THE HYPERSPHERICAL MODEL OF THE UNIVERSE by Graham K. Glover Copyright 1995, G. K. Glover. All rights reserved. Abstract This paper documents my theory on the nature and shape of the universe. My theory considers gravity, the principles of general relativity, observation, fourspace, and the mathematics associated with these concepts. Implications of my theory are included in this paper. Preface, September 1996 This version contains minor changes from the original edition. Equations are now represented in a "mathematical" form versus a computer program form required with the word processor used in the original version. Highlighting is also slightly changed; italics were used and/or substituted where appropriate. The most significant change was the addition of Equation 4-2, a change that simply shows a different form of Equation 4-1. This was done to more clearly show the interesting potential link between Euclidean and non-Euclidean space. 1 INTRODUCTION 1.1 Background Sometime around the year 1987, I began work that eventually lead to the theory described in this paper regarding the nature of the physical universe. This work was performed on my own time with my own motivation to investigate the validity of a dominant theory that did not seem logically consistent. Over the years I came to discover that this dominant theory, the Big Bang theory, did not account for either the apparent age of the observable universe, or the effects that would be described by general relativity. On September 21, 1994, I began the final act of closure on my theory of the universe, an act which was completed on the next day September 22. The day following closure, the 23rd, was the day I discovered that Einstein had developed a conceptually similar model of the universe (Einstein 1917), a model all but ignored today. So be it. Einstein's model is more physically based. My model is more mathematically based. Now another model can be ignored; perhaps that model will be the Big Bang model. 1.2 Scope This paper will discuss my theory regarding the nature of the universe on the bases of gravity and general relativity, and the mathematics associated with these bases. It will show direct similarities between relativity and a mathematical model of a hyperspherical surface. 1.3 Outline Section 2 discusses gravity in a mass field, and how it relates to the observable universe. Section 3 describes the shape of the universe, and includes equations that show an expected correlation between the effects of general relativity and the geometry of a hyperspherical surface. Implications of a universe described by my theory are included in Section 4. Conclusions from this work are given in Section 5. 2 GRAVITY IN MASS FIELD 2.1 Force The force of gravity outside a spherical mass behaves in a manner defined by the equation [2.1-1] where f is the gravitational force G is the gravitational constant m1 is the mass of the spherical object m2 is the mass of the object under the gravitational influence of m1 r is the distance from the center of m1 The force of gravity inside a spherical mass behaves in a different manner. Within a sphere of uniform density, the gravitational force increases linearly from the center. At any point within the sphere, a concentric subsphere may be defined by the range from the center of the sphere. Only that mass within the subsphere will yield a resultant gravitational force; that mass beyond the concentric sphere yields no net resultant force. (Newton 1686; Klein 1959) The mass of a sphere of uniform density is defined by [2.1-2] where m is the mass of the sphere d is the density of the sphere in mass/unit volume pi is the constant pi rm is the radius of the sphere Since [2.1-3] it follows that [2.1-4] for the force of gravity as it applies to a sphere of uniform density. For the case when gravity is measured either within the sphere or on the surface of the sphere, [2.1-5] and thus the gravitational force simplifies to [2.1-6] showing that in fact the force is linear with the range from the center. 2.2 Energy and General Relativity The effects of gravity on light radiating from an object are described by the principles of general relativity (Einstein 1911; 1916). As light leaves an object, it loses energy such that its apparent wavelength increases. For visible light, this appears as a shift towards the red end of the spectrum (versus the blue end). Potential energy in a gravitational field is defined by the equation [2.2-1] where Ep is the potential energy difference between r0 and r1 r is the range of the m2 object r0 is the starting range for the m2 object r1 is the ending range for the m2 object Kinetic energy is defined as [2.2-2] where Ek is the kinetic energy of the object m m is the mass of the object v is the velocity of the object If one equates m2 and m from the energy equations and compares the maximum potential energy against the maximum kinetic energy, [2.2-3] then [2.2-4] Simplifying (remembering m2 = m), [2.2-5] With this equation, v equates to the escape velocity of an object starting at a range r from m1. Of course, there is a limiting factor to v. When v = c, the speed of light or 3 x 10^8 m/s, m1 has become a black hole, preventing the aforementioned object from escaping. For instance, for the Earth to become a black hole, its mass of 5.98 x 10^24 Kg would need to be shrunk to a sphere with a radius of 8.9 x 10^-3 m, or just under 1 cm. Of course at this radius, the event horizon for Earth would be at infinity; reducing the radius of the Earth further would bring the event horizon closer to the Earth itself, giving meaning to it as a black hole. The effects of gravity on light within an object are also defined through general relativity. The potential energy within a spherical object of uniform density is defined by the equation [2.2-6] where Ep is the potential energy between r0 and r1 As before, kinetic energy is defined as [2.2-7] Equating Ep and Ek, and m2 and m as before, [2.2-8] Simplifying, [2.2-9] or [2.2-10] Here, v equates to the velocity of an object that is lost in traveling from the center of the sphere out to some radius r. A light source at the center of a sphere of uniform mass density will lose energy as it radiates toward the surface of the sphere. If the sphere is sufficiently large, light will never escape it; the event horizon relative to the center of the mass will be located at a position within the sphere. If the Earth were expanded at a density equivalent to its current average mass density of 5500 Kg/m^3, light from the center would be prevented from escaping to the surface if the planet had a radius of 2.4 x 10^11 m. This would be similar to having a planet centered at the sun, extending just beyond the current orbit of the planet Mars. Consider the following idea. Define the universe to be an enormous mass field with no discernible center and no preferential reference frame. While this universe may not be infinite in size, it appears to be so to any observer. Define the density of the universe to be uniform over large volumes. While the universe would be considered "lumpy" with respect to structures such as the "Local Group" or the "Great Wall", volumes on the order of hundreds of cubic megaparsecs (Mpcs) would be of similar density. Finally, define the universe to be relatively static in terms of size and internal motion. Although there may be some motion of galaxies within the universe as can be observed with those in the Local Group, there is not a great outward movement of galaxies, nor is there a great expansion of space. Given this defined universe, what could one expect to see? Take a galaxy within this defined universe. Light from the stars in this galaxy radiates outward in all directions. Since this universe has neither a preferential reference frame nor a discernible center, to the light leaving the galaxy it is radiating from a local center. As it moves away from the galaxy over large distances, a considerable amount of mass in the form of a sphere is accumulating behind the light, as though the light were traveling through an object such as the Earth (only larger). Since this spherical mass can be considered to be exerting an increasing force on the light, it therefore is reducing the energy of the light. As the light passes through the universe, observers in subsequent galaxies would see the light as being shifted increasingly toward the red end of the visible light spectrum, until all of the energy is lost. At what range would all of the energy be lost? At an average density of 10-26 Kg/m3, a density value that might account for nonluminous and undetected matter in the universe (Kaufmann 1991), all energy would be lost at a range of 1.90 x 10^10 light years, or almost 20 billion light years. This range of 20 billion light years represents the limit of the observable universe (Kaufmann 1991). (By my theory, this also equates to a Hubble constant equivalent to 52 km/s/Mpc.) At an average density of 5 x 10^-28 Kg/m3, the density of the observable matter in the universe (Kaufmann 1991), all energy would be lost at a range of 8.48 x 10^10 light years, or almost 85 billion light years. But of course, this information is based on the aforedefined universe. 3 SPHERES AND THE SHAPE OF THE UNIVERSE Two works cited in the bibliography of this paper-Reichenbach 1927, and Rucker 1977-are major sources I used regarding the conceptualization of fourspace models. As I do not directly apply their models, I do not cite them specifically. However, their work was of such influence on me that I think it is only proper to credit these two men here. 3.1 Sphere, the Conceptual Three Dimensional Model To a person G on a sphere, how does his world appear? His is a two dimensional world where he can move horizontally, either forward and backward, or sideways. Yet, this seeming two dimensional world is actually a world in threespace. From a starting point P, he can travel in any direction and, assuming environmental conditions support such a journey, can return to P without having deviated from his course. If our intrepid traveler G were to try this feat again but in a different direction, he would return again to his starting point. He would also cross his path from his first journey at the halfway point. (Think of lines of longitude and their convergence at the north and south poles.) Consider a tangent line starting at P, extending out from the surface of the sphere in the direction of our traveler's journey. This line is one unit long. Our traveler takes his own tangent line with him, also one unit long. As he travels away from P, his tangent line is projected along that one at P. While he walks, there is an angle formed by the vectors P and G with their origins at the center of the sphere. As this angle grows, the amount of G's tangent line that is projected onto that at P is reduced. [3.1-1] where X' is the projection of the tangent at G onto the tangent at P X is the length of the tangent at G, one unit x is the angle between vectors P and G When G reaches a point where x is 90 , X' goes to zero. Yet, while X' has been getting smaller and smaller, G observes nothing unusual other than a similar reduction of the projection of the tangent at P onto his tangent at G. 3.2 Hypersphere, the Four Dimensional Model Take the spherical model one dimension further. This fourth dimension is a physical dimension (versus "time" so often abused) that is at right angles to the three dimensional world in which we live. If i, j, and k denote the vectors along the X, Y, and Z axes, then l denotes this fourth dimension vector along the W axis. To a person S on a hypersphere, a four dimensional sphere, how does her world appear? Hers is a three dimensional world where she can move both horizontally and vertically. Yet this seeming three dimensional world is actually a world in fourspace. From a starting point P, she can travel in any direction and, assuming environmental conditions support such a journey, can return to P without having deviated from her course. In fact, it would appear to an observer at P as though she were following a straight line and yet was becoming dimmer and dimmer until she disappeared from view, only to reappear much later coming from the opposite direction. If our intrepid traveler S were to try this feat again but in a different direction, she would return again to her starting point. She would also cross her path from her first journey at the halfway point. (Think of lines of longitude and their convergence at the north and south poles.) Consider a tangent line starting at P, extending out from the surface of the hypersphere in the direction of our traveler's journey. This line is one unit long. Our traveler takes her own tangent line with her, also one unit long. In three dimensions, these two tangent lines appear to be collinear, but in fact are not since this is actually a three dimensional "flat" surface curved in fourspace. As she travels away from P, her tangent line is projected along that one at P. While she travels, there is an angle formed by the vectors P and S with their origins at the center of the hypersphere. As this angle grows, the amount of S's tangent line that is projected onto that at P is reduced. [3.2-1] where X' is the projection of the tangent at S onto the tangent at P X is the length of the tangent at S, one unit x is the angle between vectors P and S When S reaches a point where x is 90 , X' goes to zero. Yet, while X' has been getting smaller and smaller, S observes nothing unusual other than a similar reduction of the projection of the tangent at P onto her tangent at S. 3.3 The Hyperspherical Universe of General Relativity The Lorentz transform states that the length of an object is proportional to Lp where [3.3-1] and can also be written as [3.3-2] Considering the nature of this equation in these two forms, note that [3.3-3] and can be rewritten as [3.3-4] and [3.3-5] Since [3.3-6] and from Equation 2.2-10 [3.3-7] then [3.3-8] or rewritten [3.3-9] Finally, [3.3-10] If the effects of gravity in the universe are modeled by general relativity, and if our universe is indeed a three dimensional surface in fourspace as is suggested by theory and possibly observation, then the preceding equations model this universe. Lp from the Lorentz transform corresponds to the cosine of the angle x formed between vectors defining two points, where the two vectors have their origins in the center of the hypersphere and the points are located in our visible universe, the "surface" of the hypersphere. Furthermore, v/c from the Lorentz transform corresponds to the sine of angle x. This set of equations is consistent with a universe of stars, galaxies, and dust, with no discernible center and no preferential reference frame. This universe has generally a uniform density over large volumes of space. Light emitted from an object is subject to the effects of gravity as described by general relativity, given that the gravitational potential in this universe increases linearly with distance. As such, light traveling from an object is subject to energy loss as a direct function of range traveled. This energy loss is observed as a "red shift" that is not an effect of an expanding universe, but as an apparent velocity related to the velocity term in the Lorentz transform. 4 IMPLICATIONS My theory of the nature of the universe has many implications, some of which are highlighted below. Size of the Local Universe. Assuming that our observable universe is, in fact, the surface of a hypersphere, this surface on (in?) which we exist may be twice the size of the observable universe. The equations from Section 3 would imply that our range of visibility corresponds to an ability to see objects just under 90 of arc in any given direction, or roughly 180 total-90 in one direction and 90 in the opposite direction. In three dimensions, that would correspond to a hemisphere; twice that is a sphere. Special Relativity. Special relativity effects may be a consequence of the nature and shape of space. Special relativity accounts for physical compression in the direction of motion, as well as the accompanying time dilation, as per the Lorentz transform. Although we have known of nature's speed limit of c, the reason for it has been unknown. Could the natural speed limit of light in fact be an effect of the shape and nature of the universe? Does fourspace somehow tilt physical objects that move at relativistic speed? Could this imply that there may exist a way to circumvent the Light Police? Gravity. Space may not be definable in quantum terms. The curvature of space may only be a consequence of mass. Furthermore, gravity may only be a consequence of the nature and shape of space, and thus may not be definable in quantum terms. Gravitons may move from being theoretical particles to historical theoretical particles. Size of the Complete Universe. The universe may be orders of magnitude larger than ever before imagined. If our universe is indeed merely the surface of a single four dimensional object, albeit a very large object, the size of the actual universe has grown incomprehensibly again. How many hyperobjects exist? How large is hyperspace? To imagine a four dimensional universe, is it fair to make the following analogy: Flatland, a two dimensional world (Abbott 1884) is to our Universe as our Universe is to Hyperworld, my name for a four dimensional world? Age of the Universe. Our universe may be orders of magnitude older than ever before imagined. Today we cannot explain large scale structures in the universe such as the Great Wall, given the Big Bang theory and the accompanying value for the age of the universe (Lerner 1991). Yet such structures exist. The Great Wall is described as a "huge sheet of galaxies ... stretching more than two hundred million light-years across and seven hundred million light-years long, but only about twenty million light years thick". Over five thousand galaxies were mapped in this region that lies within six hundred million light years of Earth. (Lerner 1991) That such structures require more time to form than our universe is theorized to have existed raises a question of the validity of the Big Bang model. Furthermore, the expansion characteristics and age of the universe based on the Big Bang are even today unresolved. Consider recent research on developing a more accurate Hubble constant (Pierce et al. 1994). Despite their reasoned confidence in their value, they offer this statement for balance: "This high value, in conjunction with most fashionable cosmological models, conflicts with the ages of the oldest known stars." (Pierce et al. 1994) This discrepancy is further echoed, based on recent findings resulting in a significantly different value for the Hubble constant (Hogan 1994). A Doppler explanation for galactic red shifts implies an error in either our calculation of the age of the universe or in the existence of large scale structures in the universe. A relativistic explanation for galactic red shifts accommodates both an older universe and the existence of large scale structures. Perhaps it is time to seriously consider a new universe model. Link Between Flat and Curved Geometries. Do the equations that define the Lorentz transform and trigonometry, [4-1] -which may be rewritten as [4-2] -and [4-3] show a physical link between Euclidean and non-Euclidean space? 5 CONCLUSIONS My theory of the nature of the universe is that the universe is a volume of space containing matter-galaxies, nebulae, quasars, dust. It is not experiencing expansion as per the "Big Bang" theory, but is in fact generally steady-state in size and internal motion. The observed red Doppler shift of the galaxies is a function of the effects described by general relativity. The shape of the observable universe is the apparent three dimensional surface of a four dimensional sphere, the hypersphere. The size of this "surface" may be two times the volume of our observable universe. This surface may be a minuscule object in a universe that is much larger and much older than credited by accepted theory. BIBLIOGRAPHY Abbott, Edwin A. "Flatland: A Romance of Many Dimensions". 1884. Revised. 1963. Forward by Isaac Asimov. Reprint of 5th edition, New York: Harper & Row, publishers, 1983. Einstein, Albert. "Relativity: The Special and the General Theory". 1916. Translated by Robert W. Lawson. New York: Crown Publishers, 1961. Einstein, Albert. "On the Influence of Gravitation on the Propagation of Light". 1911. "Cosmological Considerations on the General Theory of Relativity". 1917. With: H. A. Lorentz; H. Weyl; and H. Minkowski. In The Principle of Relativity. Translated by W. Perrett and G. B. Jeffery with notes by A. Sommerfeld. 1923. Reprint, New York: Dover Publications, 1952. Hogan, Craig J. "Cosmological conflict". Nature. 371 (29 September 1994): 374-375. Kaufmann, William J. "Universe". 3rd ed. New York: W. H. Freeman and Company, 1991. Kline, Morris. "Mathematics and the Physical World". 1959. Reprint, slightly corrected, New York: Dover Publications, 1981. Lerner, Eric J. "The Big Bang Never Happened". Hardcover, New York: time Books, 1991; paperback with new preface by author, New York: Vintage Books, 1992. Newton, Sir Isaac. "Principia". 1686. Translated by Andrew Motte. 1729. Revised by Florian Cajori. 1934. 2 vols. Berkeley: University of California Press, 1962. Pierce, Michael J.; Douglas L. Welsh; Robert D. McClure; Sidney van den Bergh; Rene Racine; and Peter B. Stetson. "The Hubble constant and Virgo cluster distance from observations of Cepheid variables". Nature. 371 (29 September 1994): 385-389. Reichenbach, Hans. "The Philosophy of Space & Time". 1927. Translated by Maria Reichenbach and John Freund with introduction by Rudolph Carnap. Reprint, New York: Dover Publications, 1958. Rucker, Rudolph v. B. "Geometry, Relativity, and the Fourth Dimension". New York: Dover Publications, 1977.