Einstein Tensor Research Articles

Gauge inflation by kinetic coupled gravity

Recently, a new class of inflationary models, so-called gauge-flation or non-Abelian gauge field inflation has been introduced where the slow-roll inflation is driven by a non-Abelian gauge field A with the field strength F. This class of models are based on a gauge field theory having F2 and F4 terms with a non-Abelian gauge group minimally coupled to gravity. Here, we present a new class of such inflationary models based on a gauge field theory having only F2 term with non-Abelian gauge fields non-minimally coupled to gravity. The non-minimal coupling is set up by introducing the Einstein tensor besides the metric tensor within the F2 term, which is called kinetic coupled gravity. A perturbation analysis is performed to confront the inflation under consideration with Planck and BICEP2 results

Black hole quasinormal modes in a scalar-tensor theory with field derivative coupling to the Einstein tensor

We investigate the quasinormal modes of a test massless, minimally coupled scalar field on a static and spherically symmetric black hole in the scalar-tensor theory with field derivative coupling to the Einstein tensor, which is a part of the Horndeski theory with the shift symmetry. In our solution, the spacetime is asymptotically AdS (anti-de Sitter), where the effective AdS curvature scale is determined solely by the derivative coupling constant. The metric approaches the AdS spacetime in the asymptotic infinity limit and precisely recovers the Schwarzschild-AdS solution in general relativity if the coupling constant is tuned to the inverse of the cosmological constant. We numerically find the lowest lying quasinormal frequency for the perturbation about a test massless, minimally coupled scalar field. The quasinormal frequency agrees with that of the Schwarzschild-AdS solution for the tuned case. For other parameters, in the large black hole limit, as the metric coincides with that of the Schwarzschild-AdS black hole, the quasinormal frequency almost agrees with that of the Schwarzschild-AdS black hole and is insensitive to the value of the cosmological constant. On the other hand, for a small back hole the real part of the quasinormal frequency decreases as the absolute value of the cosmological constant increases.

Vector inflation by kinetic coupled gravity

Vector inflation is a newly established model where inflation is driven by nonminimally coupled massive vector fields with a potential term. This model is similar to the model of chaotic inflation with scalar fields, except that for vector fields the isotropy of expansion is achieved either by considering a triplet of orthogonal vector fields or N randomly oriented independent vector fields. We introduce a new model of vector inflation where the vector field has no potential term and is nonminimally coupled to gravity through the kinetic term. This model shows that vector inflation without a potential term is possible and helps to avoid the fine tuning problem in inflationary models having a potential. The nonminimal coupling is established by introducing the Einstein tensor besides the metric tensor within the kinetic term of the vector field. A perturbation analysis is performed to confront the inflation under discussion with Planck and BICEP2 results.

On the conservation of superenergy and its applications

In this work we present a geometric identity involving the Bel–Robinson tensor which is formally similar to the Sparling identity (which involves the Einstein tensor through the Einstein 3-form). In our identity the Bel–Robinson tensor enters through the Bel–Robinson 3-form which, we believe, is introduced in the literature for the first time. The meaning of this identity is that it is possible to formulate a generic conservation law for the quantity represented by the Bel–Robinson tensor (superenergy). We also show how one can use the Bel–Robinson 3-form to estimate the components of the Bel–Robinson tensor which are computed with respect to the causal elements of a frame. This estimate could be useful in a global existence proof of the solutions of a theory of gravitation in dimension four.

On singular semi-Riemannian manifolds

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

Asymptotically locally AdS and flat black holes in Horndeski theory

In this paper we construct asymptotically locally AdS and flat black holes in the presence of a scalar field whose kinetic term is constructed out from a linear combination of the metric and the Einstein tensor. The field equations as well as the energy-momentum tensor are second order in the metric and the field, therefore the theory belongs to the ones defined by Horndeski. We show that in the presence of a cosmological term in the action, it is possible to have a real scalar field in the region outside the event horizon. The solutions are characterized by a single integration constant, the scalar field vanishes at the horizon and it contributes to the effective cosmological constant at infinity. We extend these results to the topological case. The solution is disconnected from the maximally symmetric AdS background, however, within this family there exits a gravitational soliton which is everywhere regular. This soliton is therefore used as a background to define a finite Euclidean action and to obtain the thermodynamics of the black holes. For a certain region in the space of parameters, the thermodynamic analysis reveals a critical temperature at which a Hawking-Page phase transition between the black hole and the soliton occurs. We extend the solution to arbitrary dimensions grater than four and show that the presence of a cosmological term in the action allows to consider the case in which the standard kinetic term for the scalar it's not present. In such scenario, the solution reduces to an asymptotically flat black hole.

Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario

Asymptotically locally anti--de Sitter and asymptotically flat black hole solutions are found for a particular case of the Horndeski action. The action contains the Einstein-Hilbert term with a cosmological constant, a real scalar field with a nonminimal kinetic coupling given by the Einstein tensor, the minimal kinetic coupling, and the Maxwell term. There is no scalar potential. The solution has two integration constants related with the mass and the electric charge. The solution is given for all dimensions. A new class of asymptotically locally flat spherically symmetric black holes is found when the minimal kinetic coupling vanishes and the cosmological constant is present. In this case, we get a solution which represents an electric universe. The electric field at infinity is only supported by $\mathrm{\ensuremath{\Lambda}}$. When the cosmological constant vanishes, the black hole is asymptotically flat.

Braneworlds with field derivative coupling to the Einstein tensor

In this paper, we investigate the Randall-Sundrum type braneworld models in the scalar-tensor theory with field derivative coupling to the Einstein tensor. We first formulate the generalized junction conditions of the metric and the scalar field on the timelike codimension-one hypersurface (=brane). With the use of these junction conditions, we then derive the Minkowski and de Sitter brane solutions embedded into the $Z_2$-symmetric five-dimensional anti-de Sitter (AdS) bulk spacetime. The configuration of the scalar field depends on the slice of the AdS spacetime. These branes are supported by the tension and not coupled to the scalar field. The Minkowski brane solution can be obtained when the brane tension is tuned to the bulk contributions. Once this tuning relation is broken, the de Sitter brane solutions are obtained. The de Sitter brane solutions have two branches. One has the smooth limit to the case where the scalar field becomes trivial, while the other branch does not. The latter branch has the upper bound on the brane tension, where the expansion rate vanishes. Finally, we investigate the low energy effective gravitational theory realized on the brane, which is given by the four-dimensional Einstein-scalar theory with the corrections from the bulk. In the case that there is no generation of the dark radiation from the bulk scalar field, we recover the the four-dimensional Einstein-scalar theory.

Solutions in the scalar-tensor theory with nonminimal derivative coupling

We present black hole type solutions in the scalar-tensor theory with nonminimal derivative coupling to the Einstein tensor. The effects of the nonminimal derivative coupling appear in the large scales, while the solutions approach those in the Einstein gravity in the small scales. For the particular coupling constant tuned to the inverse of the cosmological constant, the scalar field becomes trivial and the solutions in the Einstein gravity are recovered. For the other coupling constant, more general solutions can be obtained. If the two-dimensional maximally symmetric space is a two-sphere, the spacetime structure approaches anti-de Sitter spacetime in the large scales. On the other hand, if the two-dimensional space is a two-hyperboloid, the spacetime approaches de Sitter (AdS) spacetime. If it is a flat space, the cosmological constant affects only the amplitude of the scalar field. The extension to the higher-dimensional case is also straightforward. For a certain range of the negative cosmological constant, thermodynamic properties of a black hole are very similar to those of the Schwarzschild-AdS black hole in the Einstein gravity.

Gravity induced from quantum spacetime

We show that tensoriality constraints in noncommutative Riemannian geometry in the two-dimensional bicrossproduct model quantum spacetime algebra [x, t] = λx drastically reduce the moduli of possible metrics g up to normalization to a single real parameter, which we interpret as a time in the past from which all timelike geodesics emerge and a corresponding time in the future at which they all converge. Our analysis also implies a reduction of moduli in n-dimensions and we study a suggested spherically symmetric classical geometry in n = 4 in detail, identifying two one-parameter subcases where the Einstein tensor matches that of a perfect fluid for (a) positive pressure, zero density and (b) negative pressure and positive density with ratio . The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the quantum spacetime model as a quadratic algebra. The noncommutative Riemannian geometry is fully solved for n = 2 and includes the quantum Levi-Civita connection and a second, nonperturbative, Levi-Civita connection which blows up as λ → 0. We also propose a ‘quantum Einstein tensor’ which is identically zero for the main part of the moduli space of connections (as classically in 2D). However, when the quantum Ricci tensor and metric are viewed as deformations of their classical counterparts there would be an O(λ2) correction to the classical Einstein tensor and an O(λ) correction to the classical metric.

Instability of Schwarzschild–AdS black hole in Einstein–Weyl gravity

We investigate the classical stability of Schwarzschild–AdS black hole in a massive gravity theory of the Einstein–Weyl gravity. It turns out that the linearized Einstein tensor perturbations exhibit unstable modes featuring the Gregory–Laflamme instability of five-dimensional AdS black string, in contrast to the stable Schwarzschild–AdS black hole in Einstein gravity. We point out that the instability of the black hole in the Einstein–Weyl gravity arises from the massiveness but not a feature of fourth-order derivative theory giving ghost states.

Phase transition to a hairy black hole in asymptotically flat spacetime

We discuss a phase transition of a Reissner-Nordström black hole to a hairy black hole in asymptotically flat spacetime. The hair is due to a massive charged scalar field. The no-hair theorem is evaded thanks to a derivative coupling of the scalar field to the Einstein tensor. The resulting hairy configuration is spherically symmetric. We solve the equations analytically near the transition temperature and show that the hair is concentrated near the horizon decaying exponentially away from it.

Screening the fifth force in the Horndeski's most general scalar-tensor theories

We study how the Vainshtein mechanism operates in the most general scalar-tensor theories with second-order equations of motion. The field equations of motion, which can be also applicable to most of other screening scenarios proposed in literature, are generally derived in a spherically symmetric space-time with a matter source. In the presence of a field coupling to the Ricci scalar, we clarify conditions under which the Vainshtein mechanism is at work in a weak gravitational background. We also obtain the solutions of the field equation inside a spherically symmetric body and show how they can be connected to exterior solutions that accommodate the Vainshtein mechanism. We apply our general results to a number of concrete models such as the covariant/extended Galileons and the DBI Galileons with Gauss-Bonnet and other terms. In these models the fifth force can be suppressed to be compatible with solar-system constraints, provided that non-linear field kinetic terms coupled to the Einstein tensor do not dominate over other non-linear field self-interactions.

Gravitational particle production in gravity theories with non-minimal derivative couplings

We study the gravitational production of heavy X-particles of mass of the order of the inflaton mass, produced after the end of inflation. We find that, in the presence of a derivative coupling of the inflaton field or of the X-field to the Einstein tensor, the number of gravitationally produced particles is suppressed as the strength of the coupling is increased.

Third Order Lagrangians, Weyl Invariants and Classical Trace Anomaly in Six Dimensions

We have proceeded the analogy (represented in our previous works) of the Einstein tensor and the alternative form of the Einstein field equations for the generic coefficients of the eight terms in the third order of the Lovelock Lagrangian. We have found the constraint between the coefficients into two forms, an independent and a dimensional dependent versions. Each form has three degrees of freedom, and not only the exact coefficients of the third order Lovelock Lagrangian do satisfy the two forms of the constraints, but also the two independent cubic of the Weyl tensor satisfy the independent constraint in six dimensions and yield the dimensional dependent version identically independent of the dimension. Then, we have introduced the most general effective expression for a total third order type Lagrangian with the homogeneity degree number three which includes the previous eight terms plus the new three ones among the all seventeen independent terms. We have proceeded the analogy for this combination, and have achieved the relevant constraint. We have shown that the expressions given in the literature as the third Weyl-invariant combination in six dimensions do satisfy this constraint. Thus, we suggest that these constraint relations to be considered as the necessary consistency conditions on the numerical coefficients that a Weyl-invariant in six dimensions should satisfy. Finally, we have calculated the “classical” trace anomaly (an approach that was presented in our previous works) for the introduced total third order type Lagrangian and have achieved a general expression with four degrees of freedom in more than six dimensions (three degrees in six dimensions). Then, we have demonstrated that the resulted expression contains exactly the relevant coefficient of the Schwinger–DeWitt proper time method (that linked with the relevant heat kernel coefficient) in six dimensions, as a particular case. Of course, this result is a necessary consistency check, nevertheless our approach can be regarded as an alternative (perhaps simpler, and classical) derivation of the trace anomaly which also gives a general expression with the relevant degrees of freedom.

Planck constraints on single-field inflation

We place observational constraints on slow-variation single-field inflationary models by carrying out the cosmological Monte Carlo simulation with the recent data of Planck combined with the WMAP large-angle polarization, baryon acoustic oscillations, and ACT/SPT temperature data. Our analysis covers a wide variety of models with second-order equations of motion-- including potential-driven slow-roll inflation, non-minimally coupled models, running kinetic couplings, Brans-Dicke theories, potential-driven Galileon inflation, field-derivative couplings to the Einstein tensor, and k-inflation. In the presence of running kinetic exponential couplings, covariant Galileon terms, and field-derivative couplings, the tensor-to-scalar ratio of the self-coupling potential gets smaller relative to that in standard slow-roll inflation, but the models lie outside the 68 % CL observational contour. We also show that k-inflation models can be tightly constrained by adding the bounds from the scalar non-Gaussianities. The small-field inflationary models with asymptotic flat Einstein-frame potentials in the regime phi >> M_{pl} generally fit the data very well. These include the models such as Kahler-moduli inflation, non-minimally coupled Higgs inflation, and inflation in Brans-Dicke theories in the presence of the potential V(phi)=3M^2 (phi-M_{pl})^2/4 with the Brans-Dicke parameter omega_{BD}<O(1) (which covers the Starobinsky's model f(R)=R+R^2/(6M^2) as a special case).

Cosmological implications of the effective field theory of cosmic acceleration

We consider cosmological constraints arising from the background expansion history on the effective field theory of cosmic acceleration, a theoretical framework that allows for a unified way to classify both models of dark energy and modified gravity within the linear regime. In the Einstein frame, the most general action for the background can be written in terms of a canonical scalar field which is nonminimally coupled to matter. The leading corrections to the action are expressible through a quartic kinetic term, and scalar couplings to a Gauss-Bonnet curvature term and the Einstein tensor. We determine the implications of the terms in this general action for the predicted expansion history in the context of dynamical attractors. We find that each modifies the matter dominated and/or accelerative eras in ways that allow us to place cosmological constraints on them. We present current constraints on the effective action using the latest Type Ia supernovae, cosmic microwave background, and baryon acoustic oscillation data. This includes finding that the scalar field effective field theory with a coupled Gauss-Bonnet term and the data are significantly discrepant.

A novel mechanism to generate FFLO states in holographic superconductors

We discuss a novel mechanism to set up a gravity dual of FFLO states in strongly coupled superconductors. The gravitational theory utilizes two U(1) gauge fields and a scalar field coupled to a charged AdS black hole. The first gauge field couples with the scalar sourcing a charge condensate below a critical temperature, and the second gauge field provides a coupling to spin in the boundary theory. The scalar is neutral under the second gauge field. By turning on an interaction between the Einstein tensor and the scalar, it is shown that, in the low temperature limit, an inhomogeneous solution possesses a higher critical temperature than the homogeneous case, giving rise to FFLO states.

Causality-violating Higgs singlets at the LHC

We construct a simple class of compactified five-dimensional metrics which admits closed timelike curves (CTCs), and derive the resulting CTCs as analytic solutions to the geodesic equations of motion. The associated Einstein tensor satisfies all the null, weak, strong and dominant energy conditions. In particular, no negative-energy ``tachyonic'' matter is required. In extra-dimensional models where gauge charges are bound to our brane, it is the Kaluza-Klein modes of gauge singlets that may travel through the CTCs. From our brane point of view, many of these Kaluza-Klein modes would appear to travel backward in time. We give a simple model in which time-traveling Higgs singlets can be produced by the LHC, either from decay of the Standard Model Higgs or through mixing with the Standard Model Higgs. The signature of these time-traveling singlets is a secondary decay vertex preappearing before the primary vertex which produced them. The two vertices are correlated by momentum conservation. We demonstrate that preappearing vertices in the Higgs singlet-doublet mixing model may well be observable at the LHC.

Absolute Maximum Proper Time to an Initial Event, the Curvature of Its Gradient along Conflict Strings and Matter

Einstein equation of gravity has on one side the momentum energy density tensor and on the other, Einstein tensor which is derived from Ricci curvature tensor. A better theory of gravity will have both sides geometric. One way to achieve this goal is to develop a new measure of time that will be independent of the choice of coordinates. One natural nominee for such time is the upper limit of measurable time form an event back to the big bang singularity. This limit should exist despite the singularity, otherwise the cosmos age would be unbounded. By this, the author constructs a scalar field of time. Time, however, is measured by material clocks. What is the maximal time that can be measured by a small microscopic clock when our curve starts at near the “big bang” event and ends at an event within the nucleus of an atom? Will our tiny clock move along geodesic curves or will it move in a non geodesic curve within matter? It is almost paradoxical that a test particle in General Relativity will always move along geodesic curves but the motion of matter within the particle may not be geodesic at all. For example, the ground of the Earth does not move at geodesic velocity. Where there is no matter, we choose a curve from near “big bang” to an event such that the time measured is maximal. Without assuming force fields, the gravitational field which causes that two or more such curves intersect at events, would cause discontinuity of the gradient of the upper limit of measurable time scalar field. The discontinuity can be avoided only if we give up on measurement along geodesic curves where there is matter. In other words, our tiny test particle clock will experience force when it travels within matter or near matter.