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Continuum Mechanics of Space Seen from the Aspect of General Relativity—An Interpretation of the Gravity Mechanism

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Continuum Mechanics of Space Seen from the Aspect of General Relativity—An Interpretation of the Gravity Mechanism Journal of Earth Science and Engineering 5 (2015) 188-202 doi: 10.17265/2159-581X/2015. 03. 004 D DAVID PUBLISHING Continuum Mechanics of Space Seen from the Aspect of General Relativity—An Interpretation of the Gravity Mechanism Yoshinari Minami Advanced Science-Technology Research Organization (Formerly NEC Space Development Division), Yokohama 220-0062, Japan Received: February 15, 2015 / Accepted: March 05, 2015 / Published: March 20, 2015. Abstract: A mechanical structure of space is suggested. On the supposition that a space as vacuum has a physical fine structure like continuum, it enables us to apply a continuum mechanics to the so-called “vacuum” of space. A space is an infinite continuum and its structure is determined by Riemannian geometry. Assuming that space is an infinite continuum, the pressure field derived from the geometrical structure of space is newly obtained by applying both continuum mechanics and General Relativity to space. A fundamental concept of space-time is described that focuses on theoretically innate properties of space including strain and curvature. As a trial consideration, gravity can be explained as a pressure field induced by the curvature of space. Key words: General Relativity, continuum mechanics, curvature, strain, space-time, gravity. 1. Introduction expansion, contraction, elongation, torsion and bending. The latest expanding universe theories Given a priori assumption that space as a vacuum (Friedmann, de Sitter, inflationary cosmological has a physical fine structure like continuum, it enables model) support this assumption. Space can be us to apply a continuum mechanics to the so-called regarded as an elastic body like rubber. This “vacuum” of space. Minami proposed a hypothesis for conveniently coincides with the precondition of a mechanical property of space-time in 1988 [1]. A mechanical structure of space. primary motive was research in the realm of space General relativity implies that space is curved by propulsion theory. His propulsion principle using the the existence of energy (mass energy or substantial physical structure of space-time is based electromagnetic energy etc.). General relativity is on this hypothesis [2-10]. based on Riemannian geometry. If we admit this space In this paper, a fundamental concept of space-time curvature, space is assumed as an elastic body. is described that focuses on theoretically innate According to continuum mechanics, the elastic body properties of space including strain and curvature. has the property of the motion of deformation such as Assuming that space as vacuum is an infinite expansion, contraction, elongation, torsion and continuum, space can be considered as a kind of bending. General relativity uses only the curvature of transparent elastic field. That is, space as a vacuum space. Expansion and contraction of space are used in performs the motions of deformation such as cosmology, and a theory using torsion has also been studied by Hayasaka [11] and twistor theory as Corresponding author: Yoshinari Minami, administrative proposed by Roger Penrose [12] to the torsion of director, research fields: satellite design and engineering, propulsion theory and propulsion physics, laser propulsion, space. interstellar navigation theory. E-mail: [email protected]. Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 189 Interpretation of the Gravity Mechanism 2. Mechanical Concept of Space (2) The spatial strain is defined as a localized geometrical structural change of space. It implies a When we make a comparison between the space on change from flat space involved in zero curvature the Earth and outer space throughout the universe, components to curved Riemann space involved in although there seems to be no difference, obviously a non-zero curvature components. different phenomenon occurs. Simply put, an object (3) Space has the only strain-free natural state, and moves radially inward, that is, drops straight down on space always returns to the strain-free natural state, the Earth, but in the universe, the object floats and i.e., flat space, when an external physical action does not move. causing spatial strain is removed. The difference between the two phenomena can be (4) Spatial strain means some kinds of structural explained as whether space is curved or not, that is, deformation of space, and a body filling up space is whether 20 independent components of a Riemann affected by the action from its spatial strain. We must curvature tensor is zero or not. In essence, the distinguish space from an isolated body. An isolated existence of spatial curvature (and curved extent) body occupies an area of space by its movement. determines whether the object drops straight down or Basically, an isolated body can move in space and also not. Although the spatial curvature at the surface of can change its position. −23 2 the Earth is very small value, i.e.,3.42 ×10 (1/ m ) , (5) In order to keep the continuity of space, the 2 it is of enough value to produce 1G (9.8 m/s ) velocity of body filling up space cannot exceed the acceleration. Conversely, the spatial curvature in the strain rate of space itself. universe is zero, therefore any acceleration is not Since the subject of our study is a four-dimensional produced. Accordingly, if the spatial curvature of a Riemann space as a curved space, we ascribe a great localized area including object is controlled to deal of importance to the curvature of space. We a −23 2 curvature 3.42 ×10 (1/ m )with an extent, the object priori accept that the nature of actual physical space is moves and receives 1G acceleration in the universe. a four-dimensional Riemann space, that is, three Of course, we are required to control both the dimensional space (x=x1, y=x2, z=x3) and one magnitude and extent of curvature. dimensional time (w=ct=x0), where c is the velocity of light. These four coordinate axes are denoted as xi (i = 2.1 Fundamental Concept of Space 0, 1, 2, 3). Space is an infinite continuum and its structure is The square of the infinitesimal distance “ds” determined by Riemannian geometry. Space satisfies between two infinitely proximate points xi and xi+dxi the following conditions: is given by equation of the form: (1) When the infinitesimal distance regulating the ds 2 = g dxi dx j distance between the two points changes by a certain ij (1) physical action, the change is continuous, and the where gij is a metric tensor. space maintains a continuum even after its change. The metric tensor gij determines all the geometrical Now, the concept of strain of continuum mechanics is properties of space and it is a function of this space ij very important in order to relate a spatial curvature to coordinate. In Riemann space, the metric tensor g i a practical force. Because the spatial curvature is a determines a Riemannian connection coefficient Γ jk , purely geometrical quantity. A strain field is required and furthermore determines the Riemann curvature p for the conversion of geometrical quantity to a tensor R ijk or Rpijk , thus the geometry of space is practical force. determined by a metric tensor. 190 Continuum Mechanics of Space Seen from the Aspect of General Relativity—An Interpretation of the Gravity Mechanism i Riemannian geometry is a geometry which provides original vase vector g and the gij′ is the a tool to describe curved Riemann space, therefore a transformed metric tensor from the original metric Riemann curvature tensor is the principal quantity. All tensor g ij . Since the degree of deformation can be the components of Riemann curvature tensor are zero expressed as the change of distance between the two for flat space and non-zero for curved space. If a points, we get: non-zero component of Riemann curvature tensor 22 ij ij ds′′−= ds gij dx dx − g ij dx dx exists, the space is not flat space, but curved space. In (4) =−()g′ g dxij dx = r dx ij dx curved space, it is well known that the result of the ij ij ij parallel displacement of vector depends on the choice Hence the degree of geometrical and structural of the path. Further, the components of a vector differ deformation can be expressed by the quantity denoted from the initial value, after we displace a vector change of metric tensor, i.e. parallel along a closed curve until it returns to the r = g′ − g (5) starting point. ij ij ij An external physical action such as the existence of On the other hand, the state of deformation can be mass energy or electromagnetic energy yields the also expressed by the displacement vector “u” (Fig. 1). structural deformation of space. In the deformed space From the continuum mechanics [13-16], using the region, the infinitesimal distance is given by: following equations: ′2 ′ i j i j ds = gij dx dx (2) du = g ui: j dx (6) i j where gij′ the metric tensor of deformed space ds′ = ds + du = ds + g ui: j dx (7) region, and we use the convected coordinates We use the usual notation “:” for covariant ( x′i = x i ). As shown in Fig. 1, if the line element between the differentiation. From the usual continuum mechanics, arbitrary two near points (A and B) in space region S the infinitesimal distance after deformation becomes (before structural deformation) is defined as [13] (see Appendix A): i 22 ij ds = gi dx , the infinitesimal distance between the ds′ −= ds r dx dx ij (8) two near points is given by Eq. (1): kij =++()uij:::: u ji u i u kj dx dx 2 i j ds = gij dx dx k The terms of higher order than second u :i u k: j Let us assume that a space region S is structurally can be neglected if the displacement is of small deformed by an external physical action and enough value.
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