Links to sections:

- All the world's a kinema.
- Time and motion
- The shape space of triangle land.
- Shortest paths in triangle land.

Suppose the audience of a picture house are immortal souls and that the happenings on screen relate to this world of ours. A holographic movie would be even more like three-dimensional reality. The audience of souls become so absorbed in the goings-on, on screen, that they forget themselves and identify with the actors, some more than others, and perhaps one in particular, who becomes one's ( mortal ) self. Consciousness has shifted from timelessness to time.

Our holographic lives, in the kinema, are apparently kinematic or moving in time. But God, the great movie maker, knows better. A movie reel is made up of a lot of static images. The projector runs them too fast for us to see the jumps between them, giving an impression of flowing motion. God's cuttings floor is strewn with rejected images, immensely more than left out by any human director, because the divine director works on a grand scale.

This stupendous totality of imaginary realities, from which a miniscule number
are selected to become our conscious reality, is called Platonia by Julian
Barbour.

This is after Plato's belief in an underlying reality of perfect forms, to which our world only approximates. We are likened to cave-dwellers, round a fire, who see only shadows of a real world outside.

Platonia is a timeless jumble of practically infinite possibilities. The
images most likely to come together in an appearance of time's motion are the
kinematic slides best matched to each other.

If you cut a movie reel into its individual slides, you could put them back
together in sequence by comparing those which were best matched to run
continuously. In platonia, tho, there is an embarrassment of choice from every
conceivable possibility of image, tho the vast majority are so ill-matched
that they are easily dismissed from any probable historical sequence.

The explanatory success of quantum theory has made physicists take seriously the notion that there is a graded potential for all logical possibilities of existence to become reality. Barbour makes the point that these possibilities could include unimaginable heaven, purgatory and hell. ( He has a pantheistic belief in platonia, rather than in a personal god. )

Julian Barbour's 'The End Of Time' ( 1999 ) is about how such a timeless nature of reality might work. It is about physics not the metaphysics of God and immortal souls. Tho not Barbour's view, it is perhaps worth mentioning that karma, for example, is a sort of probability theory of reincarnation, whereby the moral conditions of souls best-matches them to a succession of mortal bodies.

I attempt to explain Barbour's book to myself, but dont pretend to fully understand it, and apologise to readers for errors or infelicities. Barbour gives us passengers a privileged tour of the engine room to the ship of physics. This review is meant to be no more than a guide to a guide.

Julian Barbour is a physicist, who forsook main-stream university life, to concentrate on the meaning of time. He decided, as a young man, that time could be reduced to terms of 'movement'. This was the motive for, to quote the sub-title of his book: The next revolution in our understanding of the universe.

This is rather as if a clock was considered not as telling 'the time' but as its 'movement', which is the name for the mechanics of a clock. Mechanical toy trains were described as ( running on ) 'clock-work'.

The heavenly bodies regulate living bodies, so that they have their own 'biological clocks'. Human beings even regulate themselves with an abstract concept of time. Where does this notion come from? And from what does time ultimately derive?

Astronomers used the earth's rotation for their clock. About the turn of the twentieth century, they found earth's rotation rate was not quite regular enough, because of the moon's gravitational effects. The earth may be put out of sync by the moon. But it is possible to take the sun and its planets, as a whole, as a more regular clock-work, because they are isolated from any such intrusive influences.

( This change to a newer heavenly time-piece may be compared to man's making a more regular mechanism by designing a longer train of cog-wheels, that slows down the full force of the main-spring to unwind more gradually and uniformly. That way, the clock is less liable to gain too much when fully wound up and to lose too much as it runs down. )

So, the solar system was adopted as a more regular natural clock. This is called ephemeris time. Ephemeris tables give the positions of the planets at given times. The most convenient planetary pointer for so telling the time, but not the most accurate, is the moon. Thus, time is a convention that depends on the most regular available time-piece.

Ephemeris time was soon superceded by a convention based on atomic periodicity. ( On the very small scale of the atom, gravitational disturbances are negligible. ) But it remains a pointer in Julian Barbour's quest for the true nature of time. The solar system is a little universe in itself. Indeed, by the start of the third millenium, man's technical ability scarcely reaches to its limits. Barbour suggests that if time is more accurately measured by a graduation from diurnal time to ephemeris time, then the ultimate time-piece is nothing less than the universe itself.

Mach's principle says time only makes sense in terms of motion and motion is relative to the motions of all the masses of the universe, on which time, therefore, ultimately depends.

Universal gravity is its 'main-spring' which spins whole galaxies in its train. ( The actual train of the stellar clock-work mechanism, or which cogs connect with which, also might be used as a fanciful analogy to Barbour's idea of 'best matching'. )

By definition, the universe is everything that there is. It would be
illogical to think of a god, outside it, timing its run, with a stop-watch.
But that is essentially what the notion of 'absolute time' is, in classical
physics. Barbour wishes to promote Ernst Mach's belief in a self-referential
universe, from which time is a convenient derivation but no more than that.
The very convenience of the concept of time may make it a most powerful
illusion. But basically time is just an illusion, Barbour thinks. Hence, the
title of his book.

The obvious precedent for this way of thinking is how physics over-turned the
convenient fiction that the sun travels round the earth, rather than vice
versa.

Barbour sets out to show how the notion of time may emerge from a universe considered merely as the relations between all its objects. To do this, he imagines a model of the simplest of universes, consisting of three objects. Three objects have a triangular relationship and he calls this universe 'triangle land'. If these are massive objects, their relations will be governed by the law of gravity.

But there is no-one running a stop-watch to gauge the time it takes for the
three bodies to change their positions relative to each other. There is no
absolute time. Nor is there an 'absolute space': these bodies are *not*
spatially measured with respect to the sides of a box of co-ordinates, the
length, breadth and height of some cube-shaped room.

Barbour's triangle land is a universe sufficient to itself. So, it must fashion any co-ordinates, such as it may have, in its own terms. In practise, this means taking the three bodies in one configuration at a time. That is one differently shaped triangle at a time, with the bodies at its three corners. Barbour calls each of these triangles 'time-instants'. They are like photographs or snaps of the bodies at one instant of time.

All the possible configurations for three bodies can be represented by all
the possible shapes of a triangle. The general geometry of triangles is, in
effect, the structure of the space, or the 'configuration space', within which
these three bodies must move.

For the possible relations between four bodies,
the geometry of a tetrahedron land, in six dimensions, would apply,
analgously. And so on, for greater numbers of bodies, in hugely
multi-dimensional configuration spaces.

Happily, the simple 'universe', of positions for three bodies, can be visualised in three dimensions. Their three co-ordinates are AB, BC and CA. These represent the lengths of each side of a given triangle. Each triangle is then pin-pointed accordingly within this box of co-ordinates. One corner of this box is taken as the origin, meaning the point representing a triangle, all three of whose sides are zero. This is the unique point at which all three bodies, configuring a triangle, meet. Barbour suggestively calls this the alpha point.

From that corner, the three room-edges of length, breadth and height extend. A 'triangle' pin-pointed exactly on co-ordinate AB, BC or CA is just a greater or shorter line AB, BC or CA, respectively.

The geometric properties of triangles limit the area of the box that may be
filled with positions for possible triangle shapes. One triangle side cannot
be longer than the sum of the other two. This limitation removes some points in the
box as possible positions to represent triangles. The borders of this limit
are described by a regular three-cornered pyramid, whose apex is at the origin
of the box.

The pyramid's three edges, to its base corners, extend at 45 degrees to the
three co-ordinates AB, BC, CA. These pyramid-edges represent triangles, two of
whose three corners coincide, one of their sides being zero length. This stands
for two out of three bodies meeting. They are the next most unique positions
to the origin or alpha point. ( Diagrams are given in Barbour's book! )

It does not matter how far the edges of the pyramid extend or how large is the base triangle of the pyramid. You can cut the pyramid into triangular cross-sections. Each cross-section is, in effect, a more or less broad base to the pyramid. These cross-sections all reveal the same pattern of information about the geometrical nature of the triangles that all the positions on their surfaces represent. These cross-sections are called 'shape space'. Independently of scale, shape space contains positions for every possible shape of triangle.

Notice that the geometric meaning, of the pyramid's edges, is retained in
the cross-sections of shape space, considered as the base corners, that the
edges extend-to from the pyramid apex. The base lines of the pyramid, which
are also the sides of shape space, are the positions for all triangles, in
which the length of one side equals the sum of the other two sides.

Also, the very centre of each cross-section is the one position in shape space
marking an equilateral triangle. It is perpendicular to the pyramid apex,
where three bodies are zero distance from each other. The apex or alpha point
is like an equal-sided triangle, where all three sides are zero length.

( A potential energy contour map can be drawn over shape space. Potential energy is inversely proportional to separation, so the potential energy rises like canyon walls over the shape space's three corners, each representing two of the three masses coinciding. Indeed, the potential energy only depends on the relative configuration of masses. It is independent of a frame-work of absolute space and absolute time, and so is a suitable measure of changes from place to place, in platonia, as mentioned, below. )

A fuller analysis of the shape space of triangle land revealed the
following. Three lines, that cross at the centre to make perpendicular
bisections with the three sides of each cross-section, mark the only positions
for isosceles triangles ( where two of the sides are equal ).

It turned-out that right-angled triangles' positions were found only on three
lines concave to the three sides of shape space. The three 'lens' areas, in
between, represented positions for triangles with an obtuse angle ( greater
than 90 degrees ). The remaining central area of shape space designated
acute-angled triangles: all their angles less than 90 degrees.

The moral is that even the simplest of platonias or relative configuration spaces, consisting of three bodies' possible positions, has a geometric structure. Whereas, absolute space is treated as absolutely uniform. No point in absolute space is regarded as different from any other. It is essentially a transparent abstraction. Julian Balfour believes that the extremely complex structure of a platonia of the real universe will be shown to guide the apparent 'arrow of time', which gives us the impression of being caught on a present flow of time out of the past into the future. But the mathematical demonstration of Barbour's conjecture is liable to be extremely difficult.

The notes of his book mention collaboration on a dynamic geometry to remove completely the concept of absolute distance. This would be analgous to the way shape space removes over-all scale from triangle land. But, as well as that, the ratios of lengths of sides would no longer be relevant. ( Barbour's web sites promised news: www.julianbarbour.com or platonia.com ).

We must remember that triangle land has no outside observer timing a sequence that these triangles might make, as a means to determining the distances between them. The so-called 'relative configuration space', that makes up triangle land, consists of nothing more than a jumble of triangles. 'Platonia' is Barbour's name for all such relative configuration spaces for any given number of bodies, up to the totality of the universe itself.

But it is still possible to measure a 'distance', in platonia, between neighboring triangles, without reference to an outside frame-work.

( The distance measure for triangle land, used in 'The End Of Time', is
like Pythogoras' theorem in three dimensions, but with the hypotenuse
transformed into the platonia 'distance' and the three perpendicular sides of
a three-dimensional triangle transformed into the distances, AA*, BB*, CC*,
between the respective corners of two triangles in triangle land.

If the three bodies, at the corners, have different masses, this may be
allowed-for by weighting this calculation.

In other words, if *a, b, c,* are the masses of three objects, in two
different triangular configurations, ABC and A*B*C*, then the platonia
distance, squared, equals *a* times (AA*) squared plus *b* times
(BB*) squared plus *c* times (CC*) squared. )

Any two such triangles can be moved relative to each other, so that their distance, represented by neighboring points in platonia, is at a minimum. This minimum distance is termed their 'intrinsic difference' and, as such, is said to represent their best-matching position. ( The formula for intrinsic difference is modified by a function of potential energy, actually the square root of minus the potential, according to Barbour. But this modification, 'the action', does not affect the argument. ) It can be found for any two distributions of matter, such as astronomers might observe, at any time, in any arbitrary relation to each other.

These shortest paths, in platonia, are like the geodesics in general relativity. In that theory, light appears to bend or curve under gravitational attraction, because it is actually following a shortest path, determined by the geometrical curving of space around gravitational masses. Given that platonia can have its own geometrically determined 'lines of least resistance' or geodesics, these are the most probable paths for a configuration of masses, plotted in platonia.

Richard Lung.

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