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To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.

(William Blake - Auguries of Innocence)

Complete Relativity


(The theory of the physical world)


Abstract


         This paper presents the basic structure of Complete Relativity, a theory that subsumes General Relativity. In this theory the fundamental form of the motion of matter, or ‘motion’ for short, consists of four pairs of eigen velocity vectors from the tangent vector space of a point in a base manifold of spacetime. The two vectors of a pair of eigen velocity vectors are orthogonal with respect to a general tensor the symmetric component of which is a pseudo-Riemannian tensor. Consequently, a fundamental feature of ‘motion’ is a unification of energy and an angular momentum due to the angular separation of the two vectors. The mean sum and the mean difference of the two vectors work out to a time-like and a space-like velocity vector. These two are orthogonal with respect to the pseudo-Riemannian tensor and together they constitute the observational form of the motion of matter. Of the four space-like velocity vectors, one is translational and the remaining three are rotational. Therefore, translation and rotation are integrated features of the motion of matter, the simplest form of which is a uniformly rotating circular ring as a 2-dimensional cylindrical embedding in Minkowski spacetime.


1 Introduction


         General relativity, which has integrated space and time in terms of a null-motion, has so far failed to achieve a similar integration of matter with spacetime. According to this paper, this failure is due to the primary concept of General Relativity being the base manifold of a tangent bundle rather than the tangent vector space. In Complete Relativity, a theory that reduces to General Relativity in empty spacetime, primary concept is the tangent vector space. The basic structure of this theory is the subject of this paper.


         Consider a tangent bundle with a base manifold of spacetime. A point P in the base manifold is the common point of intersection of world lines of which a minimum of two will determine P. Thus pairings of intersecting world lines in the base manifold can be considered as a fundamental feature of the tangent bundle. Next, consider that all the tangent vectors of the tangent vector space at P and the space of all derivatives along world lines at P are in a 1-1 correspondence (Ref. 1). Hence, because of the basic pairings of world lines at P, a corresponding pairings of tangent velocity vectors, say v and u, is characteristic of the tangent vector space at P. The concept of pairings becomes precise if v and u represent pairs of eigen vectors and, as such pairs do, satisfy the following Eqns (Ref. 2).


                                                        

In these equations ß is an eigen value, A is a tensor of type (1,1) and à is its transpose. In 4-spacetime the solution of these equations consists of 4 values of ß² and 4 corresponding pairs of eigen velocity vectors v and u. Thus, in general 4 pairs of eigen velocity vectors are present at P.


         Eigen values, ß², are real and both column vectors |v> and |u> can act as orthogonal sets of base vectors. Therefore, the four pairs of eigen velocity vectors also constitute two sets of mutually referencing vectors. Hence not only each pair of eigen vectors are mutually referencing according to Eqns. (1a) and (2a), but also the two sets of four vectors of the four pairs of eigen vectors are so also.


         Formulation of the pair of Eqns (1a) and (2a) is carried out in §2 where it becomes apparent that, along with a pseudo-Riemannian symmetric metric tensor that characterises the base manifold an antisymmetric tensor of the same type exists which tends to disappear as v and u tend to coincide. The tensor A in Eqns (1a) and (2a) are functions of these symmetric and antisymmetric tensors.


         Of the above two tensors, the antisymmetric tensor, because it depends on v and u existing as a pair, can act as a window to the basic structure of the 4 pairs of eigen velocity vectors v and u. To create this window, for convenience, let spacetime be confined to the flat infinitesimal neighbourhood of P complete with a Minkowski metric. Next let each of the elements of the antisymmetric tensor be represented symbolically by its corresponding pair of coordinates. If the coordinate system is (t, x ,y, z), then this symbolic representation of the antisymmetric tensor is as follows.


                                                             

Accordingly, the antisymmetric tensor has three principal spacetime cross-sections. These are { t, x - y, z }, { t, y- x, z } and { t, z - x, y } and each of these is able to stand on its own as a non-trivial antisymmetric tensor with a non-zero determinant. In addition, each cross-section supports a rotational motion of its own that corresponds to a translatory motion related to the Minkowski metric. For example in the { t, x - y, z } cross-section the rotational motion is in the {y, z} plane with its axis of rotation as the x-axis. The corresponding translational motion related to the Minkowski metric is in the x -direction.


         When the antisymmetric tensor, complete with the three cross-sections, form a general tensor with the pseudo-Riemannian symmetric tensor, the three translational motions become compounded into a single translational motion. A similar compounding is not possible with the three rotational motions and they remain as three suitably modified rotational motions according to prevailing physical conditions. A good example is the motion of the Earth around the sun. Any point P in the Earth has a translatory motion around the sun. It also has two corresponding rotational motions one of which has a rotational period of about 24 hours, and the other, the precession of the equinoxes, has a rotational period of about 25,920 years. Perhaps a third rotational motion with a longer period exists which is yet to be discovered.


         The fundamental forms of the above four observable motions, one of translation and three of rotation, are what the four pairs of eigen velocity vectors represent. In §3 the mean sum and the mean difference of a pair of eigen velocity vectors work out to a time-like velocity and a space-like velocity that are orthogonal with respect to the pseudo-Riemannian symmetric metric tensor and they are the observable motions that correspond to the pair of eigen velocity vectors. Just like the latter, they are also meaningful only when together.
Of the four space-like velocity vectors, one is translational and the remaining three are rotational. Therefore, translation and rotation are integrated features of the motion of matter,


         Accordingly, in Complete Relativity, the point-particle form of matter, which has no rotational velocity but possesses still an angular momentum of magnitude 1/2, corresponds to the state of degeneration of the concept of a pair of eigen velocity vectors to the concept of a single translational velocity vector which in turn reduces the corresponding space-like velocity to the zero-vector.


         The full participation of all four pairs of eigen velocity vectors not only in the motion of fundamental particles of matter in the sub atomic region but also in the motion of material bodies in the cosmic region is likely to be the ultimate reality of the physical world. In that case Complete Relativity has the capacity to integrate the two extremes of the physical world, atomic and cosmic.


         The symmetric and antisymmetric tensors, outlined above, form a general tensor with respect to which the two vectors of a pair of eigen velocity vectors become orthogonal to each other. Because of this orthogonality, energy, and an implicit angular momentum associated with the angular separation of the two velocities, becomes equal in magnitude and opposite in sign causing their sum to become null in perpetuity. This state of nothingness in perpetuity of a pair of eigen velocity vectors has the potential to integrate matter and spacetime in terms of, once again, a null motion that subsumes the null-motion of General Relativity. In this paper on just the basic structure of Complete Relativity, suffice here to say that if the antisymmetric tensor corresponds to matter, then the general tensor corresponds to a state of integration of matter and spacetime.


         The field equation that the symmetric and antisymmetric tensor fields satisfy has been formulated separately in a follow up paper along with an application of this field equation that yields a structure of dark matter the predictions of which match observations perfectly.

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