TORSION AND CURVATURE GOVERN SPACETIME DEFECTS


Copyright 1985-2015 John Bruce Davies


This research is based on and extends the following published paper:


Davies, J.B., 1988: "New Curvature-Torsion Relations through Decomposition of the Bianchi Identities", Foundations of Physics, 18, 5, 63.





Abstract


The fundamental geometric properties of the 3-dimensional space that we are all familiar with have both translational and rotational defects manifesting in the most general U3 space. We can thus expect and demand that such defects should be manifest in the most general 4-dimensional U4 spacetime. Such a U4 spacetime geometrically possesses non-symmetric metric, curvature and torsion. Einstein recognized it was necessary to expand his general theory of relativity to incorporate more than energy-linear momentum as symmetric curvature. However, lack of knowledge at that time of many of the properties of the operating force fields, stymied these more encompassing identifications.


We approach the geometry through obtaining the conservation laws for 2-tensor linearization of the Bianchi Identities, Davies (1988). We obtain 2-tensor equations relating the non-symmetric metric to curvature and torsion terms and their respective gradients. A subset of this new set of equations governs the usual contribution of symmetric curvature to the gravitational field though with an added contribution due to torsion gradients. As in the previous 3-dimensional case, the antisymmetric curvature 2-tensor is shown to be related to the divergence of the torsion.


Using a model of particle-antiparticle pair production, identification of certain torsion components with electroweak fields is proposed. These components obey equations, similar to Maxwell's, that are subsets of these linear Bianchi Identities. These results are shown to be consistent with gauge and other previous analyses. Other evidence for torsion gradient effects is added to the published research.




Introduction


In order to encompass more force fields than just the gravitational field, the simple 4D space-time with only symmetric curvature must be extended. This was understood by Einstein and Cartan who incorporated torsion as well as non-symmetric curvature in this most general space-time, known as U4. The fundamental equations that relate non-symmetric curvature and torsion are the Bianchi Identities, which, for the most general cases, are highly non-linear, Schouten (1954), Larner (1973).


We have derived the linearized Bianchi Identities that govern solely second order curvature tensors, Davies (1988). We use second order tensors because they can be uniquely separated into symmetric and antisymmetric components, which are independent. By examining a subset, it is shown that these linearized Bianchi Identities yield the usual relationship of symmetric curvature to the gravitational field gradients. However, we find that an additional component, due to torsion gradients, also contributes to the gravitational field.


The antisymmetric curvature component is obtained as the divergence of a modified dual of the torsion tensor. The time-like components of the torsion tensor can also be split into symmetric and antisymmetric components both of which, after taking the covariant divergence, will yield two independent currents, related to the antisymmetric curvature.


Many researchers, since Einstein, have proposed that torsion terms are related to electromagnetic fields and/or spin. By applying a model of particle-antiparticle pair production, curvature is identified with the massive, charged particle currents while the forces intermediating the particle and anti-particle are the electroweak fields which are incorporated in the torsion tensor. Using this simple fundamental model, similar equations to Maxwell's for both the electromagnetic and the weak interactions are found to be incorporated in these linearised Bianchi Identities. We show how the conservation of angular momentum is equivalent to our linearized Bianchi-Davies Identity wherein the antisymmetric curvature is conserved. Strong fields governed by Yang-Mills equations, relevant to intra-nuclear quark behavior, have been proven to be equivalent to the non-linear Bianchi Identities.


Definitions


The space-time of interest contains torsion as well as curvature and assumes a non-symmetric metric tensor. This has been termed a Riemann-Cartan U4 space-time, Schouten (1954). The fundamental fields for measuring change in a vector around a circuit C in the body of U4 are called the coefficients of affine connection  .


Consider a closed circuit C in the body and construct a continuous local reference configuration (tetrad) for the points of C. If, upon completing the circuit, the orientation of the reference configuration is rotated from its original orientation, then C encloses angular defects represented by the curvature tensor. Also, the curve C in its local reference configuration will not, in general, be a closed curve. This closure failure is due to translational defects and is represented by the torsion tensor to leading order in the limit as C shrinks to zero. However, for general curves C, the closure failure may be caused by both curvature and torsion, Schouten (1973).


In terms of this affine connection, the usual definition of the curvature tensor is:


R = , - , +   +  


where we have not yet incorporated covariant differentiation. Note the linearity of the first two terms and the quadratic nature of the second two terms on the right hand side in this definition. Thus sufficiently small connections will allow the linear terms to dominate with the curvature tensor also constrained to be small.


The usual definitions of the related curvature objects are:


L = g 


L = g R


L = g g g R


where g is the most general non-symmetric metric tensor.


We examine the case of sufficiently small fields such that linearization is valid and only terms of first order are incorporated. To this order, L is antisymmetric between the pair of indices (  ), as well as (  ). Thus it suffices to consider, to this order, the related non-symmetric curvature tensor.


(1) L = -1/4  L


whose symmetric part is the usual Einstein tensor, and is the usual alternating tensor. Because is a "dummy" index, this non-symmetric curvature tensor can be equivalently expressed as composed of time-like and space-like components that are independent of each other.


(2) L = -1/4 00 L -1/4 ii L


Where 0 represents the time-like component and i represents the space-like components.


Torsion Q is the antisymmetric part of the connection, where the antisymmetry occurs only in two of the indices:


(3) Q = 1/2 [ L - L ]


We define the modified dual of the torsion as


(4a) A L


or, alternatively,


(4b) A Q


where the alternating tensor only operates on the two antisymmetry indices and where A is also known as the dislocation density tensor in the theory of defect-containing media. Hence we also have the inverse relation:


(4c) Q 1/2 A


Consequently the torsion and dislocation density tensors are equivalent quantities in that each determines the other.


Bianchi-Davies Identities


We derive the linearized First Bianchi Identity by extending the 3-dimensional case, Davies (1988), and incorporating covariant differentiation in the linearized continuity equation:


(5) L = L - L


where the fundamental relation used is:


(6) L = 1/2[g g g] + Q + Q - Q


On substitution of these equations (5,6) into the definition of the second order curvature tensor (1) and after some manipulation, we obtained the linearized First Identity in the limit of small fields and in terms of these curvature and modified torsion tensors:


(7) LA; - 1/2gA; - 1/2g,,


where the semi-colon implies covariant differentiation. This linearized formulation of the First Identity is complemented by the usual Second Bianchi Identity:


(8) L = 0


where, however, the curvature 2-tensor is non-symmetric in our more general approach.


We used second order tensors as they can be uniquely decomposed into independent symmetric and antisymmetric components. We can thus uniquely split the second order curvature tensor L into its symmetric Einstein curvature tensor G and the antisymmetric Davies curvature tensor D.


L = G + D



From the First Bianchi-Davies Identity,(7), we obtain the relation connecting the metric gradients and the symmetric curvature component:


(9a) 1/2g = -G+[A ]s


where the subscript s indicates that only the symmetric component of these torsion gradient terms is taken. Using (4c) we have:


(9b) 1/2g = -G+[1/2 Q ]s


Apart from the torsion dependent term, this equation, on expansion, is equivalent to that derived by Weinberg (1972) in his classic text (eq. 7.2.6), whose approach was based on the assumptions of maximally general metric structure, weak i.e. linear fields, the conservation of symmetric curvature and the non-relativistic limit of Einstein's field equation giving Newton's Law.


This fundamental agreement on the production of gravitational field by the symmetric curvature two-tensor yields added significance to the torsion term in (9). This implies that torsion gradients also contribute to gravitational fields.


Removing (9) from (1) shows that the antisymmetric curvature two-tensor D is related solely to torsion gradient terms. Multiplying the first Bianchi-Davies Identity,(7), with the alternating tensor, we get the fundamental equation:


(10) ADg


where only the antisymmetric component of curvature is involved.


Taking the divergence of Equation (9), we obtain:


(11) G = 0


which is the usual conservation of symmetric Einstein curvature. Removing this from the Second Bianchi Identity (8) gives:


(12) D = 0


which implies conservation of antisymmetric curvature as well.


Pair Production


In order to understand concepts and mathematical relationships in a four dimensional space-time, it is often useful to extrapolate from similar situations in lower dimensional spaces. In 2 and 3 dimensional space, curvature and torsion manifest as disclinations and dislocations respectively which govern the structure and properties of cracks. Metric gradients correspond to generalised strain terms in the medium surrounding the crack. The simplest fundamental 2 dimensional crack consists of a translational defect along a line, the dislocation, whose ends are comprised of equal and opposite angular defects termed disclinations, Nabarro (1967).


A 2 or 3 dimensional crack can be formed due to excessive strain in the medium. In 4-dimensional spacetime, crack formation will correspond to production of a particle-antiparticle pair which is the fundamental mass-containing entity that can be formed from sufficiently energetic photon interaction. The massive particle and the equal and opposite anti-particle thus correspond to the angular defects at the ends of the crack and are represented by non-symmetric curvature. The fields that intermediate the particle and its anti-particle must therefore correspond with the dislocation, or translational defect, and must be represented by torsion terms.


Einstein made the identification of symmetric curvature with energy and linear momentum where the Gravitational Constant is the effective unit of proportionality. The concept of mass energy-momentum being represented by angular defects has previously been used in Regge calculus where symmetric curvature is related to the angular defect in a skeleton space-time. We shall examine the quantization of such a discrete space-time in a later chapter.


Electroweak Fields


The time-like fields through which the particle and anti-particle interact are, in the linear regime, the electromagnetic and weak fields. These fields act along the time-like line between the pair so that, as in the 2 and 3 dimensional cracks, one index of the modified dual of the torsion will correspond to the direction of the dislocation which will be the "0" component in this U4 space-time. It is therefore to be expected that the fundamental equations of electromagnetism and of the weak interaction should be embedded in these Bianchi-Davies Identities.


On this basis, we use the Davies-Bianchi Identity subset, eqn.(10), by examining the case when the directional index is the time-like index 0.


(13) A00Dg


The right-hand side of eqn.(13) contains the antisymmetric curvature two-tensor and can be represented by a current:


(14) I0 = 0Dg = J + K


where the current term I has been split into the two independent currents J and K.


The dislocation density tensor, A0, may be uniquely split into its independent symmetric component, W, and antisymmetric component, F.


(15) A0 = W + F


Equation (13) then splits into two independent relations, namely:


(16) F J


and


(17) W K


Equation (16) is identical to Maxwell's "electric" equation. Maxwell's magnetic equation follows as a consequence of the antisymmetry of F and the gauge condition, the antisymmetry also implying the usual conservation of the electromagnetic current, i.e. J= 0. In this case, identifying these antisymmetric torsion 2-tensor fields and the corresponding antisymmetric curvature with electromagnetic fields and currents implies that the proportionality constant is just the fundamental unit of charge,’e’.


Equation (17) is of similar form to Maxwell’s equation but with symmetry of the W field. We argue that this equation governs the other field that mediates the particles in pair production, the so-called ‘Weak’ field.


Electromagnetic and Weak Field Correlations

Einstein (1956) originally argued that torsion should be related to the electromagnetic field. Gogala (1980) has demonstrated that the electromagnetic field can be related to a time-like modification of the torsion. Also, Borschenius (1976a,b) has identified a modified torsion tensor with electroweak fields using Lagrangian and gauge methods. Similar techniques have been used by Rumpf (1979) to show that a step-gradient in torsion produces fermion currents through a pair production example. Edelen (1980) has examined the gauge properties of the linear Bianchi Identities and argues for a complete identification with those of electromagnetism.


The other linear torsion field is W which is governed by equation (16). Because of the symmetry of W the current K is not conserved. This non-conservation is a fundamental property of the so called weak current, Taylor (1976). Rauch (1982) has also shown that torsion can produce symmetry breaking. Such symmetry breaking is readily observed in 2 dimensional cracks where spatial symmetry is maximally broken in "en echelon" cracks represented by on-diagonal dislocation components, Davies (1980). Other researchers have also shown the similarity between torsion and the weak field, Kaempffer (1976).


Strong Fields


The linear Bianchi Identities are based on the assumption of fields sufficiently small that only first order terms are necessary. When the fields are too strong to satisfy this condition, then the non-linear full Bianchi Identities must be used. The main additional terms in the non-linear situation are products of torsion and curvature. The non-Abelian gauge properties of these non-linear Bianchi Identities have parallels, Kadic and Edelen (1983), with those of Yang-Mills equations.


The non-linear Bianchi Identities and the Yang-Mills equations possess similar solutions of non-linear character due to the presence of Klein-Gordon operators, Altimirano and Villaroel (1981). Similar non-linear behavior for defects is well-known from examining their physics at the atomic level. Dislocations in crystals, subject to sinusoidal atomic potentials, obey the non-linear Sine-Gordon wave equation. Solutions manifest behavior, Fischbach (1986), such as wave superposition and splitting (bifurcation) due to the amplitude dependence of the applicable dispersion relation. Other interesting phenomena include the production of solitary waves of constant shape and energy that have been used in the modeling of particle behavior.


References


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Davies, J.B., 1988: "New Curvature-Torsion Relations through Decomposition of the Bianchi Identities", Foundations of Physics, 18, 5, 63.


Davies, J.B., 1980: "The Geometry of Defects in Faults", E.O.S., 61, 1051.


Edelen, D.G.B., 1980, ‘A Four-Dimensional Formulation of Defect Dynamics’, Int. J. Eng. Sci., 18, 1095.


Edelen, D.G.B. and D.C. Lagoudas, 1988,

Dispersion Relations of the Linearized Field Equations of Dislocation Dynamics’, Int. J. Eng. Sci., 26, 8, 837.


Einstein, A., 1952, ‘Relativity, The Special and General Theory’, Bonanza Books, Crown Publishing Inc., New York.


Fischbach, E., et al., 1986, Phys Rev. Letters, 56, 1, 3.


Gogala, B., 1980, Int. J. Theor. Phys., 19, 8, 587.


Kaempffer, F., 1976, Gen. Rel. Grav., 7,

327.


Kadic, A. and D.G.B. Edelen, 1983, ‘A Gauge Theory of Dislocations and Disclinations’, Springer-Verlag, N.Y.


Lardner, R.W., 1973, ‘Foundations of the Theory of Disclinations’, Archives of Mechanics, Poland, 25, 6, 911.


Lardner, R.W., 1974, ‘Mathematical Theory of

Dislocations and Fracture’, University of Toronto Press.


Nabarro, F.R.N., 1967, ‘Theory of Crystal Dislocations’, Oxford University Press.


Nabarro, F.R.N., 1970, ‘Disclinations in Surfaces’; in ‘Fundamental Aspects of Dislocation Theory’, N.B.S. (U.S.) Spec. Publ. 317, 1, 593.


Rauch, R.T., 1982, Phys. Rev. D, 25, 2, 577.


Rumpf, H., 1979, Gen. Rel. Grav., 10, 8, 647.


Schouten, J.A., 1954, ‘Ricci-Calculus’, 2nd edn., Springer-Verlag Press, Berlin.


Taylor, J.C., 1976, ‘Gauge theories of weak interactions’, Cambridge University Press, Cambridge, U.K.


Vargas, J.C. and D.G. Torr, 1999, Foundations of physics, v.29, 2, 145.


Weinberg, S., 1972, ‘Gravitation and Cosmology’, John Wiley & Sons, Inc., N.Y.





EVIDENCE FOR TORSION GRADIENT EFFECTS


The following analyses are unpublished and not peer reviewed.


Background


We have shown above that the new set of linearized Bianchi-Davies Identities affects the observed metric in a number of ways. Because the additional torsion gradient term in Einstein’s equation contributes to the gravitational field, we would expect that the electromagnetic field gradient would affect measurements of the Gravitational Constant. We shall show that such an effect is observed by examining variations in its value when measurements and experiments are performed in high electric field gradients.


This fundamental additional contribution is in the form of a gradient while the metric term is of double derivative form and curvature term is independent of gradients. We show that this combination implies time asymmetry in Einstein’s equation; a necessary and sufficient condition for satisfying the most important observation of our Universe.


Recently, observation of satellite flybys of planets has found time delays in their transit that are functions of incidence and exit angles with respect to the planets’ rotation axes. We show that this result is consistent with our identification of antisymmetric curvature with angular momentum fields, as torsion gradients equivalent to antisymmetric curvature affect the local metric.


A 3-dimensional explosion can be described by disclination and dislocation fields. Thus, extrapolating to 4-dimensional spacetime, an explosion will have similar properties such as an initial matter zone obeying non-linear Bianchi Identities or, equivalently, strong fields obeying Yang-Mills equations. Subsequently, the explosion reduces the crack, i.e. particle-antiparticle pair, density as it expands so that the linearized Bianchi-Davies conservation laws then apply. We see this in the Big-Bang model where now electromagnetic and weak fields dominate except in the high curvature nuclei of baryons.


The Big-Bang is an explosion outward and, on a smaller scale, is essentially a Whitehole, c.f. Davies (2014). This is the opposite disclination to the implosion, the well-known Blackhole. For the conservation laws of energy and momentum to be satisfied, we argue that for every blackhole sink there is an equal and opposite whitehole source. As the present observable Universe is dominated by particles, we argue that the antiparticles equally formed in the very early Universe have collapsed inward as opposed to particles expanding outward. Thus, conserving the fundamental laws.


Variations in Measurement of the Gravitational Constant


The relative gravitational field is measured through observation of the local force of gravity mainly produced by the Earth with small tidal effects due to moon, planets and sun. The absolute amplitude of gravitational fields is measured through the Gravitational Constant G.


The best measurements of the Gravitational Constant G have accuracy for the individual experiments of about 1 part in 10,000. This error range is due to the noise in the measuring system, this noise even being correlated with environmental activity, Michaelis et al. (1996), sometimes to an excessive level. The range of the older and more recent experimental values is shown therein and the inter-experiment accuracy is, from these results, about 1 part in 1000.


Our unified model shows, from equation (9) in the previous chapter, that torsion, and thus electromagnetic, gradients may also contribute to the gravitational field. As did Weinberg (1972), we take the non-relativistic limit of this equation and get Newton's equation with an added torsion gradient term.


(1) Δ2ψ = 4πGρ + ΔF


where ψ is the gravitational potential, ρ the mass-energy density and F the electric and/or magnetic field.


If these torsion gradients of sufficient magnitude are present, measurements of G using mass densities can be affected. Constant and variable torsion gradients may thus manifest as noise and error in measurements. We roughly estimate the relative values of such terms where the error ratio R is:


(2) R = e.ΔE /4πGρ


where e is the unit electric charge and E the electric field near the apparatus.


The electric field near the surface of the earth is usually about 1 volt/cm, Feynman (1964). In storms, this field can be hundreds of times greater and can vary over centimeter distances. With sharp local topography effects, we can expect a range of electric field gradients of 1 to 10 volt/cm/cm. Using these values and assuming a density of 10 gm/cc for the experimental masses, we get a range for the error ratio from about 0.5 parts in 10,000 to 0.5 parts in 1000. These lower values are consistent with observed noise values in the quieter individual experiments reported. The higher values are consistent with inter-experiment differences and in the higher levels of noise observed in certain experiments. That a fundamental constant such as G is only measurable to such a poor accuracy is thus telling us that other forces are at work.


Experimental Evidence


Gillies (1990) has documented the various historical experimental measurements of gravitation; also described are the searches for variations in G values due to time, electro/magnetic fields, temperature, radioactivity and other unknown forces. Vargas and Torr (1999), and previous papers, have used Davies (1988) model and examined Einstein and Cartan's historical work. They also evaluated experiments for measuring G but concluded with incorrect evaluation of the amplitude of torsion contributions.


Experiments have been attempted to measure effects of electrical fields on the gravitational constant, the original one being by the well-respected experimental physicist Nipher (1916). He showed a definite effect, of the same magnitude as predicted above, of electrical fields on the balance measurement system even when the apparatus was completely shielded. Woodward (1982) has experimented with rotating charged disks and argues that the results show an effect consistent with electro-gravitational induction.


Shielding and anisotropy of gravitational fields have also been investigated; Allais (1959) found an anisotropy of 5 ppm in G. More recently, using the new high-temperature superconductors to shield electromagnetic fields, Podkletnov and Nieminen (1992) showed a decrease in gravitational fields above a levitated superconducting rotating disk. Other experiments have found anomalous effects due to topography and location with respect to the earth's surface


Time Asymmetry in the Universe


"The moving finger writes, and having writ, moves on." - Rubaiyat of Omar Khayam


The Universe is time asymmetric in a number of fundamental processes and phenomena. We grow older. The Universe is expanding. Radioactive nuclei emit particles. In closed systems, entropy is constant or increasing. All these processes have one fundamental all-encompassing element: the governing space-time metric changes irreversibly along the world-line of the process. The arrow of time has only one direction: forward. Yet the traditional equations of space-time physics are symmetrical in time, being equally applicable in one direction as the other. We shall prove that our simple and necessary extension of general relativity brings irreversibility to the fundamental equation governing the space-time metric and a direction to time.


From Newton's laws to Einstein's general relativity, all of the traditional governing differential equations are time reversible. But irreversibility manifests even in non-relativistic systems, which implies that the traditional equations governing the physics of everyday phenomena are incomplete. Electromagnetic, weak and velocity fields govern the atomic structure of these material phenomena. Thus, intuitively, we would expect changes in these fundamental fields to be related to time irreversibility. For example, highly-ordered life-forms decay into low-order systems after death with corresponding changes in their electromagnetic field structure. Radioactive particles decay in time with accompanying changes in their weak fields. Rotating planets slow down. All such phenomena are geometrically represented by space-time metric structures moving and changing irreversibly along their world-lines.


Einstein, in his original derivation of the theory of general relativity, restricted the space-time under consideration to possess only a symmetric metric and a symmetric curvature field. Under these restrictions, Einstein intuitively produced a field equation through which he related the tensors of energy-momentum and symmetric curvature. Weinberg (1972) has shown how to derive this fundamental relation based on the fundamental tenet that the weak static non-relativistic limit of Einstein's field equation must yield Newton's Law relating gravitational potential to mass density. Weinberg's further assumption is that Einstein's curvature tensor is dependent solely on terms of quadratic order in derivatives of the metric structure. Together with conservation of energy-momentum and, therefore, symmetric curvature, Weinberg derives Einstein's field equation. Simultaneously, he shows that Einstein's curvature tensor is governed by quadratic derivatives of the symmetric metric tensor.


Einstein, later in life, realized that, in order to represent physical reality as a field, a generalized relativity must incorporate a non-symmetric metric (Appendix V, 1952). In the same exposition, by examining a spinning disk, he deftly demonstrates, using clocks and measuring rods, that general relativity has an intrinsic problem with angular velocity. Changes in the angular momentum of a body are caused by torques, which are moments of forces. Everyday phenomena are dependent on torques, inherent in such human actions as turning around or breaking bread. Torques in space and time are represented in space-time by torsion fields, which must be included in any generalized relativity.


The fundamental relation, equation (9) from the previous chapter, namely:


1/2g = -G + [A ]s


implies that torsion gradients contribute to the non-symmetric metric in this generalized space-time. Einstein (1952) believed that such generalizations would allow physical reality to be described by fields, but was unable, in his allotted time on earth, to prove this hypothesis against the facts of experience. However, one of these most fundamental facts is the direction of time. As Penrose has remarked: "it seems to be clearly the case that whatever physics is operating, it must have an essentially time-asymmetrical ingredient, i.e. it must make a distinction between past and future".


If a partial differential equation contains either single or second-order derivatives in time, but not both, its solutions manifest only reversibility. A partial differential equation incorporating both types of time derivatives has solutions that are time irreversible. Thus, the above equation, with second-order derivatives of the metric and first-order time derivatives of torsion has that essential time-asymmetric ingredient.


Newton's and Einstein's equations without this torsion-gradient term are symmetric in positive and negative time. Radioactive decay implies weak field gradients. Entropy increase implies the generation of heat, which is the production of thermal electromagnetic field gradients. The slowing down of rotating solid bodies by frictional forces produces heat, eventually emitted as electromagnetic radiation.


Thus, time asymmetry in the Universe is due to the effect of torsion, i.e. electroweak, gradients on the non-symmetric metric in a generalized four-dimensional space-time. Yet, as Einstein commented: "People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion".


Torsion and Angular Momentum


Einstein's breakthrough was the relating of the symmetric curvature tensor with that of the matter density energy-momentum tensor, T, which is derived from special relativity and assumes only linear momentum.


(18) G = T


where is the effective gravitational constant. For any particular system, giving the components of T in some frame defines it completely. This energy-momentum tensor incorporates mass-energy and linear momentum and their fluxes and is symmetric. When these components for the energy-momentum tensor are equated to the symmetric curvature tensor, application of the conservation law, G = 0, yields the conservation laws of mass-energy and of linear momentum.


The usual definition of the energy-momentum tensor is derived from special relativity and assumes only linear momentum. In this case, T is the component of the linear 4-momentum through the surface x. Consider first T00. This is defined as the flux of 0- momentum, energy divided by c, across a surface of constant time t. This is just the energy density. Similarly, T0i is the flux of energy divided by c across a surface of constant xi. Then Ti0 is the flux of i- momentum across a surface of constant t, the density of i- momentum multiplied by c. Finally Tij is the j- flux of i- momentum. For any particular fluid-dynamical system, giving the components of T in some frame, defines it completely.


We have only discussed linear momentum in the development of the energy-momentum tensor and the appropriate conservation laws. There is also, in our space and time, the fundamental law of conservation of angular momentum of matter. Changes in the angular momentum of a body are caused by torques, which are moments of forces, i.e. Torque is equal to Force times Distance. We are thus led to similarly relate the antisymmetric curvature 2-tensor with that of the antisymmetric angular momentum 2-tensor, M , viz:


(19) D = M


where the M time-like components are the angular momentum vectors ( r x ) and the space-like components are the angular momentum fluxes. Thus, the conservation equation (11) of the antisymmetric curvature 2-tensor implies the conservation of angular momentum through the relation (19). Our unified model shows, from eqn.(9), that torsion gradients, and thus antisymmetric curvature, also contribute to the gravitational field. From (19), we therefore conclude that gradients of the angular momentum fields can also affect observed gravitational field measurements.


Anomalous Orbital Velocity Changes


A recent analysis of anomalous orbital velocity changes observed during spacecraft flybys of Earth has shown the dependence of the velocity increases on the incoming and outgoing geocentric latitudes of the asymptotic spacecraft velocity vectors (Anderson et al., 2008). All potential sources of systematic error were considered and none can account for the observed anomalies. It is thus concluded that the standard law of gravitation is not sufficient to explain this phenomena.


Anderson et al. (2008) determined a prediction formula for the anomalous orbital velocity/energy changes observed for 6 flybys of Earth. Their formula accurately determines the velocity changes as proportional to .R.(cos - cos) where is the Earth's angular rotational velocity, R is its mean radius, is the declination (latitude) of the incoming osculating asymptotic velocity vector and is that for the outgoing velocity vector. We see that this formula expresses the difference between the incoming and outgoing angular velocity projection components perpendicular to the Earth's rotation axis. They were unable to explain why this empirical formula was valid.


We have shown above that this phenomena and formula can be explained using our extension of the gravitational Einstein equation that proves torsion gradients, and, thus, antisymmetric curvature and its associated angular momentum fields, affect the gravitational field. Our theoretical argument therefore explains the dependence of the anomalous orbital velocity changes on the gradient of the angular momentum field of the orbited rotating astrophysical body, as Anderson et al. (2008) have demonstrated.


The Big-Bang Explosion


Our 3-dimensional explosion was shown to be described by disclination and dislocation fields. Thus, extrapolating to 4-dimensional spacetime, an explosion will have similar properties such as an initial inflation of the empty space outward. Guth and others have shown that in the first 10-35 seconds of the Big Bang rapid inflation by a factor of 1050 occurs with high energy densities and high temperatures prevailing. We consider that this represents the primordial disclination/curvature defect. As spacetime expands, and the defect density decreases, there is a zone dominated by the Strong force which is obeying non-linear Bianchi Identities or, equivalently, the Yang-Mills equations.


We have seen that, in the 3-D explosion, the defect density reduces as the radius expands so that the linearized Bianchi-Davies conservation laws become more applicable. We see this in the Big-Bang model where, at about 10-9 seconds, the strong force is becoming less important and electroweak fields dominate except in the high curvature region of the baryons. At this time, the temperature is 1015 oK and the electroweak symmetry breaks into their separate components. At 10-3 seconds and 1014 oK temperature, quarks begin to condense into neutrons and protons. For the next 3 minutes, and subsequently, these protons and neutrons start to form nuclei. These nuclei are highly ionized and are opaque to light.


At about 300,000 years, and temperatures near 104 oK, these nuclei combined with electrons to form stable atoms that were comparatively unaffected by and thus transparent to the prevailing black-body photon radiation. We shall see in future chapters that the onset of production of stable massive atoms was a crucial time in the evolution of the Universe. This standard model of the Big Bang shows that time asymmetry governs Universe expansion of curvature manifesting as massive charged particles.


In the 3-dimensional explosion, the disclination formed at the immediate center by the explosion has certain curvature components. Conservation of curvature demands that the curvature sum of the ends of the radial cracks emanating from such a curvature intrusion must be of opposite sign but equal magnitude. This phenomenon has a similar 4-dimensional spacetime effect. There is a total preponderance of particles with respect to anti-particles that is observed in our expanding Universe. When pair-production occurs in the early Universe, both particle and its anti-particle are formed. In order for there to be an equal and opposite reaction in this pair-production, and thereby satisfy the conservation of curvature, the anti-particles contracted in spacetime as opposed to the observed particle expansion. Thus, from our viewpoint the original Big Bang singularity is the location where the missing antimatter resides. This concept agrees with Wheeler and Feynman's arguments that a particle in positive time is identical to an anti-particle in negative time.


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