关于纯引力共形反常

本文采用几何和拓扑的方法,讨论弯曲时空中的纯引力共形反常,并得到了纯引力共形反常的新的表达式αε^abcdΩ_abΛΩ_cd+βε^abcd H_abΛH_cd,其中α,β是任意常数,Ω_ab与H_ab分别是Riemann曲率2-形式与Thomas引入的共形不变曲率2-形式。这里,第一项正比于Euler类,第二项除了包含通常熟知的Wegl张量平方项以外,还含有其它共形不变的不变量。 关键词：     

Conformal transformations in metric‐affine gravity and ghosts

Conformal transformations play a widespread role in gravity theories in regard to their cosmological and other implications. In the pure metric theory of gravity, conformal transformations change the frame to a new one wherein one obtains a conformal‐invariant scalar–tensor theory such that the scalar field, deriving from the conformal factor, is a ghost. In this work, conformal transformations and ghosts will be analyzed in the framework of the metric‐affine theory of gravity. Within this framework, metric and connection are independent variables, and, hence, transform independently under conformal transformations. It will be shown that, if affine connection is invariant under conformal transformations, then the scalar field of concern is a non‐ghost, non‐dynamical field. It is an auxiliary field at the classical level, and might develop a kinetic term at the quantum level. Alternatively, if connection transforms additively with a structure similar to yet more general than that of the Levi‐Civita connection, the resulting action describes the gravitational dynamics correctly, and, more importantly, the scalar field becomes a dynamical non‐ghost field. The equations of motion, for generic geometrical and matter‐sector variables, do not reduce connection to the Levi‐Civita connection, and, hence, independence of connection from metric is maintained. Therefore, metric‐affine gravity provides an arena in which ghosts arising from the conformal factor are avoided thanks to the independence of connection from the metric.     

Dark energy and gravity

I review the problem of dark energy focussing on cosmological constant as the candidate and discuss what it tells us regarding the nature of gravity. Section 1 briefly overviews the currently popular “concordance cosmology” and summarizes the evidence for dark energy. It also provides the observational and theoretical arguments in favour of the cosmological constant as a candidate and emphasizes why no other approach really solves the conceptual problems usually attributed to cosmological constant. Section 2 describes some of the approaches to understand the nature of the cosmological constant and attempts to extract certain key ingredients which must be present in any viable solution. In the conventional approach, the equations of motion for matter fields are invariant under the shift of the matter Lagrangian by a constant while gravity breaks this symmetry. I argue that until the gravity is made to respect this symmetry, one cannot obtain a satisfactory solution to the cosmological constant problem. Hence cosmological constant problem essentially has to do with our understanding of the nature of gravity. Section 3 discusses such an alternative perspective on gravity in which the gravitational interaction—described in terms of a metric on a smooth spacetime—is an emergent, long wavelength phenomenon, and can be described in terms of an effective theory using an action associated with normalized vectors in the spacetime. This action is explicitly invariant under the shift of the matter energy momentum tensor T _ab → T _ab + Λ _gab and any bulk cosmological constant can be gauged away. Extremizing this action leads to an equation determining the background geometry which gives Einstein’s theory at the lowest order with Lanczos–Lovelock type corrections. In this approach, the observed value of the cosmological constant has to arise from the energy fluctuations of degrees of freedom located in the boundary of a spacetime region.     

Radiatively induced spontaneous symmetry breaking and phase transitions in curved spacetime

Local gauge symmetries which are spontaneously broken in flat spacetime are shown to be restored for large spacetime curvatures. The case of symmetry breaking due to radiative quantum corrections in gauge theories with elementary scalar fields is considered explicitly. In spacetimes with a positive Ricci curvature scalar R and a cosmological event horizon, the critical curvature R_C is of O(m_H²) or O(m_W²), depending on whether the theory is formulated with conformal or minimal scalar fields. In Ricci flat spacetimes with a conventional event horizon the symmetry is expected to be restored when the temperature of the Hawking thermal radiation is of O(m_W). This phenomenon is described in detail, using functional integral methods and dimensional renormalization, for massless scalar electro-dynamics in de-Sitter spacetime. For conformal scalars, the symmetry restoring phase transition is first order, the critical curvature being R_C = 0.910 m_H². For minimal scalars, an anomalous, curvature dependent mass counterterm is required. The phase transition in this case is second order, and occurs at R_C = 83.57 m_W². Symmetry restoration at finite temperature in flat spacetime is considered in an appendix. The critical temperature at which a first-order phase transition occurs in the Weinberg-Salam model is found to be T_C = 0.329 m_W.     

Weyl geometry

We develop the properties of Weyl geometry, beginning with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime. We generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, enlarging the symmetry from Poincaré to Weyl, we develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We present a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that it is conformally equivalent to general relativity. This theory is invariant under local dilatations, but not the full conformal group.     

引力理论的场方程与自由粒子运动方程的一般形式

由物质场和引力场的较为广泛的拉氏函数密度及其在(ε^mn,ξ^μ)变换下的不变性出发,导出了引力场方程和自由粒子运动方程的一般形式。它们有着较为广泛的适用性。已表明广义相对论、ECSK理论及R+R²+T²型有挠引力理论等均可作为特殊情况纳入这个体系之中。 关键词：     

Minimal coupling of electromagnetic field in Riemann-Cartan spacetime for perfect fluids

We minimally couple the electromagnetic field to gravity in Riemann-Cartan spacetime in the self-consistent formalism for perfect fluids by treating the internal energy of matter as a function of the electromagnetic field. The overall Lagrangian of the gravitational field, perfect fluid, and the electromagnetic field is constrained to be gauge invariant under gauge transformations of the vector potential. The theory preserves both charge conservation and particle number conservation, and gives the usual form of the free field equations.     

Nonscalar singularities in spatially homogeneous cosmologies

Singularities in vacuum spatially homogeneous cosmological models are investigated. It is shown that in general the curvature scalarR _* ^abcd R^*abcddiverge and that the only solutions which have curvature singularities at which this scalar does not diverge describe certain plane-wave space-times. It is argued that with matter present these nonscalar singularities are even less likely to occur. The exceptional case of Bianchi type VI_–1/9 is not considered.     

Conformal compacifications from spinor geometry

Compactified Minkowski spacetime is suggested by conformal covariance of Maxwell equations, while E. Cartan's definition of simple spinors leads to the idea of compactified momentum space. Assuming both diffeomorphic to (S ₃ × S ₁ )/Z ₂ , one may obtain in the conformally flat stereographic projection field theories both infrared and ultraviolet regularized. On the compact manifold themselves instead, Fourier integrals of wave-field oscillations would have to be replaced by Fourier series summed over indices of spherical eigenfunctions: n, l, m, m. Tentatively identifying those wave structures with spacetime itself (in the frame of Big-Bang) and/or with matter and radiation distribution, some large-scale (hydrogenic) and small-scale (lattice) space structures are conjectured.     

Clifford代数中的双曲相位变换群及其在四维相对论时空中的应用

将Clifford代数所定义的双曲复空间R_H和作用在双曲复空间R_H上的双曲相位变换群U₄(H)赋予了明确的物理意义. 双曲复空间R_H同构于四维Minkowski时空，而其上的双曲相位变换群U₄(H)就是四维相对论时空中的洛仑兹(Lorentz)变换群. 进一步，利用U₄(H)群的复合变换性质，自然导出了四维Minkowski时空中Lorentz变换和速度变换的一般表达式. 由此，将狭义相对论中的特殊Lorentz变换作为特例包含其中. 关键词： 双曲复数 双曲相位变换 Minkowski时空 Clifford代数     

Action and reaction in the theories of direct interparticle action

Newton's third law of motion is examined in the context of the theories of direct interparticle action. In such theories, interactions between particles travel backward and forward in time at speeds not exceeding the speed of light. It is shown that while in the flat spacetime the equality of action and reaction can be clearly demonstrated, the situation is considerably more complicated in the curved spacetime. The phenomenon of gravitational scattering intervenes to destroy the equality of action and reaction. Nevertheless, when gravitation is taken into account, there is no violation of the conservation law of energy and momentum. These results are discussed in the framework of general relativity for the case of the electromagnetic interaction.     

The Blackbody Radiation Spectrum Follows from Zero-Point Radiation and the Structure of Relativistic Spacetime in Classical Physics

The analysis of this article is entirely within classical physics. Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. Furthermore, conformal symmetry carries solutions of Maxwell’s equations into solutions. In an inertial frame, conformal symmetry leaves zero-point radiation invariant and does not connect it to non-zero-temperature; time-dilating conformal transformations carry the Lorentz-invariant zero-point radiation spectrum into zero-point radiation and carry the thermal radiation spectrum at non-zero temperature into thermal radiation at a different non-zero temperature. However, in a non-inertial frame, a time-dilating conformal transformation carries classical zero-point radiation into thermal radiation at a finite non-zero-temperature. By taking the no-acceleration limit, one can obtain the Planck radiation spectrum for blackbody radiation in an inertial frame from the thermal radiation spectrum in an accelerating frame. Here this connection between zero-point radiation and thermal radiation is illustrated for a scalar radiation field in a Rindler frame undergoing relativistic uniform proper acceleration through flat spacetime in two spacetime dimensions. The analysis indicates that the Planck radiation spectrum for thermal radiation follows from zero-point radiation and the structure of relativistic spacetime in classical physics.     

Instantons in conformal gravity

We study extrema of the general conformally invariant action:

$\text{S}{}_{\mathrm{c}}\text{= ∫}\text{1}\text{α}{}^2\text{C}{}^{\mathrm{abcd}}\text{C}{}_{\mathrm{abcd}}\text{+γR}{}^{\mathrm{abcd}}{}^1\text{R}{}_{\mathrm{abcd}}{}^1\text{+iθR}{}^{\mathrm{abcd}}\text{1R}{}_{\mathrm{abcd}}$

.We find the first examples in four dimensions of asymptotically euclidean gravitational instantons. These have arbitrary Euler number and Hirzebruch signature. Some of these instantons represent tunneling between zero-curvature vacua that are not related by small gauge transformations. Others represent tunneling between flat space and topologically non-trivial zero-energy initial data. A general formula for the one-loop determinant is derived in terms of the renormalization group invariant masses, the volume of space-time, the Euler number and the Hirzebruch signature.     

The conformal status of o = ‐3/2 Brans‐Dicke cosmology

Following recent fit of supernovae data to Brans‐Dicke theory which favours the model with o = ‐ 3/2 [1] we discuss the status of this special case of Brans‐Dicke cosmology in both isotropic and anisotropic framework. It emerges that the limit o = ‐3/2 is consistent only with the vacuum field equations and it makes such a Brans‐Dicke theory conformally invariant. Then it is an example of the conformal relativity theory which allows the invariance with respect to conformal transformations of the metric. Besides, Brans‐Dicke theory with o = ‐3/2 gives a border between a standard scalar field model and a ghost/phantom model. In this paper we show that in o = ‐3/2 Brans‐Dicke theory, i.e., in the conformal relativity there are no isotropic Friedmann solutions of non‐zero spatial curvature except for k=‐1 case. Further we show that this k=‐1 case, after the conformal transformation into the Einstein frame, is just the Milne universe and, as such, it is equivalent to Minkowski spacetime. It generally means that only flat models are fully consistent with the field equations. On the other hand, it is shown explicitly that the anisotropic non‐zero spatial curvature models of Kantowski‐Sachs type are admissible in o = ‐3/2 Brans‐Dicke theory. It then seems that an additional scale factor which appears in anisotropic models gives an extra deegre of freedom and makes it less restrictive than in an isotropic Friedmann case.     

Effective action in higher derivative (Super)gravities

'The one-loop effective action (EA) with an accuracy up to linear curvature terms ind=4R ²-gravity, conformal gravity, andN=1,d=4 conformal supergravity on the backgroundR ₄×T_4–k,k=1, 2, 3 is calculated. (Here,R _k is thek-dimensional curved space, T_n is then-dimensional torus). The one-loop EA in multidimensionalR ²-gravity and ind=10 conformal supergravity on the backgroundR ₄ ×T _d–4 is also obtained. The mechanism of inducing the Einstein gravity from the EA of considered theories of higher derivative (super)gravity is presented.We are grateful to I. L. Bukhbinder for the numerous discussions of considered questions.     

The gravitational anomaly: An elementary,coordinate-space approach

Alvarez-Gaumé and Witten have shown that energy-momentum conservation must be violated in certain parity-violating quantum field theories involving gravity. In two dimensions this effect can be studied without the aid of Feynman diagrams or calculations in momentum space. The arguments parallel those for the conformal (trace) anomaly; as in that case, there are two kinds of arguments, one based on the conservation equations themselves with some very general assumptions, and the other based on explicit calculations and renormalization in a model theory with a linear field equation. The basic point is that if matter is created at all by the gravitational field, it must appear in both left-moving and right-moving modes if the conservation law is to hold always.This essay received an honourable mention (1985) by the Gravity Research Foundation—Ed.     

The momentum constraints of general relativity and spatial conformal isometries

Transverse-tracefree (TT-) tensors on (R ³,g _ab), withg _ab an asymptotically flat metric of fast decay at infinity, are studied. When the source tensor from which these TT tensors are constructed has fast fall-off at infinity, TT tensors allow a multipole-type expansion. Wheng _ab has no conformal Killing vectors (CKV's) it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable TT tensor. When CKV's exist there are obstructions — certain (combinations of) moments have to vanish — which we study.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P9376-PHY.Partially supported by Forbairt Grant SC/94/225.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

The volume of the past light-cone and the Paneitz operator

We study a conjecture involving the invariant volume of the past light-cone from an arbitrary observation point back to a fixed initial value surface. The conjecture is that a fourth order differential operator which occurs in the theory of conformal anomalies gives 8π when acted upon the invariant volume of the past light-cone. We show that an extended version of the conjecture is valid for an arbitrary homogeneous and isotropic geometry. First order perturbation theory about flat spacetime reveals a violation of the conjecture which, however, vanishes for any vacuum solution of the Einstein equation. These results may be significant for constructing quantum gravitational observables, for quantifying the back-reaction on spacetime expansion and for alternate gravity models which feature a timelike vector field.     

Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.