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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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