Brian Greene: The Elegant Universe.
Part 2, hiden dimensions.

Super-strings, Hiden Dimensions, and the Quest for the Ultimat Theory.

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Hiden dimensions.

In 1919, Theodor Kaluza show'd Einstein that jeneral relativity kud unIt his equation of gravity with Maxwell's elektro-magnetik equations, by asuming a forth dimension of spas. ( With tIm, this mayd fIv dimensions in al. ) Oskar Klein sujested the forth dimension kud exist as a kurl'd-up spas tu smal to be observabl, perhaps being only of Planck lenth. A simpl analojy is that a garden hos lwks lIk a singl dimension from a distans. But klos up, the lIn has thiknes admiting of an other sirkular dimension that kan be travel'd round by an insekt.

Kaluza's findings didnt fit the experimental data about the elektron's mas and charj. Eventualy, as mor partikls and the strong and wyk forses bekaim nown, theorists wonder'd wether the fault with Kaluza-Klein theory had byn tu fw dimensions rather than tu many.

This turn'd out to be the kais for string theory. It had resolv'd the infinit probabilitys, thrown up by elementary point partikls, in an atempted quantum gravity theory. But negativ probabilitys also kept turning up. And thys kud only be remov'd by leting the strings vibrat in nIn dimensions. ( A tenth spatial dimension was leiter infer'd, making eleven, inkluding tIm. )

Just as an ordinary string may be alow'd to vibrat in thry independent direktions, a theoretikal string may vibrat in nIn independent direktions. The kurl'd-up six dimensions, that fulfil the equations of string theory, ar kal'd Calabi-Yau spases ( or shaips ).
Thes shaips may be lIken'd to musikal instruments that kreat partikular vibration paterns. The testing question is: how wel do thys paterns match with the elementary partikls found, or kapabl of being found, by experiment?

Calabi-Yau shaips kontain varius hols, wich themselvs hav varius dimensions, analjus to a do-nut and a dubl or tripl do-nut. A family of lowest enerjy string paterns is asociated with such hols. Multipl hols implI multipl familys, lIk the thry familys of elementary partikls. Just the rIt shaips ar kurently stil being sot.

String theory predikts other fraktional charjes than thos of the quarks. And experiments finding super-partners wud also be relevant to super-strings.

Posibl behavior of strings kan be deskrib'd in a simplify'd form of the larj and the hiden dimensions, the afor-mention'd 'garden hos' univers, with uon familiar lIn dimension and a hiden, kurl'd-up dimension. The univers may kolaps bak on itself, from a big bang to a big krunch. This depends on wether ther is enuf mas in the kosmos to pul it bak. The big krunch may resembl the formation of a singularity at the hart of a blak hol. Al the kosmik mas may be krunching into a singl linear strym. It lwks to be of uon dimension only but has a sylindrikal dimension also, lIk a garden hos.

The diferens from point partikl fysiks is that strings kan not only mov about on this sylinder. They kan also rap around it: they hav a wInding moud. So, strings hav tw sorses of enerjy, wInding enerjy, as wel as vibrational motion. The later konsist of uniform vibrations and ordinary vibrations. Ordinary vibrations ar the kind of osilations konsider'd abov, and ar not desisiv in this kontext. Uniform vibrations ar 'the over-al motion of a string as it slIds from uon position to another without chanjing its shaip.'

Uniform vibrations' string enerjys ar inversly proportional to the radius of the sirkular dimension. The unsertainty relation ensurs that a konstrikting hos radius, konfining the string, inkryses its enerjy.
But the wInding enerjy is proportional to the radius. The greiter radius and sirkumferens, the longer the string and the greiter its mas, wen rap'd around the 'hos', and akording to how many tIms rap'd round, giving the 'wInding number'. Ther ar also multipl vibration numbers.
The units ar on the Planck skail of lenth and enerjy.

The wInding enerjys and vibration enerjys of the strings kompensat ych other. Yu kud hav a tabl of wInding numbers and vibration numbers for a given radius, and an other tabl of the saim, for its invers radius, giving an over-al korespondens of entrys. Yu kud hav uon univers with a smal radius and larj vibration enerjy that koresponded exaktly in total enerjy with an other univers, having a larj radius and a smal wInding enerjy. The tw universes ar efektivly the saim, having the saim alow'd quantum partikl enerjys and charjes.

We do not no wether our own univers has a hiden kurvatur, in the sens that it is tu larj, rather than tu smal, for us to sy. Spas mIt be travers'd as Magellan's expedition sirkum-navigated the gloub. If the univers has a 15 bilion lIt yer's expansion aij to put a radius to, say 10, to the power of 61, Planck lenths, then string theory provids an alternativ invers radius of the univers ( 10 to the power of minus 61 ), a radius that is miniskul and kontrakting, but just as valid in its own terms.

Mesuring distans, the familiar way, by lIt amounts to yusing lIt ( myning not-hevy ) string mouds as proubs. In prinsipl, if they wer teknikly abl, astronomers mIt equaly wel mesur distans by hevy wound-string mouds. But such proubs, being proportional to a kosmik radius wud hav to be inkredibly masiv.

Wich-ever string moud hapen'd to be the lIt or 'ysy' moud, it never mesurs below Planck lenth. Even if the non-standard mesur of distans wer adopted, so the radius is below Planck lenth, the fysiks is the saim as for the komplementary tabl in wich the radius is mor than Planck length in the konventional mesur of distans.

Having diskover'd that jeometrikal forms kud difer in sIz, yet be fysikly indistinguishabl, fysisists, inkluding Greene, found that the saim kud be tru of diferent shaips, by orbifolding Calabi-Yau spases. The number of od-dimensional hols equal'd the number of even uons, in the orijinal, and vice versa. Ther totals of hols is equal, implIing the saim number of partikl familys, tho ther shaips and strukturs difer.

The shaips agry'd on the rest of ther fysikal propertys. The byuty of thys 'miror manifolds' was that uon mIt be chosen as the posibl hiden dimensions kreating the sot-after partikl mases and fors charjes. The kalkulations involv'd had often byn imposibl. This had also byn the kais for the pyur mathematikal study of Calabi-Yau spases. But it turns out that the 'miror' partners, figurativly spyking, ar often ysy to kalkulat, a sors of progres in string theory, and a return by fysisists for wat they'd lern'd from pyur mathematiks.

In 1987, Calabi-Yau spases wer found to be transformabl into ych other, akording to a mathematikal patern of punkturing and sowing ther surfases. ( This was a spas-tering 'flop transition', wich is som-tIms 'topolojikly distinkt'. ) Konsidering such proseses as posibl fysikal terings of spas, miror symetry of Calabi-Yau spases was yus'd to giv fuler grounds for the suspicion. An absens of katastrofy in the 'miror' partner wud mak the spas-tering orijinal fysikly alowabl.

Edward Witten show'd that traveling strings, unlIk point partikls, kud protektivly ensirkl spatial ters, with relativ posibilitys ( kalkulated from Feynmann's sum-over-paths ) that wud kansel out a 'kosmik kalamity'.
He and kolygs, inkluding Brian Greene, also show'd that spatial ters wud lyv tIps and familys of partikls unafekted. But the enerjys of the posibl string vibration paterns, myning the individual partikl mases, kud chanj. Experiment shows thys to be stabl. If ther is any spatial tering in the univers at larj, it is tu slow to be notisabl.

Spas tering opens the way for the posibility of worm-hols, the kreation of nw spas joining previusly unkonekted parts of the univers.

Beyond strings: M-theory.

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Up til 1995, fIv string theorys sym'd to be at ods with ych other. Only aproximat string equations kud be found and ych of the fIv theorys difer'd from ych other. Ther difikulty ment that perturbation theory had to be yus'd, that is a method of suksesiv aproximations. A klasik exampl of this is how the gravitational inter-aktions of the solar system ar work'd out. The sun is by far the most important gravitational mas. So, its efekt in relation to the erth is kalkulated first. This result has to tak into akount the next most important efekt, the mwn in gravitational relation to the erth, and so on, until al the signifikant planetary mases hav byn alow'd for.

But the sukses of perturbation theory depends on being abl to order the importans of the efekts. Then, dyling with ych in turn, uon has som idea of how the marjin of eror shud diminish in ych suksesiv aproximation. Yusing Feynmann diagrams, Richard Feynmann's popular lekturs, QED, giv exampls of this proses of ading suksesivly smaler korektions for al the posibl ways a given partikl inter-aktion mIt tak plas. Experiment konfirm'd this quantum elektro-dynamiks as the most akurat theory in history.

String theory has Feynmann diagrams for strings insted of point partikls. The Heisenberg unsertainty prinsipl's alowans for the kreation and anihilation of virtual string pairs, in a string inter-aktion, is diagram'd as a serys of lwps betwyn in-koming and out-going strings.

The lIklihud of such temporary enerjy inkursions is mesur'd by the sIz of a 'string kupling konstant'. It wud determin mases and charjes of string vibrations. Strongly or wykly kupl'd valus, abov and below unity, respektivly, determin wether it is inkrysingly lIkly or unlIkly for mor and mor virtual partikls to apyr. Therfor, valus, abov uon, for any of the fIv string theorys, wud invalidat the yus of perturbation theory.

In 1995, Witten introdus'd 'duality' to get beyond perturbation theory. Of the fIv string theorys, tw pairs of them get exchanj'd by the larj / smal radius duality, diskus'd in the previus sektion.
Insted of asuming the fIv theorys wer independent kompetitors, al amenabl to perturbation theory, by being wykly kupl'd, it was found that tw of the theorys kud be transmuted into ych other, bekaus of a strong-wyk duality. Ther fysiks apyr'd the saim, wen uon theory was wykly kupl'd and the other strongly kupl'd.

To this end, yus was mayd of super-symetry konstraints and minimum mas konstraints to giv klus about partikl staits ( BPS staits ), for the string theory with a strong kupling konstant.

An other of the fIv theorys apyr'd to korespond to itself wen wykly and strongly kupl'd: it was self-dual.

To komplyt the link-up of the fIv theorys requir'd a further in-sIt. Super-gravity theorys atempted to yus super-symetry to unify quantum fyld theorys with jeneral relativity. It turns out that thys point partikl theorys wer aproximations to varius of the fIv string theorys.

Uon of the super-gravity theorys was in eleven dimensions, rather than ten, and didnt fit in with the existing 10-D string theorys. But a string theory, by gradualy inkrysing its kupling konstant, and with respekt to its BPS staits, show'd 11-D super-gravity to be a low-enerjy aproximation. The extra dimension emerjes with the inkrysing kupling konstant and a string lwp turns into a tw dimensional sylindrikal membrain or a hwp, depending on the string theory.

HIer dimensional membrains, than tw, ar also posibl. But with wyk string kupling, al but the strings wud be tu masiv to be produs'd without enormus enerjys.

Witten provisionaly naim'd the 11-D theory as M-theory, stil som-thing of a mystery, but the supos'd under-lIing theory to the fIv string theorys, inkorporating 11-D super-gravity. Wat was previusly an embarasment of theorys, as to the truth, has bekom an inter-related variety of aproches to mak the problems of theoretikal prediktion mor traktabl.

Witten demonstrated a primal emerjens of the gravitational fors from the other thry forses, akording to ther varying strenths wen the string kupling konstant nyd not be smal.

Blak hols as elementary partikls.

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Elementary partikls and blak hols hav in komon that they ar distinguish'd only by ther mas, fors charjes and spin. A blak hol mIt be a huj elementary partikl. A smal enuf blak hol shud resembl an elementary partikl. But this brot into play the big versus smal theory inkompatibility betwen jeneral relativity and quantum mekaniks - until string theory, or M-theory.

In the kontext of spas-tering flop transitions ( diskus'd abov ) string equations show thry-dimensional surfases, as wel as bych-bal-lIk tw-dimensional surfases, embeded in a Calabi-Yau shaip, ar lIkly to vanishingly kolaps. A uon-dimensional string, moving in tIm, kud 'laso' a 2-D sfyr, preventing a kataklysmik spatial ter. In this respekt, at any instant, a 1-D string ( or uon-brain ) kud only suround a sirkl; a 2-D string membrain, or 'tw-brain' rap round a tw-dimensional sfyr ( lIk an oranj ); and a thry-brain rap round a thry-dimensional sfyr.

Folowing-up the flop transition for the 3-D sfyr ( kal'd a konifold transition ), it was found that the sfyr repairs and re-inflats only as a 2-D sfyr. This kan only be imajin'd in lower dimensions. A tw-dimensional sfyr is 'a kolektion of points in thry-dimensional spas that ar the saim distans from a chosen senter.' Its reduktion, to a uon-dimensional sfyr, wud be to the points making up the sirkumferens of a sirkl, wich is in tw spatial dimensions.
A further reduktion wud be to a zero-dimensional sfyr, 'the kolektion of points in a uon-dimensional spas ( a lIn ) that ar the saim distans from a chosen senter.'

The replasment of a 1-D sfyr ( a sirkl ) with a 0-D sfyr ( tw points ) kan kreat a diferent topolojikal shaip. A do-nut has a sirkl, round its leser diameter, wich is pinch'd to nothing. The do-nut turns into a kresent or banana-shaip, with the tw end-points repair'd by the tw points of a zero-dimensional sfyr. The torus kum kresent kan now transform into a bal, without further tering.
This is as if Klein's hiden extra dimensions of spas transform'd from the uon kurl'd-up shaip to an other, komparably to the normal extended thry dimensions chanjing the shaip of the univers from a torus to a bal.
The evolution of the univers may involv such transmutations betwyn kurl'd-up Calabi-Yau spases.

Equations governing the '-brains' show'd that, from our limited thry-dimensional viw-point, the thry-brayn "smyr'd" around a thry-dimensional sfyr, within a ( kurl'd-up ) Calabi-Yau spas, sets up a gravitational fyld lIk a blak hol.
The spas tering konifold transition from thry to tw dimensional sfyr hapens to inkrys the number of hols by uon. Thys hols determin the number of low mas partikls, konsider'd as low enerjy string vibration paterns. The shrinking volum of the 3-D sfyr gos with a proportionat mas dekrys to zero: a masles blak hol.

The blak hol is konsider'd to hav under-gon a feiz transition to a masles elementary partikl, lIk a foton. String theory has identify'd them as being mayd of the saim 'stringy material'. Much as Is under-gos a feiz transition to water, they lwk diferent but ther mak-up is the saim.

'Hawking radiation' establish'd the 'entropy' of blak hols. To solv wat this disorder was of, string theorists theoretikly bilt

certain extremal black holes by starting with a particular collection of BPS branes ( of certain specified dimensions ) and binding them together according to a precise mathematical blueprint...

Strominger and Vafa could easily and directly count the number of rearrangements of the black hole's microscopic constituents that would leave its overall observable properties, its mass and force charges, unchanged. They could then compare this number with the area of the black hole's horizon -- the entropy predicted by Bekenstein and Hawking.

Hawking radiation implIs the eventual evaporation of blak hols. With the gradual shrinking of ther areas, ther entropy dekryses. A kurent reserch question is wether order or 'information', lost to the blak hol's gravitational suktion, kud be rekover'd from the surounding area that the shrinking event-horizon has given up.

If the anser is 'no', this wud further tak the ej off a deterministik fysiks. Quantum mekaniks had mayd, only probabilistik, Laplace's totaly determinist konseption of mekaniks.

How-ever, the author dosnt mention kaos theory, wich requirs infinit akurasy in initial konditions, to predikt the osilations of so simpl a klasikal system as the fors-driven pendulum. Laplace's skwl thot thys information fyd-ins, to aplI fysikal laws to partikular sirkumstanses, wud, in prinsipl, determin the evolution of the univers.

Brian Greene diskuses other questions, mainly to do with the nw subjekt of super-string kosmolojy. Alredy, som posibl ansers hav byn put forward. Pending the big bang, al the eleven dimensions of spas and tIm wer supos'd to be kurl'd up in a univers of Planck skail sIz. WI did only thry dimensions of spas extend ( thru 'inflation' and so forth )?
As related, abov, strings kan rap round thys dimensions. But ther ar anti-strings, raping round 'the other way', wich anihilat them on kontakt, produsing an unrap'd string, and relysing the dimension to expand. Thys relysing kolisions ar most lIkly in uon dimension. At diferent spyds, tw marbls, konfIn'd to a lIn, ar swner or leiter going to hit. This is les lIkly of tw objekts moving fryly on a surfas, and les lIkly stil for objekts moving fryly in thry dimensions. Thus, the chanses wer that the forth and hIer dimensions of spas wer not relys'd from ther string rapings by string pair anihilations.

Alan Guth's popular bwk, 'The Inflationary Universe', mentions ther being about fifty versions of inflation theory, wich explains several diskrepansys in the erlier big bang model. Greene refers to a kontroversial pre-big bang version, deriv'd from string theory, by Gasperini and Veneziano, wich they houp presajes a mor inevitabl development to inflation.

Klosing remark.

Brian Greene's explanation of string theory may be lIken'd to a serys of bykon hils that trail off to hw nos wer. They giv yu an idea of the jeneral direktion string theory is going but they lyv in darknes the maiz of valys below, wich only wiz mathematicians and fysisists kan folow, to lIt up mor bykons.

Post-skript: Paralel universes.

BBC tv's Horizon ( 14 feb. 2002 ) fytur'd an astonishing development in linking string theory to kosmolojy via the konsept of paralel universes. The program folow'd the implikation of a unify'd string theory or M-theory fyturing an eleventh dimension and, beyond strings, the existens of membrains of varius dimensions.

Uon of the sientists involv'd deskrib'd the arival of an Italian liner in New York, damaj'd by a roug wav. It so hapen'd that a study of the mathematikal posibilitys of wat mIt hapen wen the membrains kolId in ther hyper-spas also yilded katastrofik results of the order of the Big Bang itself, or inumerabl big bangs.

Klasikal kosmolojy kloses off posibl events, befor the big bang, with an infinitly smal begining, a singularity. But quantum theory of the Planck skail of events transends the big bang, as the out-kom of thys kolIding membrains. As they mov, they ripl, so that kolisions yild the klumps of mater after the big bang. That is the material univers.

This implIs that tIm presyds the big bang, wich is indyd uon of an infinit number of diferent big bangs resulting in an infinit number of posibl universes, with diferent laws of fysiks.

Hens, string theory has theoretikly explain'd the orijin of the big bang by implIing paralel universes.
At the tIm of rIting, this is a nw theory, wich the fysiks komunity has yet to desId wether to aksept or not. As yet, paralel universes hav not byn the majority viw.

Richard Lung.

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