我们将先前工作 [arXiv:2302.12209] 的结果推广到了具有两个空间坐标和一个时间坐标的三维（3D）时空情形。我们发现，平坦的 Minkowski 3D 时空是“良好缝合的”，这意味着它拥有一种由 12 个事件之间的 24 种因果关系所描述的结构。我们证明了，一个 3D 时空是“良好缝合的”，当且仅当它是共形平坦的。“良好缝合”时空的概念不依赖于关于长度、时间等的度量信息，也不属于度量几何，而是属于关联几何。因此，我们将共形平坦时空这一重要概念从 Riemannian 几何的“度量”语言“翻译”为关联几何的“非度量”语言。本文的结果提供了一种工具，只要时空不是共形平坦的，就可以仅基于因果关系来探测 3D 时空的曲率，而无需任何诸如尺子和时钟之类的测量仪器。
英文摘要：
We have generalized the results of the previous work [arXiv:2302.12209] to the case of three-dimensional (3D) spacetime with two spatial and one temporal coordinates. We have found that the flat Minkowski 3D spacetime is "well-stitched", which means that it possesses a structure described by 24 causal relations between 12 events. We have proved that a 3D spacetime is "well-stitched" if and only if it is conformally flat. The concept of a "well-stitched" spacetime does not rely on metrical information about lengths, times, etc., and does not belong to the metric geometry, but rather to geometry of incidence. We therefore have "translated" an important concept of a conformally-flat spacetime from the "metric" language of Riemannian geometry to the "non-metric" language of the geometry of incidence. The results of this paper provide a tool for detecting the curvature of the 3D spacetime on the basis of causal relations only, without any measurement instruments like rulers and clocks, provided that the spacetime is not conformally flat.
We demonstrate how one can distinguish a curved 4-dimensional spacetime from a flat one, when it is possible, relying only on the causality relations between events. It is known that it is possible only for spacetimes that are not conformally flat. We prove that if a spacetime is not conformally flat, then its non-flatness can be verified by only a few (sixteen) measurements of causal relations. Therefore the results of this paper clarify what can be said about flatness or non-flatness of the spacetime after a finite number of measurements of causal relations.
Special Relativity (SR) kinematics is derived from very intuitive assumptions. Contrary to standard Einstein's derivation, no light signal is used in the construction nor it is assumed to exist. Instead we postulate the existence of two equivalence classes of physical objects: proportional clocks and proportional rulers. Simple considerations lead to Lorentz kinematics as one of three generic cases. The Lorentz case is characterized by the maximal relative speed of physical objects. The two others are the Galilean and the Euclidean cases.
We study the accelerated expansion of the universe by using the kinematic approach. In this context, we parameterize the deceleration parameter, q(z), in a model independent way. Assuming three simple parameterizations we reconstruct q(z). We do the joint analysis with combination of latest cosmological data consisting of standard candles (Supernovae Union2 sample), standard ruler (CMB/BAO), cosmic clocks (age of passively evolving galaxies) and Hubble (H(z)) data. Our results support the accelerated expansion of the universe.
We present a stochastic framework for emergent quantum gravity coupled to matter. The Hamiltonian constraint in diffeomorphism-invariant theories demands the identification of a clock relative to which dynamics may be defined, and other degrees of freedom can play the role of rulers. However, a global system of clock and rulers is generally not available. We provide evidence that stochasticity associated with critical points of clock and ruler fields can be related to the emergence of both a probabilistic description consistent with ordinary quantum theory, and gravitation described by general…
Field theories that are generally covariant but nongravitational at tree level typically give rise to an emergent gravitational interaction whose strength depends on a physical regulator. We consider emergent gravity models in which scalar fields assume the role of clock and rulers, addressing the problem of time in quantum gravity. We discuss the possibility of nontrivial dynamics for clock and ruler fields, and describe some of the consequences of those dynamics for the emergent gravitational theory.
Gravitomagnetic clock effects for circularly rotating orbits in black hole spacetimes are studied from a relative observer point of view, clarifying the roles played by special observer families.
The precision of optical atomic clocks is approaching a regime where they resolve gravitational time dilation on smaller scales than their own extensions. Hence, an accurate description of quantum clocks has to take their spatial extension into account. In this article, as a first step towards a fully relativistic description of extended quantum clocks, we investigate a quantized version of Einstein's light clock fixed at a constant distance from a large massive object like the Earth. The model consists of a quantum light field in a one-dimensional cavity in Schwarzschild spacetime, where the …
Stationary axisymmetric spacetimes containing a pair of oppositely-rotating periodically-intersecting circular geodesics allow the study of various so-called `clock effects' by comparing either observer or geodesic proper time periods of orbital circuits defined by the observer or the geodesic crossing points. This can be extended from a comparison of clocks to a comparison of parallel transported vectors, leading to the study of special elements of the spacetime holonomy group. The band of holonomy invariance found for a dense subset of special geodesic orbits outside a certain radius in the …
We derive the functional Schrodinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrodinger equation that we obtain exhibits a functional time derivative that completes the usual definition of WKB time for curved spacetime, and the usual Schrodinger-type evolution is recovered in Minkowski spacetime.
Einstein's theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.
This paper is a generalization of earlier papers [Nucl. Phys. B 884, 344 (2014) (arXiv:1312.2759) and JHEP 6, 63 (2015) (arXiv:1401.2488)]. We generalize the idea of quantum clock time to quantum spacetime reference frame via physical realization of a reference system by quantum rulers and clocks. Omitting the internal degrees of freedom (such as spins) of the physical rulers and clocks, only considering their metric properties, the spacetime reference frame is described by a bosonic non-linear sigma model (NLSM). We study the quantum behavior of the system under approximations, and obtain (1)…
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 …
We study the behaviour of spinning test particles in the Schwarzschild spacetime. Using Mathisson-Papapetrou equations of motion we confine our attention to spatially circular orbits and search for observable effects which could eventually discriminate among the standard supplementary conditions namely the Corinaldesi-Papapetrou, Pirani and Tulczyjew. We find that if the world line chosen for the multipole reduction and whose unit tangent we denote as $U$ is a circular orbit then also the generalized momentum $P$ of the spinning test particle is tangent to a circular orbit even though $P$ and …
What do simple clocks, simple computers, black holes, space-time foam, and holographic principle have in common? I will show that the physics behind them is inter-related, linking together our concepts of information, gravity, and quantum uncertainty. Thus, the physics that sets the limits to computation and clock precision also yields Hawking radiation of black holes and the holographic principle. Moreover, the latter two strongly imply that space-time undergoes much larger quantum fluctuations than what the folklore suggests --- large enough to be detected with modern gravitational-wave inte…
We provide a setup by which one can recover the geometry of spacetime from local measurements of quantum particle detectors coupled to a quantum field. Concretely, we show how one can recover the field's correlation function from measurements on the detectors. Then, we are able to recover the invariant spacetime interval from the measurement outcomes, and hence reconstruct a notion of spacetime metric. This suggest that quantum particle detectors are the experimentally accessible devices that could replace the classical 'rulers' and 'clocks' of general relativity.
We critically discuss the measure of very short time intervals. By means of a Gedankenexperiment, we describe an ideal clock based on the occurrence of completely random events. Many previous thought experiments have suggested fundamental Planck-scale limits on measurements of distance and time. Here we present a new type of thought experiment, based on a different type of clock, that provide further support for the existence of such limits. We show that the minimum time interval $Δt$ that this clock can measure scales as the inverse of its size $Δr$. This implies an uncertainty relation betwe…
We discuss the question of time in a Bianchi I quantum cosmology in the framework of singularity avoidance. We show that time parameters fall into two distinct classes, that are such that the time development of the wave function either always leads to the appearance of a singularity (fast-gauge time) or that always prevents it from occurring (slow-gauge time). Furthermore, we find that, in the latter case, there exists an asymptotic regime, independent of the clock choice. This may point to a possible solution of the clock issue in quantum cosmology if there exists a suitable class of clocks …
A recent paper (gr-qc/9909017) criticizes our work on the structure of spacetime foam. Its authors argue that the quantum uncertainty limit for the position of the quantum clock in a gedanken timing experiment, obtained by Wigner and used by us, is based on unrealistic assumptions. Here we point out some flaws in their argument. We also discuss their other comments and some other issues related to our work, including a simple connection to the holographic principle. We see no reason to change our cautious optimism on the detectability of spacetime foam with future refinements of modern gravita…
We study the motion of spinning test particles in Kerr spacetime using the Mathisson-Papapetrou equations; we impose different supplementary conditions among the well known Corinaldesi-Papapetrou, Pirani and Tulczyjew's and analyze their physical implications in order to decide which is the most natural to use. We find that if the particle's center of mass world line, namely the one chosen for the multipole reduction, is a spatially circular orbit (sustained by the tidal forces due to the spin) then the generalized momentum $P$ of the test particle is also tangent to a spatially circular orbit…
If a clock, mathematically modeled by a parametrized timelike curve in a general-relativistic spacetime, is given, the radar method assigns a time and a distance to every event which is sufficiently close to the clock. Several geometric aspects of this method are reviewed and their physical interpretation is discussed.
We derive exact expressions for the relativistic redshift between an Earth-bound observer, that is meant to model a standard clock on the Earth's surface, and various (geodesic) observers in the Schwarzschild spacetime. We assume that the observers exchange radial light signals to compare the frequencies of standard clocks, which they transport along their respective worldlines. We calculate the redshift between an Earth-bound clock and static observers, observers in radial free fall, on circular geodesics, and on arbitrary bound quasi-elliptical orbits. For the latter case, we consider as exa…
In the article we present explicit expressions for quantum fluctuations of spacetime in the case of $(4+n)$-dimensional spacetimes, and consider their holographic properties and some implications for clocks, black holes and computation. We also consider quantum fluctuations and their holographic properties in ADD model and estimate the typical size and mass of the clock to be used in precise measurements of spacetime fluctuations. Numerical estimations of phase incoherence of light from extra-galactic sources in ADD model are also presented.
Lecture at BOSS2011 on relativistic metrology, on clock synchronization, relativistic dynamics and non-inertial frames in Minkowski spacetime, on relativistic atomic physics, on ADM canonical tetrad gravity in asymptotically Minkowskian spacetimes, on the York canonical basis identifying the inertial (gauge) and tidal degrees of freedom of the gravitational field, on the Post-Minkowskian linearization in 3-orthogonal gauges, on the Post-Newtonian limit of matter Hamilton equations, on the possibility to interpret dark matter as a relativistic inertial effect connected with relativistic metrolo…
The conflict between quantum theory and the theory of relativity is exemplified in their treatment of time. We examine the ways in which their conceptions differ, and describe a semiclassical clock model combining elements of both theories. The results obtained with this clock model in flat spacetime are reviewed, and the problem of generalizing the model to curved spacetime is discussed, before briefly describing an experimental setup which could be used to test of the model. Taking an operationalist view, where time is that which is measured by a clock, we discuss the conclusions that can be…
General Relativistic Chronometry with Clocks on Ground and in Space
Center of Applied Space Technology and Microgravity (ZARM), University of Bremen（不莱梅大学应用空间技术与微重力中心（ZARM））；Gauss-Olbers-Center, University of Bremen（不莱梅大学高斯-奥尔伯斯中心）；Institute for Theoretical Physics, University of Bremen（不莱梅大学理论物理研究所）
In an apparently unexplored region of relativistic spacetime, a simple thought experiment demonstrates that conjoined Lorentz transformations predict a proper clock at rest will run backwards and that prediction violates the logical principle of causality. Shown first in a modification of the standard clock paradox thought experiment, this fault carries over to finite accelerations of the moving observer. After re-examination of the standard clock paradox, a logical fault was also found in the concept of spacetime. A two-dimensional treatment of the Earth orbit predicts that our astronomers sh…
On the basis of the Woodhouse causal axiomatics, we show that conformal proper times and an extra variable in addition to those of space and time, precisely and physically identified from experimental examples, together give a physical justification for the `chronometric hypothesis' of general relativity. Indeed, we show that, with a lack of these latter two ingredients, no clock paradox solution exists in which the clock and message functions are solely at the origin of the asymmetry. These proper times originate from a given conformal structure of the spacetime when ascribing different compa…
Using gyroscopes we generalize results, obtained for the gravitomagnetic clock effect in the particular case when the exterior spacetime is produced by a rotating dust cylinder, to the case when the vacuum spacetime is described by the general cylindrically symmetric Lewis spacetime. Results are contrasted with those obtained for the Kerr spacetime.
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…
The historical origins of Fermi-Walker transport and Fermi coordinates and the construction of Fermi-Walker transported frames in black hole spacetimes are reviewed. For geodesics this transport reduces to parallel transport and these frames can be explicitly constructed using a Killing-Yano tensor as shown by Marck. For accelerated or geodesic circular orbits in such spacetimes, both parallel and Fermi-Walker transported frames can be given, and allow one to study circular holonomy and related clock and spin transport effects. In particular the total angle of rotation that a spin vector under…
We discuss a recent provocative suggestion by Amelino-Camelia and others that classical spacetime may break down into ``quantum foam'' on distance scales many orders of magnitude larger than the Planck length, leading to effects which could be detected using large gravitational wave interferometers. This suggestion is based on a quantum uncertainty limit obtained by Wigner using a quantum clock in a gedanken timing experiment. Wigner's limit, however, is based on two unrealistic and unneccessary assumptions: that the clock is free to move, and that it does not interact with the environment. Re…
Shape Dynamics is a metric theory of pure gravity, equivalent to General Relativity, but formulated as a gauge theory of spatial diffeomporphisms and local spatial conformal transformations. In this paper we extend the construction of Shape Dynamics form pure gravity to gravity-matter systems and find that there is no obstruction for the coupling of gravity to standard matter. We use the matter gravity system to construct a clock and rod model for Shape Dynamics which allows us to recover a spacetime interpretation of Shape Dynamics trajectories.
We consider the geometry of spacetime based on a non-metric, Finslerian, length measure, which, in terms of physics, represents a generalized clock. Our defnition of Finsler spacetimes ensure a well defined notion of causality, a precise description of observers and a geometric background for field theories. Moreover we present our Finsler geometric extension of the Einstein equations, which determine the geometry of Finsler spacetimes dynamically.
The Lorentzian length of a timelike curve connecting both endpoints of a classical computation is a function of the path taken through Minkowski spacetime. The associated runtime difference is due to time-dilation: the phenomenon whereby an observer finds that another's physically identical ideal clock has ticked at a different rate than their own clock. Using ideas appearing in the framework of computational complexity theory, time-dilation is quantified as an algorithmic resource by relating relativistic energy to an $n$th order polynomial time reduction at the completion of an observer's jo…
In Einstein's gedankenexperiment for measuring space and time, an ensemble of clocks moving through curved spacetime measures geometry by sending signals back and forth, as in the global positioning system (GPS). Combining well-known quantum limits to measurement with the requirement that the energy density of clocks and signals be be no greater than the black hole density leads to the quantum geometric limit: the total number of ticks of clocks and clicks of detectors that can be contained in a four volume of spacetime of radius r and temporal extent t is less than or equal to rt/πl_P t_P, wh…
Ideal rods and clocks are defined as an infinitesimal symmetry of the spacetime. Since no a priori geometric structure is considered, all the possible models of spacetime are obtained.
Finsler geometry naturally appears in the description of various physical systems. In this review I divide the emergence of Finsler geometry in physics into three categories: as dual description of dispersion relations, as most general geometric clock and as geometry being compatible with the relevant Ehlers-Pirani-Schild axioms. As Finsler geometry is a straightforward generalisation of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalisation is subtle due to the existence of non-trivial null directions. I review how a…
The purpose of this paper is to evaluate the `Lorentzian pedagogy' defended by J.S. Bell in his essay ``How to teach special relativity'', and to explore its consistency with Einstein's thinking from 1905 to 1952. Some remarks are also made in this context on Weyl's philosophy of relativity and his 1918 gauge theory. Finally, it is argued that the Lorentzian pedagogy - which stresses the important connection between kinematics and dynamics - clarifies the role of rods and clocks in general relativity.
We extend de la Fuente and Romero's defining equation for uniform acceleration in a general curved spacetime from linear acceleration to the full Lorentz covariant uniform acceleration. In a flat spacetime background, we have explicit solutions. We use generalized Fermi-Walker transport to parallel transport the Frenet basis along the trajectory. In flat spacetime, we obtain velocity and acceleration transformations from a uniformly accelerated system to an inertial system. We obtain the time dilation between accelerated clocks. We apply our acceleration transformations to the motion of a char…
I stress that spacetime is a redundant abstraction, since describing the physical content of all so-called "space-time measurements" only requires timing (by a physical/material clock) of particle detections (at a physical/material detector). It is interesting then to establish which aspects of our current theories afford us the convenient abstraction of a spacetime. I emphasize the role played by the assumed triviality of the geometry of momentum space, which makes room for an observer-independent notion of locality. This is relevant for some recent studies of the quantum-gravity problem that…
Atomic clocks and gravitational waves as probes of non-metricity
Institute for Research in Fundamental Sciences (IPM)（基础科学研究院）；Damghan University（达姆甘大学）；Khazar University（哈扎尔大学）；National Observatory of Athens（雅典国家天文台）；Universidad Católica del Norte（天主教主教大学）；University of Science and Technology of China（中国科学技术大学）
We analyse the Einstein equivalence principle (EEP) for a Hubble observer in Friedmann-Lemaitre-Robertson-Walker spacetime. We show that the affine structure of light cone in the FLRW spacetime should be treated locally in terms of the optical metric which is not reduced to the Minkowski metric due to the non-uniform parametrization of the local equations of light propagation with the proper time of the observer's clock. The physical consequence of this difference is that the Doppler shift of radio waves measured locally, is affected by the Hubble expansion.
We investigate a relativistic positioning system where the coordinates of the users are determined by the proper times broadcasted by clocks in motion in spacetime: these are the so-called emission coordinates. In particular, we focus on emission coordinates in flat spacetime: in this case, we show that these coordinates can be defined using an approach based on simple geometrical properties of geodesic triangles. We analyse the 2-dimensional case and then we show how the whole procedure can be applied to 4 dimensions, also in terms of ordinary three-dimensional spheres. An analytic solution f…
The geometrical formulation of gravity is not unique and can be set up in a variety of spacetimes. Even though the gravitational sector enjoys this freedom of different geometrical interpretations, consistent matter couplings have to be assured for a steady foundation of gravity. In generalised geometries, further ambiguities arise in the matter couplings unless the minimal coupling principle (MCP) is adopted that is compatible with the principles of relativity, universality and inertia. In this work, MCP is applied to all Standard Model gauge fields and matter fields in a completely general (…
Junctions, strings, clocks and gravitational memory in three dimensional dS space
Universidad Andres Bello（安德烈斯贝略大学）；Independent University, Bangladesh（孟加拉独立大学）；Center for Computational and Data Sciences (CCDS), Independent University, Bangladesh（孟加拉独立大学计算与数据科学中心）；Universidad Técnica Federico Santa María（费德里科圣塔马利亚技术大学）；Pontificia Universidad Católica de Valparaíso（瓦尔帕莱索天主教 Pontificia 大学）
Einstein Equivalence Principle (EEP) requires all matter components to universally couple to gravity via a single common geometry: that of spacetime. This relates quantum theory with geometry as soon as interactions with gravity are considered. In this work, I study the geometric theory of coupling a spin-1/2 particle to gravity in a twofold expansion scheme: First with respect to the distance based on Fermi normal coordinates around a preferred worldline (e.g., that of a clock in the laboratory), second with respect to 1/c (post-Newtonian expansion). I consider the one-particle sector of a ma…
