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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)

Extended Relativity


(The theory of the physical world)


INTRODUCTION


      Theoretical physics came into its own with Newton’s discovery of the three laws of motion and the law of universal gravitation and eventually this fledgling physics became known as Newtonian Mechanics. In 1697, Newton published a detailed exposition of his laws and their applications in a book titled ‘Philosophiae Naturalis Principia Mathematica’ (or simply The Principia). Of this book it can be justly said that, with the possible exception of Euclid’s ‘Elements’ and the English naturalist Charles Darwin’s ‘The Origin of Species’, no greater scientific work has been produced by the human intellect, before or since.


      In a ‘Scholium’ at the beginning of The Principia Newton postulates the existence of an absolute space and an absolute time. A paraphrasing of these two postulates is as follows.


      Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.


      Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external.


      These postulates of Newton held their ground for over 200 years. Then in 1905, Einstein published a paper titled ‘Zur Elektrodynamik bewegter Körper‘(On the Electrodynamics of Moving Bodies) based on two postulates, English translations of verbatim statements of which are as follows.


      The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.


      Any ray of light moves in the ‘stationary’ system of co-ordinates with the determined velocity c, whether the ray is emitted by a stationary or by a moving body.


      Earlier in the paper, Einstein had defined ‘stationary’ system in the second postulate as a system of coordinates in which the equations of Newtonian mechanics hold good to the first approximation.


      Einstein was unaware that his paper had the potential to reveal the true nature of space and time. However, Minkowski, Einstein’s former professor, not only saw this potential but also realised it and laid the foundation for the future of physics when he deduced that, according to Einstein’s paper, space and time do not exist independently as Newton had imagined but exist only in a state of integration as a continuum of 4-dimensions. Minkowski announced this momentous discovery of his on September 21, 1908 at the 80th Assembly of German Natural Scientists and Physicians as follows.


      The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, is doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.


      Eventually assisted by efforts made by notably Lorentz and Planck, Einstein’s two postulates led to Special Relativity according to which the following are true of the nature of space and time.


      (a) Space and time exist in a fundamental state of integration, often referred to as spacetime, in the form of space-like and time-like regions that join at an interface occupied by the motion of light in which space and time unify completely.


      (b) In the time-like region, where speeds at which information on material phenomena travel are subluminal, if two phenomena are linked causally with respect to one observer, they are also linked causally with respect to all other observers. However, in the space-like region where speeds are superluminal, if a causal link exists between two phenomena with respect to one observer, they may not be linked causally with respect to all other observers and hence paradoxical situations may occur.


      (c) The two postulates of Newton on space and time, stated earlier, are applicable only to those physical phenomena that involve matter moving at speeds negligible in comparison to the speed of light.


      Special Relativity is limited only to 4-dimensional Minkowski spacetime which is comparable to a flat surface as opposed to a spherical or any other topologically curved surface. In 1915, Einstein formulated General Relativity which is a geometrical theory of curved spacetime that reduces to Special Relativity for flat spacetime or for any sufficiently small region of curved spacetime. In General Relativity, a symmetric metric tensor, which is a cluster of ten numbers in 4-dimensional spacetime, replaces the single number of the Newtonian gravitational potential. The resulting symmetric metric tensor field, just like the Newtonian gravitational potential field, is a function of the mass distribution in spacetime. Einstein was able to establish that the curvature of spacetime as determined by the symmetric metric tensor field is the cause of all physical phenomena of purely gravitational origin. However, General Relativity, in its fundamental form applies only to empty spacetime. More precisely, General Relativity has just one fundamental field equation conjectured by Einstein and it determines the symmetric metric tensor field only in spacetime empty of matter.


      The field equation in General Relativity, mentioned above, has produced two exact solutions. Schwarzschild found the first of these and it corresponds to a spherically symmetric massive object at spatial rest, Kerr found the second and it corresponds to an axially symmetric massive object rotating uniformly in space. In each of these two cases, a spherical spatial surface co-centric with the massive object, called event horizon, exists at which the fields can become singular depending on the system of coordinates used. In addition, the roles of radial and time coordinates in a spherical polar coordinate system, or their equivalents in some other coordinate system such as that formulated by Kruskal and Szekeres in respect of the Schwarzschild symmetric metric tensor field, reverse their roles on one side of the event horizon in comparison to their roles on the other side.


      Singularities at the event horizons, mentioned above, are of a purely coordinate origin. Therefore, they may be deemed physically inconsequential. However, reversal of the roles of space and time coordinates, or their equivalents, mentioned above, is not of coordinate origin. Therefore, the event horizon may be interpreted as an interface which maintains physical continuity between two regions of spacetime which could, however, be fundamentally different from each other. That the region outside the event horizon has the nature of an empty spacetime stretching to infinity has been well established, but what of the region inside the event horizon? That this region is non-empty and fundamentally different from the region outside the event horizon would be established in due course in this book. In the meantime, it would be assumed as a temporary measure that the region inside the event horizon is empty of matter and hence in this region Schwarzschild and Kerr solutions remain valid.


      Schwarzschild symmetric metric tensor field expressed in terms of spherical polar coordinates has a singularity that occurs at the centre of the system of coordinates which is also the centre of the massive body. According to the corresponding Schwarzschild line element, this singularity is such that at this centre time stands still. Therefore, this centre has the character of a point frozen in time, or pure spatial point. In other words, at the centre of the system of coordinates where the spherically symmetric object may be found as a point-particle, space and time separate and lose their integrity as a continuum of spacetime.


      Kerr symmetric metric tensor field, expressed in terms of spherical polar coordinates, has a singularity that occurs at the centre of the equatorial plane of rotation. However, Kerr line element, unlike Schwarzschild line element, places no restrictions on any of the physical dimensions anywhere in spacetime. In other words, Kerr line element preserves spacetime integrity, particularly at the centre of the spherical coordinate system where the axially symmetric rotating body may be found as a point-particle. A reasonable inference that can be drawn from this comparison between Schwarzschild and Kerr line elements at the point of occupation by a particle of matter is that linear and rotational motions are a fundamental feature of the motion of even a point-particle of matter. Thus, it seems that General Relativity, in spite of being a theory of only empty spacetime, is able to provide fundamental information on the nature of a point-particle of matter, albeit under limiting circumstances of the Schwarzschild and Kerr line elements, rather like through a keyhole.
Linear and rotational motions occur together naturally in the cosmic region of the physical world since, as a rule, a cosmic body moves with a linear velocity as a whole whilst also rotating about an axis passing through its centre of mass. The rotational motion with its direction preserving quality in this case is merely the result of the finite size of the cosmic body brought about by a complex assembly in spacetime of a vast number of fundamental particles of matter.


      This book presents a theory of the physical world according to which the directional invariance of the axis of rotation of the rotational motions of a cosmic body, mentioned above, is a reconstruction in the time-like region of spacetime, of a fundamental rotational motion of matter which takes place in the space-like region of spacetime, a region that is at present inaccessible to physics. This comparison of the fundamental space-like rotational motion of matter with the time-like rotational motion of a cosmic body is in principle the same as Rutherford’s comparison of an atom with the solar system. Both these comparisons reflect a primal unity that exists between the cosmic and atomic extremes of the physical world, or more precisely between the whole of a cosmic body and its smallest constituent part, all of which are profound illustrations of Newton’s vision that nature is very consonant and conformable to herself.


      The theory begins in Chapter 1 with a formulation of the fundamental form of the motion of matter as a state of integration between two components, the one linear and the other rotational, which remain confined to time-like and space-like regions of spacetime. Due to the intrinsic nature of the space-like region, the rotational component of motion occurs at speeds that are superluminal, but principal tenets of relativity are not violated, as these superluminal speeds remain confined to the space-like region only. Owing to the presence of this superluminal rotational motion in the space-like region of spacetime as a fundamental feature of the physical world, the complete arena of space-time becomes accessible to physical phenomena. Consequently, the existing loophole in physics, which renders the space-like region of spacetime superfluous to physics and thereby prevent General Relativity from reaching its full potential, is now closed.


      In Chapter1 it would become clear that the time-like linear motion and the space-like rotational motion, discussed above, are orthogonal with respect to the symmetric metric tensor field which characterises General Relativity. During the course of this book it would become clear that just this state of orthogonality between the linear and rotational motions of the fundamental form of the motion of matter is all that is needed to extend general relativity to the state of a unique theory of the physical world that retains all that has been well established not only in general relativity, but also in quantum mechanics. Henceforth in this book, this theory will be referred to as Extended Relativity or ER for short

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