Cosmology in Flat Space-Time

Flat space-time has not heretofore been thought a suitable locus in which to construct model universes because of the presumed necessity of incorporating gravitation in such models and because of the historical lack of a theory of gravitation in flat spacetime. It is here shown that a Lorentz-invariant theory of gravitation can be formulated by incorporating in it the mass-energy relation. Such a theory correctly predicts the well-known relativistic effects (advance of perihelion, gravitational refraction, gravitational red shift, echo delay of sun-grazing radio signals and others). The equations of motion, properly stated, are also seen to be identical with those of electromagnetism and lead to the correct prediction of gravitational radiation. Therefore Milne's kinematic model of the universe, mappable into his dynamical (or Newtonian) model, offers a unique alternative to the general relativistic models which are encumbered with both theoretically and observationally objectionable features. 1. Absolute Space and Time Cosmology is the study of the entire universe as a physical system. Therefore in principle every physical application is a cosmological application. In a narrower usage, however, most of current cosmology concerns itself with world-wide or ?global” problems such as model universes. In such universes, one may frame specific special problems, such as the conditions or processes in the ?early universe”. In any case, these problems are subject to the constraints characteristic of the model universe chosen; the universes of general relativity and of the late steady state theory or of kinematic relativity are salient examples. Let us illustrate some of the problems and methods characteristic of cosmology To keep the discussion within tractable bounds, let us confine our attention to a universe which in its essential properties is as simple as is consistent with both interest and correspondence to the real world of physical observation. We begin, therefore, by appealing to Helmholtz's Theorem that free mobility of rigid bodies of non-infinitesimal extension requires that space be Riemannian and of constant curvature. The manifest existence and mobility of rigid bodies of more than infinitesimal extent makes this a highly desirable, if not necessary prerequisite; only if the more general consequences of such an assumption make it appear untenable would one discard it. Similarly, by Schur's Theorem, a space is isotropic only if the curvature invariant R is constant. Since the large-scale universe appears highly isotropic as observed in the primeval fireball radiation, this is a second cogent reason to seek a model universe in a Riemannian space of constant curvature. Inasmuch as the number of model universes of constant curvature is currently none, there is therefore also an element of novelty in exploring this possibility. A model universe is, simply, a map in space-time. It specifies a density distribution defined through all space and for an infinite range of time. To possess credibility, it must also incorporate a law of gravitation compatible with its continued Cosmology in Flat Space-Time 2 existence, present state and/or presumed history. It must further satisfy some form of relativity, so that it is not modified in its essential characteristics by an exchange of observers (spatial locations) or epochs (times). To construct a map in space-time, it is first necessary to have a clock.Therefore the logical first step in recovering absolute time is to define a clock in some acceptable operational manner; traditional discussions simply assume ?identical clocks” and ?standard meter sticks”. E. A. Milne was the first to define a generalized clock as a monotonic single-valued parametrization of the sequence of events at a single observer. Thus any observer O sees a succession of events at himself and monotonically assigns real numbers to those events according to the earlier-later ordering relation. These numbers are the ?times” or epochs of the respective events at himself. (The observer does not at this stage presume to designate the epochs of events elsewhere.) This generalized clock may be given a conventional mechanical form with face and hands, which provide as events to parametrize the coincidences of the hands with a scale of numbers on the face. Another observer may do the same. The results will in general be entirely different, for the parametrizations are arbitrary. By what means can two arbitrary clocks be tested to determine whether or not they qualify as ?identical (congruent) clocks” of traditional special relativity theory, and if they are found not to be identical, how may one or both be altered – regraduated – so that they will so qualify? Milne used the reflexive property of clock equivalence to prescribe an operational test. Suppose sends a light signal to . The signal is sent forth when 's clock reads When the signal arrives at 's clock reads . By sending a modulated signal during some non-zero interval of time, observers and can establish that If returns the signal to by reflection, and if receives it back at time according to his own clock, they can further establish that If the clocks used by and are ?identical”, it is by definition both necessary and sufficient that the functions and be identical. Milne and Whitrow have shown (Zeitschrift für Astrophysik, v. 15, p. 270 (1938)) that it is always possible to find a transformation of one or the other of the clocks so that the clocks of and may be made congruent in an operationally precise sense. With light signals and a clock, each observer can assign space-time coordinates to any observable event E. Such coordinates are conventional to the extent that their definition is at the discretion of the observer at . The convention which offers the appeal of greatest simplicity and the closest correspondence to everyday experience is the radar-ranging convention (adopted by Milne before the advent of radar) that a distant event E occurs at a time 1. Absolute Space and Time 3