This paper reframes Riemannian geometry as a generalized Lie algebra allowing the equations of both RG and then General Relativity to be expressed as commutation relations among fundamental operators. We begin with an Abelian Lie algebra of n operators, X, whose simultaneous eigenvalues, y, define a real n-dimensional space. Then with n new operators defined as independent functions, we define contravariant and covariant tensors in terms of their eigenvalues, on a Hilbert space representation. We then define n additional operators, D, whose exponential map is to translate X as defined by a noncommutative algebra of operators (observables) where the structure constants are shown to be the metric functions of the X operators thus allowing for spatial curvature resulting in a noncommutativity among the D operators. The D operators then have a Hilbert space position-diagonal representation as generalized differential operators plus an arbitrary vector function A(X), which, with the metric, written as a commutator, can express the Christoffel symbols, and the Riemann, Ricci and other tensors as commutators in this representation. Traditional RG and GR are obtained in a position diagonal representation of this noncommutative algebra of 2n+1 operators. We seek to provide a more general framework for RG to support an integration of GR, QT, and the SM by generalizing Lie algebras as described.
Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics
Instituto de Cosmologia Relatividade e Astrofisica (ICRA/CBPF)（宇宙学相对论与天体物理研究所）；CAPES Foundation, Ministry of Education of Brazil（巴西教育部CAPES基金会）；Physics Department, “La Sapienza” University of Rome（罗马大学物理系）
The LARES (LAser RElativity Satellite) satellite was successfully launched in February 2012. The LARES space experiment is based on the orbital determinations of the laser ranged satellites LARES, LAGEOS (LAser GEOdynamics Satellite) and LAGEOS 2 together with the determination of the Earth's gravity field by the GRACE (Gravity Recovery And Climate Experiment) mission. It will test some fundamental physics predictions and provide accurate measurements of the framedragging effect predicted by Einstein's theory of General Relativity. By one hundred Monte Carlo simulations of the LARES experiment…
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.
In complex general relativity, Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold, with holomorphic connection and holomorphic curvature tensor. A multisymplectic analysis shows that the Hamiltonian constraint is replaced by a geometric structure linear in the holomorphic multimomenta, providing some boundary conditions are imposed on two-complex-dimensional surfaces. On studying such boundary conditions, a link with the Penrose twistor programme is found. Moreover, in the case of real Riemannian four-manifolds, the local theory of primary and secondary…
In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature. As a step towards resolving the problem of time in quantum gravity, we attempt to merge the Ricci flow equation with the Hamilton-Jacobi equation for general relativity.
We analyze some extensions of General Relativity. Within the framework of modified gravity, the Newtonian limit of a class of gravitational actions is discussed on the basis of the corresponding scalar-tensor model. For a generalized asymmetric metric, autoparallel trajectories are defined under suitable conditions at first approximation order.
We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for $\le 9$ points based on the known atlas of regular graphs. We also discuss aspects of the quantum theory defined by functional integration.
We argue that all Einstein-Maxwell or Einstein-Proca solutions to general relativity may be used to construct a large class of solutions (involving torsion and non-metricity) to theories of non-Riemannian gravitation that have been recently discussed in the literature.
The Barrett-Crane intertwiner for the Riemannian general relativity is systematically derived by solving the quantum Barrett-Crane constraints corresponding to a tetrahedron (except for the non-degeneracy condition). It was shown by Reisenberger that the Barrett-Crane intertwiner is the unique solution. The systematic derivation can be considered as an alternative proof of the uniqueness. The new element in the derivation is the rigorous imposition of the cross-simplicity constraint.
W.Lawvere suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E, particular in some elementary topos. The synthetic differential geometry (SDG) is the theory developed by A.Kock in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a …
A new variable in the Riemannian geometry is introduced by the tetrad and the Ricci's coefficients of rotation, the characters of curve of the Riemannian geometry are determined completely by the new variable; for general relativity, all the Einstein-Hilbert action, the Einstein equation in general relativity and the Dirac equation in curved spacetime can be expressed by the new variable, and, further, as well the action of the theory on the interaction of gravitational, electromagnetic and spinor field (TGESF). All the characters of transformations of the new variable, the Einstein-Hilbert ac…
We give a justification for generalizing the notion of the energy momentum stress tensor in general relativity which results in the Jacobi curvature tensor being the embodiment of total energy momentum stress including gravity in general relativity.
We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure equations and their symmetries in terms of a new product (the co-multiplication). It is showen how the Cartan - Grassmann algebra can be endowed with the structure of a Hopf algebra.
A surface theoretic view of non-perturbative quantum gravity as "spin-foams" was proposed by Baez. A possibility of constructing such a model was studied some time ago based on (2+1) dimensional general relativity as a reformulation of the Ponzano-Regge model in Riemannian spacetime. In the present work, a model based on (3+1) dimensional general relativity in Riemannian spacetime is presented. The construction is explicit and calculable in details. For a physical application, a computation formula for spacetime volume density correlations is presented. Remarks for further investigations are m…
We start from the pure Einstein-Hilbert action in Metric Affine Gravity, with the orthonormal metric. We get an effective Levi-Civita Dilaton gravity theory in which the Dilaton field is related to the scaling of the gravitational coupling. When the Weyl symmetry is broken the resulting Einstein-Hilbert term is equivalent to the Levi-Civita one, using the projective invariance of the model, the non-metricity and torsion may be removed, so that we get a theory perfectly equivalent to general relativity. This may explain why low energy gravity is described by a Riemannian connection.
Less explored than their metric (Riemannian) counterparts, metric-affine (or Palatini) theories bring an unexpected phenomenology for gravitational physics beyond General Relativity. Lessons of crystalline structures, where the presence of defects in their microstructure requires the use of non-Riemannian geometry for the proper description of their properties in the macroscopic continuum level, are discussed. In this analogy, concepts such as wormholes and geons play a fundamental role. Applications of the metric-affine formalism developed by the authors in the last three years are reviewed.
The configuration space of general relativity is superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. Taking this conformal nature seriously leads to a new theory of gravity which although very similar to general relativity has some very different features particularly in cosmology and quantisation. It should reproduce the standard tests of general relativity. The cosmology is stud…
We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry ca…
The Killing equation for a general Finsler space is set up. It is showed that the Killing equation of $(α,β)$ space can be divided into two parts. One is the same with Killing equation of a Riemannian metric, another equation can be regarded as a constraint. The solutions of Killing equations present explicitly the isometric symmetry of Finsler space. We find that the isometric group of a special case of $(α,β)$ space is the same with the symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk space are given. Unlike Riemannian constant curvature space, the 4 dimensional…
We study, in this paper, curvature inheritance symmetry (CI), $£_ξR_{bcd}^{a}=2αR_{bcd}^{a}$, where $α$ is a scalar function, for string cloud and string fluid in the context of general relativity. Also, we have obtained some result when a proper CI (i.e., $α\neq 0$) is also a conformal Killing vector.
General Very Special Relativity (GVSR) is the curved space-time of Very Special Relativity (VSR) proposed by Cohen and Glashow. The geometry of GVSR possesses a line element of Finsler Geometry proposed by Bogoslovsky. We calculate the Einstein field equations and derive a modified FRW cosmology, for an osculating Riemannian space. The Friedman equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial spurionic vector field introduced into the metric, gives back an estimation of t…
The configuration space of general relativity is superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. Recently a manifestly 3-dimensional theory was constructed with conformal superspace as the configuration space. Here a fully 4-dimensional action is constructed so as to be invariant under conformal transformations of the 4-metric using general relativity as a guide. This action i…
We show that the concept of the ADM mass in general relativity can be understood as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large Riemannian polyhedra. We also express the $n$-dimensional mass as a suitable integral of geometric quantities that determine the $(n-1)$-dimensional mass.
The notion of geometrical duality is discussed in the context of both Brans-Dicke theory and general relativity. It is shown that, in some particular solutions, the spacetime singularities that arise in usual Riemannian general relativity may be avoided in its dual representation (Weyl-type general relativity). This dual representation provides a singularity-free picture of the World that is physicaly equivalent to the canonical general relativistic one.
The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion. The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula. Our approach is a local one: …
A generalization of the notion of ellipsoids to curved Riemannian spaces is given and the possibility to use it in describing the shapes of rotating bodies in general relativity is examined. As an illustrative example, stationary, axisymmetric perfect-fluid spacetimes with a so-called confocal inside ellipsoidal symmetry are investigated in detail under the assumption that the 4-velocity of the fluid is parallel to a time-like Killing vector field. A class of perfect-fluid metrics representing interior NUT-spacetimes is obtained along with a vacuum solution with a non-zero cosmological constan…
We present results relevant to the relation between quantum effects in a Riemannian space and on the surface appearing as a result of its isometric embedding in a flat space of a higher dimension. We discuss the mapping between the Hawking effect fixed by an observer in the Riemannian space with a horizon and the Unruh effect related to an accelerated motion of this observer in the ambient space. We present examples for which this mapping holds and examples for which there is no mapping. We describe the general form of the hyperbolic embedding of the metric with a horizon smoothly covering the…
In this paper the causality issues are discussed in a non-Riemannian geometry, called Lyra geometry. It is a non-Riemannian geometry originated from Weyl geometry. In order to compare this geometry with the Riemannian geometry, the Einstein field equations are considered. It is verified that the Gödel and Gödel-type metric are consistent with this non-Riemannian geometry. A non-trivial solution for Gödel universe in the absence of matter sources is determined without analogue in general relativity. Different sources are considered and then different conditions for causal and non-causal solutio…
Computer algebra programs are presented for application in general relativity, in electrodynamics, and in gauge theories of gravity. The mathematical formalism used is the calculus of exterior differential forms, the computer algebra system applied Hearn's Reduce with Schruefer's exterior form package Excalc. As a non-trivial example we discuss a metric of Plebanski & Demianski (of Petrov type D) together with an electromagnetic potential and a triplet of post-Riemannian one-forms. This whole geometrical construct represents an exact solution of a metric-affine gauge theory of gravity. We desc…
This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M - the Cheeger-Gromov theory - and extensions thereof to Ricci curvature in place of full curvature. This theory is then applied to study a collection of different issues in mathematical aapects of General Relativity.
Admitting non-Riemannian geometries, Double Field Theory extends the notion of spacetime beyond the Riemannian paradigm. We identify a class of singular spacetimes known in General Relativity with regular non-Riemannian geometries. The former divergences merely correspond to coordinate singularities of the generalised metric for the latter. Computed in string frame, they feature an impenetrable non-Riemannian sphere outside of which geodesics are complete with no singular deviation. Approaching the non-Riemannian points, particles freeze and strings become (anti-)chiral.
An alternative to the representation of complex relativity by self-dual complex 2-forms on the spacetime manifold is presented by assuming that that the bundle of real 2-forms is given an almost-complex structure. From this, one can define a complex orthogonal structure on the bundle of 2-forms, which results in a more direct representation of the complex orthogonal group in three complex dimensions. The geometrical foundations of general relativity are then presented in terms of the bundle of oriented complex orthogonal 3-frames on the bundle of 2-forms in a manner that essentially parallels …
Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds $(Σ\cong \mathbb{S}^2,g)$, with $g$ satisfying $λ_1 = λ_1(-Δ_g + K(g))>0$, where $λ_1$ is the first eigenvalue of the operator $-Δ_g+K(g)$ and $K(g)$ is the Gaussian curvature of $g$, with control on the ADM mass of the extension. Remarkably, this procedure allowed them to compute the Bartnik mass in this so-called minimal case; the Bartnik mass is a notion of quasi-local mass in General Relativity which is very challenging to compute. In this survey, we describe the Mantoulid…
All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in 3, 4 and 6 dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in 3-dimensional spaces because of the link to General Relativity. In particular, all 3-dimensional constant negative scalar curvature spheri…
We analyse the quantum geometry of 3-dimensional deformed special relativity (DSR) and the notion of spacetime points in such a context, identified with coherent states that minimize the uncertainty relations among spacetime coordinates operators. We construct this system of coherent states in both the Riemannian and Lorentzian case, and study their properties and their geometric interpretation.
In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the photon spheres of Schwarzschild, Reissner-Nordstrom, as well as continuous mass models solving the Tolman-Oppenheimer-Volkoff equation. Moreover, we derive an isoperimetric inequality for gravitational lensing in Riemannian optical geometry, using curve shortening flow and the Gaus…
Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of VTM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this approach is unsatisfactory when there is an indefinite metric on M because the corresponding Finsler fundamental function would not be differentiable over TM (even without its zero section) and therefore TM cannot serve as the base space. We then make the simple observation that for any differentiable Lorentzian metric on a smooth space-time, the corresponding…
More then forty years ago R.I. Pimenov introduced a new geometry -- semi-Riemannian one -- as a set of geometrical objects consistent with a fibering $ pr: M_n \to M_m.$ He suggested the heuristic principle according to which the physically different quantities (meter, second, coulomb etc.) are geometrically modelled as space coordinates that are not superposed by automorphisms. As there is only one type of coordinates in Riemannian geometry and only three types of coordinates in pseudo-Riemannian one, a multiple fibered semi-Riemannian geometry is the most appropriate one for the treatment of…
The four-dimensional gauge group of general relativity corresponds to arbitrary coordinate transformations on a four-manifold. Theories of gravity with a dynamical structure remarkably like Einstein's theory can be obtained on the basis of a four-dimensional gauge group of arbitrary coordinate and conformal transformations of riemannian metrics defined on a three-manifold. This new symmetry is more restrictive and hence more predictive. Many of the difficulties that have plagued the canonical quantization of general relativity seem to vanish.
We present the general theory of relativity in the language of a non-Riemannian geometry, namely, Weyl geometry. We show that the new mathematical formalism may lead to different pictures of the same gravitational phenomena, by making use of the concept of Weyl frames. We show that, in this formalism, it is possible to construct a scalar-tensor gravitational theory that is invariant with respect to the so-called Weyl tranformations and reduces to general relativity in a particular frame, the Riemann frame. In this approach the Weyl geometry plays a fundamental role since it appears as the natu…
The now-famous Majumdar-Papapetrou exact solution of the Einstein-Maxwell equations describes, in general, $N$ static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When $N=2$, this solution defines the two-black-hole spacetime, and the relativistic two-center problem is the problem of geodesic motion on this static background. Contopoulos and a number of other workers have recently discovered through numerical experiments that in contrast with the Newtonian two-center problem, where the dynamics is completely integrable, relativistic null-geo…
A recent analysis of real general relativity based on multisymplectic techniques has shown that boundary terms may occur in the constraint equations, unless some boundary conditions are imposed. This paper studies the corresponding form of such boundary terms in complex general relativity, where space-time is a four-complex-dimensional complex-Riemannian manifold. A complex Ricci-flat space-time is recovered providing some boundary conditions are imposed on two-complex-dimensional surfaces. One then finds that the holomorphic multimomenta should vanish on an arbitrary three-complex-dimensional…
Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-)Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativit…
Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct such foliations. For this purpose we define a special family of topological two-spheres, which we call "rigid spheres". We prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold (in case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere). The rigi…
This letter describes a novel derivation of general relativity by considering the (non)self-consistency of theories whose Hamiltonians are constraints. The constraints, from Hamilton's equations, generate the evolution, while the evolution, in turn, must preserve the constraints. This closure requirement can be used as a selection mechanism for general relativity starting from a very simple set of assumptions. The configuration space is chosen to be a family of $3 \times 3$ positive definite symmetric matrices on some bare 3-manifold. A general Hamiltonian is constructed on this space of matri…
A new approach to the dynamics of the universe based on work by O Murchadha, Foster, Anderson and the author is presented. The only kinematics presupposed is the spatial geometry needed to define configuration spaces in purely relational terms. A new formulation of the relativity principle based on Poincare's analysis of the problem of absolute and relative motion (Mach's principle) is given. The enire dynamics is based on shape and nothing else. It leads to much stronger predictions than standard Newtonian theory. For the dynamics of Riemannian 3-geometries on which matter fields also evolve,…
Considering a five-dimensional (5D) Riemannian spacetime with a particular stationary Ricci-flat metric, we have applied the recently developed Extended General Relativistic theory to calculate the lower limit for the size of some relevant galaxies: the Milky Way, Andromeda and M87. Our results are in very good agreement with observations.
We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology.
Standard approaches to quantum gravity start with a pre-spacetime structure and attempt, in accordance with Bohr's correspondence principle, to recover the pseudo-Riemannian manifold in the low energy limit. These approaches assume there is a smooth transition from quantum gravity to general relativity common to successful quantum theories. However, as gravitational field, and hence spacetime, cannot be considered in isolation from physical fields, discontinuities in properties of physical fields, such as loss of mass at the electroweak symmetry breaking scale, may result in change of spacetim…
An approach to the discrete quantum gravity based on the Regge calculus is discussed which was developed in a number of our papers. Regge calculus is general relativity for the subclass of general Riemannian manifolds called piecewise flat ones. Regge calculus deals with the discrete set of variables, triangulation lengths, and contains continuous general relativity as a particular limiting case when the lengths tend to zero. In our approach the quantum length expectations are nonzero and of the order of Plank scale $10^{-33}cm$. This means the discrete spacetime structure on these scales.
In the framework of non-riemannian geometry, we derive exact static vacuum solutions of the field equations obtained from the full equivalent version of the Einstein-Hilbert action when torsion degrees of freedom are taken into account. By imposing spherical symmetry and a suitable choice for the contorsion degrees of freedom, the static geometry provides deviations on the predictions of the observational tests predicted by General Relativity -- namely on the advance of planetary perihelia and the bending of light rays -- which we infer. The analytical extension is built in two particular doma…
The classical and continuum limit of a quantum gravitational setting could lead, at mesoscopic regimes, to a very different notion of geometry w.r.t. the pseudo-Riemannian one of special and general relativity. A possible way to characterize this modified space-time notion is by a momentum dependent metric, in such a way that particles with different energies could probe different spacetimes. Indeed, doubly special relativity theories, deforming the special relativistic kinematics while maintaining a relativity principle, have been understood within a geometrical context, by considering a curv…
Between the microscopic domain ruled by quantum gravity, and the macroscopic scales described by general relativity, there might be an intermediate, "mesoscopic" regime, where spacetime can still be approximately treated as a differentiable pseudo-Riemannian manifold, with small corrections of quantum gravitational origin. We argue that, unless one accepts to give up the relativity principle, either such a regime does not exist at all (hence, the quantum-to-classical transition is sharp), or the only mesoscopic, tiny corrections conceivable are on the behaviour of physical fields, rather than …
It is shown that, with some reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the field equations of general relativity to construct a Robertson-Walker metric for a quantum particle so that the line element of the particle can be transformed entirely to that of the Minkowski spacetime, which is assumed by a quantum observer, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. Spacetime structure of a quantum particle may have either positive or negative curvature. However…
Some pairs of neutron star models can intuitively be thought of as being `closer' together than others, in the sense that more precise observations might be required to distinguish between them than would be necessary for other pairs. In this paper, borrowing ideas from the study of geometrodynamics, we introduce a mathematical formalism to define a geometric distance between stellar models, to provide a quantitative meaning to this notion of closeness. In particular, it is known that the set of all metrics on a Riemannian manifold itself admits the structure of a Riemannian manifold (`configu…
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Later, their decay assumptions were weakened by Metzger, Huang, Eichmair-Metzger, and the author. In this work, we prove the reverse implication, i.e. any three-dimensional Riemannian manifold is asymptotically flat if it possesses a CMC-cover satisfying certain geometric curvature estimates, a uniqueness property, a weak foliation property, and each s…
We show that the general theory of relativity can be formulated in the language of Weyl geometry. We develop the concept of Weyl frames and point out that the new mathematical formalism may lead to different pictures of the same gravitational phenomena. We show that in an arbitrary Weyl frame general relativity, which takes the form of a scalar-tensor gravitational theory, is invariant with respect to Weyl tranformations. A kew point in the development of the formalism is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in term…
