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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 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:

  1. Newtonian Causality is entirely compatible with Indeterminism.
  2. Mechanical systems are frequently characterized by singular behaviour.
  3. This is generally masked by thermal fluctuations and other forms of noise.
  4. The Lagrangian Principal of Least Action is the Metaphysicist's idea of "Final Causation" realized in the Physical world.
  5. 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 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 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 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 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.

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 co-ordinates, 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 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 )  =  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.
    • This construct is numerically independent of any co-ordinate transformation. Such single numbers are called scalars.

    Matrix multiplication is a trivial extension of this rule:

    [( A~) ( B ) ]pq  = Sj = 1,D  Ajp Bjq
     
  • Bold, underlined, letters signify matrices.
    • An Rth-rank tensor, T is defined as any linear operation defined for an R product domain of vectors that yields a scalar:
    K  =  Sj1=1,D ;....; jR = 1,D    Tj1....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 D-dimensional linear vector space, it is possible: by use of a suitable linear co-ordinate 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.

    If L is independent of position, the following quadratic form is significant:

    (p~ ) L ( q )  =  Sj = 1,D  pj  Lij q j
    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:
    L(P,O)  =  [ ( p~) L ( p ) ]½  =  [ Si=1,D ; j = 1,D  pi Lij p j ]½  = [ Sj=1,D   pj Ljj pj ]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)    =  [ 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 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 sub-set.

    Minkowskian space-time

    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: 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:
    || 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 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 thing 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 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 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.

    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 exchange rates.

    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 ]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

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