Julian Barbour's The End Of Time - in klasikal fysiks:

(2) jeometrik dynamiks.

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( Kapital-i, in 'I, myself', now spels Il as in isle or aisle.
Leter y spels sym for seem or seam and partys for parties.
Leter w spels swn for soon. )

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Deriving Newton's laws from Mach's prinsipl.

Bruno Bertotti and Julian Barbour konstrukted a theory of jeodesiks to determin the shortest path betwyn any tw fix'd points in platonia. They found that the unyk history this produs'd, from an initial point and direktion, koresponds to uon of many such historys in Newton's fraim-work of absolut spas and tIm. This korespondens was to the special kais of a Newtonian history, with zero enerjy and angular momentum, wich solvs in terms of 'a simpler tImles and fraimles theory.'

Barbour and Bertotti produs'd the mathematiks that put Newton's paradIm in the kontext of Mach's prinsipl.
Asuming the univers is finit, the total enerjy and angular momentum of its sub-systems ad up to zero. But this wil not be tru of most sub-systems, themselvs. So, they wil produs the much mor komon Newtonian solutions for galaxys or solar systems with non-zero enerjys and angular momentums, as if they wer in absolut spas and tIm.

Even to his kontemporary, Leibniz, Newton's absolut spas and tIm sym'd a kumbersom fraim of mesurment. In determining the evolution of a konfiguration of, say, thry mases, from its initial konditions and direktion to a sekond konfiguration, fortyn dimensions ( alowing fortyn 'degrys of frydom' ) ar involv'd. But only for of them mak any diferens to the result.

The ten absolut dimensions that mak no diferens ar: thry spatial dimensions ych for the first and sekond konfigurations, from the points of viw of ther senters of mas; the starting tIm, an other dimension; and the thry-dimensional orientation of the first triangl.

The for remaining dimensions that do mater ar the orientation of the sekond triangular konfiguration and the absolut tIm elapsing betwyn the tw konfigurations' positions, or to put it an other way, the angular momentum and the kinetik enerjy, respektivly. The angular momentum of the thry-mas konfiguration is lIk a spining top with an imajinary pivot, thru its senter of mas, pointing skIwards over tw dimensions, with a third dimension from the axial rotation of the triangl perpendikular to the pivot.
Spiral galaxys and the rings of Saturn are spektakular astronomikal exampls of angular momentum.

In relativ konfiguration spas or platonia, the proses of best matching, from uon konfiguration to an other, kreats a determinat relation, of al the konfigurations to the first chosen konfiguration, wich givs an apyrans of a rijid fraim-work, lIk absolut spas and tIm.

Barbour komplyts a derivation of Newton's world viw from Mach's, with respekt to the 'spasings in tIm':

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.

Prominent rol of tIm in Relativity.

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Having deriv'd a simulakrum of tIm from a dynamik jeometry of gravitational mases, in Newton's system, the next step was to do lIkwIs for the theory that super-syded it, Einstein's jeneral theory of relativity. This was a daunting task, sins tIm plays such a prominent rol in both special and jeneral relativity.

A first set-bak is that Barbour's platonia is a kolektion of tIm-instants or Nows. But special relativity syms to do away with the konsept of ( wat tIm is ) Now, or Simultaneity, as wat we ar agryd is the saim tIm for al of us. Simultaneity turn'd out not to be a universal tIm but lokaly mesur'd tIms, that difer'd as to wat tIm is now, from ther diferent fraims of referens, in uniform relativ motion to ych other. Only observers at rest in relation to ych other wud agry wen som-thing hapen'd, in hI velosity or hI enerjy fysiks.

Galileo's relativity prinsipl observ'd that the laws of motion hold within the kabin of a galy, wether it is moving or at rest. Yu kudnt tel, say, from tw pepl playing ping-pong on the kaptain's tabl, wether the ship was krusing or at ankor. Therfor, we kan not say wether the galy is at rest or moving relativ to som supos'd 'absolut spas'. As a so-kal'd 'inertial fraim of referens', the galy myrly maintains the inertia of staying wer it is or kontinuing to mov stedily in a strait lIn, unles akted upon by out-sId forses.
It is Galileo's relativity prinsipl wich maks redundant so much of the afor-mention'd absolut spas and tIm fraim-work for mesuring the lawful motion of bodys from nown initial konditions.

In Einstein's special relativity, Galileo's relativity is kombin'd, insted, with a nw absolut, in the limiting spyd of lIt. Wen the motions of bodys, being konsider'd, ar very slow kompar'd with lIt, as they usualy ar on erth, lIt's spyd is efektivly a stand-in for absolut spas and tIm. And diferent observers kan match ych other's fraim of mesurment, in a komon-sens maner, kal'd the Galilean transformations ( of ther respektiv ko-ordinats, to agry with ych other, as to wat they'v both syn ).

But hevenly bodys, resyding at astronomikal spyds, or the basik konstituents of mater, in partikl akselerators, aproch to a signifikant fraktion of the spyd of lIt. Observers in uniform relativ motion nyd mor jeneral formulas to mak ther spas and tIm mesurments korespond to a given event. Thys ar kal'd the Lorentz transformations, wich redus to the Galilean transformations for motions not signifikant kompar'd to lIt spyd.

The Lorentz transformations ajust the observers' spas and tIm mesurs, or rods and kloks, so that neither observer kan send a lIt signal, in relativ motion away from the other, that wud mov faster than the limiting spyd of lIt. Ther folow aparent paradoxes of tIm and spas. This inkluds an inability to agry on the Now that som-thing hapen'd: simultaneity is lost.

However, Hermann Minkowski kaim up with a quantity kal'd 'The Interval', by wich al observers, in uniform relativ motion, agryd wat they had mesur'd, in terms of the saim 'spas-tIm' event. Al the observers noted ther respektiv distanses and tIms, from an event. But wen yu tryted tIm as akin to a forth dimension of spas, the totality of ych observer's spas and tIm rydings always aded up to the saim amount, konsider'd as a jeometrikly integrated spas-tIm quantity.

Most fysisists prIz this formalism for its unexpekted unifikation of tw of klasikal fysiks' basik konsepts. But it syms at kros-purposes to Julian Barbour's program, as he syks to de-mystifI tIm, render'd in terms pyurly of chanjing spatial relationships or a dynamik jeometry.

Barbour gos bak to Henri Poincaré's 1898 paper that pos'd tw problems in the definition of tIm. Uon was simultaneity, wich Einstein solv'd in 1905. The other was duration: how do we no a sekond today is the saim as a sekond tomorow? How do we no that the hands or pointers of diferent klokks wil alow us literaly to kyp appointments?

An inertial klok of only thry identifI'd partikls moving inertialy kud hav 4 snap-shots taken to show the distanses betwyn them. Peter Tait show'd, in 1883, how this information of triangular positions provided enuf nown quantitys, obeying the inertial law, to deriv the unown quantitys of the tIms the snap-shots wer taken and ther positions in an absolut spas.

Yusing uon of the partikls as a referens point or orijin, it turns out that spas kan be found in wich the triangl korners mov on mutualy uniform strait lIns. In other words, either of the other partikls kan serv as a klok 'hand' for the motion of the other tw partikls, konsider'd as the 'movment' or mekanism of the inertial klok.

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 saim aplIs for mekanisms of much larjer numbers of bodys. The astronomers' efemeris tIm, yusing the fairly isolated solar system is an inertial klok. The univers is the ultimat inertial klok, bekaus, by definition, ther ar no out-sId forses akting on the univers.

For mor than fIv partikls, thry snap-shots ar enuf to solv Tait's problem but tw ar never enuf to tel about relativ orientations and separation in tIm. Thys variabls requir the for out of fortyn dimensions not redundant in Newton's absolut spas and tIm fraim-work, that wer mention'd in the previus sektion.
Angular momentum akounts for thry of thos for dimensions. Even if the rotation of a mas is uniform, the velosity of that rotation involvs a ( sentripetal or normal ) komponent of akseleration, du to the chanj in direktion of the velosity. And Newton justifI'd his absolut fraim-work with regard to akseleration, not velosity.

A tImles jeometrik dynamiks within jeneral relativity.

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Jeneral relativity is usualy konsider'd in terms of Minkowski's for-dimensional spas-tIm jeneralIs'd from Euclid's flat surfases to Riemann's jeometry of kurv'd spas. The special theory mayd yus of Galileo's relativity prinsipl for the frydom it konfer'd to observers irespektiv of ( uniform ) relativ velosity betwyn ther ko-ordinat systems. Einstein's jeneral theory extended this relativ frydom of observation with ko-ordinat systems ( of jeometrik kurvatur ), irespektiv of ( uniform ) relativ akseleration betwyn observers.

Einstein's equivalens prinsipl was akin to Galileo's relativity prinsipl in that it imajin'd observing law-lIk fysikal efekts, wich wer reprodus'd in either of tw aparently diferent konditions. For instans, the aparent bending or kurving of lIt, pasing thru a spas-ship's portals, wud not tel the krw wether they wer in tow under uniform akseleration in empty spas or wether they wer in a gravitational fyld.

This reviwer is no expert on fysiks and Barbour's akount of relativity is mainly konsern'd to relat relativity to his own program that tIm dos not exist. SufIs it to say that Barbour and Bertotti wer told by Karel Kuchar that the math of ther ideas of best matching and duration was implisit in a les wel nown dynamik trytment of jeneral relativity.

Orijinaly, Barbour and Bertotti didnt no that a dynamik jeometry ( or jeometro-dynamiks ) of kurv'd thry-dimensional spases, bais'd on Mach's prinsipl, is implisit in jeneral relativity. ( Einstein never nw it. ) So, they set about kreating uon. Uon insentiv 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 defin'd as 'any class of objects that differ intrinsically but are all constructed according to the same rule'. We saw this of sets of thry partikls akording to the jeometrik ruls of triangls. LIkwIs, Riemann spases kan be tryted as tIm-instants, rejistering as points in a platonia, if uon of infinitly many dimensions.
As mention'd ( on previus web paj, at end of sektion, the shaip spas of triangl land ) the problem of infinit spases mIt be remov'd. At any rait, it is konvenient to konsider finit Riemann spases. How a thry-dimensional spas folds up on itself is hard to imajin. But a finit tw-dimensional spas is lIk the surfas of the erth or an eg.

Al the ( topolojikal ) shaips, that kan not be transform'd into ych other without stretching, kount as a separat point in platonia, as is any number of superficial variations on thys shaips. This platonia of posibl empty spases is itself vastly inkrys'd by 'painted paterns' on the surfases, to represent mater, elektro-magnetik and other fylds in the univers. Indyd, jeneral relativity related gravitational mas to spatial kurvatur. And the platonia of thry-dimensional Riemann spases is nown as 'super-spas' in the formalism, given to jeneral relativity, by John Wheeler and kolygs.

Best matching in this Riemannian platonia is much mor komplikated than for triangl land. Barbour ilustrated this at lekturs, yusing tw konvoluted fungi, of rather diferent sIz and markings, wich he label'd Tristan and Isolde. A first ges, at ther koresponding positions, is mark'd by pins, with matching numbers 1, 2, 3, etc. This 'trial pairing' servs as a basis to establish a 'provisional diferens', an averaj of al the diferenses of kurvatur at ych pair of points.

Kyping the pins fix'd on Tristan, the pins ar re-aranj'd on Isolde, as many ways as posibl, in a kontinuus fashon. The best matching pairing and koresponding intrinsik diferens is the transition, betwyn any ever so slItly difering pairings, on this kontinuum, wer the provisional diferens remains unchanj'd. ( That is the 'stationary point' ).

'Now' apyrs even mor arbitrary in jeneral relativity's spas-tIm than Minkowski spas-tIm. But jeneral relativity exaktly positions tw thry-spases in for-dimensional spas-tIm, hws jeodesiks ar folow'd as the world-lIns of bodys. Indyd, kloks, traveling the lIns, mesur the proper tIms. The world lIns, lIk a serys of 'struts' kompar to the 'pin'-matching of pairs of thry-spases.
The Einstein equation staits a best-matching kondition betwyn tw thry-spases, wich also fytur as tIm-instants in platonia. As just thys tw 'nows' ar nyded, jeneral relativity turns out to justifI Mach's belyf in the redundansy of a third 'now', thot nesesary for mesurment in the kontext of Newton's absolut fraim-work of spas and tIm.

At the end of the abov sektion, deriving Newton's laws from Mach's prinsipl, Barbour was quoted on the distinguish'd simplifIer, as kreating the saim 'tIm separation' akros al spas. But, to quot 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.

Quantum gravity's konflikt over tIm.

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TrIing to giv a deskription of jeneral relativity in terms of quantum theory expos'd ther konflikting konsepts of tIm. The theoretikal lIkneses betwyn Maxwell's elektro-magnetism and Einstein's spas-tIm ( especialy wen almost flat lIk Minkowski spas-tIm ) led fysisists to belyv that just as the foton is the quantum konsept asociated with the elektro-magnetik wav fyld, an analjus masles partikl, the graviton, kud be konjektur'd for the fyld of gravitational wavs, tw wyk to be detekted by kontemporary devises.

A relativistik efekt of the maslesnes of the foton nulifIs ( longitudinal ) wavs that wud mov along its direktion. At rIt angls to the direktion, tw perpendikular ( transvers ) wavs, giving tw tru degrys of frydom, konstitut tw independent polarisations of lIt. Thys ar wat bys kan sy to orient themselvs.

Paul Dirac and the ADM tym ( Arnowitt, Deser and Misner ) wanted a jeneral quantum gravity, inkluding for kurv'd spas. But the graviton and gravitational fyld's implI'd tw tru degrys of frydom did not match jeneral relativity's thry degrys of frydom of the thry-spases, hws 'geometry - the way in which they are curved - is described by three numbers at each point of space.'

Fysisists atempted to match the tw theorys' difering degrys of frydom. Sins tIm was thot to be nyded for the quantum deskription of gravity, it was belyv'd tIm mIt be identifI'd with uon of the thry-spases. But this wud go against the equivalens of al ko-ordinats in relativity theory, denIing a definit tIm.

To aplI the ( Schrödinger ) equation of quantum mekaniks to jeometrik dynamiks, the Wheeler-DeWitt equation fel bak on Dirac's method of defering a chois of tIm dimension from thry spatial dimensions. Barbour advokats a 'naiv' interpretation, of this version, as the stationary stait Schrödinger equation for a zero sum enerjy of the univers.

The text-bwk bal-and-strut models of molekuls ar only the most probabl konfigurations of the struktur of mikro-skopik mater, this equation normaly deskribs. The Wheeler-DeWitt equation is a teleskopik version, with the univers as uon 'monster molekul', wich also has its huj number of other posibl konfigurations. Thys ar the kolektions of 'tIm-instants' or nows, that mak up the points of a tImles land-skaip, that is Barbour's platonia, at the other extrym of komplexity to triangl land.

Richard Lung.

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Barbour on quantum mekaniks: making wavs.

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