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Journal of Earth Science and 5 (2015) 188-202 doi: 10.17265/2159-581X/2015. 03. 004 D DAVID PUBLISHING

Continuum of Seen from the Aspect of —An Interpretation of the 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 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- 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 . 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 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 of 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 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 , 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 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 . −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 ) of body filling up space cannot exceed the . Conversely, the spatial curvature in the 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 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 . 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 of this space very important in order to relate a spatial curvature to coordinate. In Riemann space, the metric tensor gij i a practical . 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 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 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. As the actual physical space can be transformed to space region T. In the deformed space dealt with the minute displacement from the trial region T, the line element between the identical two calculation of strain, we get: near point (A’ and B’) of the identical space region r = u + u newly changes, differs from the length and direction, ij i: j j:i (9) and becomes i . ds′ = gi′dx Whereas, according to the continuum mechanics [13], Therefore, the infinitesimal distance between the the strain tensor eij is given by: two near points using the convected coordinate 1 1 i i x′ x eij = ⋅ rij = ⋅ (ui: j + u j:i ) (10) ( = ) is given by: 2 2 2 i j So, we get: ds′ = gij′ dx dx (3) ds′2 − ds 2 = (g′ − g )dxi dx j = 2e dxi dx j (11) The gi′ is the transformed base vector from the ij ij ij

Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 191 Interpretation of the Gravity Mechanism

Fig. 1 Fundamental structure of space.

where gij′ , gij is a metric tensor, eij is a strain kl ωμν = Rμνkl dA (13) ′2 2 tensor, and ds − ds is the square of the kl where ω μν is rotation tensor, dA is infinitesimal infinitesimal distance between two infinitely areal element. proximate points xi and xi+dxi. According to the nature of Riemann curvature Eq. (11) indicates that a certain geometrical tensor Rμνkl , ω μν indicates the rotation of structural deformation of space is shown by the displacement field. Eq. (13) indicates that a curved concept of strain. In essence, the change of metric space produces the rotation of displacement field in tensor (gij′ − gij ) due to the existence of mass energy the region of space. Now, the rotation tensor ω μν or electromagnetic energy tensor produces the strain and strain tensor eij satisfy the following differential field eij . equation in continuum mechanics: Since space-time is distorted, the infinitesimal i distance between two infinitely proximate points x ωμν , j = eνj,μ − eμj,ν (14) and xi+dxi is important in our understanding of the This equation is true on condition that the order of geometry of the space-time; the physical strain is differential can be exchanged in a flat space. To generated by the difference of a geometrical metric of expand above equation into a curved Riemann space, space-time. Namely, a certain structural deformation the equation shall be transformed to covariant is described by strain tensor eij . From Eq. (11), the differentiation and it is possible on condition of strain of space is described as follows: α α Γjν eμα = Γjμeνα . Thus, we obtain eij = 1/ 2 ⋅ (gij '−gij ) (12)

ωμν: j = eνj:μ − eμj:ν (15) It is also worth noting that this result yields the principle of constancy of light velocity in Special Here we use the usual notation “:” for covariant Relativity. differentiation. As is well known, the partial ∂u derivative u = i is not tensor equation. The 2.2 Mechanics of Space i, j ∂x j u = u −u Γk Expanding the concept of vector parallel i: j i, j k ij is tensor displacement in Riemann space, the following equation and can be carried over into all coordinate equation has newly been obtained (see Appendix B): systems.

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

Eq. (15) indicates that the displacement gradient of 3 3330 30 FFERA== ()3030 :3 rotation tensor corresponds to difference of the (21) =⋅E3330∂ ()/RA 30 ∂ r displacement gradient of strain tensor. 3030 Here, if we multiply both sides of Eq. (15) by 2.3 Generation of Surface Force Induced by Spatial fourth order tensor denoted the nature of space E ijμν Curvature formally, we obtain On the supposition that space is an infinite E ijμν ω = E ijμν (R dAkl ) = E ijμν R dAkl μν: j μνkl : j μνkl: j continuum, continuum mechanics can be applied to (16) the so-called “vacuum” of space. This means that and space can be considered as a kind of transparent field

ijμν ij μν with elastic properties. That is, space as a vacuum has Eeνμjj::−= Ee μν (17) the elastic properties of expansion, contraction, ()()Eeijμν−=−=Δ Ee ij μνσ i μσσ i ν ir νμj ::::: μjr ν μ ν elongation, torsion and bending. The latest expanding As is well known in the continuum mechanics universe theory (Friedmann, de Sitter, inflationary [13-16], the relationship between tensor σ ij cosmological model) supports this assumption. We and strain tensor eml is given by can regard the space of the cosmos as an infinite elastic body like rubber. σ ij = E ijml e (18) ml If space curves, then an inward normal stress “－P” Furthermore, the relationship between is generated. This normal stress, i.e. surface force i F and stress tensor σ ij is given by serves as a sort of pressure field.

i ij 00 1/ 2 : j (19) (22) F = σ − P = N ⋅(2R ) = N ⋅(1/ R1 +1/ R2 ) from the equilibrium conditions of continuum. That is, where N is the line stress, R1 , R2 are the radius of the elastic force F i is given by the gradient of stress principal curvature of curved surface, and R00 is the tensor σ ij . major component of spatial curvature. Therefore, Eq. (17) indicates the difference of body A large number of curved thin layers form the force ΔF i . Accordingly, from Eqs. (16) and (17), the unidirectional surface force, i.e. acceleration field. i ir 00 change of body force ΔF (= Δσ :r ) becomes Accordingly, the spatial curvature R produces the i ijμν kl acceleration field α . ΔF = E Rμνkl: j dA (20) The fundamental three-dimensional space structure Here, we assume that E ijμν is constant for is determined by quadratic surface structure. covariant differentiation, A kl is area element. Therefore, a Gaussian curvature K in two-dimensional i The stress tensor σ ij is a surface force and F is Riemann space is significant. The relationship a body force. The body force is an equivalent between K and the major component of spatial gravitational action because of acting all elements of curvature R 00 is given by: space uniformly. R1212 1 00 Eq. (20) indicates that the gradient of Riemann K = = ⋅ R (23) 2 2 curvature tensor implying space curvature produces (g11g 22 − g12 ) the body force as a space strain force. The non-zero where R1212 is non-zero component of Riemann component of Eq. (20) is just only one equation as curvature tensor. follows: It is now understood that the membrane force on

Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 193 Interpretation of the Gravity Mechanism the curved surface and each principal curvature Therefore we get: generates the normal stress “-P” with its direction Nbαβ== NKg αβ αβ()i αβ (26) normal to the curved surface as a surface force. The αβ α gαβNK()iii= NK α ()=⋅ NK () normal stress –P acts towards the inside of the surface From Eq. (24) and Eq. (26), we get: as shown in Fig. 2a. α A thin-layer of curved surface will take into Nα K(i) = −P (27) consideration within a spherical space having a radius As for the quadratic surface, the indices α and i of R and the principal radii of curvature that are equal take two different values, i.e. 1 and 2, therefore Eq. to the radius (R1=R2=R). Since the membrane force N (27) becomes: (serving as the line stress) can be assumed to have a 1 2 constant value, Eq. (22) indicates that the curvature N1 K(1) + N2 K(2) = −P (28) R 00 generates the inward normal stress P of the where K(1) and K(2) are principal curvature of curved surface. The inwardly directed normal stress curved surface and are inverse number of radius of serves as a pressure field. principal curvature (i.e. 1/R and 1/R ). When the curved surfaces are included in a great 1 2 The Gaussian curvature K is represented as: number, some type of unidirectional pressure field is formed. A region of curved space is made of a large K = K (1) ⋅ K (2) = (1/ R1 ) ⋅ (1/ R2 ) (29) number of curved surfaces and they form the field as a 1 2 Accordingly, suppose N = N = N , we get: unidirectional surface force (i.e. normal stress). Since 1 2 the field of the surface force is the field of a kind of N ⋅ (1/ R1 +1/ R2 ) = −P (30) force, the force accelerates matter in the field, i.e. we It is now understood that the membrane force on the can regard the field of the surface force as the curved surface and each principal curvature generate acceleration field. A large number of curved thin the normal stress “–P” with its direction normal to the layers form the unidirectional acceleration field (Fig. curved surface as a surface force. The normal stress 2b). Accordingly, the spatial curvature 00 R –P is towards the inside of surface as showing in Fig. 2. produces the acceleration field α . Therefore, the A thin-layer of curved surface will be taken into curvature of space plays a significant role to generate consideration within a spherical space having a radius pressure field. of R and the principal radii of curvature which are Applying membrane theory, the following equal to the radius (R1=R2=R). From Eqs. (23) and equilibrium conditions are obtained in quadratic (29), we then get: surface, given by: 1 1 1 R00 αβ (31) N bαβ + P = 0 (24) K = ⋅ = 2 = R1 R2 R 2 where N αβ is a membrane force, i.e. line stress of Considering N ⋅ (2 / R) = −P of Eq. (30), and curved space, b is second fundamental metric of αβ substituting Eq. (31) into Eq. (30), the following curved surface, and P is the normal stress on curved equation is obtained: surface [13]. 00 The second fundamental metric of curved space − P = N ⋅ 2R (32) b and principal curvature K has the following αβ (i) Since the membrane force N (serving as the line relationship using the metric tensor g , αβ stress) can be assumed to have a constant value, Eq. 00 bαβ = K (i) gαβ (25) (32) indicates that the curvature R generates the

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

(a)

(b) Fig. 2 Curvature of Space: (a) curvature of space plays a significant role. If space curves, then inward stress (surface force) “P” is generated ⇒ A sort of pressure field; (b) a large number of curved thin layers form the unidirectional surface force, i.e. acceleration field α . inward normal stress P of the curved surface. The where Rμν is the Ricci tensor, R is the scalar inwardly directed normal stress serves as a kind of curvature, G is the gravitational constant, c is the pressure field. When the curved surfaces are included velocity of light, T μν is the energy in great number, some type of unidirectional pressure tensor. Furthermore, we have the following relation field is formed. A region of curved space is made of a for scalar curvature R : large number of curved surfaces and they form the ααβ RR==ααβ gR, field of unidirectional surface force (i.e. normal stress). RggRμν= μα νβ , (34) Since the field of surface force is the field of a kind of αβ RR==jij gR force, a body in the field is accelerated by the force, αβ α jβ ijα β i.e. we can regard the field of surface force as the Ricci tensor Rμν is represented by: acceleration field. Accordingly, the cumulated curved α α α β α β Rμν = Γμα,ν −Γμν,α −ΓμνΓαβ +ΓμβΓνα (= Rνμ ) region of curvature R00 produces the acceleration (35) field α . i where Γ jk is Riemannian connection coefficient. Here, we give an account of curvature 00 in R If the curvature of space is very small, the term of advance. The solution of metric tensor gμν is found higher order than the second can be neglected, and by gravitational as the following: Ricci tensor becomes: μν 1 μν 8πG μν α α R − ⋅ g R = − ⋅T (33) R = Γ , −Γ , (36) 2 c 4 μν μα ν μν α

Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 195 Interpretation of the Gravity Mechanism

The major curvature of Ricci tensor ( μ = ν = 0 ) is curved space region, the massive body “m (kg)” calculated as follows: existing in the acceleration field is subjected to the i 00 00 00 following force F (N) : R = g g R00 = −1×−1×R00 = R00 (37) jk iidx dx As previously mentioned, Riemannian geometry is Fm= Γ⋅jk ⋅ = ddττ (41) a geometry that deals with a curved Riemann space, 2 ijk i mgcuum−Γ00 jk =α therefore a Riemann curvature tensor is the principal j k i quantity. All components of Riemann curvature tensor where u , u are the four velocity, Г jk is the are zero for flat space and non-zero for curved space. Riemannian connection coefficient, and τ is the proper If an only non-zero component of Riemann curvature time. tensor exists, the space is not flat space but curved From Eqs. (40) and (41), we obtain: space. Therefore, the curvature of space plays a 2 ijk iidx dx dx significant role. α ==−Γ⋅⋅2 jk dddτ ττ (42) 2 ijk 2.4 Acceleration Induced by Spatial Curvature =− −gc00 Γ jk uu

A massive body causes the curvature of space-time Eq. (42) yields a more simple equation from the around it, and a free particle responds by moving condition of linear approximation, that is, weak-field, along a in that space-time. The path of free quasi-static, and slow motion ( v << speed of particle is a geodesic line in space-time and is given light c:u 0 ≈1): by the following geodesic equation; i 2 i α = − − g00 ⋅c Γ00 (43) 2 i j k d x i dx dx + Γ ⋅ ⋅ = 0 (38) On the other hand, the major component of spatial dτ 2 jk dτ dτ curvature R 00 in the weak field is given by i where Γ jk is Riemannian connection coefficient, RRR00 ≈=μ = i 00 0μ 0 τ is proper time, x is four-dimensional Riemann (44) ∂ Γ−∂Γ+ΓΓ−ΓΓμ μνμνμ space, that is, three dimensional space (x=x1, y=x2, 00μ μμννμ 00 0 0 00 z=x3) and one dimensional time (w=ct=x0), c is the In the nearly Cartesian , the value μ velocity of light. These four coordinate axes are of Γνρ are small, so we can neglect the last two denoted as xi (i = 0, 1, 2, 3). terms in Eq. (44), and using the quasi-static condition Proper time is the time to be measured in a clock we get resting for a coordinate system. We have the 00 μ i R ≈ −∂ Γ = −∂ Γ (45) following relation derived from an invariant line μ 00 i 00 element ds 2 between (flat space) From Eq. (45), we get formally and General Relativity (curved space): Γ i = − R 00 (x i )dx i (46) 00 ∫ 0 dτ = − g dx = − g cdt (39) 00 00 Substituting Eq. (46) into Eq. (43), we obtain From Eq. (38), the acceleration of free particle is α i = − g c 2 R 00 (x i )dx i (47) obtained by 00 ∫ d 2 xi dx j dxk Accordingly, from the following linear α i = = −Γi ⋅ ⋅ (40) dτ 2 jk dτ dτ approximation scheme for the As is well known in General Relativity, in the equation: (1) weak gravitational field, i.e. small

196 Continuum Mechanics of Space Seen from the Aspect of General Relativity—An Interpretation of the Gravity Mechanism curvature limit, (2) quasi-static, (3) slow-motion be advantageous. approximation (i.e., v / c <<1 ), and considering 3. Consideration of Gravity range of curved region, we get the following relation between acceleration of curved space and curvature of Let us consider about gravity. Why does apple fall space: in the Earth? A well-known answer is that there exists b iii200 gravity between Earth and apple. Apple is because it’s α =−gc00 R() xdx (48) ∫a Mm pulled by a law of universal gravitation i 2 F = G 2 where α : acceleration (m/s ), g 00 : time component r of metric tensor, a-b: range of curved space region (m), to the Earth. Here, M is the mass of Earth, m is the xi: components of coordinate (i = 0, 1, 2, 3), c: mass of apple, G is the gravitational constant, r is the velocity of light, R00 : major component of spatial distance between Earth and apple, F is the curvature (1/m2). gravitational force. From a phenomenological Eq. (48) indicates that the acceleration field α i is standpoint, it is a sufficient explanation. produced in curved space. The intensity of However, what is the mechanism? According to acceleration produced in curved space is proportional General Relativity, it is said that apple moves to the product of spatial curvature R00 and the geodesic line formed by curved space near the Earth. length of curved region. This is seen as lacking in sufficient explanation. The Eq. (41) yields more simple equation from following explanation may allow someone to above-stated linear approximation (u 0 ≈1), understand the mechanism of gravity. Fmgcii=−2 Γ= If we were to visualize the curvature of space 00 00 (49) b around the Earth (M), we would describe it as mmgcRxdxα iii=− 200() 00 ∫a having an aggregation of curved surface. A great Setting i = 3(i.e., direction of radius of curvature: r), number of thin curved surfaces are arranged in a we get Newton’s second law: spherical concentric pattern. This curvature would FFm3 ==α = gradually become smaller as we moved away from the b (50) Earth in what we could imagine as layers of an onion. mgcRrdrmgc−=−Γ200() 23 00∫a 00 00 The surrounding space becomes a flat space of The acceleration ( α ) of curved space and its curvature 0 at an imagined immense distance from the 3 Riemannian connection coefficient ( Γ00 ) are given Earth (Fig. 3). by: In the following thought experiment, an apple of mass m positioned at a distance r apart from the Earth 2 3 3 − g00, 3 α =−gcΓΓ, = (51) would receive a pressure of the field formed by an 00 00 00 2g 33 accumulation of the normal stress (Fig. 3). As was where c: velocity of light, g00 and g33: component of described earlier, with reference to Fig. 2, the 3 metric tensor, g00,3 : ∂g00 ∂ x = ∂g00 ∂r . We membrane force on the curved surface and each choose the spherical coordinates “ct=x0, r=x3, θ=x1, principal curvature generates the normal stress “–P” ϕ=x2” in space-time. The acceleration α is with its direction normal to the curved surface as a represented by the equation both in the differential surface force. The normal stress –P acts towards the form and in the integral form. Practically, since the inside of the surface as shown in Fig. 2a. metric is usually given by the solution of gravitational A thin-layer of curved surface will take into field equation, the has been found to consideration within a spherical space having a radius

Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 197 Interpretation of the Gravity Mechanism of R and the principal radii of curvature that are equal of light. to the radius (R1=R2=R). Since the membrane force N However, in our thought experiment, the apple (serving as the line stress) can be assumed to have a would still receive pressure from the surrounding field constant value, the inwardly directed normal stress by the accumulation of the normal stress. serves as a pressure field. When the curved surfaces Because, since there still exists the curved region are included in a great number, some type of behind the apple from a to b (the remote flat unidirectional pressure field is formed. space), the apple continues falling. The pressure That is, a sort of graduated pressure field is continues to push the apple to the center of the Earth generated by the curved range from an imaginary (Fig. 4). point “a” in curved space to a point “b” (the point at However, as soon as the change from a curved which space is absent of curvature, i.e., flat space of surface to a flat surface passes through the point of the curvature 0) (Fig. 3). Then apple moves directly apple (i.e., “a” point), the pressure at point “a” towards the center of the Earth, that is, the apple falls. disappears and the apple would only float without Falling acceleration of apple in curved space is falling (Fig. 5). proportional to both the value of spatial curvature and The above discussion provides a to consider the size of curved space. the following thought experriment. Even if the Sun If the Earth (M) were to disappear instantly, the instantly disappeared, the Earth would still continue to curved surface of space close to the Earth would revolve around the Sun until 8 minutes 32 seconds, or return to the flat surface. Because an external action the time at which it takes light to advance between the causing curvature (i.e., mass energy) disappears. The Sun and Earth. However, as soon as the change from change from a curved surface to a flat surface would curved surface to flat surface passes through the point advance the position r of the apple at the speed of light of the Earth, or at 8 minutes 32 seconds after the event, (i.e., the strain rate of space-time). The propagation the pressure pushing the Eaarth would disappear, and velocity of the change from flat space to curved space the Earth would fly away in a direction tangential to and the propagation velocity of change from curved the revolution of its orbit. space to flat space are both the same, i.e. the velocity

Fig. 3 Apple falls receiving a pressure of the field.

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

Fig. 4 Apple still continues falling receiving a pressure of the field.

Fig. 5 Apple only floats without falling due to lack of pressure of the field.

In view of this, gravity may be considered as a As a result, a fundamental concept of space-time is pressure generated in a region of curved space. described that focuses on theoretically innate properties of space including strain and curvature. As 4. Conclusion a trial consideration, gravity can be explained as a Assuming that space is an infinite continuum, a pressure field induced by the curvature of space. mechanical concept of space became identified. Space References can be considered as a kind of transparent elastic field. [1] Minami, Y. 1988. “Space Strain Propulsion System.” In The pressure field derived from the geometrical Proceedings of 16th International Symposium on Space structure of space is newly obtained by applying both Technology and Science, 125-36. continuum mechanics and General Relativity to space. [2] Minami, Y. 1994. “Possibility of Space Drive Propulsion.”

Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 199 Interpretation of the Gravity Mechanism In Proceedings of 45th Congress of the International Given by a Spacetime Featuring Imaginary Time.” Astronautical Federation, 658. Journal of the British Interplanetary Society 56: [3] Minami, Y. 1997. “Space faring to the Farthest 205-11. Shores-Theory and Technology of a Space Drive [10] Minami, Y., 2014. “A Space Propulsion Principle Propulsion System.” Journal of the British Interplanetary Brought about by Locally-Expanded Space.” Journal of Society 50: 263-76. Earth Science and Engineering 4: 490-9. [4] Minami, Y. 2003. “An Introduction to Concepts of Field [11] Hayasaka, H. 1994. “Parity Breaking of Gravity and Propulsion.” Journal of the British Interplanetary Society Generation of Antigravity due to the de Rham 56: 350-9. Cohomology Effect on Object’s Spinning.” In [5] Minami, Y. 2013. “Space Drive Propulsion Principle Proceedings of 3rd International Conference on from the Aspect of Cosmology.” Journal of Earth Problems of Space, Time, Gravitation, Science and Engineering 3: 379-92. [12] Huggett, S. A., and Tod, K. P. 1985. An Introduction to [6] Minami, Y. 2013. “Basic concepts of space drive Twistor Theory. UK: Cambridge University Press. propulsion―Another view (Cosmology) of propulsion [13] Flügge, W. 1972. Tensor Analysis and Continuun principle.” Journal of Space Exploration 2: 106-15. Mechanics. New York: Springer-Verlag. [7] Minami, Y., 2014. A Journey to the Stars—By Means of [14] Fung, Y. C. 2001. Classical and Computational Space Drive Propulsion and Time-Hole Navigation. Mechanics. New Jersey: World Scientific Publishing Co. Berlin: LAMBERT Academic Publishing. Pre. Ltd.. [8] Minami, Y. 1993. “Hyper-Space Navigation Hypothesis [15] Ziegler, H. 1977. An Introduction to Thermomechanics. for Interstellar Exploration.” In Proceedings of 44th Zürich: North-Holland Publishing Company. Congress of the International Astronautical Federation, [16] Borg, S. F. 1963. -Tensor Methods in Continuum [9] Minami, Y. 2003. “Travelling to the Stars: Possibilities Mechanics. New York: D. Van Nostrand Company.

Appendix A: Derivation of Eq. (8)

Let us consider two adjacent spatial points A and B in the unreformed space, Fig. 1, which are the end points of a line element vector ds. During the deformation, point A undergoes the displacement u and moves point A’, while point B experiences a slightly different displacement u+du when moving to point B’. From Fig. 1 we read the simple vector equation

i j i i j ds′ = ds + du = ds + g ui: jdx = gidx + g ui: j dx (A1)

We may now write the square of the deformed line element. Since all indices are dummies, they have been chosen so that the final result looks best. When we multiply the two factors term by term and switch the notation for some dummy pairs, we obtain:

i k i j l j l kl i l ds'⋅ds'= (g dx + g u dx ) ⋅(g dx + g u dx ) = (g + 2g ⋅ g u + g u u )dx dx (A2) i k:i j l: j ij i l: j k:i l: j Using Eq. (3) and Eq. (1), from Eq. (A2), we get:

2 i j l kl i l ds' −gijdx dx = (2gi ⋅ g ul: j + g uk:iul: j )dx dx (A3)

Left side of Eq. (A3):

i j i j i j i j g'ij dx dx − gij dx dx = (g'ij −gij )dx dx = rij dx dx (A4)

l l kl l k Right side of Eq. (A3): considering 2gi g ul:j = 2δi ul:j = 2ui:j and g uk:iul:j = u :iul: j = u :iuk:j , (changes of the dummy indices l→k), then,

l kl i l k (2gi ⋅ g ul: j + g uk:iul: j )dx dx = (2ui: j + u :iuk: j ) (A5)

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

Since , considering , rij = rji 2ui: j = ui: j + u j:i

i j i j k i j (g' −g )dx dx = r dx dx = (u + u + u :iu )dx dx ij ij ij i:j j:i k: j Finally we obtain:

2 2 i j k i j (A6) ds′ − ds = rij dx dx = (ui: j + u j:i + u :iuk: j )dx dx

Appendix B: Derivation of Eq. (13)

Let us suppose that covariant vector at point P(x) is transported parallel to Q(x+d1x+d2x) via path Ι is . On the Ai (P) Ai (Q)Ι while, covariant vector at point P(x) is transported parallel to Q(x+d1x+d2x) via path Ⅱ is A (Q) . (Fig. B1). Ai (P) i C

Fig. B1 Parallel transport of vector via two paths.

The result of these parallel transport of vector via two paths does not differ. The difference is indicated by:

ρ k σ A (Q) − A (Q) = R ikσ (P)A (P)d x d x (B1) i Ι i C ρ 1 2

ρ where R ikσ ( p) is Riemann curvature tensor at point P.

Considering A (Q) + (−A (Q) ) , Eq. (B1) indicates the quantum not returned of vector A (P) in case that parallel transport i Ι i C i of a vector Ai (P) along a closed path has been employed at each segment of the loop Ｐ→ＲΙ→Ｑ→ＲⅡ→Ｐ and ultimately leads back to the point of departure ,i.e., original point P(x) (Fig. B2). Mathematically, Riemann curvature tensor is the result of a difference that changed the order of the covariant derivatives as seen Eq. (B2), and its non-commutative part is represented by the Riemann curvature tensor.

p X i: jk − X i:kj = R ijk X p (B2)

Let us consider point R adjacent spatial point P, Fig. B2, which are the end points of a line element vector ds. If vector ds at P(x) is transported parallel to RΙ(x+d1x) and thence to Q(x+d1x+d2x), then parallel transport from Q(x+d1x+d2x) to original point P(x) via RⅡ(x+d2x), the result is the new vector ds’. Parallel transport of a vector ds along a closed path that ultimately leads back to the point of departure will result in a new vector ds’ at the original point P(x); the new vector ds’ differs from the original vector ds, even though the proper procedure for parallel transport has been employed at each segment of the loop. ds’-ds indicates the quantum not returned of vector, also is denoted by

Continuum Mechanics of Space Seen from the Aspect of General Relativity—An 201 Interpretation of the Gravity Mechanism displacement vector du. This arises from nonzero curvature of space. Another interpretation, two adjacent spatial points P and R in the unreformed space, Fig. B2, which are the end points of a line element vector ds. During the deformation, P under goes the displacement u and moves P’, while R exxperiences a slightly different displacement u+du when moving to R’. These phenomena are equivalent, it is not possible to identify them.

Fig. B2 Parallel transport of vector along a closed path and displacement vector.

From above, we get:

ρ k σ A (Q) − A (Q) = ds'−ds = R ikσ (P)A (P)d x d x (B3) i Ι i C ρ 1 2

Since ds’-ds = du, infinitesimal displacement vector du is described in

j du = dui = ui:j dx (B4)

A (Q) − A (Q) = ds'−ds = du = u dx j i Ι i C i: j (B5)

Apply a vector Aρ (P) in Eq. (B3) to a line element vector ds = dsρ, from Eq. (B5),

j ρ k σ ui:j dx = R ikσ dsρ d1x d2 x (B6)

Now let us multiply both sides of Eq. (B6) by grρ , we get following:

rρ j ρ r k σ g ui:j dx = R ikσ ds d1x d2 x , then,

r r rρ ρ ds k σ ρ dx k σ ρ r k σ g u = R ikσ d x d x = R ikσ d x d x = R ikσ δ d x d x , finally i: j dx j 1 2 dx j 1 2 j 1 2

jρ ρ k σ g ui: j = R ikσ d1x d2 x (B7)

Multiplying both sides of Eq. (B7) by g ρm ,

jρ ρ k σ jρ j gρm g ui: j = gρm R ikσ d1x d2 x , then applying, gρm g = δm , we get:

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

u = R d xk d xσ i:m mikσ 1 2 . (B8)

For convenience, returns the index m to j,

k σ ui: j = R jikσ d1x d2 x (B9)

Interchanging index i, j, we get:

k σ u j:i = Rijkσ d1x d2 x (B10)

On the other hand, using the nature of the Riemann curvature tensor,

(B11) R jikσ = −Rijkσ

Subtracting Eq. (B10) from Eq. (B9), we obtain:

k σ k σ k σ ui: j − u j:i = −Rijkσ d1x d2 x − Rijkσ d1x d2 x = −2Rijkσ d1x d2 x (B12)

By continuum mechanics, anti-symmetric part of the displacement gradient tensor represents the rotation tensor ωij, 1 ω = (u −u ) (B13) ij 2 j:i i:j

Accordingly, we obtain:

1 k σ kσ ω = (u −u ) = R d x d x = R dA (B14) ij 2 j:i i:j ijkσ 1 2 ijkσ

kσ k σ where dA is the area element enclosed by the vector d1x and vector d2x . Thus, changing the dummy of incidences i, j,k,σ → μ,ν,k,l , we get finally: ω = R dAkl μν μνkl (B15) kl where ωμν is rotation tensor, dA is infinitesimal areal element.