Three Roads to Quantum Gravity

THERE IS NOTHING OUTSIDE THE UNIVERSE

We humans are the species that makes things. So when wefind something that appears to be beautifully and intricatelystructured, our almost instinctive response is to ask, `Whomade that?' The most important lesson to be learned if weare to prepare ourselves to approach the universe scientificallyis that this is not the right question to ask. It is truethat the universe is as beautiful as it is intricately structured.But it cannot have been made by anything that existsoutside it, for by definition the universe is all there is, andthere can be nothing outside it. And, by definition, neithercan there have been anything before the universe thatcaused it, for if anything existed it must have been part ofthe universe. So the first principle of cosmology must be`There is nothing outside the universe'.

    This is not to exclude religion or mysticism, for there isalways room for those sources of inspiration for those whoseek them. But if it is knowledge we desire, if we wish tounderstand what the universe is and how it came to bethat way, we need to seek answers to questions about thethings we see when we look around us. And the answerscan involve only things that exist in the universe.

    This first principle means that we take the universe to be,by definition, a closed system. It means that the explanationfor anything in the universe can involve only other things thatalso exist in the universe. This has very important consequences,each of which will be reflected many times in thepages that follow. One of the most important is that thedefinitionor description of any entity inside the universe canrefer only to other things in the universe. If something has aposition, that position can be defined only with respect to theother things in the universe. If it has a motion, that motion canbe discerned only by looking for changes in its position withrespect to other things in the universe.

    So, there is no meaning to space that is independent ofthe relationships among real things in the world. Space is nota stage, which might be either empty or full, onto whichthings come and go. Space is nothing apart from the thingsthat exist; it is only an aspect of the relationships that holdbetween things. Space, then, is something like a sentence. Itis absurd to talk of a sentence with no words in it. Eachsentence has a grammatical structure that is defined byrelationships that hold between the words in it, relationshipslike subject-object or adjective-noun. If we take out all thewords we are not left with an empty sentence, we are leftwith nothing. Moreover, there are many different grammaticalstructures, catering for different arrangements of wordsand the various relationships between them. There isno such thing as an absolute sentence structure that holdsfor all sentences independent of their particular words andmeanings.

    The geometry of a universe is very like the grammaticalstructure of a sentence. Just as a sentence has no structure andno existence apart from the relationships between the words,space has no existence apart from the relationships that holdbetween the things in the universe. If you change a sentenceby taking some words out, or changing their order, itsgrammatical structure changes. Similarly, the geometry ofspace changes when the things in the universe change theirrelationships to one another.

    As we understand it now, it is simply absurd to speak of auniverse with nothing in it. That is as absurd as a sentencewith no words. It is even absurd to speak of a space with onlyone thing in it, for then there would be no relationships todefine where that one thing is. (Here the analogy breaks downbecause there do exist sentences of one word only. However,they usually get their meaning from their relationships withadjacent sentences.)

    The view of space as something that exists independent ofany relationships is called the absolute view. It was Newton'sview, but it has been definitively repudiated by the experimentsthat have verified Einstein's theory of general relativity.This has radical implications, which take a lot of thinking toget used to. There are unfortunately not a few good professionalphysicists who still think about the world as if spaceand time had an absolute meaning.

    Of course, it does seem as though the geometry of space isnot affected by things moving around. When I walk from oneside of a room to the other, the geometry of the room does notseem to change. After I have crossed the room, the spacewithin it still seems to satisfy the rules of Euclidean geometrythat we learned in school, as it did before I started to move.Were Euclidean geometry not a good approximation to whatwe see around us, Newton would not have had a chance. Butthe apparent Euclidean geometry of space turns out to be asmuch an illusion as the apparent flatness of the Earth. TheEarth seems flat only when we can't see the horizon.Whenever we can see far enough, from an aircraft or whenwe gaze out to sea, we can easily see that this is mistaken.Similarly, the geometry of the room you are in seems to satisfythe rules of Euclidean geometry only because the departuresfrom those rules are very small. But if you could make veryprecise measurements you would find that the angles oftriangles in your room do not sum to exactly 180 degrees.Moreover, the sum actually depends on the relation of thetriangle to the stuff in the room. If you could measureprecisely enough you would see that the geometries of allthe triangles in the room do change when you move from oneside of it to the other.

    It may be that each science has one main thing to teachhumanity, to help us shape our story of who we are andwhat we are doing here. Biology's lesson is natural selection,as its exponents such as Richard Dawkins and LynnMargulis have so eloquently taught us. I believe that themain lesson of relativity and quantum theory is that theworld is nothing but an evolving network of relationships. Ihave not the eloquence to be the Dawkins or Margulis ofrelativity, but I do hope that after reading this book you willhave come to understand that the relational picture of spaceand time has implications that are as radical as those ofnatural selection, not only for science but for our perspectiveon who we are and how we came to exist in thisevolving universe of relations.

    Charles Darwin's theory tells us that our existence was notinevitable, that there is no eternal order to the universe thatnecessarily brought us into being. We are the result ofprocesses much more complicated and unpredictable thanthe small aspects of our lives and societies over which wehave some control. The lesson that the world is at root anetwork of evolving relationships tells us that this is true to alesser or greater extent of all things. There is no fixed, eternalframe to the universe to define what may or may not exist.There is nothing beyond the world except what we see, nobackground to it except its particular history.

    This relational view of space has been around as an idea fora long time. Early in the eighteenth century, the philosopherGottfried Wilhelm Leibniz argued strongly that Newton'sphysics was fatally flawed because it was based on a logicallyimperfect absolute view of space and time. Other philosophersand scientists, such as Ernst Mach, working in Viennaat the end of the nineteenth century, were its champions.Einstein's theory of general relativity is a direct descendent ofthese views.

    A confusing aspect of this is that Einstein's theory ofgeneral relativity can consistently describe universes thatcontain no matter. This might lead one to believe that thetheory is not relational, because there is space but there is nomatter, and there are no relationships between the matter thatserve to define space. But this is wrong. The mistake is inthinking that the relationships that define space must bebetween material particles. We have known since the middleof the nineteenth century that the world is not composed onlyof particles. A contrary view, which shaped twentieth-centuryphysics, is that the world is also composed of fields.Fields are quantities that vary continuously over space, suchas electric and magnetic fields.

    The electric field is often visualized as a network of lines offorce surrounding the object generating the field, as shown inFigure 1. What makes this a field is that there is a line of forcepassing through every point (as with a contour map, onlylines at certain intervals are depicted). If we were to put acharged particle at any point in the field, it would experiencea force pushing it along the field line that goes through thatpoint.

    General relativity is a theory of fields. The field involved iscalled the gravitational field. It is more complicated than theelectric field, and is visualized as a more complicated set offield lines. It requires three sets of lines, as shown in Figure 2.We may imagine them in different colours, say red, blue andgreen. Because there are three sets of field lines, the gravitationalfield defines a network of relationships having to dowith how the three sets of lines link with one another. Theserelationships are described in terms of, for example, howmany times one of the three kinds of line knot around those ofanother kind.

    In fact, these relationships are all there is to the gravitationalfield. Two sets of field lines that link and knot in thesame way define the same set of relationships, and exactly thesame physical situation (an example is shown in Figure 3).This is why we call general relativity a relational theory.Points of space have no existence in themselves — the onlymeaning a point can have is as a name we give to a particularfeature in the network of relationships between the three setsof field lines.

    This is one of the important differences between generalrelativity and other theories such as electromagnetism. In thetheory of electric fields it is assumed that points have meaning.It makes sense to ask in which direction the field linespass at a given point. Consequently, two sets of electric fieldlines that differ only in that one is moved a metre to the left, asin Figure 4, are taken to describe different physical situations.Physicists using general relativity must work in the oppositeway. They cannot speak of a point, except by naming somefeatures of the field lines that will uniquely distinguish thatpoint. All talk in general relativity is about relationshipsamong the field lines.

    One might ask why we do not just fix the network of fieldlines, and define everything with respect to them. The reasonis that the network of relationships evolves in time. Except fora small number of idealized examples which have nothing todo with the real world, in all the worlds that general relativitydescribes the networks of field lines are constantly changing.

    This is enough for the moment about space. Let us turn nowto time. There the same lesson holds. In Newton's theory timeis assumed to have an absolute meaning. It flows, from theinfinite past to the infinite future, the same everywhere in theuniverse, without any relation to things that actually happen.Change is measured in units of time, but time is assumed tohave a meaning and existence that transcends any particularprocess of change in the universe.

    In the twentieth century we learned that this view of time isas incorrect as Newton's view of absolute space. We nowknow that time also has no absolute meaning. There is no timeapart from change. There is no such thing as a clock outsidethe network of changing relationships. So one cannot ask aquestion such as how fast, in an absolute sense, something ischanging: one can only compare how fast one thing is happeningwith the rate of some other process. Time is describedonly in terms of change in the network of relationships thatdescribes space.

    This means that it is absurd in general relativity to speak ofa universe in which nothing happens. Time is nothing but ameasure of change — it has no other meaning. Neither spacenor time has any existence outside the system of evolvingrelationships that comprises the universe. Physicists refer tothis feature of general relativity as background independence.By this we mean that there is no fixed background, or stage,that remains fixed for all time. In contrast, a theory such asNewtonian mechanics or electromagnetism is backgrounddependent because it assumes that there exists a fixed,unchanging background that provides the ultimate answer toall questions about where and when.

    One reason why it has taken so long to construct a quantumtheory of gravity is that all previous quantum theories werebackground dependent. It proved rather challenging to constructa background independent quantum theory, in whichthe mathematical structure of the quantum theory made nomention of points, except when identified through networksof relationships. The problem of how to construct a quantumtheoretic description of a world in which space and time arenothing but networks of relationships was solved over the last15 years of the twentieth century. The theory that resulted isloop quantum gravity, which is one of our three roads. I shalldescribe what it has taught us in Chapter 10. Before we getthere, we shall have to explore other implications of theprinciple that there is nothing outside the universe.

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Excerpted from THREE ROADS TO QUANTUM GRAVITY by LEE SMOLIN. Copyright © 2001 by Lee Smolin. Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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