Back to part one.

Links to sections:

- Deriving Newton's laws from Mach's principle.
- Prominent role of time in Relativity.
- A timeless geometric dynamics within general relativity.
- Quantum gravity's conflict over time.

Bruno Bertotti and Julian Barbour constructed a theory of geodesics to determine the shortest path between any two fixed points in platonia. They found that the unique history this produced, from an initial point and direction, corresponds to one of many such histories in Newton's frame-work of absolute space and time. This correspondence was to the special case of a Newtonian history, with zero energy and angular momentum, which solves in terms of 'a simpler timeless and frameless theory.'

Barbour and Bertotti produced the mathematics that put
Newton's paradigm in the context of Mach's principle.

Assuming the universe is finite, the total energy and angular momentum of its
sub-systems add up to zero. But this will not be true of most sub-systems,
themselves. So, they will produce the much more common Newtonian solutions for
galaxies or solar systems with non-zero energies and angular momentums, as if
they were in absolute space and time.

Even to his contemporary, Leibniz, Newton's absolute space and time seemed a cumbersome frame of measurement. In determining the evolution of a configuration of, say, three masses, from its initial conditions and direction to a second configuration, fourteen dimensions ( allowing fourteen 'degrees of freedom' ) are involved. But only four of them make any difference to the result.

The ten absolute dimensions that make no difference are: three spatial dimensions each for the first and second configurations, from the points of view of their centres of mass; the starting time, another dimension; and the three-dimensional orientation of the first triangle.

The four remaining dimensions that do matter are the orientation of the
second triangular configuration and the absolute time elapsing between the two
configurations' positions, or to put it another way, the angular momentum and
the kinetic energy, respectively. The angular momentum of the three-mass
configuration is like a spinning top with an imaginary pivot, thru its centre
of mass, pointing skywards over two dimensions, with a third dimension from
the axial rotation of the triangle perpendicular to the pivot.

Spiral galaxies and the rings of Saturn are spectacular astronomical examples
of angular momentum.

In relative configuration space or platonia, the process of best matching, from one configuration to another, creates a determinate relation, of all the configurations to the first chosen configuration, which gives an appearance of a rigid frame-work, like absolute space and time.

Barbour completes a derivation of Newton's world view from Mach's, with respect to the 'spacings in time':

In the equations that describe how the objects move in the framework built up by best matching, it is very convenient to measure how far each body moves by making a comparison with a certain average of all the bodies in the universe. The choice of the average is obvious, and simplifies the equations dramatically...It is directly related to the quantities used to determine the geodesic paths in Platonia. To find how much it changes as the universe passes from one configuration to another slightly different one, it is necessary only to divide their intrinsic difference by the square root of minus the potential. ( The action, by contrast, is found by multiplying it by the same quantity. ) When this distinguished simplifier is used as 'time', it turns out that each object in the universe moves in the Machian framework described above exactly as Newton's laws prescribe.

Having derived a simulacrum of time from a dynamic geometry of gravitational masses, in Newton's system, the next step was to do likewise for the theory that super-ceded it, Einstein's general theory of relativity. This was a daunting task, since time plays such a prominent role in both special and general relativity.

A first set-back is that Barbour's platonia is a collection of time-instants or Nows. But special relativity seems to do away with the concept of ( what time is ) Now, or Simultaneity, as what we are agreed is the same time for all of us. Simultaneity turned out not to be a universal time but locally measured times, that differed as to what time is now, from their different frames of reference, in uniform relative motion to each other. Only observers at rest in relation to each other would agree when something happened, in high velocity or high energy physics.

Galileo's relativity principle observed that the laws of motion hold within
the cabin of a galley, whether it is moving or at rest. You couldnt tell, say,
from two people playing ping-pong on the captain's table, whether the ship was
cruising or at anchor. Therefore, we cannot say whether the galley is at rest
or moving relative to some supposed 'absolute space'. As a so-called 'inertial
frame of reference', the galley merely maintains the inertia of staying where
it is or continuing to move steadily in a straight line, unless acted upon by
outside forces.

It is Galileo's relativity principle which makes redundant so much of the
afore-mentioned absolute space and time frame-work for measuring the lawful
motion of bodies from known initial conditions.

In Einstein's special relativity, Galileo's relativity is combined, instead, with a new absolute, in the limiting speed of light. When the motions of bodies, being considered, are very slow compared with light, as they usually are on earth, light's speed is effectively a stand-in for absolute space and time. And different observers can match each other's frame of measurement, in a common-sense manner, called the Galilean transformations ( of their respective co-ordinates, to agree with each other, as to what theyve both seen ).

But heavenly bodies, receding at astronomical speeds, or the basic constituents of matter, in particle accelerators, approach to a significant fraction of the speed of light. Observers in uniform relative motion need more general formulas to make their space and time measurements correspond to a given event. These are called the Lorentz transformations, which reduce to the Galilean transformations for motions not significant compared to light speed.

The Lorentz transformations adjust the observers' space and time measures, or rods and clocks, so that neither observer can send a light signal, in relative motion away from the other, that would move faster than the limiting speed of light. There follow apparent paradoxes of time and space. This includes an inability to agree on the Now that something happened: simultaneity is lost.

However, Hermann Minkowski came up with a quantity called 'The Interval', by which all observers, in uniform relative motion, agreed what they had measured, in terms of the same 'space-time' event. All the observers noted their respective distances and times, from an event. But when you treated time as akin to a fourth dimension of space, the totality of each observer's space and time readings always added up to the same amount, considered as a geometrically integrated space-time quantity.

Most physicists prize this formalism for its unexpected unification of two of classical physics' basic concepts. But it seems at cross-purposes to Julian Barbour's program, as he seeks to de-mystify time, rendered in terms purely of changing spatial relationships or a dynamic geometry.

Barbour goes back to Henri Poincaré's 1898 paper that posed two problems in
the definition of time. One was simultaneity, which Einstein solved in 1905.
The other was duration: how do we know a second today is the same as a second
tomorrow? How do we know that the hands or pointers of different clocks will
allow us literally to keep ap*point*ments?

An inertial clock of only three identified particles moving inertially could have four snapshots taken to show the distances between them. Peter Tait showed, in 1883, how this information of triangular positions provided enough known quantities, obeying the inertial law, to derive the unknown quantities of the times the snapshots were taken and their positions in an absolute space.

Using one of the particles as a reference point or origin, it turns out that space can be found in which the triangle corners move on mutually uniform straight lines. In other words, either of the other particles can serve as a clock 'hand' for the motion of the other two particles, considered as the 'movement' or mechanism of the inertial clock.

Duration is reduced to distance. If today or tomorrow any one of the 'hands' of the inertial clock moves through the same distance, then we can say that the 'same amount of time' has passed. The extra time dimension is redundant: everything we need to know about time can be read off from the distances.

The same applies for mechanisms of much larger numbers of bodies. The astronomers' ephemeris time, using the fairly isolated solar system is an inertial clock. The universe is the ultimate inertial clock, because, by definition, there are no outside forces acting on the universe.

For more than five particles, three snapshots are enough to solve Tait's
problem but two are never enough to tell about relative orientations and
separation in time. These variables require the four out of fourteen dimensions not
redundant in Newton's absolute space and time frame-work, that were mentioned
in the previous section.

Angular momentum accounts for three of those four dimensions. Even if the
rotation of a mass is uniform, the velocity of that rotation involves a (
centripetal or normal ) component of acceleration, due to the change in
direction of the velocity. And Newton justified his absolute frame-work with
regard to acceleration, not velocity.

General relativity is usually considered in terms of Minkowski's four-dimensional space-time generalised from Euclid's flat surfaces to Riemann's geometry of curved space. The special theory made use of Galileo's relativity principle for the freedom it confered to observers irrespective of ( uniform ) relative velocity between their co-ordinate systems. Einstein's general theory extended this relative freedom of observation with co-ordinate systems ( of geometric curvature ), irrespective of ( uniform ) relative acceleration between observers.

Einstein's equivalence principle was akin to Galileo's relativity principle in that it imagined observing law-like physical effects, which were reproduced in either of two apparently different conditions. For instance, the apparent bending or curving of light, passing thru a space-ship's portals, would not tell the crew whether they were in tow under uniform acceleration in empty space or whether they were in a gravitational field.

This reviewer is no expert on physics and Barbour's account of relativity is mainly concerned to relate relativity to his own program that time does not exist. Suffice it to say that Barbour and Bertotti were told by Karel Kuchar that the math of their ideas of best matching and duration was implicit in a less well known dynamic treatment of general relativity.

Originally, Barbour and Bertotti didnt know that a dynamic geometry ( or geometrodynamics ) of curved three-dimensional spaces, based on Mach's principle, is implicit in general relativity. ( Einstein never knew it. ) So, they set about creating one. One incentive was the work of Dirac and others:

They found that if general relativity is to be cast in dynamic form, then the 'thing that changes' is not... the four-dimensional distances within space-time, but the distances within three-dimensional spaces nested in space-time. The dynamics of general relativity is about three-dimensional things: Riemannian spaces.

A platonia is defined as 'any class of objects that differ intrinsically
but are all constructed according to the same rule'. We saw this of sets of
three particles according to the geometric rules of triangles. Likewise,
Riemann spaces can be treated as time-instants, registering as points in a platonia, if one of infinitely many dimensions.

As mentioned ( on previous web page, at end of
section, the shape space of triangle land ) the problem of infinite spaces
might be removed. At any rate, it is convenient to consider finite Riemann
spaces. How a three-dimensional space folds up on itself is hard to imagine.
But a finite two-dimensional space is like the surface of the earth or an
egg.

All the ( topological ) shapes, that cannot be transformed into each other without stretching, count as a separate point in platonia, as is any number of superficial variations on these shapes. This platonia of possible empty spaces is itself vastly increased by 'painted patterns' on the surfaces, to represent matter, electro-magnetic and other fields in the universe. Indeed, general relativity related gravitational mass to spatial curvature. And the platonia of three-dimensional Riemann spaces is known as 'superspace' in the formalism, given to general relativity, by John Wheeler and colleagues.

Best matching in this Riemannian platonia is much more complicated than for triangle land. Barbour illustrated this at lectures, using two convoluted fungi, of rather different size and markings, which he labelled Tristan and Isolde. A first guess, at their corresponding positions, is marked by pins, with matching numbers 1, 2, 3, etc. This 'trial pairing' serves as a basis to establish a 'provisional difference', an average of all the differences of curvature at each pair of points.

Keeping the pins fixed on Tristan, the pins are re-arranged on Isolde, as many ways as possible, in a continuous fashion. The best matching pairing and corresponding intrinsic difference is the transition, between any ever so slightly differing pairings, on this continuum, where the provisional difference remains unchanged. ( That is the 'stationary point' ).

'Now' appears even more arbitrary in general relativity's space-time than
Minkowski space-time. But general relativity exactly positions two
three-spaces in four-dimensional space-time, whose geodesics are followed as
the world-lines of bodies. Indeed, clocks, traveling the lines, measure the
proper times. The world lines, like a series of 'struts' compare to the
'pin'-matching of pairs of three-spaces.

The Einstein equation states a best-matching condition between two
three-spaces, which also feature as time-instants in platonia. As just these
two 'nows' are needed, general relativity turns out to justify Mach's belief
in the redundancy of a third 'now', thought necesary for measurement in the
context of Newton's absolute frame-work of space and time.

At the end of the above section, deriving Newton's laws from Mach's principle, Barbour was quoted on the distinguished simplifier, as creating the same 'time separation' across all space. But, to quote him again:

In Einstein's geometrodynamics, the separation between the 3-spaces varies from point to point, but the principle that determines it is a generalization, now applied locally, of the principle that works in the Newtonian case and explains how people can keep appointments...

Since the equivalence principle is essentially the condition that the law of inertia holds in small regions of space-time, and all clocks rely in one way or another on inertia, this is the ultimate explanation of why it is relatively easy ( nowadays at least ) to build clocks that all march in step. They all tick to the ephemeris time created by the universe through the best matching that fits it together.

Trying to give a description of general relativity in terms of quantum theory exposed their conflicting concepts of time. The theoretical likenesses between Maxwell's electro-magnetism and Einstein's space-time ( especially when almost flat like Minkowski space-time ) led physicists to believe that just as the photon is the quantum concept associated with the electro-magnetic wave field, an analgous massless particle, the graviton, could be conjectured for the field of gravitational waves, too weak to be detected by contemporary devices.

A relativistic effect of the masslessness of the photon nullifies (
longtitudinal ) waves that would move along its direction. At right angles to
the direction, two perpendicular ( transverse ) waves, giving two *true
degrees of freedom*, constitute two independent polarisations of light.
These are what bees can see to orient themselves.

Paul Dirac and the ADM team ( Arnowitt, Deser and Misner ) wanted a general quantum gravity, including for curved space. But the graviton and gravitational field's implied two true degrees of freedom did not match general relativity's three degrees of freedom of the three-spaces, whose 'geometry - the way in which they are curved - is described by three numbers at each point of space.'

Physicists attempted to match the two theories' differing degrees of freedom. Since time was thought to be needed for the quantum description of gravity, it was believed time might be identified with one of the three-spaces. But this would go against the equivalence of all co-ordinates in relativity theory, denying a definite time.

To apply the ( Schrödinger ) equation of quantum mechanics to geometric dynamics, the Wheeler-DeWitt equation fell back on Dirac's method of defering a choice of time dimension from three spatial dimensions. Barbour advocates a 'naive' interpretation, of this version, as the stationary state Schrödinger equation for a zero sum energy of the universe.

The text-book ball-and-strut models of molecules are only the most probable configurations of the structure of micro-scopic matter, this equation normally describes. The Wheeler-DeWitt equation is a telescopic version, with the universe as one 'monster molecule', which also has its huge number of other possible configurations. These are the collections of 'time-instants' or nows, that make up the points of a timeless landscape, that is Barbour's platonia, at the other extreme of complexity to triangle land.

Richard Lung.

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Barbour on quantum mechanics: making waves.

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