Injectivity Radius of Lorentzian Manifolds

Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) –where p is a point of the manifold and T is a future-oriented time-like unit vector prescribed at p only. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the reference Riemannian metric, and then express them in terms of the Lorentzian metric. In the context of general relativity, our estimate on the injectivity radius of an observer should be useful to investigate the regularity of spacetimes satisfying Einstein field equations.     

PP-waves with torsion: a metric-affine model for the massless neutrino

In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. We then introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely generalised pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a conformally invariant metric-affine model for a massless elementary particle. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely Einstein–Weyl theory and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein–Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT

Finite renormalization freedom in locally covariant quantum field theories on curved spacetime is known to be tightly constrained, under certain standard hypotheses, to the same terms as in flat spacetime up to finitely many curvature dependent terms. These hypotheses include, in particular, locality, covariance, scaling, microlocal regularity and continuous and analytic dependence on the metric and coupling parameters. The analytic dependence hypothesis is somewhat unnatural, because it requires that locally covariant observables (which are simultaneously defined on all spacetimes) depend continuously on an arbitrary metric, with the dependence strengthened to analytic on analytic metrics. Moreover the fact that analytic metrics are globally rigid makes the implementation of this requirement at the level of local \({*}\)-algebras of observables rather technically cumbersome. We show that the conditions of locality, covariance, scaling and a naturally strengthened microlocal spectral condition, are actually sufficient to constrain the allowed finite renormalizations equally strongly, thus eliminating both the continuity and the somewhat unnatural analyticity hypotheses. The key step in the proof uses the Peetre–Slovák theorem on the characterization of (in general non-linear) differential operators by their locality and regularity properties.     

Causally continuous spacetimes

Causally continuous general relativistic spacetimes are defined and analyzed. In a causally continuous spacetime, the past and future of a local observer behave continuously under small perturbations of the metric or small changes in his location. Causally simple spacetimes are causally continuous; causally continuous spacetimes are causally stable. Possible reasons for taking causal continuity as a basic postulate in macrophysics are briefly discussed.     

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.     

A classification of Einstein lightlike hypersurfaces of a Lorentzian space form

We study Einstein lightlike hypersurfaces of a semi-Riemannian manifold of constant curvature c$c$, whose shape operator is conformal to the shape operator of its screen distribution. Our main result is a classification theorem for Einstein lightlike hypersurfaces of Lorentzian space forms.     

Global hyperbolicity and completeness

We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric. We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature K are integrable. This last condition is required only for the tracefree part of K if the universe is expanding.     

The Ricci tensor of SU(3)-manifolds

Following the approach of Bryant [R. Bryant, Some remarks on G₂-structures. e-print: math.DG/0305124] we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.     

Radial asymptotics of Lemaître–Tolman–Bondi dust models

We examine the radial asymptotic behavior of spherically symmetric Lemaître–Tolman–Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length ?, orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasi-local scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular “open” LTB models whose space slices allow for a diverging ?, we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as ? → ∞. The “asymptotic state” is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne’s space), sections of the Schwarzschild–Kruskal manifold or self-similar dust solutions.     

Concircular Curvature Tensor and Fluid Spacetimes

In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.     

Rough Solutions of Einstein Vacuum Equations in CMCSH Gauge

In this paper, we consider very rough solutions to the Cauchy problem for the Einstein vacuum equations in CMC spatial harmonic gauge, and obtain the local well-posedness result in H ^s , s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric g, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation ${\square_{\bf g} \phi=0}$ directly.     

On the Backward Stability of the Schwarzschild Black Hole Singularity

We study the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface \({\Sigma_0}\) in the interior of the black hole region, tangent to the singular hypersurface \({\{r = 0\}}\) at a single sphere, we study the problem of perturbing the Schwarzschild data on \({\Sigma_0}\) and solving the Einstein vacuum equations backwards in time. We obtain a local backwards well-posedness result for small perturbations lying in certain weighted Sobolev spaces. No symmetry assumptions are imposed. The perturbed spacetimes all have a singularity at a “collapsed” sphere on \({\Sigma_0}\), where the leading asymptotics of the curvature and the metric match those of their Schwarzschild counterparts to a suitably high order. As in the Schwarzschild backward evolution, the pinched initial hypersurface \({\Sigma_0}\) ‘opens up’ instantly, becoming a regular spacelike (cylindrical) hypersurface. This result thus yields classes of examples of non-symmetric vacuum spacetimes, evolving forward-in-time from regular initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which turns out to be at the threshold that our energy methods can handle.     

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

Spatially Homogeneous Rotating Solution in f(R) Gravity and Its Energy Contents

In this paper, the metric approach of f(R) theory of gravity is used to investigate the exact vacuum solutions of spatially homogeneous rotating spacetimes. For this purpose, R is replaced by f(R) in the standard Einstein-Hilbert action and the set of modified Einstein field equations reduce to a single equation. We adopt the assumption of constant Ricci scalar which maybe zero or non-zero. Moreover, the energy density of the non-trivial solution has been evaluated by using the generalized Landau-Lifshitz energy-momentum complex in the perspective of f(R) gravity for some appropriate f(R) model, which turns out to be a constant quantity.     

Unique continuation results for Ricci curvature and applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed.     

Two 3-branes in Randall–Sundrum setup and current acceleration of the universe

Five-dimensional spacetimes of two orbifold 3-branes are studied, by assuming that the two 3-branes are spatially homogeneous, isotropic, and independent of time  , following the so-called “bulk-based” approach. The most general form of the metric is obtained, and the corresponding field equations are divided into three groups, one is valid on each of the two 3-branes, and the third is valid in the bulk. The Einstein tensor on the 3-branes is expressed in terms of the discontinuities of the first-order derivatives of the metric coefficients. Thus, once the metric is known in the bulk, the distribution of the Einstein tensor on the two 3-branes is uniquely determined. As applications, we consider two different cases, one is in which the bulk is locally AdS₅${\mathit{AdS}}_5$, and the other is where it is vacuum. In some cases, it is shown that the universe is first decelerating and then accelerating. The global structure of the bulk as well as the 3-branes is also studied, and found that in some cases the solutions may represent the collision of two orbifold 3-branes. The applications of the formulas to the studies of the cyclic universe and the cosmological constant problem are also pointed out.     

On Ricci tensors of Randers metrics

In this paper, we study Randers metrics and find a condition on the Ricci tensors of these metrics for being Berwaldian. This generalizes Shen’s Theorem which says that every R-flat complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on the Ricci tensors under which a Randers metric of scalar flag curvature is of zero flag curvature.     

Locally Rotationally Symmetric Vacuum Solutions in f(R) Gravity

This paper is devoted to find the Locally Rotationally Symmetric (LRS) vacuum solutions in the context of f(R) theory of gravity. Actually, we have considered the three metrics representing the whole family of LRS spacetimes and solved the field equations by using metric approach as well as the assumption of constant scalar curvature. It is mention here that R may be zero or non-zero. In all we found 10 different solutions.     

Scalar curvature and holomorphy potentials

A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given k$k$ holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where k=0$k=0$ and the functional is the square of the L²$L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.