Charged Black Holes In Quadratic Theories

We point out that in general the Reissner-Nordstr\"om (RN) charged black holes of general relativity are not solutions of the four dimensional quadratic gravitational theories. They are, e.g., exact solutions of the $R+R^2$ quadratic theory but not of a theory where a $R_{ab}R^{ab}$ term is present in the gravitational Lagrangian. In the case where such a non linear curvature term is present with sufficiently small coupling, we obtain an approximate solution for a charged black hole of charge $Q$ and mass $M$. For $Q\ll M$ the validity of this solution extends down to the horizon. This allows us to explore the thermodynamic properties of the quadratic charged black hole and we find that, to our approximation, its thermodynamics is identical to that of a RN black hole. However our black hole's entropy is not equal to the one fourth of the horizon area. Finally we extend our analysis to the rotating charged black hole and qualitatively similar results are obtained.Comment: 11 pages, LaTeX/RevTeX3.

The Universality of Einstein Equations

It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as independent variables, leads to ``universal'' equations. If the dimension $n$ of space--time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians $L=R^{n/2} \sqrt g$ and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2--dimensional space--time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi--Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.Comment: 15 pages, LaTeX, (Extended Version), TO-JLL-P1/9

Gravitation, electromagnetism and cosmological constant in purely affine gravity

The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the purely affine formulation of gravity by the Eddington Lagrangian. In this paper we combine the electromagnetic field and the cosmological constant in the purely affine formulation. We show that the sum of the two affine (Eddington and Ferraris-Kijowski) Lagrangians is dynamically inequivalent to the sum of the analogous ($\Lambda$CDM and Einstein-Maxwell) Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid, like the affine Einstein-Born-Infeld formulation, only for weak electromagnetic fields, on the order of the magnetic field in outer space of the Solar System. Therefore the purely affine formulation that combines gravity, electromagnetism and cosmological constant cannot be a simple sum of affine terms corresponding separately to these fields. A quite complicated form of the affine equivalent of the metric Einstein-Maxwell-$\Lambda$ Lagrangian suggests that Nature can be described by a simpler affine Lagrangian, leading to modifications of the Einstein-Maxwell-$\Lambda$CDM theory for electromagnetic fields that contribute to the spacetime curvature on the same order as the cosmological constant.Comment: 17 pages, extended and combined with gr-qc/0612193; published versio

Equivalence of black hole thermodynamics between a generalized theory of gravity and the Einstein theory

We analyze black hole thermodynamics in a generalized theory of gravity whose Lagrangian is an arbitrary function of the metric, the Ricci tensor and a scalar field. We can convert the theory into the Einstein frame via a "Legendre" transformation or a conformal transformation. We calculate thermodynamical variables both in the original frame and in the Einstein frame, following the Iyer--Wald definition which satisfies the first law of thermodynamics. We show that all thermodynamical variables defined in the original frame are the same as those in the Einstein frame, if the spacetimes in both frames are asymptotically flat, regular and possess event horizons with non-zero temperatures. This result may be useful to study whether the second law is still valid in the generalized theory of gravity.Comment: 14 pages, no figure

Topological Defects in Gravitational Theories with Non Linear Lagrangians

The gravitational field of monopoles, cosmic strings and domain walls is studied in the quadratic gravitational theory $R+\alpha R^2$ with $\alpha |R|\ll 1$, and is compared with the result in Einstein's theory. The metric aquires modifications which correspond to a short range `Newtonian' potential for gauge cosmic strings, gauge monopoles and domain walls and to a long range one for global monopoles and global cosmic strings. In this theory the corrections turn out to be attractive for all the defects. We explain, however, that the sign of these corrections in general depends on the particular higher order derivative theory and topological defect under consideration. The possible relevance of our results to the study of the evolution of topological defects in the early universe is pointed out.Comment: LaTeX (uses revrex macros), 13 page

On Physical Equivalence between Nonlinear Gravity Theories

We argue that in a nonlinear gravity theory, which according to well-known results is dynamically equivalent to a self-gravitating scalar field in General Relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical. We explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and backwards. We study energetics for asymptotically flat solutions. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space. The proof of this Positive Energy Theorem relies on the existence of the Einstein frame, since in the (Helmholtz--)Jordan frame the Dominant Energy Condition does not hold and the field variables are unrelated to the total energy of the system.Comment: 37 pp., TO-JLL-P 3/93 Dec 199

Quadratic Curvature Gravity with Second Order Trace and Massive Gravity Models in Three Dimensions

The quadratic curvature lagrangians having metric field equations with second order trace are constructed relative to an orthonormal coframe. In $n>4$ dimensions, pure quadratic curvature lagrangian having second order trace constructed contains three free parameters in the most general case. The fourth order field equations of some of these models, in arbitrary dimensions, are cast in a particular form using the Schouten tensor. As a consequence, the field equations for the New massive gravity theory are related to those of the Topologically massive gravity. In particular, the conditions under which the latter is "square root" of the former are presented.Comment: 24 pages, to appear in GR

Resonant particle production with non-minimally coupled scalar fields in preheating after inflation

We investigate a resonant particle production of a scalar field $\chi$ coupled non-minimally to a spacetime curvature $R$ ($\xi R \chi^2$) as well as to an inflaton field $\phi$ ($g^2\phi^2\chi^2$). In the case of $g < 3 \times 10^{-4}$, $\xi$ effect assists $g$-resonance in certain parameter regimes. However, for $g > 3 \times 10^{-4}$, $g$-resonance is not enhanced by $\xi$ effect because of $\xi$ suppression effect as well as a back reaction effect. If $\xi \approx -4$, the maximal fluctuation of produced $\chi$-particle is $\sqrt{}_{max} \approx 2 \times 10^{17}$ GeV for $g < 1 \times 10^{-5}$, which is larger than the minimally coupled case with $g \approx 1 \times 10^{-3}$.Comment: 33pages, 12figures. to appear in Physical Review

New constraints on multi-field inflation with nonminimal coupling

We study the dynamics and perturbations during inflation and reheating in a multi-field model where a second scalar field $\chi$ is nonminimally coupled to the scalar curvature $(\frac12 \xi R\chi^2$). When $\xi$ is positive, the usual inflationary prediction for large-scale anisotropies is hardly altered while the $\chi$ fluctuation in sub-Hubble modes can be amplified during preheating for large $\xi$. For negative values of $\xi$, however, long-wave modes of the $\chi$ fluctuation exhibit exponential increase during inflation, leading to the strong enhancement of super-Hubble metric perturbations even when $|\xi|$ is less than unity. This is because the effective $\chi$ mass becomes negative during inflation. We constrain the strength of $\xi$ and the initial $\chi$ by the amplitude of produced density perturbations. One way to avoid nonadiabatic growth of super-Hubble curvature perturbations is to stabilize the $\chi$ mass through a coupling to the inflaton. Preheating may thus be necessary in these models to protect the stability of the inflationary phase.Comment: 20 pages, 8 figures, submitted to Physical Review