Sigbjørn Hervik

Spacetimes characterized by their scalar curvature invariants

In this paper, we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an -non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is -non-degenerate. This enables us to prove our main theorem that a spacetime metric is either -non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.

Inflation with stable anisotropic hair: is it cosmologically viable?

A bstractRecently an inflationary model with a vector field coupled to the inflaton was proposed and the phenomenology studied for the Bianchi type I spacetime. It was found that the model demonstrates a counter-example to the cosmic no-hair theorem since there exists a stable anisotropically inflationary fix-point. One of the great triumphs of inflation, however, is that it explains the observed flatness and isotropy of the universe today without requiring special initial conditions. Any acceptable model for inflation should thus explain these observations in a satisfactory way. To check whether the model meets this requirement, we introduce curvature to the background geometry and consider axisymmetric spacetimes of Bianchi type II,III and the Kantowski-Sachs metric. We show that the anisotropic Bianchi type I fix-point is an attractor for the entire family of such spacetimes. The model is predictive in the sense that the universe gets close to this fix-point after a few e-folds for a wide range of initial conditions. If inflation lasts for N e-folds, the curvature at the end of inflation is typically of order ~ e−2N . The anisotropy in the expansion rate at the end of inflation, on the other hand, while being small on the one-percent level, is highly significant. We show that after the end of inflation there will be a period of isotropization lasting for

Cosmology of a scalar field coupled to matter and an isotropy-violating Maxwell field

sim frac{2}{3}N

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

e-folds. After that the shear scales as the curvature and becomes dominant around N e-folds after the end of inflation. For plausible bounds on the reheat temperature the minimum number of e-folds during inflation, required for consistency with the isotropy of the supernova Ia data, lays in the interval (21, 48). Thus the results obtained for our restricted class of spacetimes indicates that inflation with anisotropic hair is cosmologically viable.

Higher dimensional bivectors and classification of the Weyl operator

A bstractMotivated by the couplings of the dilaton in four-dimensional effective actions, we investigate the cosmological consequences of a scalar field coupled both to matter and a Maxwell-type vector field. The vector field has a background isotropy-violating component. New anisotropic scaling solutions which can be responsible for the matter and dark energy dominated epochs are identified and explored. For a large parameter region the universe expands almost isotropically. Using that the CMB quadrupole is extremely sensitive to shear, we constrain the ratio of the matter coupling to the vector coupling to be less than 10−5. Moreover, we identify a large parameter region, corresponding to a strong vector coupling regime, yielding exciting and viable cosmologies close to the ΛCDM limit.

Curvature operators and scalar curvature invariants

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (defined by a unit timelike vector field u), in any dimension. We study the cases where one of these parts vanishes in detail, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. For instance, we prove that the only permitted Weyl types are G, Ii and D, and discuss the possible relation of u with the Weyl aligned null directions (WANDs); we provide invariant conditions that characterize PE/PM spacetimes, such as Bel–Debever-like criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (such as spinning black holes) are in general neither PE nor PM. Whereas ample classes of PE spacetimes exist, PM solutions are elusive; specifically, we prove that PM Einstein spacetimes of type D do not exist, in any dimension. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor, which is useful when considering spacetimes with matter fields, and moreover leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl (or Riemann) tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem (Hervik 2011 Class. Quantum Grav. 28 215009). This in turn sheds new light on the classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the (minimal) types G/I/D from the (non-minimal) types II/III/N.

Type III and N universal spacetimes

We develop the bivector formalism in higher dimensional Lorentzian spacetimes. We define the Weyl bivector operator in a manner consistent with its boost-weight decomposition. We then algebraically classify the Weyl tensor, which gives rise to a refinement in dimensions higher than four of the usual alignment (boost-weight) classification, in terms of the irreducible representations of the spins. We are consequently able to define a number of new algebraically special cases. In particular, the classification in five dimensions is discussed in some detail. In addition, utilizing the (refined) algebraic classification, we are able to prove some interesting results when the Weyl tensor has (additional) symmetries.

Lorentzian manifolds and scalar curvature invariants

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterized by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of the alignment theory and the bivector form of the Weyl operator in higher dimensions and introduce the important notions of diagonalizability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterized by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.

A spacetime not characterized by its invariants is of aligned type II

Universal spacetimes are spacetimes for which all conserved symmetric rank-2 tensors, constructed as contractions of polynomials from the metric, the Riemann tensor and its covariant derivatives of arbitrary order, are multiples of the metric. Consequently, metrics of universal spacetimes solve vacuum equations of all gravitational theories with Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its derivatives of arbitrary order. In the literature, universal metrics are also discussed as metrics with vanishing quantum corrections and as classical solutions to string theory. Widely known examples of universal metrics are certain Ricci-flat pp waves. nIn this paper, we start a general study of geometric properties of universal metrics in arbitrary dimension and we arrive at a broader class of such metrics. In contrast with pp waves, these universal metrics also admit non-vanishing cosmological constant and in general do not have to possess a covariantly constant or recurrent null vector field. First, we show that a universal spacetime is necessarily a CSI spacetime, i.e. all curvature invariants constructed from the Riemann tensor and its derivatives are constant. Then we focus on type N spacetimes, where we arrive at a simple necessary and sufficient condition: a type N spacetime is universal if and only if it is an Einstein Kundt spacetime. A class of type III Kundt universal metrics is also found. Several explicit examples of universal metrics are presented.

Supersymmetry, holonomy and Kundt spacetimes

We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. Recently, we have shown that in four dimensions a Lorentzian spacetime metric is either -non-degenerate, and hence locally characterized by its scalar polynomial curvature invariants, or is a degenerate Kundt spacetime. We present a number of results that generalize these results to higher dimensions and discuss their consequences and potential physical applications.