Effective euclidean space-time

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

Spectroscopic observation indicates a relationship between redshift and intergalactic distance, which, interpreted as a Doppler effect, implies a rate of recession between galaxies that depends on their mutual separation. To account for this relationship, known as Hubble s law, a metric tensor, in which the time coordinate is separated from a monotonically expanding three-dimensional Euclidean space, is assumed. 

Probably the most familiar example of an unreachable limit in physical science is the absolute zero of temperature. Rather than a mathematical impossibility, 0 K implies the cessation of all motion, which renders it experimentally inaccessible. A logical explanation of the effect may well be another futile search for infinity. Energy, matter and motion can only occur in curved space-time. An eventless situation is, by definition, restricted to flat, infinite Euclidean space, which occurs nowhere in the... 

In this equation, Dx(x) and S [x(x)] are, respectively, the position space path measure and the Euclidean time action. The centroid density also formally defines a classical-like effective potential, i.e., ... 

The geometric description of the light propagation and the kinetics description of motion were closely correlated in the history of science. Among the main evidence of classical Newtonian mechanics is Euclidean geometry based on optical effects. In Newtonian physics, space has an affine structure but time is absolute. The basic idea is the inertial system, and the relations are the linear force laws. The affine structure allows linear transformations in space between the inertial coordinate systems, but not in time. This is the Galilean transformation ... 

The simplest explanation of astronomical redshift, apparently the explanation also favoured by Hubble, is the effect of curved space (Boeyens, 1995). As the curvature of a two-dimensional surface requires a third dimension non-Euclidean three-dimensional space must curve into a fourth (time) dimension. 

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