solved.     

From 2-dimensional surfaces to cosmological solutions

We construct perfect fluid metrics with two symmetries by means of a recently developed geometrical method [1]. The Einstein equations are reduced to a single equation for a conformal factor. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI₀ and VII₀. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.     

Killing vectors in empty space algebraically special metrics. I

Empty space algebraically special metrics possessing an expanding degenerate principal null vector and a Killing vector are investigated. It is shown that the Killing vector falls into one of two classes. The class containing all asymptotically timelike Killing vectors is investigated in detail and the associated metrics are identified. Several theorems concerning these metrics are given, among which is a proof that if the metric is regular and possesses an asymptotically timelike Killing vector, then it must be typeD. In addition some relations between Killing vectors in general spaces are developed along with a set of tetrad symmetry equations stronger than those of Killing.     

Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations

On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on ℝ ⁿ (n≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity. Received: 8 November 1999 / Accepted: 27 March 2000     

Shear-free spherically symmetric perfect fluid solutions with conformal symmetry

In 1987, Dyer, McVittie and Oattes determined the general relativistic field equations for a shear-free perfect fluid with spherical symmetry and a conformal Killing vector in thet-r plane, which depend on an arbitrary constantm. Two particular solutions of these equations were given recently by Maharaj, Leach and Maartens, as well as a partial solution thought to be valid for almost allm. In this paper, this solution is completed for four values ofm, and it is shown that it cannot be completed for any others by currently available techniques; however, a new solution of a different form, but also depending on a Weierstrass elliptic function, is found for a further value ofm. None of these metrics are conformally flat; one of them has a constant expansion rate.     

Dynamical mechanism for varying light velocity as a solution to cosmological problems

A dynamical model for varying light velocity in cosmology is developed, based on the idea that there are two metrics in spacetime. One metric g_μν describes the standard gravitational vacuum, and the other describes the geometry through which matter fields propagate. Matter propagating causally with respect to can provide acausal contributions to the matter stress-energy tensor in the field equations for g_μν, which, as we explicitly demonstrate with perfect fluid and scalar field matter models, provides a mechanism for the solution of the horizon, flatness and magnetic monopole problems in an FRW universe. The field equations also provide a ‘graceful exit' to the inflationary epoch since below an energy scale (related to the mass of ψ_μ) we recover exactly the standard FRW field equations.     

Complex relativity and real solutions. III. Real type-N solutions from complexN ⊗N ones

Einstein's vacuum field equations are integrated in complex relativity in a major subcase of the class whose Weyl tensor is of the type NN, i.e., when the left and right Weyl spinors and are each of typeN. The subcase is the complex equivalent of the real nontwisting case. Five separate families of solutions are found. Three of these are complexified versions of the two families of plane-fronted waves and the Robinson-Trautman real type-N metrics and two are complex solutions which do not have any real slices of Lorentz signature. Before the equations are integrated, the relevant general theory and equations are developed in a tetrad frame which is well suited to the discussion of these and a wider class of complex solutions and is called aleft quarter flat frame. The relationship between this frame and the coordinates used and some other frames and coordinates, including the complexified version of the frame often used for real type-N metrics, is discussed.     

On the Robinson,Schild, and Strauss solution of the Einstein-Maxwell equations

The Robinson, Schild, and Strauss explicit solution of the Einstein-Maxwell equations is considered. Two subclasses of this solution are determined: (i) the class of metrics for which both eigenvectors of the Maxwell field are eigenvectors of the Weyl tensor, and (ii) the class of metrics of Petrov typeD.     

Localization of the SFT inspired nonlocal linear models and exact solutions

A general class of gravitational models driven by a nonlocal scalar field with a linear or quadratic potential is considered. We study the action with an arbitrary analytic function ℱ(□ _g ), which has both simple and double roots. The way of localization of nonlocal Einstein equations is generalized on models with linear potentials. Exact solutions in the Friedmann-Robertson-Walker and Bianchi I metrics are presented.     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

Orientifolds and slumps in G2 and Spin(7) metrics

We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by , which are complete on a complex line bundle over . The principal orbits are S⁷, described as a triaxially squashed S³ bundle over S⁴. The behaviour in the S³ directions is similar to that in the Atiyah–Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S⁴. We then consider new G₂ metrics which we denote by , which are complete on an bundle over T^1,1, with principal orbits that are S³×S³. We study the metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S² cycles, and both carry magnetic charge with respect to the R–R vector field. We also discuss some four-dimensional hyper-Kähler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(∞) Toda equation, which can provide a way of studying their interior structure.     

Ricci-Flat Metrics,Harmonic Forms and Brane Resolutions

 We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S ⁿ⁺¹ . We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic self-dual (p, q)-forms in the middle dimension p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p, p)-forms are L ² -normalisable, while for (p, q)-forms the degree of divergence grows with . We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-K?hler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions. Received: 22 February 2001 / Accepted: 16 August 2002 Published online: 7 November 2002     

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Acknowledgement I would like to thank D. Pollack, J. Isenberg, and S. Dain for helpful discussions and advice. I would also like to thank an anonymous referee for suggestions that improved the papers style. This research was partially supported by NSF grant DMS-0305048.     

On the asymptotic flatness of theC metrics at spatial infinity

It is shown that a family of exact, radiating solutions of Einstein's field equations (theC metrics) is asymptotically flat at spatial infinity (in the sense of Ashtekar-Hansen). The ADM and Bondi masses are discussed.     

Complex Transformation of the kasner metric

It is known that distinct solutions of the vacuum Einstein equations may be related by complex coordinate transformations. In particular, there is the class of simple transformations where two or more coordinates are merely made imaginary as, e.g.,t iø and ø it. There also exist substantially more complicated forms of this type of transformation. In particular, it is possible to show that two distinct members of the family of Kasner metrics are so related. These are the Bianchi I and the Weyl W-2 metrics. It is convenient to refer to solutions so related ascomplex analogues.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results

We consider two single-species reaction-diffusion models on one-dimensional lattices of lengthL: the coagulation-decoagulation model and the annihilation model. For the coagulation model the system of differential equations describing the time evolution of the empty interval probabilities is derived for periodic as well as for open boundary conditions. This system of differential equations grows quadratically withL in the latter case. The equations are solved analytically and exact expressions for the concentration are derived. We investigate the finite-size behavior of the concentration and calculate the corresponding scaling functions and the leading corrections for both types of boundary conditions. We show that the scaling functions are independent of the initial conditions but do depend on the boundary conditions. A similarity transformation between the two models is derived and used to connect the corresponding scaling functions.     

Generalized Regularly Discontinuous Solutions of the Einstein Equations

The physical consistency of the match of piecewise-C ⁰ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied.     

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces,Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.     

Horizons in weyl metrics exhibiting extra symmetries

Static, axisymmetric, vacuum metrics (commonly called Weyl metrics) are classified according to the homothetic motions they admit, with all having more than two homothetic motions explicitly computed. Within the two new families of spacetimes so determined, none of the metrics is asymptotically flat. Although most of the horizons are unlike that of the Schwarzschild metric, it is shown that they nonetheless all fall within a classification scheme previously developed for two-dimensional static metrics. TheC — metric and the question of directional singularities are also briefly considered.Based on part of the author's doctoral dissertation submitted to Princeton University, 1970. This work has been assisted in part by NSF Grant No. GP7669.