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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 *Ã* 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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