In this paper we investigate a complex symmetric generalization of general relativity and in particular we investigate its linearized field equations. We begin by reviewing some basic definitions and structures in Moffat's symmetric complex metric field theory of gravity. We then move on to derive the linearized retarded complex field equations. In addition to this we also derive a linearization of Moffat's field equations based on the more rigorous Fermi coordinate approach. In conclusion it is shown that the linearized symmetric complex field equations leads to a complex form of gravitomagnetism. We also briefly review the gravitational wave equation from the source less linearized symmetric complex field equations and discuss some open problems.
In the standard scenario of the Newtonian gravity, a late-type galaxy (i.e., a spiral galaxy) is well described by a disk and a bulge embedded in a halo mainly composed by dark matter. In Brownstein-Moffat gravity, there is a claim that late-type galaxy systems would not need to have halos, avoiding as a result the dark matter problem, i.e., a modified gravity (non-Newtonian) would account for the galactic structure with no need of dark matter. In the present paper, we probe this claim via numerical simulations. Instead of using a "static galaxy," where the centrifugal equilibrium is usually a…
In a recent paper, we discussed the formation of black holes in non-symmetric gravity. That paper was then criticized by Cornish and Moffat. In the present paper, we address the arguments raised by Cornish and Moffat. In summary, we do not see any reason to doubt the validity of our former conclusions.
Scalar Tensor Vector Gravity(STVG) is a fully covariant Lorentz invariant alternative theory of gravity also called as MOdified Gravity(MOG) which modifies General Relativity(GR) with dynamical massive vector field and scalar fields. In STVG the mass $μ$ of the vector field $ϕ$ and the Universal gravitational constant G enjoys the status of a dynamical field. We use the reconstructed total cluster mass obtained from X-ray observations by XMM-Newton telescope in combination with the Sunyaev-Zel'dovich(SZ) effect observed within Planck all-sky survey to estimate the parameters $ α$ and $μ$ with …
Scalar Tensor Vector Gravity(STVG) is one of modified gravity theories developed by John Moffat(2005). MOG is abbreviated name for this theory.It can explain a galactic rotation curve and the structure formation without dark matter. It can also explain acceleration universe without dark energy.But,they obtaion only a spherically symmetric, static vacuum solution about MOG. On this theory,the gravitational field produced by two point sources is not simply the sum of their respective spherically symmetric static vacuum solutions. However,in arXiv:0805.4774, the method to adapt MOG to extended di…
We study the gravitomagnetism in the Scalar-Vector-Tensor theory or Moffat's Modified theory of Gravity(MOG). We compute the gravitomagnetic field that a slow-moving mass distribution produces in its Newtonian regime. We report that the consistency between the MOG gravitomagnetic field and that predicted by the Einstein's gravitional theory and measured by Gravity Probe B, LAGEOS and LAGEOS 2, and with a number of GRACE and Laser Lunar ranging measurements requires $|α| < 0.0013$. We provide a discussion.
J. W. Moffat and V. T. Toth submitted recently a comment (arXiv:0903.5291) on our latest paper "Modified scalar-tensor-vector gravity theory and the constraint on its parameters" [Deng, et al., Phys. Rev. D 79, 044014 (2009); arXiv:0901.3730 ]. We reply to each of their comments and justify our work and conclusions. Especially, their general STVG (MOG) theory has to be modified to fit the modern precision experiments.
We investigate the contributions of the Friedmann-Lemaître-Robertson-Walker metric of the standard cosmology as an asymptotic boundary condition on the first-order approximation of the gravitational field in Moffat's theory of modified gravity (MOG). We also consider contributions due to the fact that the MOG theory does not satisfy the shell theorem or Birkhoff's theorem, resulting in what is known as the "external field effect" (EFE). We show that while both these effects add small contributions to the radial acceleration law, the result is orders of magnitude smaller than the radial acceler…
A solution to the gravitational field equations based on a non-symmetric metric tensor is examined. Unlike Einstein's interpretation of electromagnetism, or Moffat's generalized gravity, it is shown that the non-symmetric part of the metric tensor is the potential of the spin field. This is in agreement with string theory and provides a natural coupling between gravitation and strings.
It is well known that linearized gravity in spacetimes with compact Cauchy surfaces and continuous symmetries suffers from linearization instabilities: solutions to classical linearized gravity in such a spacetime must satisfy so-called linearization stability conditions (or constraints) for them to extend to solutions in the full non-linear theory. Moncrief investigated implications of these conditions in linearized quantum gravity in such background spacetimes and found that the quantum linearization stability constraints lead to the requirement that all physical states must be invariant und…
In a nonlinear theory, such as gravity, physically relevant solutions are usually hard to find. Therefore, starting from a background exact solution with symmetries, one uses the perturbation theory, which albeit approximately, provides a lot of information regarding a physical solution. But even this approximate information comes with a price: the basic premise of a perturbative solution is that it should be improvable. Namely, by going to higher order perturbation theory, one should be able to improve and better approximate the physical problem or the solution. While this is often the case i…
Carrying out an analysis of the constraints and their linearizations on a spacelike hypersurface, we show that topologically massive gravity has a linearization instability at the chiral gravity limit about $AdS_3$. We also calculate the symplectic structure for all the known perturbative modes (including the log-mode) for the linearized field equations and find it to be degenerate (non-invertible) hence these modes do not approximate exact solutions and so do not belong to the linearized phase space of the theory. Naive perturbation theory fails: the linearized field equations are necessary b…
In general relativity, perturbation theory about a background solution fails if the background spacetime has a Killing symmetry and a compact spacelike Cauchy surface. This failure, dubbed as {\it linearization instability}, shows itself as non-integrability of the perturbative infinitesimal deformation to a finite deformation of the background. Namely, the linearized field equations have spurious solutions which cannot be obtained from the linearization of exact solutions. In practice, one can show the failure of the linear perturbation theory by showing that a certain quadratic (integral) co…
We construct the gauge-invariant electric and magnetic charges in Yang-Mills theory coupled to cosmological General Relativity (or any other geometric gravity), extending the flat spacetime construction of Abbott and Deser. For non-vanishing background gauge fields, the charges receive non-trivial contribution from the gravity part. In addition, we study the constraints on the first order perturbation theory and establish the conditions for linearization instability: that is the validity of the first order perturbation theory.
The gravitational wave solutions obtained from a perturbation about conformally flat backgrounds in Einstein gravity are investigated. A perturbation theory analysis of the Lesame, Ellis and Dunsby results, based on a covariant approach, shows that for gravitational waves interacting with irrotational dust, the equations are linearization unstable. The gravitational wave equations based on the Weyl curvature tensor must be solved by non-perturbative methods. The significance of this result for gravitational wave calculations and experiments is discussed.
We study the question of whether the linearization of the Kodama state around classical deSitter spacetime is normalizable in the inner product of the theory of linearized gravitons on deSitter spacetime. We find the answer is no in the Lorentzian theory. However, in the Euclidean theory the corresponding linearized Kodama state is delta-functional normalizable. We discuss whether this result invalidates the conjecture that the full Kodama state is a good physical state for quantum gravity with positive cosmological constant.
The status of canonical reduction for metric and tetrad gravity in space-times of the Christodoulou-Klainermann type, where the ADM energy rules the time evolution, is reviewed. Since in these space-times there is an asymptotic Minkowski metric at spatial infinity, it is possible to define a Hamiltonian linearization in a completely fixed (non harmonic) 3-orthogonal gauge without introducing a background metric. Post-Minkowskian background-independent gravitational waves are obtained as solutions of the linearized Hamilton equations.
In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of {\it non-harmonic} 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) $r_{\bar a}(τ,\vec σ)$, $π_{\bar a}(τ,\vec σ)$, $\bar a = 1,2$. We define a Hamiltonian linearization of the theory, i.e. gravitational waves, {\it without introducing any background 4-metric}…
Recently, Brownstein and Moffat proposed a gravitational mechanism to explain the Pioneer anomaly based on their scalar-tensor-vector (STVG) metric theory of gravity. In this paper we show that their model, fitted to the presently available data for the anomalous Pioneer 10/11 acceleration, is in contrast with the latest determinations of the perihelion extra-rates of Jupiter, Saturn and Uranus.
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatible to a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrary distinguished connections, we define the class of distinguished line…
In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial value problem for $f(R)$ gravity. We consider, in particular, the vacuum field equations of Metric $f(R)$ gravity in the Jordan frame, without utilising the dynamical equivalence with scalar-tensor theories. We present the full hierarchy of non-linear hypersurface and evolution equations necessary for numerical implementation in both tensorial and eth forms. Fur…
The hypersurface deformation algebra consists in a fruitful approach to derive deformed solutions of general relativity based on symmetry considerations with quantum gravity effects, whose linearization has been recently demonstrated to be connected to the DSR program by the $κ$-Poincaré symmetry. Based on this approach, we analyzed the solution derived for the interior of a black hole and we found similarities with the, so called, rainbow metrics, like a momentum-dependence of the metric functions. Moreover, we derived an effective, time-dependent Planck length and compared different regulari…
We perform the phase space analysis in terms of the linearization technique in the Hořava-Lifshitz gravity with the softly broken detailed balance condition. It can be shown that the bouncing universe appears only for the positive spatial curvature of $k=+1$, and it is possible to obtain oscillating universe with the help of the negative dark radiation and the negative cosmological constant.
A clock synchronization thought experiment is modeled by a diffeomorphism invariant "time delay" observable. In a sense, this observable probes the causal structure of the ambient Lorentzian spacetime. Thus, upon quantization, it is sensitive to the long expected smearing of the light cone by vacuum fluctuations in quantum gravity. After perturbative linearization, its mean and variance are computed in the Minkowski Fock vacuum of linearized gravity. The naïve divergence of the variance is meaningfully regularized by a length scale $μ$, the physical detector resolution. This is the first time …
Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). Since computational solution of differential equations introduces almost inevitable errors, it is clearly "more correct" to introduce a scheme which actively maintains the constraints by solution ({\it constrained} evolution). This…
We derive an initial value formulation for dynamical Chern-Simons gravity, a modification of general relativity involving parity-violating higher derivative terms. We investigate the structure of the resulting system of partial differential equations thinking about linearization around arbitrary backgrounds. This type of consideration is necessary if we are to establish well-posedness of the Cauchy problem. Treating the field equations as an effective field theory we find that weak necessary conditions for hyperbolicity are satisfied. For the full field equations we find that there are states …
Perturbative gravity about global de Sitter space is subject to linearization-stability constraints. Such constraints imply that quantum states of matter fields couple consistently to gravity {\it only} if the matter state has vanishing de Sitter charges; i.e., only if the state is invariant under the symmetries of de Sitter space. As noted by Higuchi, the usual Fock spaces for matter fields contain no de Sitter-invariant states except the vacuum, though a new Hilbert space of de Sitter invariant states can be constructed via so-called group-averaging techniques. We study this construction for…
In this second paper we define a Post-Minkowskian (PM) weak field approximation leading to a linearization of the Hamilton equations of ADM tetrad gravity in the York canonical basis in a family of non-harmonic 3-orthogonal Schwinger time gauges. The York time ${}^3K$ (the relativistic inertial gauge variable, not existing in Newtonian gravity, parametrizing the family and connected to the freedom in clock synchronization, i.e. to the definition of the instantaneous 3-spaces) is put equal to an arbitrary numerical function. A ultraviolet cutoff on the matter allows a consistent linearization, …
The pure $R^2$ gravity is equivalent to Einstein gravity with cosmological constant and a massless scalar field and it further possesses the so-called restricted Weyl symmetry which is a symmetry larger than scale symmetry. To incorporate matter, we consider a restricted Weyl invariant action composed of pure $R^2$ gravity, SU(2) Yang-Mills fields and a non-minimally coupled massless Higgs field (a triplet of scalars). When the restricted Weyl symmetry is spontaneously broken, it is equivalent to an Einstein-Yang-Mills-Higgs (EYMH) action with a cosmological constant and a massive Higgs non-mi…
Bigravity is a natural arena where a non-linear theory of massive gravity can be formulated. If the interaction between the metrics $f$ and $g$ is non-derivative, spherically symmetric exact solutions can be found. At large distances from the origin, these are generically Lorentz-breaking bi-flat solutions (provided that the corresponding vacuum energies are adjusted appropriately). The spectrum of linearized perturbations around such backgrounds contains a massless as well as a massive graviton, with {\em two} physical polarizations each. There are no propagating vectors or scalars, and the t…
We argue that the Einstein gravity theory can be reformulated in almost Kahler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation (NGT) on manifolds enabled with nonholonomic distributions is analyzed. There are considered some conditions when the fundamental geometric and physical objects are determined/ modified by nonholonomic deformations in general relativity or by contributions from Ricci flow theory and/or quantum gravity. We prove that in such NGT, for…
We demonstrate that the emergence of a curved spacetime ``effective Lorentzian geometry'' is a common and generic result of linearizing a field theory around some non-trivial background. This investigation is motivated by considering the large number of ``analog models'' of general relativity that have recently been developed based on condensed matter physics, and asking whether there is something more fundamental going on. Indeed, linearization of a classical field theory (a field theoretic ``normal mode analysis'') results in fluctuations whose propagation is governed by a Lorentzian-signatu…
The linearization of semiclassical theories of gravity is investigated in a toy model, consisting of a quantum scalar field in interaction with a second classical scalar field which plays the role of a classical background. This toy model mimics also the evolution induced by semiclassical Einstein equations, such as the one which describes the early universe in the cosmological case. The equations governing the dynamics of linear perturbations around simple exact solutions of this toy model are analyzed by constructing the corresponding retarded fundamental solutions, and by discussing the cor…
We identify a simple physical mechanism which is at the heart of Asymptotic Safety in Quantum Einstein Gravity (QEG) according to all available effective average action-based investigations. Upon linearization the gravitational field equations give rise to an inverse propagator for metric fluctuations comprising two pieces: a covariant Laplacian and a curvature dependent potential term. By analogy with elementary magnetic systems they lead to, respectively, dia- and paramagnetic-type interactions of the metric fluctuations with the background gravitational field. We show that above 3 spacetime…
We find the Hamiltonian expression in the York basis of canonical ADM tetrad gravity of the 4-Weyl tensor of the asymptotically Minkowskian space-time. Like for the 4-Riemann tensor we find a radar tensor (whose components are 4-scalars due to the use of radar 4-coordinates), which coincides with the 4-Weyl tensor on-shell on the solutions of Einstein's equations. Then, by using the Hamiltonian null tetrads, we find the Hamiltonian expression of the Weyl scalars of the Newman-Perose approach and of the four eigenvalues of the 4-Weyl tensor. After having introduced the Dirac observables of cano…
We discuss the linearization of Einstein equations in the presence of a cosmological constant, by expanding the solution for the metric around a flat Minkowski space-time. We demonstrate that one can find consistent solutions to the linearized set of equations for the metric perturbations, in the Lorentz gauge, which are not spherically symmetric, but they rather exhibit a cylindrical symmetry. We find that the components of the gravitational field satisfying the appropriate Poisson equations have the property of ensuring that a scalar potential can be constructed, in which both contributions,…
We study the coupling of N charged scalar particles plus the electro-magnetic field to ADM tetrad gravity and its canonical formulation in asymptotically Minkowskian space-times without super-translations. We make the canonical transformation to the York canonical basis, where there is a separation between the {\it inertial} (gauge) variables and the {\it tidal} ones inside the gravitational field and a special role of the Eulerian observers associated to the 3+1 splitting of space-time. The Dirac Hamiltonian is weakly equal to the weak ADM energy. The Hamilton equations in Schwinger time gaug…
In a nonlinear theory, such as General Relativity, linearized field equations around an exact solution are necessary but not sufficient conditions for linearized solutions. Therefore, the linearized field equations can have some solutions which do not come from the linearization of possible exact solutions. This fact can make the perturbation theory ill-defined, which would be a problem both at the classical and semiclassical quantization level. Here we study the first and second order perturbation theory in cosmological Einstein gravity and give the explicit form of the integral constraint, w…
