Spacetime Quantum Reference Frames and superpositions of proper times

In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.

Recycling of Planetary Proto-Atmospheres

<p>Protoplanets formed by core accretion can become massive enough to accrete gas from the disk they are born in. If the<br />planetary proto-atmosphere exceeds a critical mass, runaway gas accretion starts and the planetary atmosphere collapses into a gas<br />giant. In recent years, many close-in super-Earths have been observed which raises the question on how they avoided becoming hot<br />Jupiters. We investigate the recycling hypothesis as a possible mechanism to avoid the collapse of the atmosphere.<br />We use three-dimensional radiation-hydrodynamics to simulate the formation of proto-atmosphere in the local frame around<br />the planet. In post-processing we use tracer particles to calculate the shape of the atmosphere and determine the non-uniform recycling<br />timescale in a quantitative manner. Our simulations converge to a quasi-steady state where the velocity field of the gas does not change anymore. For the<br />parameter space explored, a = 0.1 au, m_c ∈ [1, 2, 5, 10] M_Earth, we find that recycling of the atmosphere counteracts the collapse by<br />preventing the gas from cooling efficiently.</p>

Quaternion-Based Local Frame Alignment between an Inertial Measurement Unit and a Motion Capture System

Local frame alignment between an inertial measurement unit (IMU) system and an optical motion capture system (MCS) is necessary to combine the two systems for motion analysis and to validate the accuracy of IMU-based motion data by using references obtained through the MCS. In this study, we propose a new quaternion-based local frame alignment method where equations of angular velocity transformation are used to determine the frame alignment orientation in the form of quaternion. The performance of the proposed method was compared with those of three other methods by using data with different angular velocities, noises, and alignment orientations. Furthermore, the effects of the following three factors on the estimation performance were investigated for the first time: (i) transformation concept, i.e., angular velocity transformation vs. angle transformation; (ii) orientation representations, i.e., quaternion vs. direction cosine matrix (DCM); and (iii) applied solvers, i.e., nonlinear least squares method vs. least squares method through pseudoinverse. Within our limited test data, we obtained the following results: (i) the methods using angular velocity transformation were better than the method using angle transformation; (ii) the quaternion is more suitable than the DCM; and (iii) the applied solvers were not critical in general. The proposed method performed the best among the four methods. We surmise that the fewer number of components and constraints of the quaternion in the proposed method compared to the number of components and constraints of the DCM-based methods may result in better accuracy. Owing to the high accuracy and easy setup, the proposed method can be effectively used for local frame alignment between an IMU and a motion capture system.

A New Rotation-Free Shell Formulation Using Exact Corotational Frame for Dynamic Analysis and Applications

Rotation-free shell formulations were proved to be an effective approach to speed up solving large-scaled problems. It reduces systems' degrees-of-freedom (DOF) and avoids shortages of using rotational DOF, such as singular problem and rotational interpolation. The rotation-free element can be extended for solving geometrically nonlinear problems using a corotational (CR) frame. However, its accuracy may be lost if the approach is used directly. Therefore, a new nonlinear rotation-free shell element is formulated to improve the accuracy of the local bending strain energy using a CR frame. The linear strain for bending is obtained by combining two re-derived elements, while the nonlinear part is deduced with the side rotation concept. Furthermore, a local frame is presented to correct the conventional local CR frame. An explicit tangential stiffness matrix is derived based on plane polar decomposition local frame. Simple elemental rotation tests show that the stiffness matrix and the proposed local frame are both correct. Several numerical examples and the application of drape simulations are given to verify the accuracy of nonlinear behavior of the presented element, and some of the results show that the presented method only requires few elements to obtain an accurate solution to the problem studied.