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Space Time And Eternity
Contents
Introduction
This paper is the first of a series of three [in
need of much correction and editing], in which I explore the relationships
between: Impact, Force and Potential; The Principal of Least Action; Efficient
and Final Causes; Singularities in Newtonian Mechanics; Quantum Indeterminacy;
and human FreeWill. These papers are much more physics based and mathematical
than any I have previously posted on my Web Site. I beg the indulgence
of those readers for which these fields are foreign. I shall attempt to
make the issues intelligible. I have appended a Bibliography so that those
who are interested in acquiring a deeper familiarity with the issues here
discussed may have some idea how to set about doing so.
Some of these issues have already been addressed in earlier essays.
In particular, I have argued that:

Newtonian Causality is entirely compatible with Indeterminism.

Mechanical systems are frequently characterized by singular
behaviour.

This is generally masked by thermal
fluctuations and other forms of noise.

The Lagrangian Principal of Least
Action is the Metaphysicist's idea of "Final
Causation" realized in the Physical world.

Quantum
Mechanics adds to the indeterminacy of Nature, but only quantitatively.
In this paper, I shall sketch out the philosophical basis and conceptual
structure of space and time: the background to the physics that I discuss
in the subsequent papers.
The concept of space relies on two simpler ideas: that of position
and distance. Distance, itself, is a surprisingly complicated concept.
In the next few paragraphs I sketch how something resembling the common
place conception of space can be constructed from elemental mathematical
and philosophical ideas. My purpose in doing so is to elucidate the nature
of space and then time as a backdrop to a discussion of causation, determination
and willing.
Locality and Connectivity
A point is the potentiality for some
thing
to be. If some thing is at the point, it is
occupied or
full
and has a set of nonvanishing parameters or properties.
This set of properties entirely determine and are equivalent to the thing
that is located at the point. If no thing is at the point, it is
unoccupied
or empty and all its parameters or properties are null.
Consider a set of points organized such that for any point
belonging to this set, a few others are local to that point: related
to it by being its nearest neighbours. This means that there exists
a onemany operation, called connection which maps
that single point to a small subset of points. which are therefore
called its nearest neighbours. It proves to be possible to organize sets
of points such that if point Q is a nearest neighbour of point P then so
is P of Q, for all P and Q. Such sets of organized points are called networks.
They can be thought of as collections of points linked together by connections.
A regular space lattice is a
particularly simple type of network. In it, every point has exactly the
same number of nearest neighbours, and the local topology (the pattern
of interconnectedness of each point, or identifiable repeating set of points)
is identical. As a complication, it must be remarked that it is an elementary
result of crystallography that only certain regular space lattices are
compatible with isotropic linear macroscopic properties. The fact that
the vacuum is isotropic strongly suggests that its lattice is cubic: perhaps
close packed Face Centred Cubic. An alternative is that the vacuum is not
microscopically regular: but is rather an amorphous random glass.
While I favour this possibility, because it explains why we do not regularly
encounter crystallographic
defects such as: grain boundaries; dislocations; stacking faults and
skew dislocations in the vacuum, I shall not pursue it here.
Parity and Opposition
In a manifold of such points, for which the parity or reflection
operator, P, is defined, the number of nearest
neighbours is even: 2N. For any point P of a regular invirtable
lattice, each nearest neighbour, Q, can be paired with one other,
P{Q},
to which it is opposite. N straight
lines can now be defined, passing though each point. These identify
submanifolds,
which are constructed by stringing together sequences of opposite nearest
neighbours. Such straight lines either extend throughout the entire manifold:
never terminating (unless it is both finite in extent and bounded,
and then twice: on its boundary), or (if the mahifold is finite
and
unbounded) they close back in on themselves forming loops.
Translation and Directionality
In a manifold for which the translation operator is defined, the
pairs of opposite nearest neighbours of each point of a regular invirtable
lattice can be associated into N sets. Each of these sets defines a basic
lattice direction: so a 2N manifold has N lattice directions only.
It is possible to assign nearest neighbour connections to basic lattice
directions: such that no two straight lines oriented in the same basic
direction have any points in common. Such lines are said to be parallel.
Any basic direction for which there exists a straight line segment that
cannot be closed as a loop without that loop containing another straight
line segment parallel to it is said to be an independent direction.
The number of independent directions, D £
N, specifies the dimensionality of the manifold.
It may be that some dimensions are local only: folded up into tiny loops,
as is contemplated in String and Nbrane theory.
It may be that the dimensionality of a space varies from place to place,
even in a fractal manner. As a trivial example, consider a plane
that intersects a solid sphere. The total set of points defined by the
union of these two figures is 3D within the sphere (including the points
on the place within the sphere) and 2D within that part of the plane that
extends beyond the sphere.
Coordinates and Vectors
Any one point, O, can arbitrarily be taken as a point of reference:
the origin. Any point, P, can be joined to any other point by no
more than D straight line segments, each oriented along an independent
direction. This set of D independent directions is said to constitute a basis
for the Dspace. They span or exhaust
the Dmanifold. This means that any point, P, can then be identified in
terms of O and the number of interpoint connections that constitute each
of these D straight line segments. These D counts of connections are called
coordinates,
and P can be conveniently labelled as (O : P_{1},P_{2},...,P_{D}).
Directions other than lattice directions can be defined in terms of
linear combinations of lattice directions. This linear mapping, which specifies
a new set of directions in terms of an old set is called a coordinate
transformation. Any construct that transforms in the same manner as
a set of position coordinates, such that relative to any specific basis
set is called a vector. Obviously, positions are trivial vectors.
It may be thought that the microscopic space lattice we have
been discussing doesn't really exist, being only a conceptual convenience.Now,
this is not obviously true: it is quite arguable that space really is quantized,
just as the present discussion assumes. Nevertheless it seems fair to presume
that macroscopic Physics is insensitive to the specifics of any
such microscopic space lattice. In particular,
it is an empirical fact that space is locally both isotropic: has no favoured
direction, and uniform: does not obviously differ from place to place.
It follows that only numbers that are independent of the basis used (and
so invariant under coordinate transformation) can appear in formulae for
observables.
Multiplication and Tensors
Vectors can be written as either rows or columns of numbers. A column vector
can be transposed into a row vector and vice versa. The scalar
or dot product, K, between two column vectors, p
and q, is:
K = p . q
= (p^{~} ) ( q
) = S_{j
= 1,D} p_{j }q _{j}
Bold, struck through, letters signify vectors.
The symbol ~, signifies transposition:
a column being written as a row, or viceversa.
Suffices label coordinates.
This construct is numerically independent of any coordinate
transformation. Such single numbers are called scalars.
Matrix multiplication is a trivial extension of this rule:
[( A^{~}) ( B ) ]_{pq}
=
S_{j = 1,D}
A_{jp }B_{jq}
Bold, underlined, letters signify matrices.
An Rthrank tensor,
T
is defined as any linear operation defined for an R product domain of vectors
that yields a scalar:
K = S_{j1=1,D
;....; jR = 1,D} T_{j1....jR }p(1)_{j1
.... }p(R)_{jR}
If the construct A q itself transforms
as a vector: that is if (p^{~ }A q
) is a scalar, then
A is a second rank tensor. Note that
the operation of transposition is only defined
for objects of rank two or less.
Length and the Metric
Note that if the network of points
is thought of as embedded in an absolute frame of reference, straight
lines are not necessarily absolutely straight, and parallel directions
might not be absolutely parallel. This is because the space
lattice might be subject to local stretching and shearing. Nowhere
has it yet been asserted that the length of any connection between
nearest neighbours, P and Q: L[P,Q] exits: still less that it is
a fixed or transferable or universal quantity.
If L[P,Q] does exist, and is given by
L[P,Q] = Q[P]^{~}L[P]
Q[P]
where Q[P] is the vector from P to Q: the manifold
is called a linear vector space.
The tensor L[P] is called the metric of the space. In such a Ddimensional
linear vector space, it is possible: by use of a suitable linear
coordinate
transformation, to erect a set of D orthogonal
directions, such that at every point:
( e_{p}^{~} ) L[P]
(
e_{q} ) = S_{i=1,D
; j = 1,D} _{ }e_{ip} L_{ij}[P]
e_{jq} = L_{pp
}d_{pq}

p and q indicate the orthogonal basis vectors.

i and j signify the coordinates of these vectors (perhaps in some other
basis).

d_{pq}, the Kroneger Delta,

unless p = q:

in which case it takes the value "1".
In this basis, the metric is said to be diagonal. If L_{jj }is
independent of j, the metric is said to be isotropic.
If L is independent of position, the following quadratic
form is significant:
(p^{~} ) L (
q
) = S_{j
= 1,D} p_{j }L_{ij} q _{j}
Where p_{j} and q_{j} are integer counts of interpoint
connections along the D basis line segments used to link the points P and
Q to the origin, O. When the basis is orthogonal:
L(P,O) = [ ( p^{~})
L
( p ) ]^{½} = [ S_{i=1,D
; j = 1,D} p_{i
}L_{ij
}p _{j
}]^{½}
= [ S_{j=1,D}
p_{j
}L_{jj
}p_{j }]^{½}
is the length of the position vector p.
This implies that lengths along straight lines add. If L is not independent
of position, then a sum has to replace the simple product of integers:
L(Pn,O) = [ S_{j
= 1,n } L(P_{j},P_{j1}) ]^{½}
Grids and Gauges
As previously remarked, Physics
is presumed to be insensitive to the specifics of the space lattice. On
the other hand, the absolute frame of reference that may be pictured behind
the space lattice itself has no physical status. Hence, the formulae of
Physics must also be independent of any continuous or smooth
deformation (stretching or twisting) of the lattice. Such deformations
are called gauge transformations:
the idea being that the standards of measurement (the
gauges, such
as the "meter rule", "protractor", "atomic clock" or "gyroscopic compass")
might vary in their characteristics from place to place and time to time.
The fact that two meter rules agree whenever they are brought next
to each other, gives us no reason to assert that they are the same length
when they are
remote from each other. Similar considerations apply
to all other measuring instruments. Presently, it seems that Physics is
gauge
invariant. This means that its equations make no reference to any absolute
frame of reference, but only to measurements made with the aid of concrete
objects
substantiated on and defined in terms of the space
grid of points itself. This is a prime feature of Relativity Theory,
of which Maxwellian Electromagnetism is a subset.
Minkowskian spacetime
According to the formalism of Relativity, the three dimensions of space
and one of time together constitute a single fourdimensional manifold.
The three space dimensions can usefully be considered as forming a 3D subspace
and time a 1D subspace. The distances in the two subspaces are given
by s^{2 } = S_{j }(s_{j
}.
s_{j}), where "j" signifies 1...3, the three space coordinates
and (c.t)^{2} = s_{4 }. s_{4}, with "4" signifying
the time coordinate. The time dimension is not exactly equivalent to the
others. This is because the metric tensor, although diagonal, has a minus
sign in its timetime element. This means that the square of the fourdistance
or displacement is given by:
x^{2} = s^{2}  (c.t)^{2}

The square of a fourdistance is not necessarily positive, and can be zero.

If s^{2}  (c.t)^{2} is positive, the magnitude
of the fourdistance is given the value [ s^{2}  (c.t)^{2}
]^{½},
and the fourdistance is said to be spacelike, rather than
real.

If s^{2}  (c.t)^{2} is negative, the magnitude
of the fourdistance is given the value [ (c.t)^{2} 
s^{2} ]^{½},
and the fourdistance is said to be timelike, rather than imaginary.
An alternative formulation of relativity theory makes the metric the identity
matrix. In effect this removes it from the formalism. The cost is that
time must now be viewed as an imaginary variable, in order to reproduce
the minus sign that features in the formula for the square of the fourdistance,
x. This itself is anomalous, as generally speaking when vectors are allowed
to have complex components (in Hilbert
spaces, as feature prominently in Quantum Mechanics), the formula
for the norm (a generalization of length) is:
 p ^{2 } =
( p^{~} )( p* )
Note the additional complex conjugation operation, signified by
"*". This is included in the formalism precisely to make the norm both
real and positive definite.
Trajectories and Velocity
In Minkowskian spacetime, the notion of before
and after has much less significance than commonsense gives it. Certainly,
it is divorced from the notion of causation.
SpaceTime is radically symmetric and all points within it have an equal
status. There is no motive, within the formalism, for saying that a thing
at one place and time is contingent on other things (even the selfsame
thing) at earlier times. This conception: in which the Cosmos is altogether
given, fixed and selfconsistent when viewed from a Godlike extratemporal
perspective is called "The Block Universe".
The trajectory of a particle is a lifeline traced in
four dimensions. From the perspective of the formalism, "before and after"
is no more significant than "below and above". The sequence (ordered
set) of fourpoints that make up a lifeline do not necessarily generate
a sequence of time coordinates that monotonically increase: any more than
one might expect them to generate sequences of monotonic spatial coordinates.
Contingency is not lost in the Block
Universe. It emerges as the ordering of the set of fourpoints that make
up a lifeline, together with whatever interaction events: collisions
with other particles, that it features. No event can be without
events of previous ordering (if not earlier in time) being also:
this on the assumption that lifelines are unending.
The velocity of a particle (some
thing
at a point) is nothing more than the orientation
of its lifeline
with respect to the
temporal basis vector. A slowly moving particle has a lifeline almost
exactly aligned along the temporal basis vector, a relativistic (very quickly
moving) particle has a lifeline almost orthogonal to the temporal basis
vector. Note, however, that the minus sign in the metric tensor means that
no massive particle can ever have its lifeline exactly orthogonal to the
temporal basis vector.
Particles and Antiparticles
In relativistic quantum mechanics, particals can travel
backwards
in time[J. Powers "Philosophy and the New Physics"
(1982), 105]. All that this means is that sequentially passing from
one point to the next generates a decrementing sequence of times.
Such particles are more properly called antiparticles.
It should be noted that there is no obvious way to relate the sence of
"forward" or "next" of one lifeline to that of another independent lifeline.
Because of this, it seems to me that there are the following possibilities:

All lifelines are the same lifeline.

There is only OneThing.
[R. Feynman, R Leighton and M Sands "Lectures
on Physics, Vol II" (1964)].

The single lifeline of TheThing loops round and round in spacetime.

TheThing occasionally changes its mass, electric charge and other
parameters.

There is no ambiguity to resolve in the ordering of points along the lifeline
of TheThing.

The number of antiparticles must equal the number of particles.

There is more than one independent lifeline,
and:

Physics is independent of the sense of ordering
of each lifeline.

Antiparticles are practically indistinguishable from particles.

Physics is dependent on the sense of ordering
of each lifeline.
This ordering is fixed in some yet to be determined
and obscure manner.

Antiparticles are distinguishable from particles.

The number of antiparticles need not equal the number particles.

The transformation of Charge Conjugation, C,
in which particles and antiparticles are exchanged is entirely a matter
of geometry.

Charge Conjugation is indistinguishable from the operation of Parity reversal,
P.

Intrinsic Parity [L. Schiff "Quantum Mechanics" 3rd
Edition (1968), 226]
serves to label whether the conventional particle is travelling
forwards of backwards in time.

Physics is invariant under T,
the operation of time reversal.

Physics is C P invariant.

This is false, being slightly violated by the decay of Kmesons
[S. Hawking "A Brief History of Time" (1988),
77]

Physics is not exactly time symmetric.

Why should it be?

After all, the volume of the universe is expanding as time advances.

Physics is nevertheless C P T
invariant.

This is true. [S. Hawking "A Brief History
of Time" (1988), 78]
A particle trajectory might first move forward in time, then bend to travel
backward in time. There exists, therefore, a time after which the
particle does not exist. The bending would necessarily be associated with
a large transfer of energy and momentum between this life line and (an)other(s).
Experimentally, the two parts of the single trajectory would be seen as
two separate particles: the reverse time travelling part being the antiparticle
of the forward time travelling part. The bending back of the trajectory
would be seen as the annihilation
of a particleantiparticle pair with an associated release of energy. A
similar bending of a trajectory: for which there existed a time before
which the particle does not exist would represent the creation of a particleantiparticle
pair from nothing, with an associated absorption of energy.
The mass of a particle relates the temporal
curvatures of two interacting particle lifelines. In effect, particles
with different masses tradeoff space with time according to different
exchange rates.
The Lorentz Transformation
Relativistic theory tells us that coordinates in threespace grids travelling
uniformly relative to each other are related by a rotation in Minkowskian
fourspace. The rotation exchanges distance along the direction of motion
for time. It is specified by the formulae of the Lorentz transformation
[A. Einstein: "The Meaning of Relativity" 6th Edition (1956), 31].
An association similar to that of position and time exists between various
other physical observables. In each case, a Cartesian threevector is associated
with a scalar to constitute a Minkowskian fourvector, which then obeys
the Lorentz transformation. The most important examples are:

Momentum and MassEnergy.

The Magnetic Vector Potential and the Electric Scalar Potential.

Current density and Charge density.
The transformation of Momentum and MassEnergy is highly suggestive. It
is as though all objects "move" at the same speed,
that of light: but those that are perceived to be almost at rest with respect
to an observer are moving in a direction orthogonal to all three space
dimensions, that is along the time basis vector. No object with a nonzero
mass (a so called "massive" object) can have its fourvelocity entirely
aligned with a vector in the spatial subspace: some small temporal component
must always exist. The constant rate of "movement" is nothing more than
the magnitude (technically the square root of the trace of the square)
of the metric tensor: [ S_{j} (L_{jj})^{2
}]^{½}.
The Quantum World
The discussion so far has taken it for granted that the idea of a thing
is applicable to the real world: moreover that a thing exists at one point
at any one time, and at an adjacent point at the next moment of time. My
next
paper shows that it is impossible to sustain such a picture.
Bibliography

Undergraduate Physics

R.P. Feynman, R Leighton and M. Sands "Lectures on Physics" (1964)

K.R. Symon "Mechanics" (1971)

A. Einstein "The Meaning of Relativity" 6th Edition (1956)

Academic Philosophy

Plato "Timaeus"

K.R. Popper "Conjectures and Refutations" (1972)

Popular books on Physics, its Philosophy and related Theology

J. Powers "Philosophy and the New Physics" (1982)

S. Hawking "A Brief History of Time" (1988)

J.D. Barrow "Theories of Everything" (1990)

P. Davies and J. Gribbin "The Matter Myth" (1991)

P. Davies "The Mind of God" (1992)

P. Davies "God and the New Physics" (1993)

P. Landsburg "Seeking Ultimates: an intuitive guide to Physics"
(2000)
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