Hossein Farajollahi
A cosmological time variable is emerged from the hamiltonian formulation of unimodular theory of gravity to measure the evolution of dynamical observables in the theory. A set of constants of motion has been identified for the theory on the null hypersurfaces that its evolution is with respect to the volume clock introduced by the cosmological time variable.
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Unimodular relativity is a theory of gravity and space-time with a fixed absolute space-time volume element, the modulus, which we suppose is proportional to the number of microscopic modules in that volume element. In general relativity an arbitrary fixed measure can be imposed as a gauge condition, while in unimodular relativity it is determined by the events in the volume. Since this seems to break general covariance, some have suggested that it permits a non-zero covariant divergence of the material stress-energy tensor and a variable cosmological ``constant.'' In Lagrangian unimodular rel…
We show that the conclusion that matter stress-energy tensor satisfies the usual covariant continuity law, and the cosmological constant is still a constant of integration arrived at by Finkelstein et al (42, 340, 2001) is not valid.
The black hole entropy formula applied to local Rindler horizon at each spacetime point has been used in the literature to derive the Einstein field equation as an equation of state of a thermodynamical system of spacetime. In the present paper we argue that due to the key role of causal structure and discrete spacetime in this approach the natural framework is unimodular relativity rather than general relativity. It is shown that the equation of state is trace free unimodular relativity field equation that uniquely determines only the traceless stress tensor. Recent generalization to nonequil…
We outline unimodular conformal and projective relativity (UCPR), an extension of unimodular relativity in which the conformal and projective structures play central roles. Under $SL(4,\mathbb{R})$ symmetry group, the pseudo-Riemannian metric naturally decomposes into a four-volume element and a conformal metric; and the affine connection decomposes into a one-form and a trace-free projective connection. In UCPR, these four space-time structures are treated as independent fields that have clear physical interpretations. A Palatini-type variational principle for the usual general relativity Lag…
We carry out the canonical analysis of a covariant version of unimodular gravity in terms of the connection representation. We then proceed to quantize this theory by implementing the Dirac procedure. We confirm whether and how the Kodama state, which is a solution of quantum general relativity, can be extended into covariant unimodular gravity. Finally, we discuss the difference of quantum states between covariant unimodular gravity, the original unimodular gravity, and general relativity.
We investigate inflation and its scalar perturbation driven by a massive scalar field in the unimodular theory of gravity. We introduce a parameter $ξ$ with which the theory is invariant under general unimodular coordinate transformations. When the unimodular parameter is $ξ=6$, the classical picture of inflation is reproduced in the unimodular theory because it recovers the background equations of the standard theory of general relativity. We show that for $ξ=6$, the theory is equivalent to the standard theory of general relativity at the perturbation level. Unimodular gravity constrains the …
The so called unimodular theory of gravitation is compared with general relativity in the quadratic (Fierz-Pauli) regime, using a quite broad framework, and it is argued that quantum effects allow in principle to discriminate between both theories.
Unimodular gravity (UG) is an important theory which may explain the smallness of the cosmological constant. To get insight into the covariant quantization of UG, we discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the unimodular gauge. This process requires the introduction of an additional pair of BRST doublets which decouple from the physical sector together with the other three pairs of BRST doublets for the TDiff. We show that …
We study a slight variation of the unimodular gauge condition which we call "the derivative gauge condition". We show that at the classical level these conditions are completely equivalent up to a surface term. At the quantum level the derivative gauge condition can be easily implemented by analogy with gauge fixing in a quantum gauge theory. This condition might have advantages and disadvantages upon the context.
We introduce a modified divergence law for the energy-momentum tensor in the theory of unimodular relativity. Consequently, an additional equation for the measure field follows from the divergence of the field equations. The equations of motion are derived and the weak-field, low-velocity limit is investigated. It is found that the gravitational acceleration acquires a term that is proportional to the gradient of the mass density. We show that this term can provide the additional acceleration observed on astrophysical scales without the need for dark matter.
Wormholes are outstanding solutions of Einstein's General Relativity. They were worked out in the late 1980's by Morris and Thorne, who have figured out a recipe that wormholes must obey in order to be traversable, that is, safely crossed by travelers. A remarkable feature is that General Relativity Theory wormholes must be filled by {\it exotic matter}, which Morris and Thorne define as matter satisfying $-p_r>ρ$, in which $p_r$ is the radial pressure and $ρ$ is the energy density of the wormhole. In the present article, we introduce, for the first time in the literature, traversable wormhole…
We compute the lowest order gravitational UV divergent radiative corrections to the S matrix element of the $fermion + fermion\rightarrow fermion + fermion$ scattering process in the massive Yukawa theory, coupled either to Unimodular Gravity or to General Relativity. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contribution in Dimensional Regularization. This is a nontrivial result, since in the classical action of Unimodular Gravity coupled to the Yukawa theory, the graviton field does not couple neither to the mass operator nor to the Yukawa…
We perform the Hamiltonian analysis of unimodular gravity in terms of the connection representation. The unimodular condition is imposed straightforwardly into the action with a Lagrange multiplier. After classifying constraints into first class and second class, the canonical quantization is carried out. We consider the difference of the corresponding physical states between unimodular gravity and general relativity.
Unimodular gravity can be formulated so that transverse diffeomorphisms and Weyl transformations are symmetries of the theory. For this formulation of unimodular gravity, we work out the two-point and three-point $h_{μν}$ contributions to the on-shell classical gravity action in the leading approximation and for an Euclidean AdS background. We conclude that these contributions do not agree with those obtained by using General Relativity due to IR divergent contact terms. The subtraction of these IR divergent terms yields the same IR finite result for both unimodular gravity and General Relativ…
We work out the one-loop and order $κ^2 m_ϕ^2$ UV divergent contributions, coming from Unimodular Gravity and General Relativity, to the S matrix element of the scattering process $ϕ+ ϕ\rightarrow ϕ+ ϕ$ in a $λϕ^4$ theory with mass $m_ϕ$. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contributions in Dimension Regularization. This seems to be at odds with the known result that in a multiplicative MS dimensional regularization scheme the General Relativity corrections, in the de Donder gauge, to the beta function $β_λ$ of the $λ$ coupling do not …
We investigate path integral quantization of two versions of unimodular gravity. First a fully diffeomorphism-invariant theory is analyzed, which does not include a unimodular condition on the metric, while still being equivalent to other unimodular gravity theories at the classical level. The path integral has the same form as in general relativity (GR), except that the cosmological constant is an unspecified value of a variable, and it thus is unrelated to any coupling constant. When the state of the universe is a superposition of vacuum states, the path integral is extended to include an in…
The unimodular metagravity, with the graviscalar as a dark matter, is compared with General Relativity (GR) in the presence of a scalar field. The effect of the graviscalar on the static spherically symmetric metric is studied. An exact limit solution representing a new cosmic object, the (harmonic) graviscalar black hole, is given. The relation with the black hole in the environment of a scalar field in GR is discussed.
Unimodular Gravity is one of the oldest geometric gravity theory alternative to General Relativity. Essentially, it is based on the Einstein-Hilbert Lagrangian with an additional constraint on the determinant of the metric. It can be explicitly shown that Unimodular Gravity can be recast as General Relativity in presence of a cosmological constant. This fact has led to many discussions on the equivalence of both theories at classical and quantum levels. Here we present an analysis focused on the classical scalar perturbations around a cosmological background. The discussion is extended for the…
It is well known that classical Unimodular Gravity (UG) is equivalent to General Relativity, but is not clear the equivalence at the quantum level. Here we exhibit, in the UG framework, the Quantum Cosmology (QC) for a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, considering an inflationary scenario and also introducing a deformation on the commutation relations for the minisuperspace variables (NCC).
The so-called unimodular version of General Relativity is revisited. Unimodular gravity is constructed by fixing the determinant of the metric, what leads to the trace-free part of the equations instead of the usual Einstein field equations. Then, a cosmological constant naturally arises as an integration constant. While unimodular gravity turns out equivalent to General Relativity (GR) at the classical level, it provides important differences at the quantum level. Here we extend the unimodular constraint to some extensions of General Relativity that have drawn a lot of attention over the last…
In this paper we study a modified version of unimodular general relativity in the context of $f(G)$, $G$ denoting the Gauss-Bonnet invariant. We attach attention to Bianchi-type I and Friendmann-Robertson-Walker universes and search for unimodular $f(G)$ models according to the de Sitter and power-law solutions. Assuming unimodular $f(G)$ gravity as perfect fluid and making use of the slow-roll parameters, inflationary model has been reconstructed in concordance with the Planck observational data. Moreover, we investigate the realization of the bounce and loop quantum cosmological ekpyrotic pa…
Unimodular gravity is characterized by an extra condition with respect to General Relativity: the determinant of the metric is constant. This extra condition leads to a more restricted class of invariance by coordinate transformation. Even though, if the conservation of the energy-momentum tensor is imposed in unimodular gravity, the General Relativity theory is recovered with an additional integration constant which is associated to the cosmological term $Λ$. However, if the energy-momentum tensor does not conserve separately, a new geometric structure appears with potentially observational s…
A clarification on prevailing misconceptions in unimodular gravity
本文澄清了幺模引力（UG）领域的常见误解，指出UG仅具保体积微分同胚不变性的传统表述引发了如有效度规条件、微扰理论规范选择等错误结论，明确了微分同胚不变性的一般概念及其在UG中的正确应用。
Transverse Diffeomorphism (TDiff) theories are well-motivated theories of gravity from the quantum perspective, which are based upon a gauge symmetry principle. The main contribution of this work is to firmly establish a correspondence between TransverseDiff and the better-known scalar-tensor gravity --- in its more general form ---, a relation which is completely analogous to that between unimodular gravity and General Relativity. We then comment on observational aspects of TDiff. In connection with this proof, we derive a very general rule that determines under what conditions the procedure …
Unimodular Gravity (UG) is a restricted version of General Relativity (GR) in which the determinant of the metric is a fixed function and the field equations are given by the trace-free part of the full Einstein equations. The background equations in UG and GR are identical. It was recently claimed that, the first order contribution in the temperature fluctuation of the Cosmic Microwave Background (CMB) in UG is different from GR. In this work, we calculate the first order perturbation equations in UG and show that the Sachs-Wolfe effect in UG, in terms of gauge invariant variables, is identic…
It is commonly accepted that general relativity is the only solution to the consistency problem that appears when trying to build a theory of interacting gravitons (massless spin-2 particles). Padmanabhan's 2008 thought-provoking analysis raised some concerns that are having resonance in the community. In this work we present the self-coupling problem in detail and explicitly solve the infinite-iterations scheme associated with it for the simplest theory of a graviton field, which corresponds to an irreducible spin-2 representation of the Poincaré group. We make explicit the non-uniqueness pro…
Unimodular Gravity is normally assumed to be equivalent to General Relativity for all matters but the character of the Cosmological Constant. Here we discuss this equivalence in the presence of a non-minimally coupled scalar field. We show that when we consider gravitation to be dynamical in a QFT sense, quantum corrections can distinguish both theories if the non-minimal coupling is non-vanishing. In order to show this, we construct a path integral formulation of Unimodular Gravity, fixing the complicated gauge invariance of the theory and computing all one-loop divergences. We find a combina…
Gödel-type universes in unimodular gravity
本研究在幺模引力框架下检验哥德尔型解的有效性，发现该理论中宇宙学常数为积分常数，且特定引力源下因果与非因果区域均允许存在。
We study the beta functions of the quartic and Yukawa couplings of General Relativity and Unimodular Gravity coupled to the $λϕ^4$ and Yukawa theories with masses. We show that the General Relativity corrections to those beta functions as obtained from the 1PI functional by using the standard MS multiplicative renormalization scheme of Dimensional Regularization are gauge dependent and, further, that they can be removed by a non-multiplicative, though local, field redefinition. An analogous analysis is carried out when General Relativity is replaced with Unimodular Gravity. Thus we show that a…
We study cosmological perturbation theory within the framework of unimodular gravity. We show that the Lagrangian constraint on the determinant of the metric required by unimodular gravity leads to an extra constraint on the gauge freedom of the metric perturbations. Although the main equation of motion for the gravitational potential remains the same, the shift variable, which is gauge artifact in General Relativity, cannot be set to zero in unimodular gravity. This non-vanishing shift variable affects the propagation of photons throughout the cosmological evolution and therefore modifies the…
Unimodular quantum cosmology admits wavepacket solutions that evolve according to a kind of Schrödinger equation. Though this theory is equivalent to general relativity on the classical level, its canonical structure is different and the problem of time does not occur. We present an Ehrenfest theorem for the long term evolution of the expectation value of the scale factor for a spatially flat Friedmann universe with a scalar field. We find that the classical and the quantum behaviour in the asymptotic future coincide for the special case of a massless scalar field. We examine the general behav…
In our previous paper, we have presented a covariant BRST quantization of unimodular gravity which may account for the smallness of the cosmological constant, and have shown that the physical degrees of freedom in the theory are the same as general relativity. The formulation has been given by using rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we give an alternative formulation using a vector field for the antighost but keeping the same structure for the ghosts. This gives a significantly simpler covariant quantization with less ghosts and no tripole modes in the gho…
We compare the path integral for transition functions in unimodular gravity and in general relativity. In unimodular gravity the cosmological constant is a property of states that are specified at the boundaries whereas in general relativity the cosmological constant is a parameter of the action. Unimodular gravity with a nondynamical background spacetime volume element has a time variable that is canonically conjugate to the cosmological constant. Wave functions depend on time and satisfy a Schrödinger equation. On the contrary, in the covariant version of unimodular gravity with a 3-form gau…
Nonexistence of a parent theory for general relativity and unimodular gravity
本文研究广义相对论与单模引力是否存在以完整微分同胚和外尔变换为规范对称性的母理论，通过引入 Stueckelberg 场发现 UG 中宇宙学常数对应的额外全局自由度阻碍了母理论的构造。
Unimodular Gravity is an alternative to General Relativity (GR) which, however, is so closely related to the latter that one can wonder to what extent they are different. The different behavior of the cosmological constant in the semiclassical regimes of both frameworks suggests the possible existence of additional contrasting features. UG and GR are based on two different gauge symmetries: UG is based on transverse diffeomorphisms and Weyl rescalings (WTDiff transformations), whereas GR is based on the full group of diffeomorphisms. This difference is related to the existence of a fiduciary b…
Unimodular boundary time for Regge calculus
本文利用欧几里得 Regge 微积分离散化 Henneaux-Teitelboim 引力，在对称性约化的宇宙学模型中验证了连续极限可用单模时间定义，为量子引力的离散路径积分表述迈出第一步。
Unimodular gravity is classically equivalent to General Relativity. This equivalence extends to actions which are functions of the curvature scalar. At the quantum level, the dynamics could differ. Most importantly, the cosmological constant is not a coupling in the unimodular action, providing a new vantage point from which to address the cosmological constant fine-tuning problem. Here, a quantum theory based on the asymptotic safety scenario is studied, and evidence for an interacting fixed point in unimodular f(R) gravity is found. We study the fixed point and its properties, and also discu…
It is well known that the problem of the cosmological constant appears in a new light in Unimodular Gravity. In particular, the zero momentum piece of the potential does not automatically produce a corresponding cosmological constant. Here we show that quantum corrections do not renormalize the classical value of this observable.
In this conceptual paper we construct a local version of Shape Dynamics that is equivalent to General Relativity in the sense that the algebras of Dirac observables weakly coincide. This allows us to identify Shape Dynamics observables with General Relativity observables, whose observables can now be interpreted as particular representative functions of observables of a conformal theory at fixed York time. An application of the observable equivalence of General Relativity and Shape Dynamics is to define the quantization of General Relativity through quantizing Shape Dynamics and using observab…
In the context of the teleparallel equivalent of general relativity we establish the Hamiltonian formulation of the unimodular theory of gravity. Here we do not carry out the usual $3+1$ decomposition of the field quantities in terms of the lapse and shift functions, as in the ADM formalism. The corresponding Lagrange multiplier is the timelike component of the tetrad field. The dynamics is determined by the Hamiltonian constraint ${\cal H}'_0$ and a set of primary constraints. The constraints are first class and satisfy an algebra that is similar to the algebra of the Poincaré group.
Generalized Unimodular Gravity and Cosmological Perturbations
重新审视广义单模引力理论，通过纯几何方式获得宇宙的常规物质成分，为暗区研究提供新视角，并在背景和扰动分析中恢复广义相对论结果但解释不同。
It is a well known result that any formulation of unimodular gravity is classically equivalent to General Relativity (GR), however a debate exists in the literature about this equivalence at the quantum level. In this work, we investigate the UV quantum structure of a diffeomorphism invariant formulation of unimodular gravity using functional renormalisation group methods in a Wilsonian context. We show that the effective action of the unimodular theory acquires essentially the same form with that of GR in the UV, as well as that both theories share similar UV completions within the framework …
The problem of the cosmological constant appears in a new light in Unimodular Gravity. In particular, the zero momentum piece of the potential (that is, the constant piece independent of the matter fields) does not automatically produce a cosmological constant proportional to it. The aim of this paper is to give some details on a calculation showing that quantum corrections do not renormalize the classical value of this observable.
Revisiting Unimodular Quantum Cosmology
本文重新审视迷你超空间中幺模引力的量子化分析，将其扩展到幻影标量场与Brans-Dicke标量张量理论，论证仅幻影场可获非平凡演化，标度因子呈现反弹接德西特膨胀，重现广义相对论量子宇宙学场景。
Unimodular gravity is based on a modification of the usual Einstein-Hilbert action that allows one to recover general relativity with a dynamical cosmological constant. It also has the interesting property of providing, as the momentum conjugate to the cosmological constant, an emergent clock variable. In this paper we investigate the cosmological reduction of unimodular gravity, and its quantization within the framework of flat homogeneous and isotropic loop quantum cosmology. It is shown that the unimodular clock can be used to construct the physical state space, and that the fundamental fea…
Normalizing the Einstein-Hilbert action by the volume functional makes the theory invariant under constant shifts in the Lagrangian. The associated field equations then resemble unimodular gravity whose otherwise arbitrary cosmological constant is now determined as a Machian universal average. We prove that an empty space-time is necessarily Ricci tensor flat, and demonstrate the vanishing of the cosmological constant within the scalar field paradigm. The cosmological analysis, carried out at the mini-superspace level, reveals a vanishing cosmological constant for a Universe which cannot be cl…
Unimodular Plebański Gravity
本文在基于复二形式和复 SO(3) 联络的手征 Plebański 表述中构造了幺模引力的新作用量原理，使宇宙学常数成为积分常数，并给出了向 Krasnov 型纯联络形式的约化。
A proposal for the issue of time and observables in any parameterized theory such as general relativity is addressed. Introduction of a gauge potential 3-form A in the theory of relativity enables us to define a gauge-invariant quantity which can be used by observers as a clock to measure the passage of time. This dynamical variable increases monotonically and continuously along a world line. Then we define world line observables to be any covariantly defined quantity obtained from the field configurations on any such causal past with dynamical time T.
Regular Black Strings and $(2+1)$ Black Hole in Unimodular Gravity Supported by Maxwell Fields
本文通过单模引力框架获得Maxwell源的规范黑洞弦和BTZ黑洞，探讨了宇宙学常数作为场方程积分常数的作用，验证了Maxwell电动力学作为源的有效性，并计算了支持解的真空贡献。
In the report, the theory of unimodular bimode gravity built on principles of unimodular gauge invariance/relativity and general covariance is exposed. Besides the massless tensor graviton of General Relativity, the theory includes an (almost) massless scalar graviton treated as the gravitational dark matter. A spherically symmetric vacuum solution, describing the coherent scalar-graviton field for the soft-core dark halos with the asymptotically flat rotation curves, is demonstrated.
I present a complete set of gauge invariant observables, in the context of general relativity coupled with a minimal amount of realistic matter (four particles). These observables have a straightforward and realistic physical interpretation. In fact, the technology to measure them is realized by the Global Positioning System: they are defined by the physical reference system determined by GPS readings. The components of the metric tensor in this physical reference system are gauge invariant quantities and, remarkably, their evolution equations are local.
Unimodular gravity (UG) is an interesting theory that may explain why the cosmological constant is extremely small, in contrast to general relativity (GR). The theory has only the transverse diffeomorphism invariance and this causes lots of debate as to the equivalence of UG to GR in the covariant quantization. We study the covariant BRST quantization of UG by gauge fixing only the transverse diffeomorphism and show that the remaining physical degrees of freedom are two, the same number as GR. This is achieved by using antisymmetric tensor ghost fields which automatically satisfy the transvers…
Unimodular Theory of Gravity in Light of the Latest Cosmological Data
本文利用最新宇宙学数据拟合单模引力模型，确定参数ξ=6.23±0.5，并测得哈勃常数H0值，与标准ΛCDM模型存在偏差。
We investigate the interplay and connections between symmetry properties of equations, the interpretation of coordinates, the construction of observables, and the existence of physical relativity principles in spacetime theories. Using the refined notion of an event as a ``point-coincidence'' between scalar fields that completely characterise a spacetime model, we also propose a natural generalisation of the relational local observables that does not require the existence of four everywhere invertible scalar fields. The collection of all point-coincidences forms in generic situations a four-di…
We work out the description of the gauge symmetry of unimodular gravity in the constrained Hamiltonian formalism. In particular, we demonstrate how the transversality conditions restricting the diffeomorphism parameters emerge from the algebra of the Hamiltonian constraints. The alternative form is long known as parametrizing the volume preserving diffeomorphisms by unrestricted two-forms instead of the transverse vector fields. This gauge symmetry is reducible. We work out the Hamiltonian description of this form of unimodular gravity (UG) gauge symmetry. Becchi-Rouet-Stora-Tyutin--Batalin-Fr…
In this addendum we clarify a point which strengthens one of the results from [the original paper]. Namely, we show that the algebra of the observables $F(r,θ)$ is yet simpler then it was described in [the original paper]. This is an important point, because with this simplification an important subalgebra becomes canonical, allowing for a natural reduction of the phase space.
It is shown that the field equations of general relativity in the Einstein tensor form and the unimodular theory of gravity do not fulfill the correspondence principle commitment completely. The consistent formalisms are briefly discussed.
Some conceptual issues concerning general invariant theories, with special emphasis on general relativity, are analyzed. The common assertion that observables must be required to be gauge invariant is examined in the light of the role played by a system of observers. Some features of the reduction of the gauge group are discussed, including the fact that in the process of a partial gauge fixing the reduction at the level of the gauge group and the reduction at the level of the variational principle do not commute. Distinctions between the mathematical and the physical concept of gauge symmetry…
We present a new scheme of defining invariant observables for general relativistic systems. The scheme is based on the introduction of an observer which endowes the construction with a straightforward physical interpretation. The observables are invariant with respect to spatial diffeomorphisms which preserve the observer. The limited residual spatial gauge freedom is studied and fully understood. A full canonical analysis of the observables is presented: we analyze their variations, Poisson algebra and discuss their dynamics. Lastly, the observables are used to solve the vector constraint, wh…
The boundary conditions for canonical vacuum general relativity is investigated at the quasi-local level. It is shown that fixing the area element on the 2- surface S (rather than the induced 2-metric) is enough to have a well defined constraint algebra, and a well defined Poisson algebra of basic Hamiltonians parameterized by shifts that are tangent to and divergence-free on $. The evolution equations preserve these boundary conditions and the value of the basic Hamiltonian gives 2+2 covariant, gauge-invariant 2-surface observables. The meaning of these observables is also discussed.
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