Gauge invariant approach to nonmetricity theories and the second clock effect
Israel Quiros
备注：
18 pages, no figures. A new statement of the gauge invariant parallel transport law has been achieved, which allows to have a parallel transport law that applies to any vector and/or tensor fields without exception. The basic results of the manuscript have not changed
In this paper we discuss on recent attempts aimed at demonstrating that, contrary to well-known results, the second clock effect (SCE) does not take place in generalized Weyl spaces -- spaces with arbitrary nonmetricity -- denoted here as $W_4$ spaces. These attempts include Weyl gauge theories of gravity, as well as the symmetric teleparallel theories (STTs). Our approach to this issue is based on the adoption of Weyl gauge symmetry (WGS) which is a manifest symmetry of the basic laws of Weyl geometry. We shall consistently adapt mathematical and geometrical quantities and concepts so that the resulting geometrical framework be gauge invariant. This issue is of special relevance for the fate of nonmetricity theories, including a class of the STTs which is being intensively applied in the cosmological framework. As we shall show, if realize that WGS is a manifest symmetry of generalized Weyl spaces $W_4$, and identify physical vectors and tensors with corresponding hypothetical vectors and tensors living in $W_4$, neither the Weyl gauge theories nor the nonmetricity theories are free of the SCE, unless Weyl integrable geometry (WIG) spaces are considered.
We extend the class of recently formulated scalar-nonmetricity theories by coupling a five-parameter nonmetricity scalar to a scalar field and considering a mixed kinetic term between the metric and the scalar field. The symmetric teleparallel constraint is invoked by Lagrange multipliers or by inertial variation. The equivalents for the general relativity and ordinary (curvature-based) scalar-tensor theories are obtained as particular cases. We derive the field equations, discuss some technical details, e.g., debraiding, and formulate the Hamilton-like approach.
We study a generalization of the Holst action where we admit nonmetricity and torsion in manifolds with timelike boundaries (both in the metric and tetrad formalism). We prove that its space of solutions is equal to the one of the Palatini action. Therefore, we conclude that the metric sector is in fact identical to GR, which is defined by the Einstein-Hilbert action. We further prove that, despite defining the same space of solutions, the Palatini and (the generalized) Holst Lagrangians are not cohomologically equal. Thus, the presymplectic structure and charges provided by the Covariant Phas…
In this article we calculate the post-Newtonian limit of a general class of scalar-nonmetricity theories of gravity. The action is assumed to be a free function of the nonmetricity scalar, the kinetic term of the scalar field, two derivative couplings and the scalar field itself. We use the parametrized post-Newtonian formalism to solve the arising field equations for the case of a massless scalar field in order to compare several subclasses of this theory to solar system observations. In particular, we find several classes of theories which are indistinguishable from general relativity on the…
The intriguing choice to treat alternative theories of gravity by means of the Palatini approach, namely elevating the affine connection to the role of independent variable, contains the seed of some interesting (usually under-explored) generalizations of General Relativity, the metric-affine theories of gravity. The peculiar aspect of these theories is to provide a natural way for matter fields to be coupled to the independent connection through the covariant derivative built from the connection itself. Adopting a procedure borrowed from the effective field theory prescriptions, we study the …
In this work, we show that a class of metric-affine gravities can be reduced to a Riemann-Cartan one. The reduction is based on the cancelation of the nonmetricity against the symmetric components of the spin connection. A heuristic proof, in the Einstein-Cartan formalism, is performed in the special case of diagonal unitary tangent metric tensor. The result is that the nonmetric degrees of freedom decouple from the geometry. Thus, from the point of view of isometries on the tangent manifold, the equivalence might be viewed as an isometry transition from the affine group to the Lorentz group, …
In this paper we discuss on the phenomenological viability of nonmetricity theories of gravity which are based in the class of generalized Weyl spacetimes -- denoted by $W_4$ -- where arbitrary nonmetricity is allowed. This class of geometry includes the so called teleparallel spaces $Z_4$, which are the geometric basement of the symmetric teleparallel theories (STTs). The guiding principle in our discussion is Weyl gauge symmetry (WGS), which is a manifest symmetry of $W_4$ spaces. Here we derive the master equation that drives the gauge invariant variations of the length of vectors during pa…
Einstein's celebrated theory of gravitation can be presented in three forms: general relativity, teleparallel gravity, and the rarely considered before symmetric teleparallel gravity. Extending the latter, we introduce a new class of theories where a scalar field is coupled nonminimally to nonmetricity $Q$, which here encodes the gravitational effects like curvature $R$ in general relativity or torsion $T$ in teleparallel gravity. We point out the similarities and differences with analogous scalar-curvature and scalar-torsion theories by discussing the field equations, role of connection, conf…
In this paper we apply the symmetry principle in order to search for an alternative unified explanation of several cosmological puzzles such as the present stage of accelerated expansion of the Universe and the Hubble tension issue, among others. We argue that Weyl gauge symmetry, being a manifest symmetry of gauge invariant theories of gravity operating on Weyl integrable geometry spacetimes, may be an actual (unbroken) symmetry of our present Universe. This symmetry may be at the core of a phenomenologically feasible explanation of modern fundamental issues arising within the framework of ge…
In this PhD thesis we deal with several theoretical and phenomenological apsects of metric-affine theories of gravity. Concretely, we first give a broad introduction to the necessary tools to understand the framework and elaborate on some subtleties of the minimal coupling prescription between geometry and matter in presence of torsion and nonmetricity. Then we dedicate the central part of the thesis to study the structure of Ricci Based gravity (RBG) theories, which will be of later use to understand generic properties of metric-affine theories. We begin by analysing the structure of the RBG …
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…
We reconsider the status of the so-called second clock effect in Weyl gauge theories of gravity, which are invariant both under local Poincaré transformations and local changes of scale. In particular, we revisit and extend our previous demonstration that the second clock effect does not occur in such theories, in order both to clarify our argument and confirm our original findings in response to recent counterclaims in the literature.
This thesis covers several developments performed in metric-affine gravity. This alternative framework extends General Relativity by considering a more general connection than the one induced by the metric (i.e., arbitrary torsion and nonmetricity participate in the dynamics). We start by revising some mathematical aspects of the metric-affine framework, the way Lovelock and other invariants are extended to it and the construction of a gauge formalism. Then we focus on metric-affine pp-wave geometries and explore solutions of this type for the quadratic metric-affine Lagrangian (with even-pari…
We consider Weyl gauge theories of gravity (WGTs), which are invariant both under local Poincaré transformations and local changes of scale. Such theories may be interpreted as gauge theories in Minkowski spacetime, but their gravitational interactions are most often reinterpreted geometrically in terms of a Weyl--Cartan spacetime, in which any matter fields then reside. Such a spacetime is a straightforward generalisation of Weyl spacetime to include torsion. As first suggested by Einstein, Weyl spacetime is believed to exhibit a so-called second clock effect, which prevents the existence of …
The successful miniaturisation of extremely accurate atomic clocks and atom interferometers invites prospects for satellite missions to perform precision experiments. We discuss the effects predicted by general relativity and alternative theories of gravity that can be detected by a clock, which orbits the Earth. Our experiment relies on the precise tracking of the spacecraft using its observed tick-rate. The spacecraft's reconstructed four-dimensional trajectory will reveal the nature of gravitational perturbations in Earth's gravitational field, potentially differentiating between different …
In this work we study the stability of the four vector irreducible pieces of the torsion and the nonmetricity tensors in the general quadratic metric-affine Lagrangian in 4 dimensions. The goal will be to elucidate under which conditions the spin-1 modes associated to such vectors can propagate in a safe way, together with the graviton. This highly constrains the theory reducing the parameter space of the quadratic curvature part from 16 to 5 parameters. We also study the sub-case of Weyl-Cartan gravity, proving that the stability of the vector sector is only compatible with an Einstein-Proca …
In this paper we treat the so called clock paradox in an analytical way by assuming that a constant and uniform force F of finite magnitude acts continuously on the moving clock along the direction of its motion assumed to be rectilinear. No inertial motion steps are considered. The rest clock is denoted as (1), the to-and-fro moving clock is (2), the inertial frame in which (1) is at rest in its origin and (2) is seen moving is I and, finally, the accelerated frame in which (2) is at rest in its origin and (1) moves forward and backward is A. We deal with the following questions: I) What is t…
Clock synchronisation is conventional when inertial systems are involved. This statement is no longer true in accelerated systems. A demonstration is given in the case of a rotating platform. We conclude that theories based on the Einstein's clock synchronisation procedure are unable to explain, for example, the Sagnac effect on the platform. Implications on very precise clock synchronisation on earth are discussed.
A review is given of the implications of the absence of an intrinsic notion of instantaneous 3-space, so that a clock synchronization convention has to be introduced, for relativistic theories.
In a Palatini $f(\mathcal{R})$-model, we define chonodynamical effects due to the choice of atomic clocks as standard reference clocks and we develop a formalism able to quantitatively separate them from the usual effective dark sources one has in extended theories. We apply the formalism to Hubble drift and briefly discuss the issue about the physical frame. In particular, we argue that there is no physical frame in the sense one does different things in different frames and that, in a sense, is the physical characteristic of extended gravity. As an example, we discuss how Jordan frame may be…
Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；University of the Basque Country UPV/EHU（巴斯克公立大学）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Atominstitut, TU Wien（维也纳工业大学原子研究所）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Louisiana State University（路易斯安那州立大学）；Western University（韦仕敦大学）
Quantum mechanics and general relativity have been each successfully tested in numerous experiments. However, the regime where both theories are jointly required to explain physical phenomena remains untested by laboratory experiments, and is also not fully understood by theory. This contribution reviews recent ideas for a new type of experiments: quantum interference of "clocks", which aim to test novel quantum effects that arise from time dilation. "Clock" interference experiments could be realised with atoms or photons in near future laboratory experiments.
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In 1835 Lobachevski entertained the possibility of multiple (rival) geometries. This idea has reappeared on occasion (e.g., Poincaré) but didn't become key in space-time foundations prior to Brown's \emph{Physical Relativity} (at the end, the interpretive key to the book). A crucial difference between his constructivism and orthodox "space-time realism" is modal scope. Constructivism applies to all local classical field theories, including those with multiple geometries. But the orthodox view provincially assumes a unique geometry, as familiar theories (Newton, Special Relativity, Nordström, a…
