to Philosophical Primer
Space Time And Eternity
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 Free-Will. 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:
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.
Newtonian Causality is entirely compatible with Indeterminism.
Mechanical systems are frequently characterized by singular
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.
Mechanics adds to the indeterminacy of Nature, but only quantitatively.
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
Locality and Connectivity
A point is the potentiality for some
to be. If some thing is at the point, it is
and has a set of non-vanishing 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
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 one-many 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
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,
to which it is opposite. N straight
lines can now be defined, passing though each point. These identify
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
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.
Co-ordinates 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 D-space. They span or exhaust
the D-manifold. This means that any point, P, can then be identified in
terms of O and the number of inter-point connections that constitute each
of these D straight line segments. These D counts of connections are called
and P can be conveniently labelled as (O : P1,P2,...,PD).
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 co-ordinate
transformation. Any construct that transforms in the same manner as
a set of position co-ordinates, 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 co-ordinate transformation) can appear in formulae for
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,
and q, is:
This construct is numerically independent of any co-ordinate
transformation. Such single numbers are called scalars.
p . q
= ( p~ ) ( q
) = Sj
= 1,D pj q j
Bold, struck through, letters signify vectors.
The symbol ~, signifies transposition:
a column being written as a row, or vice-versa.
Suffices label co-ordinates.
Matrix multiplication is a trivial extension of this rule:
[( A~) ( B ) ]pq
Sj = 1,D
An Rth-rank tensor,
is defined as any linear operation defined for an R product domain of vectors
that yields a scalar:
Bold, underlined, letters signify matrices.
K = Sj1=1,D
;....; jR = 1,D Tj1....jR p(1)j1
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
Q[P] 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 D-dimensional
linear vector space, it is possible: by use of a suitable linear
transformation, to erect a set of D orthogonal
directions, such that at every point:
In this basis, the metric is said to be diagonal. If Ljj is
independent of j, the metric is said to be isotropic.
ep~ ) L[P]
eq ) = Si=1,D
; j = 1,D eip Lij[P]
ejq = Lpp
p and q indicate the orthogonal basis vectors.
i and j signify the co-ordinates of these vectors (perhaps in some other
dpq, the Kroneger Delta,
unless p = q:
in which case it takes the value "1".
If L is independent of position, the following quadratic
form is significant:
Where pj and qj are integer counts of inter-point
connections along the D basis line segments used to link the points P and
Q to the origin, O. When the basis is orthogonal:
(p~ ) L (
) = Sj
= 1,D pj Lij q j
L(P,O) = [ (
is the length of the position vector
( p ) ]½ = [ Si=1,D
; j = 1,D pi
= [ Sj=1,D
pj ]½ 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) = [ Sj
= 1,n L(Pj,Pj-1) ]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
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
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
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 sub-set.
According to the formalism of Relativity, the three dimensions of space
and one of time together constitute a single four-dimensional manifold.
The three space dimensions can usefully be considered as forming a 3D sub-space
and time a 1D sub-space. The distances in the two sub-spaces are given
by s2 = Sj (sj
sj), where "j" signifies 1...3, the three space co-ordinates
and (c.t)2 = s4 . s4, with "4" signifying
the time co-ordinate. The time dimension is not exactly equivalent to the
others. This is because the metric tensor, although diagonal, has a minus
sign in its time-time element. This means that the square of the four-distance
or displacement is given by:
x2 = s2 - (c.t)2
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 four-distance,
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:
The square of a four-distance is not necessarily positive, and can be zero.
If s2 - (c.t)2 is positive, the magnitude
of the four-distance is given the value [ s2 - (c.t)2
and the four-distance is said to be space-like, rather than
If s2 - (c.t)2 is negative, the magnitude
of the four-distance is given the value [ (c.t)2 -
and the four-distance is said to be time-like, rather than imaginary.
Note the additional complex conjugation operation, signified by
"*". This is included in the formalism precisely to make the norm both
real and positive definite.
p ||2 =
( p~ )( p* )
Trajectories and Velocity
In Minkowskian space-time, the notion of before
and after has much less significance than common-sense 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 self-same
thing) at earlier times. This conception: in which the Cosmos is altogether
given, fixed and self-consistent when viewed from a God-like extratemporal
perspective is called "The Block Universe".
The trajectory of a particle is a life-line 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 four-points that make up a life-line do not necessarily generate
a sequence of time co-ordinates that monotonically increase: any more than
one might expect them to generate sequences of monotonic spatial co-ordinates.
Contingency is not lost in the Block
Universe. It emerges as the ordering of the set of four-points that make
up a life-line, 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 life-lines are unending.
The velocity of a particle (some
at a point) is nothing more than the orientation
of its life-line
with respect to the
temporal basis vector. A slowly moving particle has a life-line almost
exactly aligned along the temporal basis vector, a relativistic (very quickly
moving) particle has a life-line 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 life-line exactly orthogonal to the
temporal basis vector.
Particles and Antiparticles
In relativistic quantum mechanics, particals can travel
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 anti-particles.
It should be noted that there is no obvious way to relate the sence of
"forward" or "next" of one life-line to that of another independent life-line.
Because of this, it seems to me that there are the following possibilities:
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 particle-antiparticle 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 particle-antiparticle
pair from nothing, with an associated absorption of energy.
All life-lines are the same life-line.
There is only OneThing.
[R. Feynman, R Leighton and M Sands "Lectures
on Physics, Vol II" (1964)].
The single life-line of TheThing loops round and round in space-time.
TheThing occasionally changes its mass, electric charge and other
There is no ambiguity to resolve in the ordering of points along the life-line
The number of antiparticles must equal the number of particles.
There is more than one independent life-line,
Physics is independent of the sense of ordering
of each life-line.
Antiparticles are practically indistinguishable from particles.
Physics is dependent on the sense of ordering
of each life-line.
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
Charge Conjugation is indistinguishable from the operation of Parity reversal,
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 K-mesons
[S. Hawking "A Brief History of Time" (1988),
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
This is true. [S. Hawking "A Brief History
of Time" (1988), 78]
The mass of a particle relates the temporal
curvatures of two interacting particle life-lines. In effect, particles
with different masses trade-off space with time according to different
The Lorentz Transformation
Relativistic theory tells us that co-ordinates in three-space grids travelling
uniformly relative to each other are related by a rotation in Minkowskian
four-space. 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 three-vector is associated
with a scalar to constitute a Minkowskian four-vector, which then obeys
the Lorentz transformation. The most important examples are:
The transformation of Momentum and Mass-Energy 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 non-zero
mass (a so called "massive" object) can have its four-velocity entirely
aligned with a vector in the spatial sub-space: 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: [ Sj (Ljj)2
Momentum and Mass-Energy.
The Magnetic Vector Potential and the Electric Scalar Potential.
Current density and Charge density.
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
paper shows that it is impossible to sustain such a picture.
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)
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"
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