Scalar Flat Metrics of Eguchi-Hanson Type

We use a new method to construct a class of asymptotically locally flat, scalar flat metrics. These metrics were constructed via algebraic geometry method by LeBrun before and provide counterexamples to the generalized positive action conjecture of Hawking and Pope.     

Radial conformal killing vectors in spherically symmetric shear-free space-times

The existence of “radial” conformai Killing (RCK) vector fields is discussed for metrics describing spherically symmetric, shear-free, perfect fluids. It is shown that conformally flat metrics admit three RCK fields, while nonconformally flat metrics might admit, at most, one such field. An RCK vector parallel to the 4-velocity of the fluid occurs in a subclass of conformally flat metrics containing the Friedmann-Robertson-Walker space-times as particular cases. A new class of nonconformally flat, self-similarity solutions is found. The necessary conditions for the existence of an RCK field in other nonconformally flat metrics are given in full.     

On spherically symmetric Finsler metrics of scalar curvature

Spherically symmetric Finsler metrics form a rich class of Finsler metrics. In this paper we find equations that characterize spherically symmetric Finsler metrics of scalar flag curvature. By using these equations, we construct infinitely many non-projectively flat spherically symmetric Finsler metrics of scalar curvature.     

Conformally flat pencils of metrics,Frobenius structures and a modified Saito construction

The structure of a Frobenius manifold encodes the geometry associated with a flat pencil of metrics. However, as shown in the authors’ earlier work [L. David, I.A.B. Strachan, Compatible metrics on manifolds and non-local bi-Hamiltoninan structures, Int. Math. Res. Notices 66 (2004) 3533–3557], much of the structure comes from the compatibility property of the pencil rather than from the flatness of the pencil itself. In this paper conformally flat pencils of metrics are studied and examples, based on a modification of the Saito construction, are developed.     

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     

Singular Hyperbolic Monopoles

We study the twistor theory of singular hyperbolic SU(2) monopoles following the approach taken by Kronheimer [9] in the Euclidean case. We use our results to show that the moduli space of charge 1 monopoles possesses a natural 2-sphere of scalar flat Kähler metrics. In the zero mass limit, the metrics reduce to a class of metrics first studied by LeBrun in [10].     

A Kummer Construction for Gravitational Instantons

We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat Kähler metrics asymptotic to the product of a real line with a flat 3-manifold.     

Kähler metrics generated by functions of the time-like distance in the flat Kähler–Lorentz space

We prove that every Kähler metric, whose potential is a function of the time-like distance in the flat Kähler–Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above-mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are Bochner–Kähler (especially of constant holomorphic sectional curvatures), are also described.     

Evaluation of Non-Uniform Image Quality Caused by Anode Heel Effect in Digital Radiography Using Mutual Information

Anode heel effects are known to cause non-uniform image quality, but no method has been proposed to evaluate the non-uniform image quality caused by the heel effect. Therefore, the purpose of this study was to evaluate non-uniform image quality in digital radiographs using a novel circular step-wedge (CSW) phantom and normalized mutual information (nMI). All X-ray images were acquired from a digital radiography system equipped with a CsI flat panel detector. A new acrylic CSW phantom was imaged ten times at various kVp and mAs to evaluate overall and non-uniform image quality with nMI metrics. For comparisons, a conventional contrast-detail resolution phantom was imaged ten times at identical exposure parameters to evaluate overall image quality with visible ratio (VR) metrics, and the phantom was placed in different orientations to assess non-uniform image quality. In addition, heel effect correction (HEC) was executed to elucidate the impact of its effect on image quality. The results showed that both nMI and VR metrics significantly changed with kVp and mAs, and had a significant positive correlation. The positive correlation is suggestive that the nMI metrics have a similar performance to the VR metrics in assessing the overall image quality of digital radiographs. The nMI metrics significantly changed with orientations and also significantly increased after HEC in the anode direction. However, the VR metrics did not change significantly with orientations or with HEC. The results indicate that the nMI metrics were more sensitive than the VR metrics with regards to non-uniform image quality caused by the anode heel effect. In conclusion, the proposed nMI metrics with a CSW phantom outperformed the conventional VR metrics in detecting non-uniform image quality caused by the heel effect, and thus are suitable for quantitatively evaluating non-uniform image quality in digital radiographs with and without HEC.     

Geometric Treatment of the Gravitomagnetic Clock Effect

We have developed a general geometric treatment of the GCE valid for any stationary axisymmetric metric. The method is based on the remark that the world lines of objects rotating along spacely circular trajectories are in any case, for those kind of metrics, helices drawn on the flat bidimensional surface of a cylinder. Applying the obtained formulas to the special cases of the Kerr and weak field metric for a spinning body, known results for time delays and synchrony defects are recovered.     

Self-dual metrics on non-simply connected 4-manifolds

We construct self-dual (SD) but not locally conformally flat (LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic. These metrics have negative scalar curvature. As an application, this is used to address Remark 4.79 of [1].     

Some real and complex solutions of Einstein's equations

The construction of some real solutions of Einstein's vacuum field equations from certain half flat holomorphic metrics is described.     

Flat Information Geometries in Black Hole Thermodynamics

The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein–Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent.Supported by VR.     

The Positive Action conjecture and asymptotically Euclidean metrics in quantum gravity

The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR ⁴ and a large class of more complicated topologies and for self-dual metrics. We show that ifR 0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric.     

Non-gravitating waves

It is pointed out that scalar-tensor theories of gravity admit solutions in which the metric is Minkowskian although the scalar and matter fields do not vanish. Explicit pp-wave solutions of the Brans-Dicke-Maxwell theory are presented. These include solutions with metrics that are flat or Ricci flat even though the Maxwell and scalar fields are non-zero.     

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.     

Integration in the GHP Formalism III: Finding Conformally Flat Radiation Metrics as an Example of an “Optimal Situation”

Held has proposed an integration procedure within the GHP formalism built around four real, functionally independent, zero-weighted scalars. He suggests that such a procedure would be particularly simple for the optimal situation, when the formalism directly supplies the full quota of four scalars of this type; a spacetime without any Killing vectors would be such a situation. Wils has recently obtained a conformally flat, pure radiation metric, which has been shown by Koutras to admit no Killing vectors, in general. In order to present a simple illustration of the ghp integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics, which are not plane waves. Our result shows that the conformally flat, pure radiation metrics are a larger class than Wils has obtained.     

Eisenhart lift in pseudo–Euclidean space and higher rank killing tensors

It is emphasized that the Eisenhart lift applied to integrable systems in pseudo–Euclidean space may result in Ricci–flat metrics of ultrahyperbolic signature which admit higher rank Killing tensors.     

Some type III solutions of the Einstein-Maxwell equations

A Petrov type III metric with nontwisting, degenerate Debever-Penrose direction is studied. This metric is, in general, a solution of the Einstein-Maxwell equations. Two particular cases are investigated in some detail. It is shown that the metric contains typeN, conformally flat and flat metrics as special subcases. Among these subcases, we find the metric of plane gravitational waves and the Bertotti-Robinson solution.     

Spaces of functions with asymptotic conditions and existence of non compact maximal submanifolds of a lorentzian manifold

We study some properties of weighted Sobolev spaces with asymptotic conditions appropriate to physical conditions on non compact space times. An existence and uniqueness theorem of non compact maximal space-like hypersurfaces for asymptotically flat metrics in a neighbourhood of the Minkowski's metric is proved.