Metric tensor (general relativity)

In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past. In general relativity, the metric tensor plays the role of the gravitational potential in the classical theory of gravitation, although the physical content of the associated equations is entirely different. Gutfreund and Renn say "that in general relativity the gravitational potential is represented by the metric tensor." Notation and conventions This article works with a metric signature that is mostly positive (− + + +); see sign convention. The gravitation constant G will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over. Definition Mathematically

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Scale-invariant alternatives to general relativity. II. Dilaton properties

Georgios Karananas

In the present paper, we revisit gravitational theories which are invariant under TDiffs-transverse (volume-preserving) diffeomorphisms and global scale transformations. It is known that these theories can be rewritten in an equivalent diffeomorphism-invariant form with an action including an integration constant (cosmological constant for the particular case of non-scale-invariant unimodular gravity). The presence of this integration constant, in general, breaks explicitly scale invariance and induces a runaway potential for the (otherwise massless) dilaton, associated with the determinant of the metric tensor. We show, however, that if the metric carries mass dimension GeV, the scale invariance of the system is preserved, unlike the situation in theories in which the metric has mass dimension different from -2. The dilaton remains massless and couples to other fields only through derivatives, without any conflict with observations. We observe that one can define a specific limit for fields and their derivatives (in particular, the dilaton goes to zero, potentially related to the small distance domain of the theory) in which the only singular terms in the action correspond to the Higgs mass and the cosmological constant. We speculate that the self-consistency of the theory may require the regularity of the action, leading to the absence of the bare Higgs mass and cosmological constant, whereas their small finite values may be generated by nonperturbative effects.

Amer Physical Soc2016

Multi-resolution SLAM for Real World Navigation

Kai Oliver Arras, Roland Siegwart, Adriana Tapus

In this paper a hierarchical multi-resolution approach allowing for high precision and distinctiveness is presented. The method combines topological and metric paradigm. The metric approach, based on the Kalman Filter, uses a new concept to avoid the problem of the drift in odometry. For the topological framework the fingerprint sequence approach is used. During the construction of the topological map, a communication between the two paradigms is established. The fingerprint used for topological navigation enables also the re-initialization of the metric localization. The experimentation section will validate the multi-resolution-representation maps approach and presents different steps of the method.

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