Nonlinear Gravity Theories Research Articles

Asymptotically nonlocal gravity

Asymptotically nonlocal field theories interpolate between Lee-Wick theories with multiple propagator poles, and ghost-free nonlocal theories. Previous work on asymp- totically nonlocal scalar, Abelian, and non-Abelian gauge theories has demonstrated the existence of an emergent regulator scale that is hierarchically smaller than the lightest Lee-Wick partner, in a limit where the Lee-Wick spectrum becomes dense and decoupled. We generalize this construction to linearized gravity, and demonstrate the emergent regula- tor scale in three examples: by studying the resolution of the singularity (i) at the origin in the classical solution for the metric of a point particle, and (ii) in the nonrelativistic gravitational potential computed via a one-graviton exchange amplitude; (iii) we also show how this derived scale regulates the one-loop graviton contribution to the self energy of a real scalar field. We comment briefly on the generalization of our approach to the full, nonlinear theory of gravity.

Lorentz-violating gravity and the bootstrap procedure

In conventional gravitational physics, the so-called ‘bootstrap procedure’ can be used to extrapolate from a linear model of a rank-2 tensor to a full non-linear theory of gravity (i.e. general relativity) via a coupling to the stress-energy of the model. In this work, I extend this procedure to a ‘Lorentz-violating’ gravitational model, in which the linear tensor field and the matter fields ‘see’ different metrics due to a coupling between the tensor field and a background vector field. The resulting model can be thought of as a generalized Proca theory with a non-minimal coupling to conventional matter. It has a similar linearized limit to the better-known ‘bumblebee model’, but differs at higher orders in perturbation theory. Its effects are unobservable in FRW spacetimes, but are expected to be important in anisotropic cosmological spacetimes.

Test-field limit of metric nonlinear gravity theories

In the framework of alternative metric gravity theories, it has been shown by several authors that a generic Lagrangian depending on the Riemann tensor describes a theory with 8 degrees of freedom (which reduce to 3 for f(R) Lagrangians depending only on the curvature scalar). This result is often related to a reformulation of the fourth-order equations for the metric into a set of second-order equations for a multiplet of fields, including – besides the metric – a massive scalar field and a massive spin-2 field (the latter being usually regarded as a ghost): this is commonly assumed to represent the particle spectrum of the theory. In this article we investigate an issue which does not seem to have been addressed so far: in ordinary general-relativistic field theories, all fundamental fields (i.e. fields with definite spin and mass) reduce to test fields in some appropriate limit of the model, where they cease to act as sources for the metric curvature. In this limit, each of the fundamental fields can be excited from its ground state independently from the others (which does not happen, instead, as long as the fields are coupled through the gravitational interaction). We thus address the following question: does higher-derivative gravity admit a test-field limit for its fundamental fields? It is easy to show that for a generic f(R) theory (carrying 3 d.o.f.) the test-field limit does exist; then, we consider the case of Lagrangians depending on the full Ricci tensor, relying on a previous analysis published several years ago. We show that, already for a quadratic Lagrangian, the constraint binding together the scalar field and the massive spin-2 field does not disappear in the limit where they should be expected to act as test fields. A proper test-field limit exists only for a particular choice of the coefficients in the Lagrangian, which cause the scalar field to disappear (so that the resulting model has only 7 d.o.f.). We finally consider the possible addition of an arbitrary function of the quadratic invariant of the Weyl tensor, C^{alpha }{}_{beta mu nu }C_{alpha }{}^{beta mu nu }, showing that the appearance of the Weyl tensor does not add physical degrees of freedom (in accordance to the known results for Lagrangians depending on the full Riemann tensor) and the resulting model with 8 d.o.f. still lacks a proper test-field limit: the differential constraints between the fundamental fields do not cancel out when gravitational interaction is suppressed. We argue that the lack of a test-field limit for the dynamics of the fundamental fields may constitute a serious drawback of the full 8 d.o.f. higher-order gravity models, which is not encountered in the restricted 7 d.o.f. or 3 d.o.f. cases.

Mapping nonlinear gravity into General Relativity with nonlinear electrodynamics

We show that families of nonlinear gravity theories formulated in a metric-affine approach and coupled to a nonlinear theory of electrodynamics can be mapped into general relativity (GR) coupled to another nonlinear theory of electrodynamics. This allows to generate solutions of the former from those of the latter using purely algebraic transformations. This correspondence is explicitly illustrated with the Eddington-inspired Born–Infeld theory of gravity, for which we consider a family of nonlinear electrodynamics and show that, under the map, preserve their algebraic structure. For the particular case of Maxwell electrodynamics coupled to Born–Infeld gravity we find, via this correspondence, a Born–Infeld-type nonlinear electrodynamics on the GR side. Solving the spherically symmetric electrovacuum case for the latter, we show how the map provides directly the right solutions for the former. This procedure opens a new door to explore astrophysical and cosmological scenarios in nonlinear gravity theories by exploiting the full power of the analytical and numerical methods developed within the framework of GR.

Absence of cosmological constant problem in special relativistic field theory of gravity

The principles of quantum field theory in flat spacetime suggest that gravity is mediated by a massless particle with helicity ±2, the so-called graviton. It is regarded as textbook knowledge that, when the self-coupling of a particle with these properties is considered, the long-wavelength structure of such a nonlinear theory is fixed to be that of general relativity. However, here we indicate that these arguments conceal an implicit assumption which is surreptitiously motivated by the very knowledge of general relativity. This is shown by providing a counterexample: we revisit a nonlinear theory of gravity which is not structurally equivalent to general relativity and that, in the non-interacting limit, describes a free helicity ±2 graviton. We explicitly prove that this theory, known as Weyl-transverse gravity (unimodular gravity with explicit Weyl invariance), can be understood as the result of self-coupling in complete parallelism to the well-known case of general relativity. We discuss the absence of cosmological constant problem in this theory, highlighting that it provides a particular realization of previous arguments formulated in studies of the emergence of the gravitational interaction from condensed-matter-like models. Overall, we conclude that the consideration that gravity is mediated by a massless particle with helicity ±2 does not inextricably lead to the cosmological constant problem.

G-theory: The generator of M-theory and supersymmetry

In string theory with ten dimensions, all Dp-branes are constructed from D0-branes whose action has two-dimensional brackets of Lie 2-algebra. Also, in M-theory, with 11 dimensions, all Mp-branes are built from M0-branes whose action contains three-dimensional brackets of Lie 3-algebra. In these theories, the reason for difference between bosons and fermions is unclear and especially in M-theory there is not any stable object like stable M3-branes on which our universe would be formed on it and for this reason it cannot help us to explain cosmological events. For this reason, we construct G-theory with M dimensions whose branes are formed from G0-branes with N-dimensional brackets. In this theory, we assume that at the beginning there is nothing. Then, two energies, which differ in their signs only, emerge and produce 2M degrees of freedom. Each two degrees of freedom create a new dimension and then M dimensions emerge. M-N of these degrees of freedom are removed by symmetrically compacting half of M-N dimensions to produce Lie-N-algebra. In fact, each dimension produces a degree of freedom. Consequently, by compacting M-N dimensions from M dimensions, N dimensions and N degrees of freedom is emerged. These N degrees of freedoms produce Lie-N-algebra. During this compactification, some dimensions take extra i and are different from other dimensions, which are known as time coordinates. By this compactification, two types of branes, Gp and anti-Gp-branes, are produced and rank of tensor fields which live on them changes from zero to dimension of brane. The number of time coordinates, which are produced by negative energy in anti-Gp-branes, is more sensible to number of times in Gp-branes. These branes are compactified anti-symmetrically and then fermionic superpartners of bosonic fields emerge and supersymmetry is born. Some of gauge fields play the role of graviton and gravitino and produce the supergravity. The question may arise that what is the physical reason which shows that this theory is true. We shown that G-theory can be reduced to other theories like nonlinear gravity theories in four dimensions. Also, this theory, can explain the physical properties of fermions and bosons. On the other hand, this theory explains the origin of supersymmetry. For this reason, we can prove that this theory is true. By reducing the dimension of algebra to three and dimension of world to 11 and dimension of brane to four, G-theory is reduced to F(R)-gravity.

Absence of cosmological constant problem in special relativistic field theory of gravity: one-loop renormalization group

There exists a nonlinear theory of gravity which is not structurally equivalent to general relativity and that, in the non-interacting limit, describes a free massless particle with helicity ±2. We have recently shown that this theory can be understood as the result of self-coupling, in complete parallelism to the well-known case of general relativity. This special relativistic field theory of gravity exhibits a decoupling of vacuum zero-point energies of matter and passes all the known experimental tests in gravitation. It is explicitly demonstrated that there is no flow of the effective cosmological constant under the action of the renormalization group at one-loop level, while simple symmetry arguments show that this would continue to be true for higher-loop corrections. The important lesson is that just mild local assumptions concerning the nature of the particle mediating the gravitational interactions are enough to motivate theories which are free of the cosmological constant problem.

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called “Natural Theories” and “Gauge-Natural Theories” that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincar´e–Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as “generators of Noether symmetries”, energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-calledADMlaws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation “à la Palatini” and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero–Immirzi connections).

On the abuse of gravity theories in cosmology

I present briefly some critical remarks on recent applications of gravity theories in cosmology. Einstein’s general relativity (GR) is exceptional among physical theories: it is formally merely a point in the huge space of all existing and conceivable theories of gravitation. Most interesting of them are metric nonlinear gravity (NLG) theories which differ from GR only in equations of motion: gravity is described by one unifying tensor field which in the initial formulation of the theories is interpreted as a spacetime metric with a Lagrangian L = f(gαβ ,R αβμν) ,w heref is any smooth scalar function. The simplest of them, restricted NLG theories based on L = f(R), have attracted most attention. In cosmology they are used to provide a generalized Friedmann equation for the cosmic scale function in spatially flat Robertson–Walker (R–W) spacetime to account for the accelerated expansion of the universe without resorting to dark energy. Why to employ to this aim NLG theories instead of assuming that the cosmic acceleration is driven in GR by a kind of self-interacting scalar, vector or tensor field? The standard response is that one must then introduce ad hoc some self-interaction of the field, yet applying an NLG theory one may find approximate solutions which well fit the cosmic evolution by making appropriate approximations in the underlying Lagrangian. Let L =1 /R + R + R 2 , then in the early universe with strong curvature the R 2 term is dominant giving rise to some kind of inflation [1]. In the intermediate curvature period (radiation and galactic eras) the R term dominates and the standard cosmological model is valid. At present and in future the decreasing curvature makes the 1/R term dominant what results in an accelerated expansion. This kind of argument is explicitly or implicitly present in most works on NLG theories. This line of argument, though suggestive, is actually misleading. One should investigate any NLG theory from the viewpoint of field theory and search for its predictions in the whole realm of gravitational physics. A success of a theory only in flat R–W spacetime is by far not enough.

Metric gravity theories and cosmology: II. Stability of a ground state in f(R) theories

In the second part of the investigation of metric nonlinear gravity theories, we study a fundamental criterion of viability of any gravity theory: the existence of a stable ground-state solution being either Minkowski, de Sitter or anti-de Sitter space. Stability of the ground state is independent of which frame is physical. In general, a given theory has multiple ground states and splits into independent physical sectors. The fact that all L = f(gαβ, Rμν) gravity theories (except some singular cases) are dynamically equivalent to Einstein gravity plus a massive spin-2 and a massive scalar field allows us to investigate the stability problem using methods developed in general relativity. These methods can be directly applied to L = f(R) theories wherein the spin-2 field is absent. Furthermore for these theories which have anti-de Sitter space as the ground state we prove a positive-energy theorem allowing to define the notion of conserved total gravitational energy in the Jordan frame (i.e., for the fourth-order equations of motion). As is shown in 13 examples of specific Lagrangians the stability criterion works effectively without long computations whenever the curvature of the ground state is determined. An infinite number of gravity theories have a stable ground state and further viability criteria are necessary.

NONLOCAL THEORY OF GRAVITY

We use a three-index tensor Aμνα to describe spin-2 fields. This quantity acts as the potential of the standard variable hμν. We apply the new variables to examine the coherence of a nonlocal nonlinear theory of gravity.

Physical equivalence between nonlinear gravity theories and a general-relativistic self-gravitating scalar field.

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.

Gravitational anyons, chern-simons-witten gravity and the gravitational aharonov-bohm effect

The anyonic behaviour of massive point particles in the (2 + 1)-dimensional theory of topologically massive gravity, as first suggested by Deser, is further investigated. At the linearised level, we recall that the exotic statistics of the particles may be understood in terms of the gravitational Aharonov-Bohm effect discussed by several authors in the 1960's. This approach suggests that massive spinning point particles in ordinary Einstein gravity may also exhibit exotic statistics, and that it is the Einstein term which takes on the role carried out by the Chern-Simons term in the standard U(1) anyon model. We investigate this further in the Chern-Simons-Witten formulation of 2 + 1 gravity, where the Einstein term is indeed a Chern-Simons term. In this context the exotic statistics may be understood in terms of a non-abelian Aharonov-Bohm effect, giving a description in terms of the fully non-linear theory of gravity, and making explicit the relationship between gravitational anyons and Chern-Simons anyons of (2 + 1)-dimensional U(1) gauge theory. The analogue of magnetic flux in this context is seen to be both torsion and Riemann curvature. We finally show the correspondence between the statistics phase obtained in this approach and that obtained by Deser and McCarthy in the linearised approximation of topologically massive gravity.

Physical versions of nonlinear gravity theories and positivity of energy

Brans (1988) has recently criticised claims that nonlinear metric theories of gravity are physically equivalent to general relativity with additional fields. The author agrees with his opinion that the physical metric is determined by its interaction with matter but emphasise that at present the available data are too scarce for it. He argues that for the theories which are mathematically equivalent under a conformal rescaling of the metric (this is the case of Lagrangian solely depending on the scalar curvature) a unique choice of physical variables can be based on a theoretical criterion-that of positivity of energy.