The Newtonian limit of the most general fourth order gravity is performed with metric approach in the Jordan frame with no gauge condition. The most general theory with fourth order differential equations is obtained by generalizing the $f(R)$ term in the action with a generic function containing other two curvature invariants: \emph{Ricci square} ($R_{αβ}R^{αβ}$) and \emph{Riemann square} ($R_{αβγδ}R^{αβγδ}$). The spherically symmetric solutions of metric tensor yet present Yukawa-like spatial behavior, but now one has two characteristic lengths. At Newtonian order any function of curvature invariants gives us the same outcome like the so-called \emph{Quadratic Lagrangian} of Gravity. From Gauss - Bonnet invariant one have the complete interpretation of solutions and the absence of a possible third characteristic length linked to Riemann square contribution. From analysis of metric potentials, generated by point-like source, one has a constraint condition on the derivatives of $f$ with respect to scalar invariants.
We study the stability of Schwarzschild-Tangherlini (ST) black holes in fourth-order gravity which provides a higher dimensional linearized massive equation. The linearized-Ricci tensor perturbation is employed to exhibit unstable modes featuring the Gregory-Laflamme (GL) instability of higher dimensional black strings, in comparison to the stable ST black holes in Einstein gravity. It turns out that the GL instability of the ST black holes in the fourth-order gravity originates from the massiveness, but not a nature of fourth-order derivative theories giving ghost states.
We consider the fourth-order differential theory of gravitation to treat the problem of singularity avoidance: studying the short-distance behaviour in the case of black-holes and the big-bang we are going to see a way to attack the issue from a general perspective.
We prove that for non-linear L = L(R), the Lagrangians L and \hat L give conformally equivalent fourth-order field equations being dual to each other. The proof represents a new application of the fact that the operator <D'Alembertian minus R/6> is conformally invariant.
It is argued that the solution space of the equation determining the class of $f(R)$ theories which admit an Einstein static universe should be broadened by including the algebraic roots.
By attaching basis vectors to the components of matter fields, one may render free action densities fully covariant. Both the connection and the tetrads are quadratic forms in these basis vectors. The metric of spacetime, which is quadratic in the tetrads, is then quartic in the basis vectors.
We consider cosmological dynamics in fourth order gravity with both $f(R)$ and $Φ(\mathcal {G})$ correction to the Einstein gravity ($\mathcal{G}$ is the Gauss-Bonnet term). The particular case for which both terms are equally important on power-law solutions is described. These solutions and their stability are studied using the dynamical system approach. We also discuss condition of existence and stability of de Sitter solution in a more general situation of power-law $f$ and $Φ$.
The existence of black holes is a central prediction of general relativity and thus serves as a basic consistency test for modified theories of gravity. In spherical symmetry, only two classes of dynamic solutions are compatible with the formation of an apparent horizon in finite time of a distant observer. Moreover, the formation of black holes follows a unique scenario involving both types of solutions. To be compatible with their existence, any self-consistent theory of modified gravity must satisfy several constraints. We derive properties of the modified gravity terms of f(R) and generic …
The conformal equivalence of fourth-order gravity following from a non-linear Lagrangian L(R) to theories of other types is widely known, here we report on a new conformal equivalence of these theories to theories of the same type but with different Lagrangian. For a quantization of fourth-order theories one needs a Hamiltonian formulation of them. One of the possibilities to do so goes back to Ostrogradski in 1850. Here we present another possibility: A Hamiltonian H different from Ostrogradski's one is discussed for the Lagrangian L depending on first and second order drivatives of the posit…
We discuss the dynamical system approach applied to Higher Order Theories of Gravity. We show that once the theory of gravity has been specified, the cosmological equations can be written as a first-order autonomous system and we give several examples which illustrate the utility of our method. We also discuss a number of results which have appeared recently in the literature.
We investigate the radial behavior of galactic rotation curves by a Fourth Order Gravity adding also the Dark Matter component. The Fourth Order Gravity is a Lagrangian containing the Ricci scalar, the Ricci and Riemann tensor, but the rotation curves are depending only on two free parameters. A systematic analysis of rotation curves, in the Newtonian Limit of theory, induced by all galactic sub-structures of ordinary matter is shown. This analysis is presented for Fourth Order Gravity with and without Dark Matter. The outcomes are compared with respect to classical outcomes of General Relativ…
The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e., the details how within fourth order gravity with L= R + R^2 the inflationary phase of cosmic evolution turns out to be a transient attractor. Fi…
We consider the Newtonian limit of the theory based on the Lagrangian L = R + \sum a_k R \Box^k R. The gravitational potential of a point mass turns out to be a combination of Newtonian and Yukawa terms. For sixth-order gravity the coefficients are calculated explicitly. For the general case one gets as a result: The the potential is always unbounded near the origin.
We consider an auxiliary fields formulation for the general fourth-order gravity on an arbitrary curved background. The case of a Ricci-flat background is elaborated in full details and it is shown that there is an equivalence with the standard metric formulation. At the same time, using auxiliary fields helps to make perturbations to look simpler and the results more clear. As an application we reconsider the linear perturbations for the classical Schwarzschild solution. We also briefly discuss the relation to the effect of massive unphysical ghosts in the theory.
Gravitational theories generated from Lagrangians of the form f(R) are considered. The spherically symmetric solutions to these equations are discussed, paying particular attention to features that differ from the standard Schwarzschild solution. The asymptotic form of solutions is described, as is the lack of validity of Birkhoff's theorem. Exact solutions are presented which illustrate these points and their stability and geodesics are investigated.
The post-Minkowskian limit and gravitational wave solutions for general fourth-order gravity theories are discussed. Specifically, we consider a Lagrangian with a generic function of curvature invariants $f(R, R_{αβ}R^{αβ}, R_{αβγδ}R^{αβγδ})$. It is well known that when dealing with General Relativity such an approach provides massless spin-two waves as propagating degree of freedom of the gravitational field while this theory implies other additional propagating modes in the gravity spectra. We show that, in general, fourth order gravity, besides the standard massless graviton is characterize…
We calculate the characteristics of the "black hole shadow" for a rotating, neutral black hole in fourth-order conformal Weyl gravity. It is shown that the morphology is not significantly affected by the underlying framework, except for very large masses. Conformal gravity black hole shadows would also significantly differ from their general relativistic counterparts if the values of the main conformal gravity parameters, $γ$ and $κ$, were increased by several orders of magnitude. Such increased values for $γ$ and $κ$ are currently ruled out by gravitational phenomenology. Therefore, it is unl…
For the non-tachyonic curvature squared action we show that the expanding Bianchi-type I models tend to the dust-filled Einstein-de Sitter model for t tending to infinity if the metric is averaged over the typical oscillation period. Applying a conformal equivalence between curvature squared action and a minimally coupled scalar field (which holds for all dimensions > 2) the problem is solved by discussing a massive scalar field in an anisotropic cosmological model.
In this paper, we explore classes of irrotational-fluid cosmological models in the context of f(R)-gravity in an attempt to put some theoretical and mathematical restrictions on the form of the f(R) gravitational Lagrangian. In particular, we investigate the consistency of linearised dust models for shear-free cases as well as in the limiting cases when either the gravito-magnetic or gravito-elecric components of the Weyl tensor vanish. We also discuss the existence and consistency of classes of non-expanding irrotational spacetimes in f(R)-gravity.
We present for the first time the complete matter power spectrum for $R^n$ gravity which has been derived from the fourth order scalar perturbation equations. This leads to the discovery of a characteristic signature of fourth order gravity in the matter power spectrum, the details of which have not seen before in other studies in this area and therefore provides a crucial test for fourth order gravity on cosmological scales.
A fourth order generalized f(R) gravity theory (FOG) is considered with the Einstein-Hilbert action $R+aR^{2}+bR_{μν}R^{μν},$ $R_{μν}$ being Ricci\'{}s tensor and R the curvature scalar. The field equations are applied to spherical bodies where Newtonian gravity is a good approximation. The result is that for $0\leq a\sim -b<<R^{2}$, $R$ being the body radius, the gravitational field outside the body contains two Yukawas, one attractive and the other one repulsive, in addition to the Newtonian term. For $a\sim -b>>R^{2}$ the gravitational field near the body is zero but at distances greater th…
We revisit the classical stability of Schwarzschild black hole in fourth-order theories of gravity. On the contrary of the stability of this black hole, it turns out that the linearized perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of five-dimensional black string. This shows clearly an instability of black hole in fourth-order gravity with $α=-β/3$.
We construct a compact phase space for flat FLRW spacetimes with standard matter described by a perfect fluid with a barotropic equation of state for general f(R) theories of gravity, subject to certain conditions on the function f. We then use this framework to study the behaviour of the phase space of Universes with a non-negative Ricci scalar in R + αR^n gravity. We find a number of interesting cosmological evolutions which include the possibility of an initial unstable power-law inflationary point, followed by a curvature fluid dominated phase mimicking standard radiation, then passing thr…
We study thermodynamic and classical instability of AdS black holes in fourth-order gravity. These include the BTZ black hole in new massive gravity, Schwarzschild-AdS black hole, and higher-dimensional AdS black holes in fourth-order gravity. All thermodynamic quantities which are computed using the Abbot-Deser-Tekin method are used to study thermodynamic instability of AdS black holes. On the other hand, we investigate the $s$-mode Gregory-Laflamme instability of the massive graviton propagating around the AdS black holes. We establish the connection between the thermodynamic instability and…
We investigate whether or not an Einstein Static universe is a solution to the cosmological equations in $f(R)$ gravity. It is found that only one class of $f(R)$ theories admits an Einstein Static model, and that this class is neutrally stable with respect to vector and tensor perturbations for all equations of state on all scales. Scalar perturbations are only stable on all scales if the matter fluid equation of state satisfies $c_s^2>\frac{\sqrt{5}-1}{6}\approx 0.21$. This result is remarkably similar to the GR case, where it was found that the Einstein Static model is stable for $c_s^2>{1/…
We discuss Newtonian and the post-Newtonian limits of Fourth Order Gravity Theories pointing out, in details, their resemblances and differences with respect to General Relativity. Particular emphasis is placed on the exact solutions and methods used to obtain them.
Canonical analysis of a recently proposed [1] linear+quadratic curvature gravity model in D=3 displays its pure fourth derivative quadratic branch as a ghost-free (massless) excitation. Hence it both negates an old no-go theorem and is power-counting UV finite. It is also conformal-invariant, so the metric is underdetermined. While the 2-term branch is also ghost-free, it has, as shown in [1], a second-derivative, two-tensor equivalent, akin to the second order scalar-tensor form of ostensibly fourth order, $R+R^2$, actions. This correspondence fails for the pure quadratic branch: it is irredu…
The Lambda-CDM model is the best fit to cosmological data, and to the observed galactic rotation curves. However, in the absence of a direct detection of dark matter one should explore theories such as MOND, and perhaps also modified gravity theories like fourth order gravity and Scalar-Tensor-Vector Gravity [STVG] as possible explanations for the non-Keplerian behaviour of galaxy rotation curves. STVG has a modified law for gravitational acceleration which attempts to fit data by fixing two free parameters. We show that, remarkably, the biharmonic equation which we get in the weak field limit…
We consider the consequences of describing the metric properties of space- time through a quartic line element $ds^4=G_{μνλρ}dx^μdx^νdx^λdx^ρ$. The associated "metric" is a fourth-rank tensor $G_{μνλρ}$. We construct a theory for the gravitational field based on the fourth-rank metric $G_{μνλρ}$ which is conformally invariant in four dimensions. In the absence of matter the fourth-rank metric becomes of the form $G_{μνλρ}=g_{(μν}g_{λρ)}$ therefore we recover a Riemannian behaviour of the geometry; furthermore, the theory coincides with General Relativity. In the presence of matter we can keep …
We reconsider the cosmic string perturbative solution to the classical fourth-order gravity field equations, obtained in Ref.\cite{CLA94}, and we obtain that static, cylindricaly symmetric gauge cosmic strings, with constant energy density, can contain only $β$-terms in the first order corrections to the interior gravitational field, while the exact exterior solution is a conical spacetime with deficit angle $D=8πμ$.
PPN-limit of higher order theories of gravity represents a still controversial matter of debate and no definitive answer has been provided, up to now, about this issue. By exploiting the analogy between scalar-tensor and fourth-order theories of gravity, one can generalize the PPN-limit formulation. By using the definition of the PPN-parameters $γ$ and $β$ in term of the $f(R)$ derivatives, we show that a family of third-order polynomial theories, in the Ricci scalar $R$, turns out to be compatible with the PPN-limit and the deviation from General Relativity theoretically predicted agree with …
We study a class of almost scale-invariant modified gravity theories, using a particular form of $f(R, G) = αR^2 + βG \log G$ where $R$ and $G$ are the Ricci and Gauss-Bonnet scalars, respectively and $α$, $β$ are arbitrary constants. We derive the Einstein-like field equations to first order in cosmological perturbation theory in longitudinal gauge.
The cosmology of general fourth order corrections to Einstein gravity is considered, both for a homogeneous and isotropic background and for general tensor perturbations. It is explicitly shown how the standard cosmological history can be (approximately) reproduced and under what condition the evolution of the tensor modes remain (approximately) unchanged. Requiring that the deviations from General Relativity are small during inflation sharpens the current constraints on such corrections terms by some thirty orders of magnitude. Taking a more conservative approach and requiring only that cosmo…
In the framework of extended gravity theories, we discuss the meaning of a time dependent "cosmological constant" and give a set of conditions to recover asymptotic de Sitter behaviour for a class of cosmological models independently of initial data. To this purpose we introduce a time-dependent (effective) quantity which asymptotically becomes the true cosmological constant. We will deal with scalar-tensor, fourth and higher than fourth-order theories.
While the standard and most popular explanation for the flatness of galaxy rotation curves is dark matter, one cannot at this stage rule out an explanation based on a modified law of gravitation, which agrees with Newtonian gravitation on the scale of the solar system, but differs from it on larger length scales. Examples include Modfied Newtonian Dynamics [MOND] and Scalar-Tensor-Vector Gravity [STVG]. Here we report on a fourth order modification of the Poisson equation which yields the same Yukawa type modification of Newtonian gravity as STVG, and which can explain flat galaxy rotation cur…
We investigate the connection between dark energy and fourth order gravity by analyzing the behavior of scalar perturbations around a Friedmann-Robertson-Walker background. The evolution equations for scalar perturbation are derived using the covariant and gauge invariant approach and applied to two widely studied $f(R)$ gravity models. The structure of the general fourth order perturbation equations and the analysis of scalar perturbations lead to the discovery of a characteristic signature of fourth order gravity in the matter power spectrum, the details of which have not seen before in othe…
Einstein field equations are second-order differential equations. In this paper, we propose a new gravity theory with pure fourth-order field equations, which we call 4G for brevity. We discuss the applications of 4G in cosmology, gravitational waves, and local gravitational systems. 4G predicts the scale factor $a\propto t^{4/3}$ for the matter-dominated universe, and $a\propto t$ for the radiation-dominated universe. The former can explain the late-time acceleration without suffering from the coincidence and fine-tuning problems, while the latter can solve the horizon problem. 4G is a massle…
Building on earlier work, we discuss a general framework for exploring the cosmological dynamics of Higher Order Theories of Gravity. We show that once the theory of gravity has been specified, the cosmological equations can be written as a first-order autonomous system and we give several examples which illustrate the utility of our method. We also discuss a number of results which have appeared recently in the literature.
We determine the full post-Newtonian limit of theories of gravity that extend general relativity by replacing the Ricci scalar, R, in the generating Lagrangian by some analytic function, f(R). We restrict ourselves to theories that admit Minkowski space as a suitable background, and perform a perturbative expansion in the manner prescribed by the parameterised post-Newtonian formalism. Extra potentials are found to be present that are not accounted for in the usual treatment, and a discussion is provided on how they may be used to observationally distinguished these theories from general relat…
We revisit the old (fourth-order or quadratically generated) gravity model of Starobinsky in four space-time dimensions, and derive the (inflaton) scalar potential in the equivalent scalar-tensor gravity model via a Legendre-Weyl transform. The inflaton scalar potential is used to compute the (CMB) observables of inflation associated with curvature perturbations (namely, the scalar and tensor spectral indices, and the tensor-to-scalar ratio), including the new next-to-leading-order terms with respect to the inverse number of e-foldings. The results are compared to the recent (WMAP5) experiment…
We have earlier proposed a fourth order gravity model as a possible explanation for late time cosmic acceleration, and for flattened galaxy rotation curves. The model has a free length parameter whose value depends on the scale of the system under study (e.g. the whole Universe, a galaxy, or a compact binary pulsar). In the present work, we investigate the constraints imposed on the free model parameter by Hulse-Taylor binary pulsar data: periastron advance; and emission of gravitational waves and consequent period decay. It is shown that the model is consistent with these observations, provid…
For a general class of analytic $f(R,R_{αβ}R^{αβ},R_{αβγδ}R^{αβγδ})$ we discuss the gravitational lensing in the Newtonian Limit of theory. From the properties of Gauss Bonnet invariant it is successful to consider only two curvature invariants between the Ricci and Riemann tensor. Then we analyze the dynamics of photon embedded in a gravitational field of a generic $f(R,R_{αβ}R^{αβ})$-Gravity. The metric is time independent and spherically symmetric. The metric potentials are Schwarzschild-like, but there are two additional Yukawa terms linked to derivatives of $f$ with respect to two curvatu…
A fourth-order theory of gravity is considered which in terms of dynamics has the same degrees of freedom and number of constraints as those of scalar-tensor theories. In addition it admits a canonical point-like Lagrangian description. We study the critical points of the theory and we show that it can describe the matter epoch of the universe and that two accelerated phases can be recovered one of which describes a de Sitter universe. Finally for some models exact solutions are presented.
The fourth order Weyl gravity theory of Mannheim and Kazanas is based on replacing the Einstein-Hilbert action with the square of the Weyl tensor, and on modifying the matter action of the standard model of particle physics to make it conformally invariant. This theory has been suggested as a model of both dark matter and dark energy. We argue that the conformal invariance is not a fundamental property of the theory, and instead is an artifact of the choice of variables used in its description. We deduce that in the limit of weak fields and slow motions the theory does not agree with the predi…
We present an analysis of the classic wormhole geometries based on conformal Weyl gravity, rather than standard general relativity. The main characteristics of the resulting traversable wormholes remain the same as in the seminal study by Morris and Thorne, namely, that effective super-luminal motion is a viable consequence of the metric. Improving on previous work on the subject, we show that for particular choices of the shape and redshift functions the wormhole metric in the context of conformal gravity does not violate the main energy conditions at or near the wormhole throat. Some exotic …
The Big Bang initial singularity problem can be solved by means of bouncing solutions. In the context of extended theories of gravity, we will look for covariant effective actions whose field equations contain up to fourth-order derivatives of the metric tensor. In finding such bouncing solutions, we will make use of an order reduction technique based on a perturbative approach. Reducing the order of the field equations to second-order, we are able to find solutions which are perturbatively close to General Relativity. We will build the covariant effective actions of the resulting order reduce…
An alternate Hamiltonian H different from Ostrogradski's one is found for the Lagrangian L = L(q, \dot q, \ddot q). We add a suitable divergence to L and insert a=q and b=\ddot q. Contrary to other approaches no constraint is needed because \ddot a = b is one of the canonical equations. Another canonical equation becomes equivalent to the fourth-order Euler-Lagrange equation of L. Usually, H becomes quadratic in the momenta, whereas the Ostrogradski approach has Hamiltonians always linear in the momenta. For non-linear L=F(R), G=dF/dR \ne 0 the Lagrangians L and \hat L=\hat F(\hat R) with \hat…
We review the history of the ghost problem in quantum field theory from the Pauli-Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rathe…
We review the constraints modified theories of gravity must satisfy to be compatible with the spherically symmetric black hole solutions of semiclassical gravity that describe the formation of an apparent horizon in finite time of a distant observer. The constraints are satisfied in generic modified gravity theories with up to fourth-order derivatives in the metric, indicating that the semiclassical solutions correspond to zeroth-order terms in perturbative solutions of these models. From an observational point of view, this result implies that it may not be possible to distinguish between the…
After the previous work on gravitational frequency shift, light deflection (arXiv:1003.5296) and perihelion advance (arXiv:0812.2332), we calculate carefully the fourth gravity test, i.e. radar echo delay in a central gravity field surrounded by static free quintessence matter, in this paper. Through the Lagrangian method, we find the influence of the quintessence matter on the time delay of null particle is presence by means of an additional integral term. When the quintessence field vanishes, it reduces to the usual Schwarzschild case naturally. Meanwhile, we also use the data of the Viking …
In this work we study a modified theory of gravity that contains up to fourth order spatial derivatives as a model for the Horava-Lifshitz gravity. The propagator is evaluated and, as a result, it is obtained one extra pole corresponding to a spin two nonrelativistic massless particle, an extra term which jeopardizes renormalizability, besides the unexpected general relativity unmodified propagator. Then, unitarity is proved at the tree-level, where the general relativity pole has shown to have no dynamics, remaining only the two degrees of freedom of the new pole. Next, the nonrelativistic ef…
In this paper we analyze the Kerr geometry in the context of Conformal Gravity, an alternative theory of gravitation, which is a direct extension of General Relativity. Following previous studies in the literature, we introduce an explicit expression of the Kerr metric in Conformal Gravity, which naturally reduces to the standard General Relativity Kerr geometry in the absence of Conformal Gravity effects. As in the standard case, we show that the Hamilton-Jacobi equation governing geodesic motion in a space-time based on this geometry is indeed separable and that a fourth constant of motion-s…
Recently, a strong debate has been pursued about the Newtonian limit (i.e. small velocity and weak field) of fourth order gravity models. According to some authors, the Newtonian limit of $f(R)$-gravity is equivalent to the one of Brans-Dicke gravity with $ω_{BD} = 0$, so that the PPN parameters of these models turn out to be ill defined. In this paper, we carefully discuss this point considering that fourth order gravity models are dynamically equivalent to the O'Hanlon Lagrangian. This is a special case of scalar-tensor gravity characterized only by self-interaction potential and that, in th…
We discuss weak lensing characteristics in the gravitational field of a compact object in the low-energy approximation of fourth order f(R) gravity theory. The particular solution is characterized by a gravitational strength parameter $σ$ and a distance scale $r_{c}$ much larger than the Schwarzschild radius. Above $r_{c}$ gravity is strengthened and as a consequence weak lensing features are modified compared to the Schwarzschild case. We find a critical impact parameter (depending upon $r_{c}$) for which the behavior of the deflection angle changes. Using the Virbhadra-Ellis lens equation we…
We study a broad class of isotropic vacuum cosmologies in fourth-order gravity under the condition that the gravitational Lagrangian be scale-invariant or almost scale-invariant. The gravitational Lagrangians considered will be of the form L = f(R) + k(G) where R and G are the Ricci and Gauss-Bonnet scalars respectively. Specifically we take f(R) = R^2n and k(G) = G^n or k(G) = G ln G. We find solutions in closed form for a spatially flat Friedmann space-time and interpret their asymptotic early-time and late-time behaviour as well as their inflationary stages. One unique example which we disc…
The gravitational collapse in fourth order theories of gravity defined by an arbitrary action of the scalar curvature shows significant deviations with General Relativity. The presence of a new scalar mode produces a higher initial contraction that favors the reduction of the collapsing time. However, depending on the particular model, there are fundamental differences when the modifications to the General Relativity collapse leave the linear regime. These analyses can be used to exclude an important region of the parameter space associated with alternative gravitational models.
We investigate a fourth order model of gravity, having a free length parameter, and no cosmological constant or dark energy. We consider cosmological evolution of a flat Friedmann universe in this model for the case that the length parameter is of the order of present Hubble radius. By making a suitable choice for the present value of the Hubble parameter, and value of third derivative of the scale factor (the jerk) we find that the model can explain cosmic acceleration to the same degree of accuracy as the standard concordance model. If the free length parameter is assumed to be time-dependen…
