scholarly journals EMBEDDING VERSUS IMMERSION IN GENERAL RELATIVITY

2009 ◽  
Vol 24 (08n09) ◽  
pp. 1501-1504 ◽  
Author(s):  
EDMUNDO M. MONTE
Keyword(s):  
General Relativity ◽  
Causal Structure ◽  
Space Time ◽  
Point Of View ◽  
Physical Point ◽  
Exterior Solution ◽  
Euclidean Manifold ◽  
Classical Work ◽  
Higher Dimensional

We briefly discuss the concepts of immersion and embedding of space-times in higher-dimensional spaces. We revisit the classical work by Kasner in which he constructs a model of immersion of the Schwarzschild exterior solution into a six-dimensional pseudo-Euclidean manifold. We show that, from a physical point of view, this model is not entirely satisfactory, since the causal structure of the immersed space-time is not preserved by the immersion.

scholarly journals On the methods of extending Dirac's equation of the electron to general relativity

1942 ◽  
Vol 7 (1) ◽  
pp. 39-50
Author(s):  
D Martin
Keyword(s):  
General Relativity ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Space Time ◽  
Point Of View ◽  
Dirac Equations ◽  
Usual Form ◽  
Galilean System ◽  
Second Category ◽  
Working Out

1. Introduction. The problem of extending Dirac's equation of the electron to general relativity has been attacked by many authors, by methods which fall roughly into either of two classes according as the formulation does or does not require the introduction of a local Galilean system of coordinates at each point of space-time. As examples of the former class we mention the methods of Fock (1929) and of Cartan (1938), and as representing the latter class the method described by Ruse (1937). Also, Whittaker (1937) discovered a vector whose vanishing is completely equivalent to the Dirac equations, but this method, unlike the others in the second category, does not apply the Riemannian technique to spinors but only to vectors and tensors derived from these. Now Cartan has denied the possibility of fitting a spinor into Riemannian Geometry if his point of view of spinors is adhered to, and this he argues accounts for the “choquant” properties with which they have been endowed by the geometricians in order to enable them to write down an expression of the usual form for the covariant derivative of a spinor. Consequently, doubt has been cast on the compatibility of the various methods, so in this paper an attempt is made to clarify the matter by working out explicitly the case of the general metric by some of the more important of these methods.

scholarly journals K-causal structure of space-time in general relativity

Pramana ◽  
2008 ◽  
Vol 70 (4) ◽  
pp. 587-601 ◽  
Author(s):  
Sujatha Janardhan ◽  
R. V. Saraykar
Keyword(s):  
General Relativity ◽  
Causal Structure ◽  
Space Time

scholarly journals Inflation and Topological Phase Transition Driven by Exotic Smoothness

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Torsten Asselmeyer-Maluga ◽  
Jerzy Król
Keyword(s):  
Space Time ◽  
Point Of View ◽  
Topological Phase ◽  
Physical Point ◽  
Expectation Value ◽  
Guiding Principle ◽  
Topological Phase Transition ◽  
Boundary Terms ◽  
Exponential Increase ◽  
Hilbert Action

We will discuss a model which describes the cause of inflation by a topological transition. The guiding principle is the choice of an exotic smoothness structure for the space-time. Here we consider a space-time with topologyS3×ℝ. In case of an exoticS3×ℝ, there is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure. From the physical point of view, we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of the space-time. The inclusion of the boundary terms produces fermionic contributions to the partition function. The expectation value of an area (with respect to some surface) shows an exponential increase; that is, we obtain inflationary behavior. We will calculate the amount of this increase to be a topological invariant. Then we will describe this transition by an effective model, the Starobinski orR2model which is consistent with the current measurement of the Planck satellite. The spectral index and other observables are also calculated.

scholarly journals THE CHRONO-GEOMETRICAL STRUCTURE OF SPECIAL AND GENERAL RELATIVITY: A RE-VISITATION OF CANONICAL GEOMETRODYNAMICS

2007 ◽  
Vol 04 (01) ◽  
pp. 79-114 ◽  
Author(s):  
LUCA LUSANNA
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Special Relativity ◽  
Geometrical Structure ◽  
Clock Synchronization ◽  
Space Time ◽  
Point Of View ◽  
Field Equations ◽  
Relativistic Limit ◽  
Inertial Frames

A modern re-visitation of the consequences of the lack of an intrinsic notion of instantaneous 3-space in relativistic theories leads to a reformulation of their kinematical basis emphasizing the role of non-inertial frames centered on an arbitrary accelerated observer. In special relativity the exigence of predictability implies the adoption of the 3 + 1 point of view, which leads to a well posed initial value problem for field equations in a framework where the change of the convention of synchronization of distant clocks is realized by means of a gauge transformation. This point of view is also at the heart of the canonical approach to metric and tetrad gravity in globally hyperbolic asymptotically flat space-times, where the use of Shanmugadhasan canonical transformations allows the separation of the physical degrees of freedom of the gravitational field (the tidal effects) from the arbitrary gauge variables. Since a global vision of the equivalence principle implies that only global non-inertial frames can exist in general relativity, the gauge variables are naturally interpreted as generalized relativistic inertial effects, which have to be fixed to get a deterministic evolution in a given non-inertial frame. As a consequence, in each Einstein's space-time in this class the whole chrono-geometrical structure, including also the clock synchronization convention, is dynamically determined and a new approach to the Hole Argument leads to the conclusion that "gravitational field" and "space-time" are two faces of the same entity. This view allows to get a classical scenario for the unification of the four interactions in a scheme suited to the description of the solar system or our galaxy with a deparametrization to special relativity and the subsequent possibility to take the non-relativistic limit.

GROUNDWATER TIME SERIES

1970 ◽  
Vol 1 (3) ◽  
pp. 181-205 ◽  
Author(s):  
ERIK ERIKSSON
Keyword(s):  
Time Series ◽  
Point Of View ◽  
Hydrologic Models ◽  
Physical Point ◽  
Run Off ◽  
Fort Collins ◽  
Statistical Sense ◽  
Full Benefit ◽  
Future Work ◽  
Prediction Problems

The term “stochastic hydrology” implies a statistical approach to hydrologic problems as opposed to classic hydrology which can be considered deterministic in its approach. During the International Hydrology Symposium, held 6-8 September 1967 at Fort Collins, a number of hydrology papers were presented consisting to a large extent of studies on long records of hydrological elements such as river run-off, these being treated as time series in the statistical sense. This approach is, no doubt, of importance for future work especially in relation to prediction problems, and there seems to be no fundamental difficulty for introducing the stochastic concepts into various hydrologic models. There is, however, some developmental work required – not to speak of educational in respect to hydrologists – before the full benefit of the technique is obtained. The present paper is to some extent an exercise in the statistical study of hydrological time series – far from complete – and to some extent an effort to interpret certain features of such time series from a physical point of view. The material used is 30 years of groundwater level observations in an esker south of Uppsala, the observations being discussed recently by Hallgren & Sands-borg (1968).

scholarly journals Percolation and tortuosity in heart-like cells

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
R. Rabinovitch ◽  
Y. Biton ◽  
D. Braunstein ◽  
I. Aviram ◽  
R. Thieberger ◽  
...  
Keyword(s):  
Action Potential ◽  
Forest Fire ◽  
Sinus Node ◽  
Heart Tissue ◽  
Point Of View ◽  
Physical Point ◽  
Electrical Currents ◽  
Single Wave ◽  
Heart Arrhythmia ◽  
The One

AbstractIn the last several years, quite a few papers on the joint question of transport, tortuosity and percolation have appeared in the literature, dealing with passage of miscellaneous liquids or electrical currents in different media. However, these methods have not been applied to the passage of action potential in heart fibrosis (HF), which is crucial for problems of heart arrhythmia, especially of atrial tachycardia and fibrillation. In this work we address the HF problem from these aspects. A cellular automaton model is used to analyze percolation and transport of a distributed-fibrosis inflicted heart-like tissue. Although based on a rather simple mathematical model, it leads to several important outcomes: (1) It is shown that, for a single wave front (as the one emanated by the heart's sinus node), the percolation of heart-like matrices is exactly similar to the forest fire case. (2) It is shown that, on the average, the shape of the transport (a question not dealt with in relation to forest fire, and deals with the delay of action potential when passing a fibrotic tissue) behaves like a Gaussian. (3) Moreover, it is shown that close to the percolation threshold the parameters of this Gaussian behave in a critical way. From the physical point of view, these three results are an important contribution to the general percolation investigation. The relevance of our results to cardiological issues, specifically to the question of reentry initiation, are discussed and it is shown that: (A) Without an ectopic source and under a mere sinus node operation, no arrhythmia is generated, and (B) A sufficiently high refractory period could prevent some reentry mechanisms, even in partially fibrotic heart tissue.

scholarly journals WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

2011 ◽  
Vol 03 ◽  
pp. 87-97 ◽  
Author(s):  
F. P. POULIS ◽  
J. M. SALIM
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Riemannian Geometry ◽  
Axiomatic Approach ◽  
Space Time ◽  
Weyl Geometry ◽  
Test Particles ◽  
Variational Formalism ◽  
Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

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