Note on stability in conformally connected frames

We find a black hole solution with non-Abelian field in Brans-Dicke theory.\nIt is an extension of non-Abelian black hole in general relativity. We discuss\ntwo non-Abelian fields: "SU(2)" Yang-Mills field with a mass (Proca field) and\nthe SU(2)$\\times$SU(2) Skyrme field. In both cases, as in general relativity,\nthere are two branches of solutions, i.e., two black hole solutions with the\nsame horizon radius. Masses of both black holes are always smaller than those\nin general relativity. A cusp structure in the mass-horizon radius\n($M_{g}$-$r_{h}$) diagram, which is a typical symptom of stability change in\ncatastrophe theory, does not appear in the Brans-Dicke frame but is found in\nthe Einstein conformal frame. This suggests that catastrophe theory may be\nsimply applied for a stability analysis as it is if we use the variables in the\nEinstein frame. We also discuss the effects of the Brans-Dicke scalar field on\nblack hole structure.\n

The conformal transformations presented in the previous chapter establish a mathematical equivalence between scalar-tensor theories and Einstein’s general relativity, with the modifications dictated by the anomalous coupling of the Einstein frame scalar to ordinary matter and its consequences. A priori, the mathematical equivalence need not be a physical one. The literature using conformal transformations in cosmology and in alternative theories of gravity is plagued by confusion and ambiguity on the issue of the physical equivalence of the two conformal frames, or which frame should be regarded as the physical one. As a consequence, the conformal transformation technique is often misused and its results are given erroneous interpretations. Incompatible points of view occurring in the literature support the following ideas [613, 363] : (first viewpoint) that the Jordan and the Einstein frame are physically equivalent, (second viewpoint) that the two frames are inequivalent the Jordan frame is the physical one, and the Einstein frame is unphysical, (third viewpoint) that the two frames are inequivalent — the Jordan frame is unphysical, and the Einstein frame is physical.

We revisit the problem of generality of Starobinsky and Higgs inflation. The known results obtained in the Einstein frame are generalized for the case of an arbitrary initial energy of the scalar field. These results are compared with the results obtained directly in the Jordan frame, which, to our knowledge, has not been thoroughly explored in the literature previously. We demonstrate that the qualitative picture of initial conditions zone in the $(\ensuremath{\phi},\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}})$ plane, which leads to sufficient amount of inflation, is quite similar for both the frames in the case of Higgs inflation. For Starobinsky inflation, the conformal transformation between the frames relates the geometrical variables in the Jordan frame with the properties of an effective scalar field in the Einstein frame. We show that the transformation $(H,R)\ensuremath{\rightarrow}(\ensuremath{\phi},\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}})$ is not regular everywhere, leading to some peculiarities in the zone of good initial conditions in the $(H,R)$ plane.

We discuss wormhole solution in the Brans-Dicke theory with existence of relativistic radiation field. The solutions of wormhole are obtained in the Einstein frame. When the dilaton field σ = const, the solution of wormhole should be reduced to the well-known Tolman wormhole. Euclidean wormhole solution also has been obtained in the Einstein frame when σ ≠ const. For the comparison, we also derive the wormhole solution in the Jordan frame with existence of relativistic radiation field. However, the two frames, the Einstein frame and the Jordan frame, are not physically equivalent.

An alternative interpretation of the conformal transformations of the metric is discussed according to which the latter can be viewed as a mapping among Riemannian and Weyl-integrable spaces. A novel aspect of the conformal transformation's issue is then revealed: these transformations relate complementary geometrical pictures of a same physical reality, so that, the question about which is the physical conformal frame, does not arise. In addition, arguments are given which point out that, unless a clear statement of what is understood by "equivalence of frames" is made, the issue is a semantic one. For definiteness, an intuitively "natural" statement of conformal equivalence is given, which is associated with conformal invariance of the field equations. Under this particular reading, equivalence can take place only if the metric is defined up to a conformal equivalence class. A concrete example of a conformal-invariant theory of gravity is then explored. Since Brans-Dicke theory is not conformally invariant, then the Jordan's and Einstein's frames of the theory are not equivalent. Otherwise, in view of the alternative approach proposed here, these frames represent complementary geometrical descriptions of a same phenomenon. The different points of view existing in the literature are critically scrutinized on the light of the new arguments.

The dynamics of the Brans-Dicke (BD) cosmology in the Einstein frame is studied. We implement phase-space analysis in order to investigate the role of initial conditions and model parameters in the evolution of the universe. The phase plane algorithm and stability of the model are considered in the Einstein Frame where the cosmological background is described in Λ-Cold Dark Matter (ΛCDM) and is supplemented by a Jordan-Fierz-Brans-Dicke field. Our results show: (1) The model parameter α (the coupling strength of the BD scalar field to matter sources) in the Einstein frame plays an important role in the model stability whereas the initial conditions significantly affect the cosmic dynamics. (2) Using type Ia SNIa data, we fit the parameters of the model with observation and show that the current value of the effective equation of state (EoS) parameter is within the range supported by observations. (3) The phase space analysis implies that the universe approaches an attractor in the near future.

We study charged boson stars in scalar-tensor (ST) gravitational theories. We analyze the weak field limit of the solutions and analytically show that there is a maximum charge to mass ratio for the bosons above which the weak field solutions are not stable. This charge limit can be greater than the GR limit for a wide class of ST theories. We numerically investigate strong field solutions in both the Brans-Dicke and power law ST theories. We find that the charge limit decreases with increasing central boson density. We discuss the gravitational evolution of charged and uncharged boson stars in a cosmological setting and show how, at any point in its evolution, the physical properties of the star may be calculated by a rescaling of a solution whose asymptotic value of the scalar field is equal to its initial asymptotic value. We focus on evolution in which the particle number of the star is conserved and we find that the energy and central density of the star decrease as the cosmological time increases. We also analyze the appearance of the scalarization phenomenon recently discovered for neutron stars configurations and, finally, we give a short discussion on how making the correct choice of mass influences the argument over which conformal frame, the Einstein frame or the Jordan frame, is physical.

Scalar gravitational waves and Einstein frame

An explicit proof of the vanishing of the covariant divergence of the energy–momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy–momentum is built in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories.

We interpret the Brans–Dicke gravity from entropic viewpoint. We first apply Verlinde's entropic formalism in the Einstein frame, then perform the conformal transformation which connects the Einstein frame to the Jordan frame. The transformed result yields the equation of motion of the Brans–Dicke theory in the Jordan frame.

We review the conformal equivalence in describing the background expansion of the universe by [Formula: see text] gravity both in the Jordan frame and the Einstein frame. In the Jordan frame, we present the general analytic expression for [Formula: see text] models that have the same expansion history as the [Formula: see text]CDM model. This analytic form can provide further insights on how cosmology can be used to test the [Formula: see text] gravity at the largest scales. Moreover we present a systematic and self-consistent way to construct the viable [Formula: see text] model in Jordan frame using the mass dilation rate function from the Einstein frame through the conformal transformation. In addition, we extend our study to the linear perturbation theories and we further exhibit the equivalence of the [Formula: see text] gravity presented in the Jordan frame and Einstein frame in the perturbed space–time. We argue that this equivalence has solid physics root.

Parametric resonance in the Einstein frame: The Jordan-frame Doppelgänger

We consider Brans-Dicke (BD) scalar tensor theory in the conformally transformed Einstein frame. In this frame BD theory behaves like an interacting quintessence model. We find the necessary conditions on the form of the potential $V(\ensuremath{\varphi})$ in order to have thawing behavior. Finally, by setting the BD coupling constant $\ensuremath{\omega}=0$, the metric $f(R)$ gravity has been considered in the Einstein frame. Assuming the existence of thawing solution, some necessary conditions for $f(R)$ gravity models have been derived.

The definition of the Sparling-Thirring form is extended to the Brans-Dicke theory. By writing the Brans-Dicke field equations in a formally Maxwell-like form, a superpotential and a corresponding pseudo energy-momentum form are defined. The general energy expression provided by the superpotential in the Jordan frame is discussed in relation to the corresponding expression in the Einstein frame. In order to substantiate its formal definition, the generalized Sparling-Thirring form is used to calculate the energy for the spherically symmetric vacuum solution in the Brans-Dicke theory.

Nonlinear massive spin-2 field generated by higher derivative gravity