We show that the extended cosmological equation-of-state developed starting from a Chaplygin equation-of-state, recently applied to stellar modeling, is a viable dark energy model consistent with standard scalar potentials. Moreover we find a Lagrangian formulation based on a canonical scalar field with the appropriate self-interaction potential. Finally, we fit the scalar potential obtained numerically with concrete functions well studied in the literature. Our results may be of interest to model builders and particle physicists.
We consider the general scalar field Horndeski Lagrangian coupled to matter. Within this class of models, we present two results that are independent of the particular form of the model. First, we show that in a Friedmann-Robertson-Walker metric the Horndeski Lagrangian coincides with the pressure of the scalar field. Second, we employ the previous result to identify the most general form of the Lagrangian that allows for cosmological scaling solutions, i.e. solutions where the ratio of matter to field density and the equation of state remain constant. Scaling solutions of this kind may help s…
In self-tuning brane-world models with extra dimensions, large contributions to the cosmological constant are absorbed into the curvature of extra dimensions and consistent with flat 4d geometry. In models with conventional Lagrangians fine-tuning is needed nevertheless to ensure a finite effective Planck mass. Here, we consider a class of models with non conventional Lagrangian in which known problems can be avoided. Unfortunately these models are found to suffer from tachyonic instabilities. An attempt to cure these instabilities leads to the prediction of a positive cosmological constant, w…
We work on the Lagrangian formulation of the Palatini action. We find that we must assume the metric compatibility condition for the Palatini action to describe General Relativity, which condition should hold in quantization. We find that we must also assume one of the torsion zero condition or the tetrad compatibility condition. Our results will hold for any action in terms of the tetrad and the internal connection which describes General Relativity.
We investigate a possible connection between Galileon gravity and teleparallel gravity. We also propose a new type of second order cosmological lagrangian and study a some of its properties.
We work on the Lagrangian and the Hamiltonian formulations of the Palatini action. In the Lagrangian formulation, we find that we need to assume the metric compatibility and the torsion zero or to assume the tetrad compatibility to describe General Relativity. In the Hamiltonian formulation, we obtain the Einstein's equations only with assuming the tetrad compatibility. The Hamiltonian from assuming the metric compatibility and the torsion zero should be used to quantize General Relativity.
By dimensional reduction of the Lovelock Lagrangian, effective four-dimensional field equations and expressions for the cosmological and gravitational constants are obtained. A cosmological model is discussed with the n-torus as internal space.
We study a general scalar field Lagrangian coupled with matter and linear in $\Boxϕ$ (also called KGB model). Within this class of models, we find the most general form of the Lagrangian that allows for cosmological scaling solutions, i.e. solutions where the ratio of matter to field density and the equation of state remain constant. Scaling solutions of this kind may help solving the coincidence problem since in this case the presently observed ratio of matter to dark energy does not depend on initial conditions, but rather on the theoretical parameters. Extending previous results we find tha…
A covariant Lagrangian formulation of a solution to the cosmological constant problem, based on vizualising the fluctuations of the vacuum energy as a non-equilibrium process with stochastic behaviour, is presented. The variational principle yields equations of motion for the cosmological ``constant" $Λ$, treated as a dynamical field, together with an equation for a Lagrange multiplier field $ϕ$, and the standard Einstein field equations with a variable cosmological constant term. A stochastic model of $Λ$ yields a natural explanation for the smallness or zero value of the constant in the pres…
We shall here discuss a new spacetime gauge-covariant Lagrangian formulation of General Relativity by means of the Barbero-Immirzi SU(2)-connection on spacetime. To the best of our knowledge the Lagrangian based on SU(2) spacetime fields seems to appear here for the first time.
We derive the Ashtekar-Barbero variables in Loop Quantum Gravity(LQG) starting from the Palatini action. We find that we need either the torsion-free condition or the compatibility condition for the Palatini action to describe General Relativity.
In this paper we show that the on-shell Lagrangian of a perfect fluid depends on microscopic properties of the fluid, giving specific examples of perfect fluids with different on-shell Lagrangians but with the same energy-momentum tensor. We demonstrate that if the fluid is constituted by localized concentrations of energy with fixed rest mass and structure (solitons) then the average on-shell Lagrangian of a perfect fluid is given by ${\mathcal L}_m=T$, where $T$ is the trace of the energy-momentum tensor. We show that our results have profound implications for theories of gravity where the m…
Formal analogies between the ordinary differential equations describing geophysical flows and Friedmann cosmology are developed. As a result, one obtains Lagrangian and Hamiltonian formulations of these equations, while laboratory experiments aimed at testing geophysical flows are shown to constitute analogue gravity systems for cosmology.
Starting from a knowledge of certain identities in the Lagrangian description, the diffeomorphism transformations of metric and connection are obtained for both the second order (metric) and the first order (Palatini) formulations of gravity. The transformation laws of the connection and the metric are derived independently in the Palatini formulation in contrast to the metric formulation where the gauge variation of the connection is deduced from the gauge variation of the metric.
The chiral Lagrangian including more than two spin-3/2 fields becomes complex in general when the self-dual connection satisfies its equation of motion, and the imaginary part of the chiral Lagrangian gives the additional equation for spin-3/2 fields which gives rise to the inconsistency. This inconsistency can be removed by taking the tetrad to be complex and defining the total Lagrangian which is the sum of the chiral Lagrangian and its complex conjugate. It is possible to establish the right (left)-handed supersymmetry in its total Lagrangian as in the case of the chiral Lagrangian. We also…
A formal analogy between the Friedmann equation of relativistic cosmology and models of convective-radiative cooling/heating of a body (including Newton's, Dulong-Petit's, Newton-Stefan's laws, and a generalization) is discussed. The analogy highlights Lagrangians, symmetries, and mathematical properties of the solutions of these cooling laws.
A Lagrangian formulation of perfect fluid due to a noncanonical three-form field is investigated. The thermodynamic quantities such as energy density, pressure and the four velocity are obtained and then analyzed by comparing with the k-essence scalar field. The nonrelativistic matter due to the generalized three-form field with the equation of state parameter being zero is realized while it might not be possible for the k-essence scalar field. We also found that nonadiabatic pressure perturbations can be possibly generated. The fluid dynamics of the perfect fluid due to the three-form field c…
In the standard model of cosmology, the background evolution of the Universe can in general be adequately described by general relativity and a uniform and isotropic metric minimally coupled with a collection of perfect fluids. These fluids are usually described by their energy-momentum tensor, which can be derived from the fluid's Lagrangian density. Under general relativity, the Lagrangian density is only relevant to the extent that it results in the correct energy-momentum tensor for a specific perfect fluid. This is not the case in theories that feature a nonminimal coupling (NMC) between …
We consider the generalized set of theories of gravitation whose Lagrangians contain the term $R^{2}$ : $L=\sqrt{-g}(R+βR^{2})$. Inserting the RW metric with an imposed non-singular and inflationary behaviour of the scale factor $a(t)$, and using a arbitrary perfect fluid, we study the properties of $ρ$ and $p$ in this context. By requiring the positivity of the energy density, as well as real and finite velocity of sound, we can obtain the range of values of $β$ that ensure the inflationary behaviour and absence of singularity.
A Lagrangian formulation for Rastall gravity and a covariant formulation for unimodular gravity
Faculdade de Ciências e Tecnologias, Universidade de Cabo Verde（佛得角大学科学与技术学院）；Núcleo Cosmo-ufes & Departamento de Física, Universidade Federal do Espírito Santo（圣埃斯皮里图联邦大学宇宙核与物理系）；Centro de Ciências Naturais e Humanas, Universidade Federal do ABC（ABC联邦大学自然与人文学中心）
A recent model predicting Omori's law giving the number of aftershocks per unit time following an earthquake involves a differential equation analogous to the Friedmann equation of cosmology. The before-shock phase is analogous to an accelerating universe approaching a Big Rip, the main shock to the Big Rip singularity, and the aftershock to a contracting universe. The analogy provides some physical intuition and Lagrangian and Hamiltonian formulations for Omori's law and its generalizations.
We give a Lagrangian formulation for the theory of Rastall of gravitation. After proposing a Lagrangian density that reproduces the equations of motion postulated by Rastall, we study the cosmological consequences and fit the parameters using recent data from Hubble function $H(z).$ According to two model selection criteria, one based on corrected Akaike Information Criterion (AICc) and another on Bayesian Information Criterion (BIC), known to penalize models with a greater number of parameters, particularly BIC, we obtain some competitive models relative do $ΛCDM.$ In one of these models the …
The recently proposed first-order parent formalism at the level of equations of motion is specialized to the case of Lagrangian systems. It is shown that for diffeomorphism-invariant theories the parent formulation takes the form of an AKSZ-type sigma model. The proposed formulation can be also seen as a Lagrangian version of the BV-BRST extension of the Vasiliev unfolded approach. We also discuss its possible interpretation as a multidimensional generalization of the Hamiltonian BFV--BRST formalism. The general construction is illustrated by examples of (parametrized) mechanics, relativistic …
It was claimed recently that a low order post-Newtonian (PN) Lagrangian formulation, which corresponds to the Euler-Lagrange equations up to an infinite PN order, can be identical to a PN Hamiltonian formulation at the infinite order from a theoretical point of view. This result is difficult to check because in most cases one does not know what both the Euler-Lagrange equations and the equivalent Hamiltonian are at the infinite order. However, no difficulty exists for a special 1PN Lagrangian formulation of relativistic circular restricted three-body problem, where both the Euler-Lagrange equa…
For the lagrangian L = G ln G where G is the Gauss-Bonnet curvature scalar we deduce the field equation and solve it in closed form for 3-flat Friedman models using a statefinder parametrization. Further we show, that among all lagrangians F(G) this L is the only one not having the form G^r with a real constant r but possessing a scale-invariant field equation. This turns out to be one of its analogies to f(R)-theories in 2-dimensional space-time. In the appendix, we systematically list several formulas for the decomposition of the Riemann tensor in arbitrary dimensions n, which are applied in…
A string motivated, N=1 global supersymmetric, general Lagrangian is obtained that represents an inflaton interacting with the modes of a string. Focus is placed on the Lagrangian's relevance to warm inflation, which is a cosmology based on dissipative dynamics. Several interaction schemes are identified from the general Lagrangian that may have interesting consequences for dissipative dynamics. Distributed-mass-models, which have been shown to solve the horizon problem within a warm inflation regime, are identified in the general Lagrangian.
The Eddington Lagrangian in the purely affine formulation of general relativity generates the Einstein equations with the cosmological constant. The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, which has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the Einstein-Maxwell Lagrangian in the metric formulation. We show that the sum of the two affine Lagrangians is dynamically inequivalent to the sum of the analogous Lagrangians in the metric-affine/metric formulation. We also show that s…
We describe a new method to parameterise dark energy theories including massive gravity, elastic dark energy and tensor-metric theories. We first examine a framework to describe any second order Lagrangian which depends on the variation of the metric and find new constraints on the parameters. We extend the method to Lorentz violating theories which depend on the variation of the time and spatial parts of the metric separately. We show how this can describe massive gravity and elastic dark energy, while ruling out the whole class of theories where the Lagrangian depends only on the variation o…
The leading-order spin-orbit coupling is included in a post-Newtonian Lagrangian formulation of spinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and 2PN non-spin terms and 2PN spin-spin coupling. This makes a 3PN spin-spin coupling occur in the derived Hamiltonian. The spin-spin couplings are mainly responsible for chaos in the Hamiltonians. However, the 3PN spin-spin Hamiltonian is small and has different signs, compared with the 2PN spin-spin Hamiltonian equivalent to the 2PN spin-spin Lagrangian. As a result, the probability of the occurrence of ch…
We reformulate the standard Lagrangian formalism to a reparameterisation invariant Lagrangian formalism by means of Finsler and Kawaguchi geometry. In our formalism, various types of symmetries that appears in theories of physics are expressed geometrically by symmetries of Finsler (Kawaguchi) metric, and the conservation law of energy-momentum is a part of Euler-Lagrange equations. The application to scalar field, Dirac field, electromagnetic field and general relativity coupled to perfect fluid (added: ver.3) are discussed. By this formalism, we try to propose an alternative definition of en…
The purpose of this paper is to introduce a new way to inquire the quantum cosmology for a certain gravitational theory. Normally, the quantum cosmological model is introduced as the minisuperspace theory which is obtained by reducing the superspace where the Wheel-DeWitt equation is defined on using the symmetry provided by cosmological principle. Unlike that, the key of our approach is to reinterpret the cosmology in a classical dynamical way using a point-like Lagrangian and then quantize the point-like model. We apply the method into Einstein gravity, gravity with a cosmological constant a…
We generalize the Lagrangian of N = 1 supergravity (SUGRA) by using an arbitrary parameter $ξ$, which corresponds to the inverse of Barbero's parameter $β$. This generalized Lagrangian involves the chiral one as a special case of the value $ξ= \pm i$. We show that the generalized Lagrangian gives the canonical formulation of N = 1 SUGRA with the real Ashtekar variable after the 3+1 decomposition of spacetime. This canonical formulation is also derived from those of the usual N = 1 SUGRA by performing Barbero's type canonical transformation with an arbitrary parameter $β(=ξ^{-1})$. We give some…
Using a suitable rescaling of the independent variable, a Lagrangian is found for the nonlinear Vialov equation ruling the longitudinal profiles of glaciers and ice caps in the shallow ice approximation. This leads to a formal analogy between the (rescaled) Vialov equation and the Friedmann equation of relativistic cosmology, which is explored. This context provides a new symmetry of the (rescaled) Vialov equation and gives, at least formally, all its solutions using a generating function, which is the Nye profile for the degenerate case of perfectly plastic ice.
We propose the Lagrangian formulation for describing the motion of a test particle in a general relativistic, stationary, and axially symmetric spacetime. The test particle is also affected by a radiation field, modeled as a coherent flux of photons traveling along the null geodesics of the background spacetime, including the general relativistic Poynting-Robertson effect. The innovative part of this work is to prove the existence of the potential linked to the dissipative action caused by the Poynting-Robertson effect in General Relativity through the help of an integrating factor, depending …
We present a simple method to derive the semiclassical equations of motion for a spinning particle in a gravitational field. We investigate the cases of classical, rotating particles (pole-dipole particles), as well as particles with intrinsic spin. We show that, starting with a simple Lagrangian, one can derive equations for the spin evolution and momentum propagation in the framework of metric theories of gravity and in theories based on a Riemann-Cartan geometry (Poincare gauge theory), without explicitly referring to matter current densities (spin and energy-momentum). Our results agree wi…
The variety of theories that can account for the dark energy phenomenon encourages current research to concentrate on a more in-depth examination of the potential impacts of modified gravity on both local and cosmic scales. We discuss some cosmological reconstruction in $f(Q,T)$ cosmology (where $Q$ is the non-metricity scalar, and $T$ is the trace of the energy-momentum tensor) corresponding to the evolution background in Friedmann-Laîmatre-Robertson-Walker (FLRW) universe. This helps us to determine how any FLRW cosmology can arise from a specific $f(Q,T)$ theory. We use the reconstruction t…
The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the…
We propose a relativistically covariant model of interacting dark energy based on the principle of least action. The cosmological term $Λ$ in the gravitational Lagrangian is a function of the trace of the energy--momentum tensor $T$. We find that the $Λ(T)$ gravity is more general than the Palatini $f(R)$ gravity, and reduces to the latter if we neglect the pressure of matter. We show that recent cosmological data favor a variable cosmological constant and are consistent with the $Λ(T)$ gravity, without knowing the specific function $Λ(T)$.
The Lagrangian theory of structure formation in cosmological fluids, restricted to the matter model ``dust'', provides successful models of large-scale structure in the Universe in the laminar regime, i.e., where the fluid flow is single-streamed and ``dust''-shells are smooth. Beyond the epoch of shell-crossing a qualitatively different behavior is expected, since in general anisotropic stresses powered by multi-stream forces arise in collisionless matter. In this paper we provide the basic framework for the modeling of pressure-supported fluids, thus restricting attention to isotropic stress…
We obtain Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body with given values of spin and momentum starting from Lagrangian action without auxiliary variables. MPTD-equations correspond to minimal interaction of our spinning particle with gravity. We shortly discuss some novel properties deduced from the Lagrangian form of MPTD-equations: emergence of an effective metric instead of the original one, non-commutativity of coordinates of representative point of the body, spin corrections to Newton potential due to the effective metric as well as spin corrections to the expression …
The ``Newtonian'' theory of spatially unbounded, self--gravitating, pressureless continua in Lagrangian form is reconsidered. Following a review of the pertinent kinematics, we present alternative formulations of the Lagrangian evolution equations and establish conditions for the equivalence of the Lagrangian and Eulerian representations. We then distinguish open models based on Euclidean space $\R^3$ from closed models based (without loss of generality) on a flat torus $\T^3$. Using a simple averaging method we show that the spatially averaged variables of an inhomogeneous toroidal model form…
In the standard Lagrangian and Hamiltonian approach to Maxwell's theory the potentials $A^μ$ are taken as the dynamical variables. In this paper I take the electric field $\vec{E}$ and the magnetic field $\vec{B}$ as the the dynamical variables. I find a Lagrangian that gives the dynamical Maxwell equations and include the constraint equations by using Lagrange multipliers. In passing to the Hamiltonian one finds that the canonical momenta $\vecΠ_E$ and $\vecΠ_B$ are constrained giving 6 second class constraints at each point in space. Gauss's law and $\vec{\nabla}\cdot\vec{B}=0$ can than be a…
The non-linear dynamics of irrotational dust in General Relativity is studied in synchronous and comoving coordinates. All the equations are written in terms of the metric tensor of spatial sections orthogonal to the flow, which allows an unambiguous expansion in powers of $1/c^2$. To lowest order, the Newtonian approximation in Lagrangian form is derived. At this level the evolution is governed by the Raychaudhuri equation for the Lagrangian-to-Eulerian Jacobian matrix. The Lagrangian spatial metric reduces to that of Euclidean 3-space in time-dependent curvilinear coordinates. A Lagrangian v…
We consider extensions of the Einstein-Hilbert Lagrangian to a general functional of metric and Riemann curvature tensor. A given such Lagrangian describes two different theories depending on considering connection and metric (Palatini formulation), or only the metric (metric formulation) as independent dynamical degrees of freedom. Equivalence of the Palatini and metric formulations at the level of equations of motion, which as we will argue is a manifestation of the Equivalence Principle, is the physical criterion that restricts form of the Lagrangians of modified gravity theories. We prove …
This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter extension is also called areal geometry in some references. On Finsler (Kawaguchi) manifold, we can define a reparameterisation invariant 1 (k)-dimensional area by the Hilbert (Kawaguchi) form, which we take as an action. The equation of motion obtained from such action also has the p…
