We show that the definition of proper time for Weyl-invariant space-times given by Perlick naturally extends to spaces with arbitrary non-metricity. We then discuss the relation between this generalized proper time and the Ehlers-Pirani-Schild definition of time when there is arbitrary non-metricity. Then we show how this generalized proper time suffers from a second clock effect. Assuming that muons are a device to measure this proper time, we constrain the non-metricity tensor on Earth's surface and then elaborate on the feasibility of such assumption.
A Weyl structure is usually defined by an equivalence class of pairs $({\bf g}, \boldsymbolω)$ related by Weyl transformations, which preserve the relation $\nabla {\bf g}=\boldsymbolω\otimes{\bf g}$, where ${\bf g}$ and $\boldsymbolω$ denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection $Γ_{\boldsymbolω}$, which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity…
This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian geometry. In the second Chapter we explore the MAG model building. In Chapter 3 we use a well known procedure to excite torsional degrees of freedom by coupling surface terms to scalars. Then, in Chapter 4 which seems to be the most important Chapter of the thesis, at least with regards to its use in applications, we present a step by step way to solve for …
In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a generalization of the Einstein equation is obtained, wherein the conventional tensors are replaced by their conformally invariant counterparts, living in metric measure space. The invariance of the geometrical part of the action under a diffeomorphism leads to a generalized contracted second Bianchi identity. In metric measure space, the covariant derivative …
Metric-affine geometry provides a non-trivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the space-time (with non-vanishing torsion and non-metricity). In this paper we study the generic form of action in this formalism, and then construct the Weyl-invariant version of this theory. It is shown that in Weitzenbock space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally the related field equations are obtained in the general case.
We consider conformally invariant form of the actions in Einstein, Weyl, Einstein-Cartan and Einstein-Cartan-Weyl space in general dimensions($>2$) and investigate the relations among them. In Weyl space, the observational consistency condition for the vector field determining non-metricity of the connection can be obtained from the equation of motion. In Einstein-Cartan space a similar role is played by the vector part of the torsion tensor. We consider the case where the trace part of the torsion is the Kalb-Ramond type of field. In this case, we express conformally invariant action in terms…
A study of conformally flat but non flat Bianchi type I and cylindrically symmetric static space-times according to proper projective symmetry is given by using some algebraic and direct integration techniques. It is shown that the special class of the above space-times admit proper projective vector fields.
Torsion and nonmetricity are inherent ingredients in modifications of Eintein's gravity that are based on affine spacetime geometries. In the context of pure f(R) gravity we discuss here, in some detail, the relatively unnoticed duality between torsion and nonmetricity. In particular we show that for R2 gravity torsion and nonmetricity are related by projective transformations. Since the latter correspond simply to redefining the affine parameters of autoparallels, we conclude that torsion and nonmetricity are physically equivalent properties of spacetime. As a simple example we show that both…
In this paper we considered the most general form of non conformally flat cylindrically symmetric non-static space-times to study proper conformal motions using direct integration technique. We have shown that very special classes for cylindrically symmetric space-times admit proper conformal motions. This classification also covers non-static and static plane symmetric space-times. In [9] it was claimed that non conformally flat plane symmetric static space-times do not admit proper conformal motion. Here it is also shown that static and non static plane symmetric space-times admit proper con…
Given a time symmetric initial data set for the vacuum Einstein field equations which is conformally flat near infinity, it is shown that the solutions to the regular finite initial value problem at spatial infinity extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data coincides with Schwarzschild data near infinity.
Using a representation of spatial infinity based in the properties of conformal geodesics, the first terms of an expansion for the Bondi mass for the development of time symmetric, conformally flat initial data are calculated. As it is to be expected, the Bondi mass agrees with the ADM at the sets where null infinity ``touches'' spatial infinity. The second term in the expansion is proportional to the sum of the squared norms of the Newman-Penrose constants of the spacetime. In base of this result it is argued that these constants may provide a measure of the incoming radiation contained in th…
We argue that four-dimensional quantum gravity may be essentially renormalizable if one relaxes the assumption of metricity of the theory. We work with Plebanski formulation of general relativity in which the metric (tetrad), the connection, and the curvature are all independent variables and the usual relations among these quantities are valid only on-shell. One of the Euler-Lagrange equations of this theory ensures its metricity. We show that quantum corrected action contains a counterterm that destroys this metricity property, and that no other counterterms appear, at least, at the one-loop…
The Xi-transform is a new spinor transform arising naturally in Einstein's general relativity. Here the example of conformally flat space-time is discussed in detail. In particular it is shown that for this case, the transform coincides with two other naturally defined transforms: one a two-variable transform on the Lie group SU(2, C), the other a transform on the space of null split octaves. The key properties of the transform are developed.
In general relativity g_ab;c=0 implies that the wave equation (\Box^2-M)g_ab=0 always has M=0. If the underlying geometry is generalized to include non-metricity this incurs M \neq 0, and the above wave equation can be rewritten as M(x)+\td{\na}_a Q_.^a+(\ep+\fr{d}{2}-2)Q_a Q_.^a=0, where \ep=0, 1, 2, or 3, d is the dimension of the spacetime, and Q is the object of non-metricity. The consequences of this equation and the properties of M are investigated.
In this paper, we consider the Bianchi type-I space-time with perfect fluid as matter content of the Universe in the framework of $f(Q)$ gravity (where $% Q$ is the non-metricity scalar) recently proposed by Jiménez et al. (Phys. Rev. D 98.4 (2018): 044048). We find exact solutions of the field equations using the anisotropic property of space-time for volumetric hybrid expansion. The cosmological parameters for the linear form of non-metricity scalar i.e. $f\left( Q\right) =αQ+β$ (where $α$ and $β$ are free model parameters) are discussed and compared with recent Hubble measurements. Also, we…
A closed mathematical model of the statistical self-gravitating system of scalar charged particles for conformal invariant scalar interactions is constructed on the basis of relativistic kinetics and gravitation theory. Asymptotic properties of the model are investigated in the ultrarelativistic limit. It is shown, that scalar charge density automatically generates scalar field effective mass and the value of this mass is found. In the paper it is proved the asymptotic conformal invariance of constitutive equations in case of homogenous isotropic Universe. Also it is proved the asymptotic conf…
Discretization of general relativity is a promising route towards quantum gravity. Discrete geometries have a finite number of degrees of freedom and can mimic aspects of quantum geometry. However, selection of the correct discrete freedoms and description of their dynamics has remained a challenging problem. We explore classical area Regge calculus, an alternative to standard Regge calculus where instead of lengths, the areas of a simplicial discretization are fundamental. There are a number of surprises: though the equations of motion impose flatness we show that diffeomorphism symmetry is b…
The paper is devoted to the description a measurable time-interval (``proper time'') in the Hamiltonian version of general relativity with the Dirac-ADM metric. To separate the dynamical parameter of evolution from the space metric we use the Lichnerowicz conformally invariant variables. In terms of these variables GR is equivalent to the conformally invariant Penrose-Chernicov-Tagirov theory of a scalar field the role of which is played by the scale factor multiplied on the Planck constant. Identification of this scalar field with the modulus of the Higgs field in the standard model of electr…
We study a large family of metric-affine theories with a projective symmetry, including non-minimally coupled matter fields which respect this invariance. The symmetry is straightforwardly realised by imposing that the connection only enters through the symmetric part of the Ricci tensor, even in the matter sector. We leave the connection completely free (including torsion) and obtain its general solution as the Levi-Civita connection of an auxiliary metric, showing that the torsion only appears as a projective mode. This result justifies the widely used condition of setting vanishing torsion …
We discuss conformally covariant differential operators, which under local rescalings of the metric, δ_σg^{μν} = 2 σg^{μν}, transform according to δ_σΔ= r Δσ+ (s-r) σΔfor some r if Δis s-th order. It is shown that the flat space restrictions of their associated Green functions have forms which are strongly constrained by flat space conformal invariance. The same applies to the variation of the Green functions with respect to the metric. The general results are illustrated by finding the flat space Green function and also its first variation for previously found second order conformal different…
It has recently been demonstrated (Class. Quantum Grav. 31, 085010, 2014) that the conformally invariant wave equation on a Minkowski background can be solved with a fully pseudospectral numerical method. In particular, it is possible to include spacelike infinity into the numerical domain, which is appropriately represented as a cylinder, and highly accurate numerical solutions can be obtained with a moderate number of gridpoints. In this paper, we generalise these considerations to the spherically-symmetric wave equation on a Schwarzschild background. In the Minkowski case, a logarithmic sin…
3-dimensional charged black holes in $f({Q})$ gravity
The British University（英国大学）；National Observatory of Athens（雅典国家天文台）；CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China（中国科学院星系与宇宙学重点实验室天文学系，中国科学技术大学）；Universidad Católica del Norte（北方天主教大学）
We investigate the dephasing suffered by a nonrelativistic quantum particle within a conformally fluctuating spacetime geometry. Starting from a minimally coupled massive Klein-Gordon field, the low velocity limit yields an effective Schrodinger equation where the wave function couples to gravity through an effective nonlinear potential induced by the conformal fluctuations. The quantum evolution is studied through a Dyson expansion scheme up to second order. We show that only the nonlinear part of the potential can induce dephasing. This happens through an exponential decay of the off diagona…
Just after Weyl's paper (Weyl in Gravitation und Elektrizität, Sitzungsber. Preuss. Akad., Berlin, 1918) Einstein claimed that a gravity model written in a spacetime geometry with non-metricity suffers from a phenomenon, the so-called second clock effect. We give a new prescription of parallel transport of a vector tangent to a curve which is invariant under both of local general coordinate and Weyl transformations in order to remove that effect. Thus since the length of tangent vector does not change during parallel transport along a closed curve in spacetimes with non-metricity, a second clo…
Direct integration technique is used to study the proper conformal vector fields in non conformally flat Bianchi type-1 space-times. Using the above mentioned technique we have shown that a very special class of the above space-time admits proper conformal vector fields.
Some classical aspects of Metric-Affine Gravity are reviewed in the context of the $F^{(n)}(R)$ type models (polynomials of degree $n$ in the Riemann tensor) and the topologically massive gravity. At the non-perturbative level, we explore the consistency of the field equations when the $F^{(n)}(R)$ models are reduced to a Riemann-Christoffel (RCh) space-time, either via a Riemann-Cartan (RC) space or via an Einstein-Weyl (EW) space. It is well known for the case $F^{(1)}(R)=R$ that any path or reduction "classes" via RC or EW, leads to the same field equations with the exception of the $F^{(n)…
Stable black holes in lower dimensional $f(\mathbb{Q})$ non-metric gravity
Centre for Theoretical Physics, The British University in Egypt（埃及英国大学理论物理中心）；Dipartimento di Fisica E. Pancini, Universit a di Napoli Federico II（那不勒斯费德里科二世大学E. Pancini物理系）；Scuola Superiore Meridionale（南方高等学院）；Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli（意大利国家核物理研究所那不勒斯分部）
A study of proper conformal vector field in non conformally flat cylindrically symmetric static space-times is given by using direct integration technique. Using the above mentioned technique we have shown that a very special class of the above space-time admits proper conformal vector field.
We investigate proper conformal vector fields in non conformally flat Kantowski-Sachs and Bianchi type III space-times using direct integration technique. Using the above mentioned technique we show that very special classes of the above space-times admit proper conformal vector fields.
In this paper, we present a class of rotating solutions in Gauss--Bonnet gravity in the presence of cosmological constant and conformally invariant Maxwell field and study the effects of the nonlinearity of the Maxwell source on the properties of the spacetimes. These solutions may be interpret as black brane solutions with inner and outer event horizons provide that the mass parameter $m$ is greater than an extremal value $m_{ext}$, an extreme black brane if $m=m_{ext}$ and a naked singularity otherwise. We investigate the conserved and thermodynamics quantities for asymptotically flat and as…
Extrinsic geometry and Hamiltonian analysis of symmetric teleparallel gravity
Universita di Napoli Federico II（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学院）；Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli（意大利国家核物理研究所（INFN）那不勒斯分部）；Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学）
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the $r=0$ singularity. We discuss the asymptotic properties of the embedding in the context of Penrose's theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metr…
We discuss in some detail the properties of a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension and whose dynamics significantly differs from that of the ordinary (conformally non-invariant) Nambu-Goto p-branes. We present explicit solutions of the WILL-brane (Weyl-Invariant Light-Like brane) equations of motion in various gravitational backgrounds of physical relevance exhibiting the following new phenomena: (i) In spherically symmetric static backgrounds the WILL-brane automatically positions itself on…
We introduce and study in some detail the properties of a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically lightlike branes for any odd world-volume dimension. Their dynamics significantly differs from that of the ordinary (conformally non-invariant) Nambu-Goto p-branes. We present explicit solutions of the Weyl-invariant lightlike brane- (WILL-brane) equations of motion in various gravitational models of physical relevance exhibiting various new phenomena. In D=4 the WILL-membrane serves as a material and charged source for gravity and electromagnetism i…
In this paper a hidden extra symmetry of conformally invariant Lagrangians occuring in physics is pointed out. This symmetry is most apparent in a metric independent, i.e. in a Palatini-like presentation of the variational problem. In such presentation, the usual conformal weight of fields can be encoded as local dilatation group gauge charges. The conventional conformal invariance of Lagrangians is then equivalent to dilatation gauge invariance. The claim of the paper is, that the most commonly occurring conformally invariant Lagrangians turning up in physics are not only invariant to local d…
It has been known for some time that the SL(2,R) WZWN model reduces to Liouville theory. Here we give a direct and physical derivation of this result based on the classical string equations of motion and the proper string size. This allows us to extract precisely the physical effects of the metric and antisymmetric tensor, respectively, on the {\it exact} string dynamics in the SL(2,R) background. The general solution to the proper string size is also found. We show that the antisymmetric tensor (corresponding to conformal invariance) generally gives rise to repulsion, and it precisely cancels…
An off-shell conformally invariant Galilean Weyl tensor
ens de Lyon, cral umr5574 , Université Claude Bernard Lyon 1, cnrs（里昂师范学院，CRAL UMR5574，克洛德·贝尔纳里昂大学，法国国家科学研究中心）；Institute of Theoretical Physics, Leibniz University Hannover（汉诺威莱布尼茨大学理论物理研究所）；Faculty of Philosophy, University of Oxford（牛津大学哲学系）
We study axisymmetric solution to the conformally invariant wave equation on a Kerr background by means of numerical and analytical methods. Our main focus is on the behaviour of the solutions near spacelike infinity, which is appropriately represented as a cylinder. Earlier studies of the wave equation on a Schwarzschild background have revealed important details about the regularity of the corresponding solutions. It was found that, on the cylinder, the solutions generically develop logarithmic singularities at infinitely many orders. Moreover, these singularities also `spread' to future nul…
