Conformally Flat Spacetimes and Weyl Frames

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordstr?m scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.     

A note on the physical interpretation of Weyl gauging

A possible physical justification of the Weyl gauge procedure is proposed. It is argued that in the case of a general affine connection geodesic parameters have a nontrivial functional relation to metric interval. Since geodesic parameters can also define a notion of interval (via Schild's ladder), we appear to have an inconsistency at the very foundation of any theory which incorporates a nonmetric-compatible connection. This inconsistency can be removed by postulating the Weyl transformations as a valid symmetry of nature, at least at the appropriate energy scale. The method would appear to work even in theories containing shear, provided that it too acquires a mass as a result of (or prior to) the breaking of Weyl symmetry.     

On a new condition distinguishing Weyl and Lorentz space-times

The paper is concerned with the derivability of a Lorentz instead of only a Weyl manifold as space-time structure from postulates about free fall and light propagation. For this purpose it identifies a property distinguishing both kinds of space-times. The property is one of a particular metric of the conformal class of the Weyl manifold. viz. that in suitably chosen locally geodesic coordinates theg _i4 components,i=1, 2, 3 vanish along the time axis. Although seemingly somewhat hidden, one is led to this property in looking for a metric which can play a distinguished role. We demonstrate that for a Lorentzian manifold such a condition is always given; thus it is a necessary one. It is sufficient since for a Weyl space it has the consequence that the metric connection of the selectedg is projectively equivalent to the Weyl connection. Thus, if a Weyl space-time complies with it, it is a reducible one. The results of this paper lay the ground for deriving in a second step this condition from a simple, empirically testable postulate about free-fall worldlines and “radar” measurements by light signals.     

A nonstationary generalization of the Kerr-Newman metric

A new metric depending on three arbitrary parameters is presented by the method of complex coordinate transformations. It gives the gravitational field of a radiating rotating charged body. The metric is algebraically special of Petrov type II according to classification of the Weyl tensor, with a twisting, shear-free, null congruence identical to that of the Kerr-Newman metric. The new metric bears the same relation to the Kerr-Newman metric as the Bonner-Vaidya metric does to the Reissner-Nordstrom metric.     

The interpretion of the C metric. The vacuum case

We present a simplified version of Bonnor's approach to the interpretation of the vacuum C metric as a preparation for using a similar method to investigate the charged case. The idea is to consider the Weyl form of the C metric without calculating the metric components explicitly in terms of the Weyl coordinates. A further coordinate transformation leads to a metric which describes space-time outside the horizons of two Schwarschild-type particles moving with uniform acceleration and joined by a spring.     

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR ², in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR ² the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.     

Type-III and IV interacting Weyl points

3+1-dimensional Weyl fermions in interacting systems are described by effective quasi-relativistic Green’s functions parametrized by a 16-element matrix e _α ^μ in an expansion around the Weyl point. The matrix e _α ^μ can be naturally identified as an effective tetrad field for the fermions. The correspondence between the tetrad field and an effective quasi-relativistic metric g_μν governing the Weyl fermions allows for the possibility to simulate different classes of metric fields emerging in general relativity in interacting Weyl semimetals. According to this correspondence, there can be four types of Weyl fermions, depending on the signs of the components g ⁰⁰ and g ₀₀ of the effective metric. In addition to the conventional type-I fermions with a tilted Weyl cone and type-II fermions with an overtilted Weyl cone for g ⁰⁰ > 0 and, respectively, g ₀₀ > 0 or g ₀₀ < 0, we find additional “type-III” and “type-IV” Weyl fermions with instabilities (complex frequencies) for g ⁰⁰ < 0 and g ₀₀ > 0 or g ₀₀ < 0, respectively. While the type-I and type-II Weyl points allow us to simulate the black hole event horizon at an interface where g ⁰⁰ changes sign, the type-III Weyl point leads to effective spacetimes with closed timelike curves.     

On the Fefferman class of metrics associated with a three-dimensional CR space

We derive the explicit forms of Fefferman's metric for a Cauchy-Riemann space admitting a solution of the tangential Cauchy-Riemann equation and of the corresponding Weyl tensor. We show that its Petrov type is 0 in the case of the hyperquadric or N in all other cases, and that the Fefferman class of metrics does not contain any nonflat solution of Einstein's vacuum equations with cosmological constant.Work supported in part by the Polish Ministry of Science and Higher Education, Research Problem CPBP 01.03.     

Inequivalence of Jordan and Einstein Frame: What Is the Low Energy Gravity in String Theory?

It is well-known that any scalar can be promoted to a Jordan-Brans-Dicke type scalar coupling to the Einstein-Hilbert term through a field dependent Weyl transformation of the metric. The Weyl rescaling also transforms mass terms into coupling constants between matter and the scalar. It is pointed out that there exists a distinguished metric where all scalars decouple from an arbitrary fiducial fermion, e.g. the nucleon. If bound states of this fermion are used to define distances and to probe the interior of the forward light cone, it seems reasonable to say that the metric in that particular frame defines the local geometry of space-time at low energies, as probed by experimental gravity and cosmology.     

Energy and momentum associated with solutions exhibiting directional type singularities

We obtain the energy and momentum densities of a general static axially symmetric vacuum space-time, the Weyl metric, with the help of Landau – Lifshitz and Bergmann – Thomson energy-momentum complexes. We find that these two definitions of energy-momentum complexes do not provide the same energy density for the space-time under consideration, while give the same momentum density. We show that, in the case of the Curzon metric (a particular case of the Weyl metric), these two definitions give the same energy only when R → ∞. Furthermore, we compare these results with those obtained using Einstein, Papapetrou and MØller energy momentum complexes.     

Establishment of the Riemannian structure of space-time by classical means

Efforts at providing a physical-axiomatic foundation of the space-time structure of the general theory of relativity have led, when based on simple empirical facts about freely falling particles and light signals, in a satisfying manner only to a Weyl space-time. By adding postulates based on quantum theory, however, the usual pseudo-Riemannian space-time can be reached. We present a newclassical postulate which provides the same results. It is based upon the notion of the radar distance between freely falling particles and demands the approximate equality of the growth of the radar distance for particle pairs of equal, small initial velocities. We show that given this, a property results, as found in earlier work by the author, that distinguishes between Weyl and Lorentz space-times. The property refers to a special metric and decides whether its metric connection has the given free-fall worldlines as geodesics or not. It consists in the vanishing of the mixed spatiotemporal componentsg _i4 of this metric in suitable coordinates along the worldline of the freely falling observer, as the rest system of which the coordinates are constructed.     

Magnetic fields and the Weyl tensor in the early universe

We have solved the Einstein–Maxwell equations for a class of metrics with constant spatial curvature by considering only a primordial magnetic field as source. We assume a slight modification of the Tolman averaging relations so that the energy–momentum tensor of this field possesses an anisotropic pressure component. This inhomogeneous magnetic universe is isotropic and its time evolution is guided by the usual Friedmann equations. In the case of a flat universe, the space-time metric is free of singularities (except the well-known initial singularity at \(\text {t} = 0\) ). It is shown that the anisotropic pressure of our model has a straightforward relation to the Weyl tensor. We then analyze the effect of this new ingredient on the motion of test particles and on the geodesic deviation of the cosmic fluid.     

Wave fields in Weyl spaces and conditions for the existence of a preferred pseudo-Riemannian structure

Previous axiomatic approaches to general relativity which led to a Weylian structure of space-time are supplemented by a physical condition which implies the existence of a preferred pseudo-Riemannian structure. It is stipulated that the trajectories of the short wave limit of classical massive fields agree with the geodesics of the Weyl connection and it is shown that this is equivalent to the vanishing of the covariant derivative of a mass function of nontrivial Weyl type. This in turn is proven to be equivalent to the existence of a preferred metric of the conformal structure such that the Weyl connection is reducible to a connection of the bundle of orthonormal frames belonging to this distinguished metric.     

Conformal invariants for determinants of laplacians on Riemann surfaces

For a Riemann surface with smooth boundaries, conformal (Weyl) invariant quantities proportional to the determinant of the scalar Laplacian operator are constructed both for Dirichlet and Neumann boundary conditions. The determinants are defined by zeta function regularization. The other quantities in the invariants are determined from metric properties of the surface. As applications explicit representations for the determinants on the flat disk and the flat annulus are derived.     

Finsler gauge transformations and general relativity

The theory of gauge transformations in Finsler space is applied to general relativity. It is seen that the transformations produce new metrics which correspond to the introduction of physical fields. The geodesic equation in the transformed space is equivalent to the equation of motion in the original space where the field is included by a force term. An example is given of a transformation and resulting metric in which the electromagnetic potential is related to parameters of the gauge transformation rather than to gauge potentials. This implies that the electromagnetic field corresponds to a connection instead of a curvature. Another example is given which shows how Weyl or conformal transformations are related to a class of the gauge transformations.     

Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the (n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G:

• •there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric;
• •in that gauge, a simple constraint on the conformal scalar curvature holds;
• •a non-exact Einstein–Weyl structure may exist only if the (n−1)-dimensional homogeneous space G/H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H;
• •the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′.

The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint.We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the (n−2)-dimensional space G/(H×H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU(n/2) being, thanks to our results, invariant under U(1)×SU(n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.     

Broken Weyl Invariance and the Origin of Mass

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, W₄, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields , representing the D(1) gauge fields, with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl symmetry is broken explicitly and the corresponding reduction of the Weyl space W₄ to a pseudo-Riemannian space V₄ is investigated assuming the breaking to be determined by an expression involving the curvature scalar R of the W₄ and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with providing a potential for the Weyl vector fields . After the Weyl-symmetry breaking, one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying V₄. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on , coupled to the energy momentum tensors of the now massive fields involved together with the (massless) radiation fields.     

Circular Photogalvanic Effect in the Weyl Semimetal TaAs

Weyl semimetal(WSM) is expected to be an ideal spintronic material owing to its spin currents carried by the bulk and surface states with spin-momentum locking. The generation of a sizable photocurrent is predicted in non-centrosymmetric WSM arising from the broken inversion symmetry and the linear energy dispersion that is unique to Weyl systems. In our recent measurements, the circular photogalvanic effect(CPGE) is discovered in the TaAs WSM. The CPGE voltage is proportional to the helicity of the incident light, reversing direction if the radiation helicity changes handedness, a periodical oscillation therefore appears following the alteration of light polarization. We herein attribute the CPGE to the asymmetric optical excitation of the Weyl cone, which could result in an asymmetric distribution of photoexcited carriers in momentum space according to an optical spin-related selection rule.