Curved space-time

Nearly two years ago, studying electrodynamics in curved space-time I found1 that Maxwell s equations impose on space-time a restriction which can be formulated by saying that a certain vector q determined by the curvature field must be the gradient of a scalar function, or... 

There is no evidence that Minkowski space is flat on the large scale. The assumption of euclidean Minkowski space could therefore be, and probably is an illusion, like the flat earth. In fact, there is compelling evidence from observed spectroscopic red shifts that space is curved over galactic distances. These red shifts are proportional to distances from the source, precisely as required by a curved space-time[52j. An alternative explanation, in terms of an expanding-universe model that ascribes the red shifts to a Doppler... 

M. W. Evans, P. K. Anastasovski, T. E. Bearden, et al., Electromagnetic energy from curved space-time, Optik (in press). 

Is the gravitational field able, in principle, to produce real, tangible particles in a void, or in the terms of general relativity theory, in curved space-time ... 

Incidentally we find that the positive operator x(r) >0 depends formally on the coordinate r of the particle m, with origin at the center of mass of M. Since the dimensions or scales x and r are subject to the description of the conjugate problem, we will on balance recover a geometry of curved space-time scales reminiscent of the classical theories, see more below. 

The precise definition of a vacuum in a curved space-time is still subject to some ambiguities. We refer the interested reader to Fulling (1979) Fulling(1989) Birrell Davis (1982) Wald(1994) and to the discussion in Chung, Notari Riotto (2003) and references therein. 

The conscious final decision to take the risk, with the current sequence, should be read as a personal conviction that the beauty of chemistry can never be fully appreciated unless viewed against the background in which all matter originates - space-time, or the vacuum. Not only matter, but all modes of interaction are shaped by the geometry of space, which at the moment remains a matter of conjecture. However, the theory of general relativity points the way by firmly demonstrating that the known material world can only exist in curved space-time. The theory of special relativity affirms that space-time has a minimum of four dimensions. Again, spaces of more dimensions are conjectural at present. 

Equation (2.11) with variable metric tensor describes the invariance in the gravitational case which is characterized by curved space-time. The summation extends over all values of y, and u, so that the sum consists of 4 x 4 terms, of which 12 are equal in pairs, hence 10 independent functions. The motion of a free material point in this field will take the form of curvilinear non-uniform motion. If the matrix of the metric tensor can be diagonalized it is independent of position and the corresponding geometry is said to be flat, which is the special case of SR. 

Note that the operator, c(r) > 0, depends on the coordinate r of the particle m, with the origin at the centre of mass of Af. It is important to distinguish between the coordinates r (and t) of a flat Euclidean space and the scales defining the curved space-time geometry defined by the (operator)-secular problem Eq. (14). We get directly... 

Extended to the four dimensions of the relativity theory of curved space-time the element of arc ds is given by... 

The assumptions on which the model rests are too crude to be realistic. In particular, the assumption of a universal time coordinate directly contradicts the basis of general relativity. To avoid the problem de Sitter repeated the calculation based on relativistically curved space-time, with the surprising result of an empty universe with variable radius. The traditional interpretation of this result as an expanding universe is not unique. It could just as well imply space-time with continuous curvature, characteristic of projective space. In the event, both solutions were soon superseded by an expanding-universe cosmology based on a Doppler interpretation of galactic redshifts. 

In om pseudo-Euclidean tangent space it is customary to distinguish between time-hke, space-like and light-like events, well aware that this is another caricatme of curved space-time. For convenience, space-like events are usually ignored as physically unreal. There is no justification for this assumption in four-dimensionally curved space-time. 

The alternative of increased space-like curvature results in relentless compression that compacts all matter into infinitesimal space with parity inversion at ZjN = 0. Our contention is that these alternatives are inseparable in curved space-time where inversion of total CPT symmetry occms on compression. 

The universal space-time geometry itself is not necessarily directly observed no apparent departures from a Euclidean model have been found by classical measurements. It might be argued that curved space-time coordinates are split into flat space and time coordinates. There is no direct observational basis for asserting that the Cosmos is Minkowskian at large distances and times. 

The isotropic lines of a projective identity mapping define the local complex Minkowski space of special relativity directly. By taking the circular points at infinity into account the global projective space of general projective relativity is obtained. No other topology reveals the transition from special to general relativity as such a simple consequence of curved space-time. 

An important aspect of quantum field theory in curved space-time is its description of Hawking radiation (seeHawking process). It is necessary to consider quantum gravity in the very early universe, just after the big bang, and the singularities associated with black holes can also be interpreted as requiting a quantum theory of gravity. 

In [208] the authors obtained a numerical scheme and code for estimating the deposition of energy and momentum due to the neutrino pair annihilation (v -f- V e + c+) in the vicinity of an accretion tori around a Kerr black hole. In order to solve the collisional Boltzmann equation in curved space-time, the authors solved approximately the so-called rendering equation along the null geodesics. They used the Runge-Kutta Fehlberg... 

Once this spatial decomposition has been stated, modeling the role of space consists of defining local variables that correspond to global state variables. Each local variable results from application of a spatial operator that depends on the geometry of space. The simplest is the Euclidean space, that is, ordinary space independent from time, also called flat space by opposition with curved space-time, which is the frame of the general relativity. 

As in wave mechanics, the simulation of chemical phenomena by number theory is characterized by the appearance of integers, in this case associated with chemical structures and transformations. An obvious conclusion is that the elementary units of matter should be viewed as wave structures rather than point particles, which is consistent with the first appearance of matter in curved space-time. Even 3D wave packets behave in a manner convincingly like ponderable matter and rationalize the equivalence of mass and energy in a natural way. There is no compelling reason why this simple model should be concealed with the notion of wave/particle duality and more so on realizing that the wave-like space-time distortions are strictly 4D structures. In response to environmental pressure, an electronic wave packet can shrink to the effective size of an elementary particle or increase to enfold a proton as a spherical standing wave. 

As a reasonable conjecture, we now propose that curved space-time, like an inflexible sheet wrapped around a curved surface, must develop persistent wrinkles—the elementary units of matter or energy. We envisage flat space-time in featureless undulation that develops elementary wave packets when curved. We recognize few types of wave packet with internal wave patterns perceived as the characteristic mass, charge, spin and chirality of the four-dimensional elementary units whose behavior is prescribed by a potential function according to Eq. (2). 

Casual interpretation of the local environment as three-dimensional space and universal time flow is not consistent with the known four-dimensional structure of space-time on a cosmic scale. Local Euclidean space is said to be tangent to the underlying four-dimensional curved space-time. 

It has been argued [8] that the transformation from curved space-time to Euclidean tangent space is described by the golden ratio. This is not an entirely unexpected conclusion, in view of the prominence of r in the operation of selfsimilar symmetries, related to equiangular logarithmic spirals. 

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... 

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