A World Without Time

An overview

The new paradigm consists of using classical hyperbolic geometry as a model of space-time. However the textbook description of this geometry is not completely adequate for the purpose and additions and modifications of it have to be justified on their own terms if the integrity of the paradigm is to be preserved. Apart from dealing with these preliminaries the burden of the paper is to translate the variables and structures of the geometry into physical terms.

One of the extensions referred to concerns the addition of a fourth dimension to represent time. It turns out that time is not a dimension in the usual sense of the word but a part of a structure enabling a transformation of space. Furthermore the geometry demands that the so-called time variable is only half the story. The reason for this is that in hyperbolic geometry a line has two parallels through a point not on it. Now under the new paradigm velocity is defined via the so-called parallel angle formula. According to this formula if the angle between two parallel lines at a given point is A and the length of the perpendicular between them there is x then:
                cos A = tanh x
So if the parallel lines represent t – time
                cos A = dx/cdt = v/c
Since a line has two parallels through a point not on it the second parallel must represent a quantity other than time. The PIRT IX paper shows that a natural extension of textbook geometry is for this quantity to represent a scale factor for space.

The geometry tells us more. Given two parallel lines only one of them can be a geodesic. The other must either be a hyper-cycle or be parabolic. A parabolic line has only one invariant point, a simple one, not two coincident points as in the Euclidean case. This means that it cannot cut the invariant surface more than once; it must end in finite space. The two parallels to a geodesic are clearly parabolic.

A hyper-cycle is a hyperbolic line but not a geodesic. It was earlier known as an equidistant line- a self-explanatory title. In the literature hyper-cycles are said to be longer than their parent geodesics by a factor of cosh d where d is the distance between them. This is a natural assumption to make in a geometry that does not admit scale as a variable. In the new paradigm where scale variation is an essential feature it is more useful to take the view that it is the scale factor on hyper-cycles that is larger.

The Lorentz transformation not only changes the velocity of the reference frame, it is also said to dilate the proper time axis and the variable s on it. The dilation factor is sech x where tanh x = v/c. This is unchanged in the new paradigm. What is new is that the other parallel to the base line defines a similar set of relations between two variables w and f representing spatial scale in reference frames at two locations in space. Whereas the dilation of proper time is a function of velocity, because dx/dt is replaced by dw/dx, the dilation of space is a function of distance from some datum position.

At the bottom of this development is the fact that the parallel angle formula is not only a symbolic definition of velocity but also a definition of the physical range of a motion set in train by a given boost in velocity. After a test body is set in motion its velocity decreases until it comes to rest. "At the same time" the distance from the point at which the boost was applied increases. Under the new paradigm as the dilation produced by velocity decreases it gives way to space dilation on a "one for one" basis.

The new paradigm depends on there being a means of detecting when a reference frame is at rest with respect to the rest of the universe. The thermal background radiation provides this means and is an essential part of the picture in other ways. Dilation of time or space is represented by the geometry as a change of curvature and in the limit when the radius of curvature of a region is infinite it is flat and, to all intents, empty. Instead of interpreting this situation as concerning some abstract property of space its seems more useful to regard curvature of the time line as the local measure of the isotropic part of the density of thermal background radiation. Curvature of a space line can then be taken as a measure of the space scale factor.

An analysis of the transmission of radiation strengthens the plausibility of this conjecture. Since light travels at the extremal velocity one would not have thought that time dilation in the usual sense would occur.

However let us consider radiation from a point source spreading out in concentric spheres. The surface area of a sphere in hyperbolic geometry is
                4R²sinh²(r/R)
where R is the space constant. If the energy density of the radiation falls off as the inverse square of the distance from the source the total energy would be increasing outwards. Clearly the energy must fall off as to satisfy the criterion of energy conservation. A factor of cosech(r/R) can be accounted for by an increase of wavelength for this is the value of Rdf/dx. To account for the other factor of Rcosech(r/R) it is easiest to recall the quantum-mechanical picture of light as a hail of photons each endowed with a wavelength value, a part of the so-called complementarity principle. If the wavelength is increased there has to be a proportionate reduction of the density of photons. Thus a combination of wavelength increase and reduction of photon density assures the conservation of energy. The PIRT IX paper shows in detail that this geometrical effect applied to the case of a moving reference frame is a better explanation of the Hubble wavelength shift than the received one that it is due to physical recession and an expansion of the universe.

Discussion of implications

The PIRT IX paper is entitled "The basis of space-time" because it describes a space filled with radiation but is empty of gravitating matter. This space is the arena in which the actions of matter and gravitation are played out. Since its properties account for the Hubble wavelength shift we can be sure that this effect is part of the geometry and is not due to a physical expansion, as the term is ordinarily understood. Moreover the space is demonstrated to be duplex in character with the two halves representing respectively the wave and particle aspects of radiation. Complementarity is thus also part of the geometry and not a property of matter. It is notable that in fact time, again as we understand it, does not enter the picture. The time dilation of received theory becomes the reduction of photon density due to a dilation of space. Nevertheless the mechanism of which time is a part does require a fourth dimension for its operation.

One is entitled to ask what this basic geometry does represent. Recalling the fact that the amount of luminous matter is apparently smaller by an order of magnitude at least to account for the curvature of large-scale space, it is clear to the author that it represents inter-galactic space. The Machian conjecture that matter determines the properties of space can apply only within galaxies. Of course most of our knowledge of space derives from observations of objects within our own galaxy so we have little first-hand experience of the basic space.

What we have is firstly the Hubble shift because this concerns whole galaxies outside the Milky Way. We also have many anomalous observations about stars in the outer reaches of galaxies that have led to speculations about the possible existence of dark matter. The present paper offers a new avenue of enquiry into these phenomena, particularly those concerned with motion near the outer boundaries of galaxies. The basic geometry outside galaxies is hyperbolic whereas the geometry created by the gravitational fields due to the matter inside galaxies has positive curvature. There must be a zone of transition that is urgently in need of investigation. The same can be said about the anomalous movement of galaxies in clusters.

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