2 / 2 0 0 6 M A X P L A N C K R E S E A R C H 4948 M A X P L A N C K R E S E A R C H 2 / 2 0 0 6

FIRSTHAND KNOWLEDGE GRAVITATIONAL PHYSICS

Since the beginning of the 20thcentury, physics has celebrated

some brilliant successes. Not onlyhas it developed quantum theoryand the theory of relativity, it has al-so demonstrated their validity in nu-merous experiments. Gradually, the-ories were developed that describethe electromagnetic force, the nu-clear force and weak interaction, aswell as their connection with fields

A Universe of Bubbling Loopsquantum gravity were developed to-tally isolated from each other. Thenthe two theories slowly converged,but even today, they are still funda-mentally incompatible. Nevertheless,scientists take both very seriously,and an increasing number – current-ly around 300 worldwide – are work-ing on loop theory and similar ap-proaches to quantum gravity.

GRAVITY NEEDS

NO FIXED TIME

Along with the Perimeter Institute inWaterloo, Canada and the Institutefor Gravitational Physics and Geom-etry in the US, the Max Planck Insti-tute for Gravitational Physics inPotsdam is one of the centers for thisresearch. Around 150 loop theoreti-cians, including all the leaders in thefield, met last October at the Loops05 conference in Potsdam to sharetheir latest ideas and progress re-ports. They also discussed how thetheory might be put to the test in ex-periments.

Of the four known forces – elec-tromagnetic, weak and strong forcesand gravity – gravity is the least un-derstood. Unlike the other forces, itresists quantization. Quantization ofthe other forces has been describedby theories such as quantum electro-dynamics, quantum flavor dynamicsand quantum chromodynamics. Eachof these theories, summarized in the Standard Model of matter, issubject to physical predictions thathave been measured very preciselyand confirmed by experiments in accelerators, such as CERN, DESYand SLAC. There is, therefore, nodoubt that quantum theory is thecorrect description for these forces.So-called interaction mediators – theparticles that carry the forces – playa part in it.

What is so different about gravitythat it resists quantization? Does it

overcome this, the most significantones being string theory and loopquantum gravity.

String theory assumes that alltypes of elementary particles, fromleptons to electrons to quarks, whichmake up protons and neutrons,manifest themselves as differentstates of excitement of a single typeof object: strings. These strings vi-brate like the strings on a guitar(hence the name) in a space that has10 or 11 dimensions.

Loop quantum gravity does notneed this multidimensionality. It re-mains within the 4 familiar dimen-sions of space and time and postu-lates that they are not continuous,but consist of quanta (the smallestindivisible units) – the loops. Usingthis imagery, it is possible to picturethe world as having a web-likestructure: space is not smooth like a

plastic film, but textured like terrycloth. The same also applies to time:it consists of tiny steps whose fur-ther breakdown is neither possiblenor practical.

Why didn’t we ever notice thisstructure before? Because it is ex-tremely fine. The smallest unit, aloop, is approximately 10-33 cen-timeters long. This is called thePlanck length, calculated from threenatural constants: the speed of light,Planck’s constant and the gravita-tional constant. Similarly, a timequantum is the time a particle trav-eling at the speed of light would taketo move through a space quantum.In other words, the surface of an A4sheet of paper would be made up of1068 – or 1 followed by 68 zeros – ofthese loops.

For more than 10 years, from 1984to 1996, string theory and loop

A theory of everything is the Holy Grail of physicists. Yet, in the search for this theory that unites

all forces, they are brought up short by the limits of what the human mind can actually grasp.

Today, loop quantum gravity is considered a promising candidate for the solution to this problem.

THOMAS THIEMANN from the MAX PLANCK INSTITUTE FOR GRAVITATIONAL PHYSICS in

Potsdam is one of its leading proponents worldwide. In the following article he reports on his work.

This image, as well as that on the left above, shows an artist’s impression of quantized space. They are snapshots of a dynamic geometry consisting of loops.Matter can be present only where the geometry is activated.

A Universe of Bubbling Loops

PHO

TO: M

PI F

OR

GRA

VITA

TIO

NAL

PHYS

ICS

PHO

TO: S

CIEN

CEPH

OTO

LIBR

ARY

and particles. Today, we have physi-cal theories that correctly describenatural processes over a huge rangeof distances – from the diameter of aproton to the expanse of the visibleuniverse. And all of that with mathe-matical formulas that fit on a sheetof paper!

However, despite these great feats,physics remains essentially a patch-work. There is still no theory of

everything, one that takes into ac-count and explains every force andevery particle. The biggest names inphysics have tried, but none of themhas yet had any success. Particularlythe attempt to reconcile the phe-nomena of the fourth interaction,namely gravity, with quantum theo-ry always comes up against an un-yielding mathematical limit. Todaythere are various approaches to

2 / 2 0 0 6 M A X P L A N C K R E S E A R C H 51

taken to its logical limits. This guar-antees that either the right theory isconstructed or, in the course ofanalysis, it is systematically deter-mined which new structures are re-quired, rather than having to guessat them.

We assess a philosophy of this na-ture, which might look like a highlyintelligent game, based on whether itprogresses the description of naturebeyond the existing theories – andloop theory does just that. It re-quires, as I already mentioned, onlyour four dimensions, and it alsomanages to avoid a shortcoming ofthe general theory of relativity: ap-plying its equations logically givesrise to bizarre physical situations,known as spacetime singularities.These are the points in space wherethe curvature, and thus the concen-tration, of matter is infinitely large.Black holes are such singularities, asis the Big Bang.

Quantization of space helps toavoid such singularities. This exam-ple from atomic physics is an ana-logy: according to the classicalMaxwell theory of electromagne-tism, atoms should not exist at all.This is because electrons, whichmove in an orbit, normally emitbremsstrahlung, thus losing energy.Each electron that circles the atomic

nucleus would therefore fall into thenucleus after a short time. The atomswould be unstable and we wouldpositively burn up. Quantum theoryexplains why this does not happen:the permitted energy levels in theatom are discrete, that is, quantized.Experiments also allow this to beread from the atomic spectra: thereare no bremsstrahlung lines becausethe atom emits none – the classicsingularity disappears.

Quantizing spacetime achieves asimilar situation. Martin Bojowaldfrom Pennsylvania State Universityin the US has applied my work onthe dynamics of loop quantum grav-ity to a quantum cosmology modeland discovered that the singularitypredicted by the general theory ofrelativity as the Big Bang no longerfigures in loop theory – quite like theexample of the atom with discreteenergy levels. Although this is notproof that there are no singularitiesat all in loop quantum gravity, it canbe considered an initial first indica-tion. Instead of singularities, thereare values that might become big,but remain finite. Calculating thelargest possible curvature yields avalue of one divided by the square ofthe Planck length, or 1066 cm-2.

What does this mean for the BigBang? It is not yet fully understood,but it might mean that the universehad no initial singularity at all. In

GRAVITATIONAL PHYSICS

50 M A X P L A N C K R E S E A R C H 2 / 2 0 0 6

even need to be quantized? The an-swer can be found by carefully con-sidering Einstein’s theory of relativi-ty. According to this theory, gravityis not a force in the traditional sense,but is equivalent to geometry. Aplanet does not orbit a star becauseinteraction-carrying particles (gravi-tons) facilitate an exchange of forcesbetween them; rather, the planet or-bits the star because the star bendsthe geometry of spacetime. In thesame way a ball in a mini-golf gameis diverted by a dip and rolls towardthe lowest point, the dent in spacecaused by the star influences theplanet’s path and makes it circular.In other words, matter curves space.

A MODEL FROM

CLASSICAL PHYSICS

In this sense, the general theory ofrelativity is much more than a theo-ry of gravity – it also realizes Ein-stein’s basic idea of the geometriza-tion of physics. For him, gravity issynonymous with geometry and istherefore independent of fixedspacetime. It is this principle of theunderlying independence of the gen-eral theory of relativity that cannotbe reconciled with quantum theory.Proponents of loop quantum gravityhave implemented this idea logicallyand have quantized, not gravity, butthe geometry of space itself.

But how can a world be construct-ed only from quanta? An analogyfrom classical physics can help ushere: in the Maxwell theory of elec-

tromagnetic fields, the interestingquantities are electrical and magneticfluxes. For example, magnetic fluxcan be shown as an integral over asurface or as a linear integral over theedge of this surface. This is how theimage of loops is created – they arethe edges of the flux surfaces. Thesequantities are interesting as they areinvariant (that is, unchanging) in theface of movement of any kind.

In the Maxwell theory, for exam-ple, it is possible to rescale the po-tential – in other words, to move thezero point – without any physicalchanges being visible. The loops areinvariant in the face of such rescal-ing. As quantum gravity theoreti-cians also deal with loop integrals,these objects of the theory havecommonly been dubbed loops sincethe mid-1980s. With many compli-cated mathematical operations, it isnow possible to construct a dynamicspacetime from these loops. To dothis, they are combined, in the formof spatial structures, into graphs or –in mathematical terms – into func-tions. Each loop is always linked toall of its neighbors. In this way, aninfinite number of these graphsform, to put it simply, a kind of crys-tal. However, this crystal is not rigid,but honeycombed because, like mostobjects in the quantum world, all ofits tiny elements are constantlychanging. Space thus no longer hasthe smooth, Euclidian geometry that,in our experience, describes the visi-ble world, but rather consists of a

seething mass of tiny loops. Similarto the way electrons incessantly or-bit the nucleus of an atom, here, thegeometry of space dances andsurges.

The edges between the crystal sur-faces are particularly important; likethe surfaces themselves, they are la-beled with quantum numbers. As theloops are all linked together, every-thing is vastly interconnected – sononlinear. Nevertheless, the theoristsuse only those mathematical toolswith which they are already familiar.They exploit mathematics to the log-ical extreme. They also use every-thing that the quantum field theo-rists have come up with in the lastcentury. Even when they put some-thing new together, they use new va-rieties of familiar constructions.

Accordingly, gravity creates thespace in which everything happens.The theorists also want to bring mat-ter into the game. However, as looptheory creates its own spacetime, theparticle terminology from quantummechanics, which is based on a fixedspacetime, cannot be used here. Theparticle term from other theories cannevertheless build on loop theory, asit is possible to develop a similarlanguage for matter and, as it were,pack it on top of what has previous-ly been said. At the same time, theforces develop from this. Photons aredescribed by loops; quarks and elec-trons, on the other hand, sit at spe-cific points on this structure.

LOOPS REQUIRE

ONLY FOUR DIMENSIONS

The loop quantum gravity approachis consciously conservative: it isbased solely on facts that are sup-ported by experiment; it does not re-quire additional assumptions; and itanalyzes the consequences of the in-terplay of the principles of the gen-eral theory of relativity and quantumtheory. Aiming for rigorous mathe-matical consistency, the theory is

FIRSTHAND KNOWLEDGE

THE LEGO BRICKS

OF LOOP THEORY

In loop quantum gravity, the basic structure of spacetime turns out to be discrete: the values of ar-eas and volumes can be changed in (naturally tiny)steps and there are minimum areas and space vol-umes below which the space cannot be further di-vided – in the same way that it is not possible tobuild an object with a Lego set that is smaller thanthe smallest Lego brick available. The basic structure

of space is a “spin network” of linked knots as shownin the illustration. The smallest conceivable volume, a space

quantum, contains one knot. Any volume, therefore, consists of a certain number of knots.If another knot is added, the space content increases by one characteristic space quantum.

Such images illustrate the construction of quantized space. On the left, a spin network of polyhedrons, on the right, the development of spins over time. The knots in the left-hand image can split open and create a kind of foam structure.

According to Einstein’s general theory of relativity, gravity is synonymous with geometry. A “dent” caused by a star in space, for instance, affects the path of a planet and shapes it into an orbit. Matter thus curves space.

ILLU

STRA

TIO

NS:

FRA

NKF

URT

ERAL

LGEM

EIN

ESO

NN

TAG

SZEI

TUN

G, O

CT. 2

0, 2

005/

F.A.

Z.-G

RAFI

KD

ÖRI

NG

PHO

TO: S

CIEN

CEPH

OTO

LIBR

ARY

GRA

PHIC

: MPI

FO

RG

RAVI

TATI

ON

ALPH

YSIC

S

2 / 2 0 0 6 M A X P L A N C K R E S E A R C H 53

A TABLE HAS STATES

The hydrogen atom has precisely defined levels of energy. The same can be posited for a table surface, for example. According to loop theory, it also has a spectrum. The table canassume only certain values for its area, not just any random values, because its geometry is quantized. If the length of the table could be measured precisely to the Planck length,then that would be visible, depending on the table’s eigenstate. If a loop is removed, thearea of the table would change by certain units of the Planck area (10-66 cm2). This isanalogous to the excitation states of the hydrogen atom. Each has a different energy.However, the excitation states of the table are not expressed by energy levels, but by theloop variables, or more precisely, their spin quantum numbers. The area of the table alsohas a discrete spectrum because its geometry is excited.

the person who falls into it. Seenfrom the outside, the black holelooks exactly the same as it did be-fore. The person falling is still stuckin it, but is not crushed into a point.

Whether loop theory is correctwith regard to these statements mustfirst be proven through experiments.“If you have a theoretical structurethat explains nothing and predictsnothing, you are no longer doingscience,” says Lee Smolin from thePerimeter Institute.

With a theory like loop gravity,which deals with Planck scales of 10-33 centimeters, experimental test-ing is difficult because the objectsare so tiny. It is not, however, as ab-surd as it sounds: if loop theory isright, then our conception of space iswrong – it is not a smooth, flawlessstructure, but a crystal on the Planckscale that changes constantly. Andthis is where an experiment couldstart out: in our normal world, lightdoes not pass undisturbed through acrystal, but is scattered widely. As aresult, the speed of light changes inthe crystal and the change dependson the frequency of the light. This is reflected in experiments by thedifferent transit times for photonspassing through the crystal.

A similar thing could happen inthe quantized spacetime geometry ofloop quantum gravity: the speed oflight in the vacuum would no longerbe an absolute value, but would varyover time. The more energy the pho-ton has, the more strongly it feelsthe presence of discrete structures,the more its propagation is disrupted– and the more it slows down.

However, these transit time differ-ences, which are a result of the im-pact of gravity, would be so small(even if they were linear) that theywould be impossible to measure,even with the greatest accuracy – atleast that is what physicists thoughtfor a long time. In 2001, however,Giovanni Amelino-Camelia from La

GRAVITATIONAL PHYSICS

52 M A X P L A N C K R E S E A R C H 2 / 2 0 0 6

other words, the universe has alwaysexisted; it had no beginning. It may,at some point, have been quite small,but before that, it may have alsobeen just as large as it is today.

THE INFINITELY LONG FALL

INTO THE BLACK HOLE

By no means does the loop theorydispute that there was a Big Bang. Atthat time, the entire universe waspacked into a very tiny volume – butnot to zero. From a philosophicalstandpoint, it makes a big differencewhether it must be assumed thateverything had a beginning orwhether it was always there. Loopquantum gravity has now also pro-vided a mathematical basis that al-lows calculations to back before theBig Bang. This did not previouslyexist with such clarity.

It works in a very similar way withblack holes. The whole theory had tobe reconstructed from scratch be-cause many propositions relating toblack holes were based on the as-sumption that they collapse into apoint, which is not possible in looptheory. In physical terms, this issomething completely different. It

means that things can indeed re-emerge from a black hole.

A black hole is a strange phenom-enon. From the outside, it is possibleto see someone fall into it, but it isnever possible to observe whetherthe person really falls into the centerbecause that would take infinitelylong – like a star, for example, thatcollapses into itself. Thus, viewedfrom the outside, a black hole shouldnot be called a black hole but rathera frozen star. From the outside, itdoes not look like a dot, but like asphere that can’t get any smallerthan the Schwarzschild radius,which marks the sphere from whichthe object could still escape if it werefaster than light – but the specialtheory of relativity forbids this.

For an observer on the inside ofthe black hole, however, things lookbizarrely different. Someone fallinginto a black hole falls in finite prop-er time precisely into the singularity.According to the theory of relativity,they wouldn’t have a chance: theywould be crushed to an infinitelysmall dot. The discrepancy betweenthe perceptions that the observer andthe intrepid traveler have of a blackhole is the result of the peculiarity ofthe concept of simultaneity. This hasalready given rise to a few brain-teasers in the special theory of rela-tivity.

In loop theory, the total matterdoesn’t collapse into a point, but isinstead stretched infinitely – also for

FIRSTHAND KNOWLEDGE

DR. THOMAS THIEMANN, 38, works at the Max Planck Institute for Gravita-tional Physics in Potsdam and is one ofthe pioneers of loop quantum gravity. He has been at Pennsylvania StateUniversity, Harvard University and thePerimeter Institute in Canada. There,as well as at Beijing Normal Univer-sity, he held a visiting professorship.

DR. BRIGITTE RÖTHLEIN has turned hercuriosity into a career. She is a physi-cist and freelance science writer, writing on everything to do with thenatural sciences and technology. She is particularly passionate aboutbasic research, where (as here) shesees herself as an interpreter for whatis often difficult subject matter.

Sapienza University in Rome andJohn Ellis from CERN in Geneva castdoubt on this assumption with somebold speculations. The trick theysuggest is “to wait long enough” inthe experiment – meaning one to tenbillion years! It is not necessary forexperimental physicists to sit outthis time themselves – the universewill do that for them; there aregalaxies from which light takes bil-lions of years to get to Earth.

Gamma ray bursts, which reach usnow and again from distances ofaround ten billion light years, areparticularly suitable for informingphysicists that the different lightwaves have left the starting line. Ifthese can be broken down with suffi-cient precision, the light they emitmight reveal a frequency-dependentshift in the transit time. Based onpredictions from loop quantumgravity, it could indeed be within themeasurable range.

PLANCK TO SUPPLY

PROOF OF THE LOOPS

At the Loops 05 conference, OliverWinkler and Stefan Hofmann fromthe Perimeter Institute put forwardanother idea for experimentally test-ing loop quantum gravity. They havebeen addressing the effects quan-tized spacetime had on quantumfluctuation when the very younguniverse was still expanding expo-nentially. Using a highly simplifiedmodel, Winkler and Hofmann calcu-lated that the frequency spectrum ofthis quantum fluctuation must havechanged constantly. This would stillbe making itself felt today as tem-perature fluctuations in the cosmicbackground radiation.

As the scale of the effect is around1 percent, it could be measured withthe Planck satellite planned forlaunch in 2007. If the model is cor-rect, this would also present a goodopportunity to experimentally testpredictions for quantum gravity. The

result of these measurements is ofenormous significance. If loop theo-ry is correct, it will have far-reach-ing consequences, and physics text-books will have to be rewritten. Oneof the conclusions would be that theworld is discrete and not continuous.This is not noticeable in everydaylife, but it plays a role in drastic situ-ations, such as black holes and theBig Bang.

The situation today is similar tothat at the beginning of the last cen-tury, when nothing was knownabout quantum mechanics. However,the new theory would not complete-ly displace the previous rules, as itfinds expression only in extreme cir-cumstances. We still use Newtonianmechanics today, although it has ac-tually been overtaken by the theoryof relativity. ●

Loop quantum gravity has provided the first mathematical basis for calculating back further than the Big Bang.

According to loop theory, space is full of tiny loops, analogous to the magnetic flowlines of Maxwell theory.

PHO

TO: S

CIEN

CEPH

OTO

LIBR

ARY

GRA

PHIC

: MPI

FO

RG

RAVI

TATI

ON

ALPH

YSIC

SPH

OTO

S: P

RIVA

TESO

URC

E