Fourth-order Gravity Research Articles

A New Duality Transformation for Fourth-order Gravity

We prove that for non-linear L = L(R), G = dL/dR ≠ 0 the Lagrangians L and ^L(^R) with ^L = 2R/G3 - 3L/G4, ^gij = G2 gij and ^R = 3R/G2 - 4L/G3 give conformally equivalent fourth-order field equations being dual to each other. The proof represents a new application of the fact that the operator □ - R/6 is conformally invariant.

Finding the Hamiltonian for Cosmological Models in Fourth-Order Gravity Theories Without Resorting to the Ostrogradski or Dirac Formalism

The Hamilton formalism of cosmological models in fourth-order theories of gravity is considered. An approach to constructing the Hamilton function is presented which starts by replacing the second order derivatives of configuration space coordinates by functions depending on these coordinates, its first order derivatives, and additional variables playing the role of configuration space coordinates. This formalism, which does not resort to the Ostrogradski or Dirac formalism, is elucidated and applied to examples. For a special class of Lagrange functions, it is demonstrated that the canonical coordinates of the considered formalism and of the Ostrogradski formalism are related via a canonical transformation. The canonical transformation is a transformation of the configuration space coordinates and a transformation of momentum components induced by the transformation of the configuration space coordinates for a special element of the class of Lagrange functions mentioned. The Wheeler-DeWitt equations belonging to this Lagrange function are related via minisuperspace coordinate transformations.

Erratum: Stability and Hamiltonian formulation of higher derivative theories

We analyze the presumptions which lead to instabilities in theories of order higher than second. That type of fourth order gravity which leads to an inflationary (quasi de Sitter) period of cosmic evolution by inclusion of one curvature squared term (i.e. the Starobinsky model) is used as an example. The corresponding Hamiltonian formulation (which is necessary for deducing the Wheeler de Witt equation) is found both in the Ostrogradski approach and in another form. As an example, a closed form solution of the Wheeler de Witt equation for a spatially flat Friedmann model and L=R\sp 2 is found. The method proposed by Simon to bring fourth order gravity to second order can be (if suitably generalized) applied to bring sixth order gravity to second order. In the Erratum we show that a spatially flat Friedmann model need not be geodesically complete even if the scale factor a(t) is positive and smooth for all real values of the synchronized time t.

Reconciling inflation with openness.

It is already understood that the increasing observational evidence for an open Universe can be reconciled with inflation if our horizon is contained inside one single huge bubble nucleated during the inflationary phase transition. In this frame of ideas, we show here that the probability of living in a bubble with the right $\Omega_0$ (now the observations require $\Omega_0 \approx .2$) can be comparable with unity, rather than infinitesimally small. For this purpose we modify both quantitatively and qualitatively an intuitive toy model based upon fourth order gravity. As this scheme can be implemented in canonical General Relativity as well (although then the inflation driving potential must be designed entirely ad hoc), inferring from the observations that $\Omega_0 < 1$ not only does not conflict with the inflationary paradigm, but rather supports therein the occurrence of a primordial phase transition.

Exact gravitational shock wave solution of higher order theories.

We find an {\it exact} pp--gravitational wave solution of the fourth order gravity field equations. Outside the (delta--like) source this {\it not} a vacuum solution of General Relativity. It represents the contribution of the massive, $m=(-\beta)^{-1/2}$, spin--two field associated to the Ricci squared term in the gravitational Lagrangian. The fourth order terms tend to make milder the singularity of the curvature at the point where the particle is located. We generalize this analysis to $D$--dimensions, extended sources, and higher than fourth order theories. We also briefly discuss the scattering of fields by this kind of plane gravitational waves.

Bubble nucleation rate in fourth-order gravity.

We calculate the bubble nucleation rate in a fourth-order gravity theory whose action contains an $R^2$ term, under the thin-wall approximation, by two different methods. First the bounce solution is found for the Euclidean version of the original action. Next, we use a conformal transformation to transform the theory into general relativity with an additional scalar field and then proceed to find the bounce solution. The same results are obtained from both calculations.

Two-dimensional higher-derivative gravity and conformal transformations

We consider the Lagrangian L=F(R) in classical (i.e. non-quantized) two-dimensional fourth-order gravity and give new relations to Einstein's theory with a non-minimally coupled scalar field. We distinguish between scale-invariant Lagrangians and scale-invariant field equations. L is scale invariant for and diverges for . The field equation is scale invariant not only for both of these, but also for . We prove this to be the only exception, and show in what sense it is the limit of as . More generally: let H be a divergence and F a scale-invariant Lagrangian, then has a scale-invariant field equation. Furthermore, we comment on the known generalized Birkhoff theorem and exact solutions including black holes.

Stability and Hamiltonian formulation of higher derivative theories.

We analyze the presuppositions leading to instabilities in theories of order higher than second. The type of fourth-order gravity which leads to an inflationary (quasi-de Sitter) period of cosmic evolution by inclusion of one curvature-squared term (i.e., the Starobinsky model) is used as an example. The corresponding Hamiltonian formulation (which is necessary for deducing the Wheeler-DeWitt equation) is found both in the Ostrogradski approach and in another form. As an example, a closed form solution of the Wheeler-DeWitt equation for a spatially flat Friedmann model and $L={R}^{2}$ is found. The method proposed by Simon to bring fourth order gravity to second order can be (if suitably generalized) applied to bring sixth-order gravity to second order.

Perturbative method to solve fourth-order gravity field equations.

We develop a method for solving the field equations of a quadratic gravitational theory coupled to matter. The quadratic terms are written as a function of the matter stress tensor and its derivatives in such a way as to have, order by order, a set of Einstein field equations with an effective ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$. We study the cosmological scenario recovering the de Sitter exact solution, and the first order (in the coupling constants $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ appearing in the gravitational Lagrangian) solution to the gauge cosmic string metric and the charged black hole. For this last solution we discuss the consequences on the thermodynamics of black holes, and, in particular, the entropy-area relation which gets additional terms to the usual $\frac{A}{4}$ value.

Mixmaster universe in fourth-order gravity theories.

We analyze the behavior of the Bianchi type-IX cosmological model in the full fourth-order theory of gravity in four space-time dimensions on approach to the singularity both analytically and numerically. By directly analyzing the full fourth-order field equations we prove explicitly that the Belinski-Khalatnikov-Lifshitz (BKL) analytic solution (and the accompanying bounce law) is also a solution to the more general case that we consider. But it appears to be nongeneric, because a perturbation analysis shows that power-law Kasner asymptotes are unstable on approach to the singularity. On the other hand, we find that the model possesses a stable isotropic monotonic power-law asymptotic solution and accordingly we show that there is a general solution which is nonchaotic near the space-time singularity. Numerical experiments confirm the existence of such a behavior. The inclusion of matter fields does not alter the evolution in the neighborhood of the singularity. The role of space-time dimensionality on the chaotic evolution of the mixmaster universe in these theories is also spelled out. In particular, we show that it is impossible to build a mixmaster universe in space-time dimensions higher than four in the full fourth-order gravity theory based on the BKL approximation scheme.

THE GAUSS-BONNET IDENTITY IN FOURTH ORDER GRAVITY

For a class of conformally flat metrics a model based in fourth order gravity is constructed to investigate the existence of quantum corrections to the Gauss-Bonnet identity. If dimensional regularization is employed we prove that anomalies spoil the quantum validity of this topological identity.

Wheeler-De Witt equations for fourth-order quantum cosmology

The author compares the methods proposed by Boulware (1984), and Buchbinder and Lyachovich (1987), to establish a Hamiltonian formalism for fourth-order gravity and studies another possibility to circumvent second-order derivatives of the metric tensor components on the level of the Lagrangian for fourth-order theories of gravity, by introduction of a scalar field. This is done in a way different from the common procedure of doing a conformal transformation of the metric. It is demonstrated how the well known fourth-order field equations result. On the basis of the Hamiltonian formalism, the author considers the quantum cosmology for homogeneous and isotropic closed models. As a first application, the WKB approximation is discussed neglecting the spatial curvature. Chosen initial conditions have the consequence that the resulting wavefunction leads to the inflationary stage of the cosmic evolution.

Conformal gravity and the flatness problem

The radiation and matter-dominated cosmologies associated with fourth-order conformal Weyl gravity, a theory which is currently being explored as a candidate alternative to the standard second-order Einstein theory, are presented. The theory naturally yields an open though nonetheless recollapsing universe which can even oscillate indefinitely, which appears to possess no flatness problem, which has a cosmology which could potentially be created out of nothing, which is singularity free, and which does not appear to require any cosmological dark matter.

EQUIVALENCE AMONG WORMHOLE INSTANTONS

Massless scalar fields coupled to gravity produces two kinds of wormhole instantons, the Giddings-Strominger wormhole and the Tolman-Hawking wormhole. A new type of coupling between gravity and a massless scalar field is discussed which produces a different wormhole-like solution, [Formula: see text]. It is seen that these three solutions can be related to each other by means of suitable conformal transformations. The Wheeler DeWitt equation corresponding to a fourth-order gravity theory minimally coupled to an axionic field is obtained and some of its particular solutions are discussed.

MODELS OF CHAOTIC INFLATION

A review of inflationary cosmological models from a geometrical point of view is made. Equivalence theorems relate different models and may be used to reduce them to a canonical form: the Einstein gravity minimally interacting with a number of coupled scalar fields (generally, coupling appears in kinetic terms, too). Double inflation appearing in the fourth-order gravity with a scalar field is considered. Some results on the attractor property of the de Sitter (inflationary) stage in various models are presented.

THE PHASE-SPACE VIEW OF INFLATION II: FOURTH-ORDER MODELS

The phase space portrait of the cosmological models deduced from fourth-order gravity theories is discussed with the analytical and numerical methods of our previous paper. A comparison is carried out between models inferred from Lagrangian densities containing powers higher than two in the Ricci scalar of curvature, and the Starobinsky model. Some peculiar structures, such as attractors and singular points, emerging neatly from both theories, have a close physical affinity, in addition to the mathematical one. Trajectories of interest in both scenarios are those undergoing an inflationary expansion and then reaching a Friedmannian asymptotic stable phase. These features are moreover discussed through a potential U(R) in RN-models. Three kinds of potential regions are recognized. They are the allowed regions (a-regions), in which trajectories can reach the Friedmannian phase after possibly undergoing an inflationary period, the disconnected regions (d-regions), in which trajectories, although physical, never reach the Friedmannian stage and the forbidden regions (f-regions), in which there are no physical solutions. A general survey of the global phase space for Starobinsky and RN-models is given via Poincaré projections of suitable variables. a-, d-, and f-regions are represented.

Solutions to the Reissner-Nordström, Kerr, and Kerr-Newman problems in fourth-order conformal Weyl gravity.

We continue our study of the general structure of fourth-order conformal Weyl gravity which we are exploring as a possible theory of gravity, and in this paper we present three new exact solutions to the theory which we have found. First we present the complete and exact solution to the Reissner-Nordstr\"om problem associated with a static, spherically symmetric point electric and/or magnetic charge coupled to fourth-order conformal Weyl gravity. We find that, unlike the familiar second-order Einstein case where the modification to the Schwarzschild metric is in the form of a term which behaves like $\frac{1}{{r}^{2}}$, in the fourth-order case the modification is found to behave like $\frac{1}{r}$, i.e., just like the Newtonian potential term itself. Additionally, we present two further exact solutions to the theory which we have found, namely, those associated with the fourth-order Kerr and Kerr-Newman problems in which a stationary, axially symmetric rotating system with or without electric and/or magnetic charge is coupled to gravity.

The Jump Problem in Fourth-Order Gravity

Using a shock wave ansatz, first, the fourth-order jump problem in gravitational theories containing fourth derivatives of the metric is investigated and, second, the obtained results are generalized for arbitrary n-order jumps. For fourth-order jumps, in particular, the dynamic modes of the field are analysed. Das Sprungproblem in Gravitationstheorien vierter Ordnung Unter Verwendung eines Stoßwellenansatzes wird zunächst in Gravitationstheorien mit vierten Ableitungen der Metrik das Sprungproblem vierter Ordnung untersucht und danach werden die dabei erzielten Ergebnisse für Sprünge beliebiger Ordnung verallgemeinert. Im Falle metrischer Sprünge vierter Ordnung werden insbesondere die dynamischen Moden des Feldes analysiert.

The Newtonian limit of fourth and higher order gravity

AbstractWe consider the Newtonian limit of the theory based on the Lagrangian \documentclass{article}\pagestyle{empty}\begin{document}$ \ell = (R + \sum\limits_{k = 0}^p {a_k R\square ^k R)\sqrt { - g.} } $\end{document}. The gravitational potential of a point mass turns out to be a combination of Newtonian and Yukawa terms. For sixth‐order gravity (p = 1) the coefficients are calculated explicitly. For general p one gets \documentclass{article}\pagestyle{empty}\begin{document}$ \phi = m/r(1 + \sum\limits_{t = 0}^p {c_i } \exp (&amp;#x02010; r/l_i))\;with\;\sum\limits_{t = 0}^p {c_i} \; = \;1/3 $\end{document}. Therefore, the potential is always unbounded near. The origin.

INFLATION IN KALUZA-KLEIN COSMOLOGY I: TRANSFORMATION TO FOURTH-ORDER GRAVITY

The higher-dimensional Einstein vacuum equation with Λ-term is shown to be conformally equivalent to the four-dimensional field equation of scale-invariant fourth-order gravity. This holds for a general warped product between space-time and internal space of arbitrary dimension m which turns out to be an Einstein space. (The limit m → ∞ makes sense!) Thus, the results concerning the attractor property of the power-law inflationary solution derived for fourth-order gravity hold for the Kaluza-Klein model, too.