The Universe: A View from Classical and Quantum Gravity E-Book

Martin Bojowald

• Herausgeber: Wiley-VCH
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2012

Seitenzahl: 923

Beschreibung

Written by a well-known author in the field, this book presents a modern understanding of the universe based on relativity, quantum physics and their elusive combination. It introduces the crucial theoretical ingredients in an accessible way, starting from the physics of Newton and developing subsequent theories all the way to the modern enigma of quantum gravity. The intermediate level presentation assumes only a general knowledge of math and physics, adopting a "two-level" approach: equations are retained throughout the chapters but set apart from the main text in boxes to allow for lay readers to understand the book. For scientists, researchers, students and lecturers in cosmology, astronomy, gravitation, quantum and theoretical physics; as well as mathematicians, students, lecturers, academics and non-experts in related fields with an interest in the subject.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Chapter 1: The Universe I

1.1 Newtonian Gravity

1.2 Planets and Stars

1.3 Cosmology

Chapter 2: Relativity

2.1 Classical Mechanics and Electrodynamics

2.2 Special Relativity

2.3 General Relativity

Chapter 3: The Universe II

3.1 Planets and Stars

3.2 Black Holes

3.3 Cosmology

Chapter 4: Quantum Physics

4.1 Waves

4.2 States

4.3 Measurements

Chapter 5: The Universe III

5.1 Stars

5.2 Elements

5.3 Particles

Chapter 6: Quantum Gravity

6.1 Quantum Cosmology

6.2 Unification

6.3 Space-Time Atoms

Chapter 7: The Universe IV

7.1 Big Bang

7.2 Black Holes

7.3 Tests

Acknowledgments

Index

Martin Bojowald

The Universe: A View from Classical and Quantum Gravity

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The Author

Martin BojowaldPennsylvania State UniversityPhysics DepartmentUniversity ParkUSA

Author photography on backcover:© Silke Weinsheim([email protected])

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ISBN 978-3-527-41018-7

for A

You delight in laying down laws,Yet you delight more in breaking them.Like children playing by the ocean who build sand-towers with constancy and then destroy them with laughter.But while you build your sand-towers the ocean brings more sand to the shore, and when you destroy them the ocean laughs with you.

Kahlil Gibran: The Prophet

Chapter 1

The Universe I

Observations of the stars and planets have always captured our imagination, in myths as well as methods. While the myths have lost much in importance and relevance, traces of inspiration from the stars can still be seen in modern thinking and technology.

1.1 Newtonian Gravity

A concept in physics that, despite its age, enjoys enormous importance, yet has often been remodeled, is the one of the force. In colloquial speech, it is associated with violence or, in personal terms, egotism: “It is a necessary and general law of nature to rule whatever one can.”1) We speak of forces of nature that cause inescapable calamity, or the personal force of despots small and large. Physicists, as human beings, know about these facets of force, but they have also developed it into a powerful and impartial method for predicting motion of all kinds.

Force In physics, forces play a more important role than realized colloquially, a role perhaps most powerful and at the same time, most innocent. Forces in physics are embodied by laws of nature. If there is a force acting on some material object, this object has to move in a certain way dictated by the force. The action is most powerful because it cannot be evaded, and it is most innocent because it happens without a purpose, without a hidden (or any) agenda. This concept of a force, although it has been somewhat eroded by quantum mechanics2) is one of the central pillars of our description of nature. It serves to explain motion of all kinds, to predict new phenomena, and to understand the workings of useful technology.

Forces govern what we do on Earth, but the sky is no exception. A force causes acceleration, or a change of velocity, in a precise and qualitative way: For any given object, the acceleration due to a force is proportional to the magnitude of the force; force equals mass times acceleration. This is a mathematical equality which tells us that its two sides, force and the product of mass times acceleration, are two sides of the same coin. Knowing the force in a given situation is as much as knowing the value which the product of mass and acceleration takes, for they can never be different.

One might think that the concept of a force seems to be redundant because we could always replace it with the product of mass times acceleration. However, the distinction of the two sides of an equation is important. We interpret the force, one side of the equation, as causing an action on the object, while the acceleration, on the other side of the equation, is the effect. Although both sides always take the same numerical value, their physical interpretations are not identical to each other. While we see and measure the effect of a force by the acceleration it causes on objects (or by deformations of their shapes), the value of the force itself is derived from the expectation we have for the strength of a certain action. This expectation may come from the physical strength of our arms which we know from experience, or it may be based on a detailed theory for a general physics notion such as the gravitational force. In this way, the distinction of the two sides of a mathematical equation becomes an important conceptual one: ingrained features of an actor in the play of nature, and effects that the action causes. Both sides of Newton’s second law can be as different as psychology and history.

As in a literary play, one often understands the characters by watching their actions and reactions. After some time, one begins to form a theory about their nature; their actions reflect back on them by the way we judge them and expect how they may behave in future scenes. In physics, forces are first understood by the effects they cause, such as making an apple fall to the ground; physicists then begin to judge the force, and try to understand what makes it pull or push. One forms a theory about the force, that is, more mathematical equations that allow one to compute the value of the force from other ingredients, independent of the accelerations caused on objects. This theory, in turn is tested by using it to compute forces in new situations and comparing with observed accelerations. New predictions can be made to see what the same force may do under hitherto unseen circumstances. The theory is being tested, and can be trusted if many tests are passed.

Acceleration Forces can be determined by the accelerations caused (or deformations, as a secondary effect of acceleration of parts of elastic materials). We feel this interplay physically when we attempt to exert a force: our muscles contract, and our body begins to move. For acceleration or general causes of forces, we have a more direct sense than for the forces themselves, reflected in the way physics determines forces by measuring accelerations of objects. We notice acceleration whenever our velocity changes. Here, we have a second relationship involving acceleration: it is the ratio of the change of velocity in a given interval of time by the magnitude of that interval, assumed small. The condition of smallness is necessary because the value of acceleration may be different at different times; if we were to compute acceleration as the ratio of velocity changes by long time intervals, we would at best obtain some measure for the average acceleration during that time. By taking small values of time intervals (and assuming that the acceleration over such short time intervals does not change rapidly), we obtain the acceleration at one instant of time.

The ratio of two vanishing numbers is undefined because its value depends on how those numbers, in a limiting process, approach zero. Dividing by zero is a dangerous operation and often produces meaningless infinities, also called divergences. If we have to divide a cake among four people, the pieces are smaller than those dividing it among two people. If there is only one person, the cake remains undivided. Fractional people are difficult to consider in this picture, but if we just continue the pattern, the ratio will continue to rise above the original size of the numerator, that is, the cake, and can be made large by making the denominator small.

Differentials and the limiting procedures used in them implement the idea of the continuum, an abstract concept of structureless space. Although the notion is not intuitive, it simplifies several mathematical constructions and equations. Derivatives and the continuum view therefore pervade many branches of mathematics and physics. Nevertheless, such limits may not be correct to describe nature at a fundamental level. We cannot be sure that time durations shorter than any number we can imagine do exist, or that objects can move in increments smaller still than any small number. We will come back to the physics behind these questions in our chapter on quantum gravity; for now, we must leave the answers open. In physical statements, we will forgo using the continuum limit and derivatives, and keep working with differences and the symbol Δ, even if it provides only approximations to instant mathematical changes of a curve.

Derivative The French amateur mathematician Pierre de Fermat, a lawyer by his main profession, is the first person known to have grasped the meaning of differentials, although he did not go as far as introducing them in precise terms. He paved the way for the development of calculus with its many applications in geometry and theoretical physics. In consistent form, differentiation and integration were introduced by Isaac Newton and Gottfried Leibniz, whereas Fermat’s assumptions and methods remained, from a strict point of view, self-contradictory. What was missing for Fermat was the notion of an infinitesimally small quantity, a quantity, as it were, squaring the circle by being smaller than any number, yet not zero.

Fermat’s crucial idea was to compute the position of an extremum of a curve, that is, a maximum, minimum or saddle point, by taking seriously the intuition that the curve at such a point is momentarily constant. When we walk over a hill, our altitude is unchanging as we pass the top. The curve may not be constant for any nonzero change of its argument, but it is “more constant” than at any other place.

Fermat developed his method further to compute the slope of tangents to curves, amounting to nonvanishing derivatives of the function defining the curve. Fermat did not publish his methods, which was not usual at his time, not only for an amateur mathematician. Instead, we know of his ideas by letters he sent to some of the eminent thinkers among his contemporaries. (Nowadays, this methods of publication is no longer encouraged. Important work could land in spam filters.) The method of finding maxima of a function was sketched by Fermat in a letter to René Descartes in 1638, a fitting recipient because the way Fermat proceeded to solving several geometrical problems relied on Descartes’ algebraization of geometry, an essential mathematical contribution which we will soon encounter.

Motion We now have two equations involving an object’s acceleration: Newton’s second law equating it to the force (divided by the mass) and the relation to the change of velocity. Just as there is a difference of perception of a force and its effect, there is an important difference between the two equations. We feel acceleration through the change of velocity it implies, and we relate it to an acting force only by experience or after some thinking. Similarly, the equation that relates acceleration to the change of velocity is considered much more evident than Newton’s second law. It is not even a law, relating two seemingly different quantities in a strict way, but rather a definition. When we talk about acceleration, we mean nothing else than the change of velocity in a given time. Unlike with Newton’s second law, the equality does not give us insights into cause-and-effect relationships. It does not make sense to speak of the change of velocity as the cause of an acceleration, or vice versa; under all circumstances, both quantities are one and the same, not just regarding their values but also their conceptual basis. Equating acceleration to the ratio of velocity change by time interval defines what we mean by acceleration in quantitative terms. It is a definition to clarify the meaning of a colloquial word, namely, acceleration, rather than a law that would give us insights into the interplay of different objects in nature.

Velocity and acceleration describe the motion of an object without regard to the forces that cause or influence the motion. Such quantities and the equations they obey amongst themselves are called kinematical, from the Greek word for motion. Laws that involve the force, or in general, the cause for motion, are called dynamical, from the Greek word for force. Kinematical equations, or kinematics for short, are easier to uncover and to formulate than dynamical ones because they do not require as much knowledge of interactions in nature. Nevertheless, they are important because they form the basis of dynamical laws so that the latter can refer to the effects caused by forces on motion. Moreover, kinematics is often far from trivial, and does not exhaust itself in definitions. In the context of special relativity, we will have an impressive example for how nontrivial, even counterintuitive, the kinematics of a physical theory can be.

If we know how an object is moving at all times, we are given the functional relationship x(t). From this position function, we compute the velocity function v(t) by taking ratios (and limits) of changes of position, and then the acceleration. If we also know the object’s mass, we can use the computed values of acceleration to compare with an idea we might have about the acting force and see if the values match. In this way, theories for different forces can be tested.

Integration The opposite of differentiation is integration. In mathematics, we differentiate the values of a function by computing how much it changes under small variations of its argument. Integration is the reverse procedure: we try to reconstruct the function from its minuscule changes between nearby points. Regarding motion, this procedure allows us to go from acceleration to velocity and then to position.

Once an understanding of a certain force has been established, the reverse of differentiation becomes more interesting. We start with an equation for the force, telling us what values the force has at different positions or at different times. Newton’s second law then identifies the acceleration of an object moving subject to the force. Knowing the acceleration, we would like to determine the corresponding velocity and position as functions of time, to give us a direct view of the way we expect the object to move. The process of taking ratios of differences can be reversed by choosing a starting value, that is, of the velocity, say, if we are to determine v(t) from a(t), and then adding up all differences as given by a(t).

Acceleration is defined not by the differences themselves but by their values obtained in the limit of small time intervals. If the time intervals are zero, it takes an infinite number of them to reach any time other than the one chosen as the starting point. The mathematical procedure to compute the velocity from acceleration thus requires us to sum up an infinite number of terms, all so small that the final value of the sum is still finite. The procedure, reversing the computation of a derivative, is called integration, and the value of the sum is called the integral of a(t).

Once we have a formula for the force independent of the one relating it to acceleration, we can compute the velocity by integration. Velocity, in turn, is defined as the ratio of position and time changes, taken in the limit of small time intervals. Once the velocity is known, we can, again choosing a starting value of x at some time t0, compute the position function x(t) by the same mathematical procedure, integrating the velocity. In this mathematical way, force laws are used to compute all details of the motion of objects, making predictions about the effects of forces.

Motion is not always local. Cars at an intersection may go straight or turn, even though they are in the same spot with about the same speed. In addition to the cars’ positions and speeds, we must know more, the prehistory and memory of their drivers. Such nonlocal motion would be much more complicated to compute than local motion, sensitive to just a few parameters.

Sometimes, nonlocal laws can be made local by introducing auxiliary variables: parameters that are not essential to describe the motion but play a helpful mathematical role. In the example of traffic, an auxiliary variable is realized by turn signals, or at intersections with turn lanes. Cars turning in different ways occupy separate lanes; the lane position is one new variable to describe motion along the road in local terms. (Then, we must know which lane a driver will choose.)

Algebra and geometry Both differentiation and integration have geometrical interpretations, which is another reason for their widespread use. The derivative of a function is related to its change, or when plotted as a graph to its slope. The integral of a function, less obviously so, determines the area under the graph: Figure 1.1.

Figure 1.1 Differentiation and integration. The derivative computes the slope Δhf(x)/h of a function f(x), the integral Ixx0 (f) the area under the curve between x0 and x. The derivative of the integral is the original function: , for small h is an area of the size f(x)h.

If integration is the opposite of differentiation, we can proof certain properties of integration by showing that they are undone by differentiation. As an example, we claim that the integral Ixx0 (f) of a function f(x), starting at one point x0 and proceeding to another x, is equal to the area under the curve between points x0 and x. If we can show that the derivative of the area is equal to the original function, our claim is proven.

Calculus The derivative of is the limit of for h small. If is equal to the area under the curve from x0 to x, is the area from x to x + h. If h is small, this area is close to hf(x), the area of a rectangle with width h and height f(x). Dividing by h and letting h go to zero produces the original function f evaluated at the endpoint x, as was to be shown. Integration can undo differentiation only when we take limits of quantities like h going to zero, one of the reasons for the mathematical power of infinitesimals. When we want to leave the question of limits open, in equations of physics, we will write an explicit sum Sumxf(x)ΔL(x) for the integral, summing over all points x separated by ΔL(x) in some interval. The continuum version uses smoothed-out notation, .

Differentiation and integration, or calculus as the subject of their study is called, have two sides: an algebraic one and a geometric one. The algebraic side appears in the definitions given, according to which the values are ratios of numbers in a derivative or sums of terms in an integral. It also shows its formal face when differentiation and integration are applied to special classes of functions. Computing a derivative according to the definition, from quotients of changes of a function, can be tedious. It is often easier to compute derivatives for certain functions that appear often, and then combine those results according to general rules until one finds the derivative of a particular function of interest. Once the derivatives of a large class of functions are known, also integration is facilitated because one no longer needs to sum up terms as the definition suggests; rather, one can use differentiation tables to look for a function whose derivative agrees with the function to be integrated.

Given an operation, interesting objects are always those that remain unchanged, or are reproduced to their original form after a finite number of repetitions. The operations are then easy to perform, even if the invariant or reproduced objects may be difficult to find or construct. Knowing that an operation does not change an object, or can be applied several more times to go back to the original form, has something reassuring.

Trigonometry Algebra and geometry overlap in many areas, also regarding trigonometric functions. The sine and cosine functions allow us to compute angles from ratios of side lengths in a right triangle. The functions can be introduced as coordinates of a point moving along the unit circle, a geometrical realization. The coordinates, by reference to a right-angled triangle in Figure 1.3a with the origin of coordinates and the point on the circle as two corners and the third on the horizontal axis, are given by (cos(, sin) if is the angle by which the point has moved from the horizontal axis.

Figure 1.3 Sine sin(x) and cosine cos(x), periodic functions. (a) Their definitions as coordinates of points on the unit circle of radius one. (b) Plots as functions of fractions x of the full circumference.

The geometrical identification of sine and cosine with coordinates of points on a circle helps us to compute derivatives of these functions. To derive changes under small variations of the angle, some trigonometric identities are useful.

The trigonometric identities allow us to demonstrate that the sine and cosine functions are reproduced by four steps of differentiation, intercommuting them after a single step. The interplay of algebra and geometry facilitates the calculations, an example for the usefulness of being able to take both the algebraic and geometric viewpoints.

Algebra and geometry The interplay of algebra and geometry has been important ever since the beginnings of modern physics; it even played a large role before physics as its own field of study was established. The philosopher René Descartes was the first to algebraize geometry in a form still used today. By describing the shape of a geometric object in terms of coordinates, that is, unique numbers assigned to all points in a plane or in space, most geometrical questions, such as those about the length of a curve, relationships between angles and side lengths of triangles, or the areas of figures, can be formulated in algebraic terms and then be answered with established methods to solve equations. Doing geometry is formalized to the extent that a direct geometrical picture, the ability to visualize geometrical objects and relations between them, becomes secondary. Only in this way has it been possible to drive mathematics to higher and higher levels of abstraction, leading for instance to the concepts of higher-dimensional and curved space. Despite their abstraction and the loss of direct imagination, these concepts can be real, as evidenced by modern theories of physics, in particular relativity.

Using the bridge between algebra and geometry both ways, geometrical visualization can also be employed to understand and derive algebraic relationships. Algebra and geometry are two sides of a coin, both essential. Intuition remains important to investigate geometry, pose interesting questions and interpret algebraic results; it was not Descartes’ intention to cut all this short by dry, formalistic manipulations. As geometry advanced over the centuries, more complicated figures had come to focus which were difficult to visualize even for trained and experienced geometers, at a time well before computer-aided visualization tools became available. There are examples in books and letters in which properties of curves were analyzed correctly even though accompanying sketches of them showed the wrong behavior.

Numbers Neither Descartes nor Fermat had a correct understanding of the shape of the Cartesian leaf, as shown by the incorrect diagrams found in their correspondence. Nevertheless, the results obtained by Fermat for tangent lines to the curve were correct, a fact which shows the importance of algebra in situations in which our intuition no longer suffices to visualize mathematical constructions. A similar sentiment was voiced about 200 years later by Carl Friedrich Gauss in a letter to Friedrich Wilhelm Bessel in 1830, in which he declared arithmetic to have primacy over geometry, saying that only the former’s laws are “necessary and true.” This correspondence happened under the impression of new versions of geometry, discovered by Gauss and others, that did not obey all the traditional rules laid out by the ancient Greeks, known as Euclid’s axioms. By using algebra, Gauss was able to perform geometry even in settings outside of the traditional realm of this field. Geometry and its rules were generalized, while algebra remained the same. By now, Gauss’ sentiment toward algebra as primary compared to geometry is no longer considered valid. Our imagination may fail, but not geometry. And just as geometry, also algebra can be formulated in different systems, depending on which of its rules one would like to be obeyed.

A set of numbers (or other objects, for what makes a number a number, in the eyes of a mathematician, is just the kind of rules it obeys) is called a group if (i) one can multiply any two of its elements, (ii) the multiplication of three elements does not depend on whether one multiplies the product of the latter two with the first or the product of the first two with the latter, (iii) there is a unit element which does not change any other number when multiplying by it, and (iv) any element can be inverted so as to provide the unit element when multiplied with this inverse number. The inverse provides the possibility of division, or of subtraction, for we could have stated the same laws with “addition” instead of “multiplication,” “sum” instead of “product,” “zero” instead of “unit element,” and “negative” instead of “inverse.” A group, however, does not offer both multiplication and addition at the same time, so its elements cannot be fully respectable numbers in the common sense.

Mathematicians call an “algebra” a set of objects any two elements of which can be added or multiplied, or a “division algebra” if there is also division (or inversion) among the objects. Commutativity, or even associativity, is often forgone in order to allow a larger variety.

Numbers that can be added, subtracted, multiplied and divided by can be defined in different ways, but there are not many options, only four. Two of them are the familiar real and less familiar complex numbers; two more are the quaternions and octonions. Complex numbers are pairs of real numbers, a two-dimensional extension. Quaternions are pairs of complex numbers, or sets of four real ones, and octonions again double the dimension. However, as we go up in dimensions, we lose laws. Quaternion multiplication is not commutative, and octonions don’t even obey the law of associativity. If we try to extend them further, no number system is possible. Our laws of numbers stand firm; they cannot be realized in arbitrary ways. Real numbers indeed seem real.

Space-time Descartes’ algebraization was important, not only for the long-established subject of geometry, but also for the emerging field of physics. It preceded calculus, as seen in the example given by Fermat, a contemporary of Descartes’ with whom he kept correspondence. And calculus, in the early days, was very much developed in parallel with laws of motion needed to address questions in physics. Stepping far ahead, much of modern physics would not be possible without an algebraic view on geometrical concepts, such as spaces of dimension larger than three, which cannot be visualized by our minds that evolved and grew up in a three-dimensional environment. Algebra, and with it an operational instead of visual view, has for a long time taken primacy in our understanding of physics, until his insights into relativity forced Albert Einstein to geometrize physics. Nowadays, geometrical formulations, even if the geometry in them may be highly abstracted and hardly recognizable, are considered more powerful and even more fundamental than algebraic ones. Most powerful, however, remains the interplay of algebra and geometry as it is needed to imagine new relations and then do computations to realize them.

One of the first instances in which the new geometrical view of physics became crucial was the understanding of space-time in relativity. In a rather different way, Immanuel Kant already associated the pair of algebra and geometry with the distinction of space and time. Geometry, according to him, is associated with space; we visualize objects in space and determine their relationships when we do geometry. Algebra, on the other hand, he associated with time; we do algebraic operations, such as manipulations of an equation in order to solve it, in sequential order, one at a time. This view of time suggests an operational interpretation, in which what happens, and ultimately the whole universe, amounts to running a computer program, a sequence of operations. It seems a small step to conclude that also our role as human beings is nothing but the outcome of a calculation, just to be fed into another part of the program, an idea which seems rather popular in modern culture. As for understanding the universe (leaving aside humans), such a view is not justified in modern physics, where not just the algebraic ideal of following rules but also intuition as embodied by geometry is central. A unified view of algebra and geometry has emerged in physics, at just about the same time when the two poles of space and time, brought in contact with the mathematical fields by Kant’s arguments, have conglomerated into a single whole: space-time.

Personal force Ideas are often driven by and associated with individuals. It was Descartes who had the thought of endowing the empty plane, the stage of geometry, with a structure of its own, laying on it a grid of numbers to name its points; for this idea, we still honor him by calling those numbers Cartesian coordinates. Much later, Einstein began contemplating how these coordinates, that is, numbers to identify points, chosen with almost as much arbitrariness as the names we assign to people (or pets), can, in some cases, be equipped with physical meaning, related, for instance, to distances between points measured by different observers. The coordinates are not unique, and two different observers need not always agree on their values. However, transformation formulas can be provided that allow one to compute the numbers assigned by one observer from those obtained by another one. After that, Einstein developed a framework by which physical predictions can be derived in a way insensitive to the arbitrariness of coordinates, even if coordinates are used to do explicit calculations. From a mathematical perspective, these two steps were crucial for the geometrization of modern physics; from a physical one, they introduced special relativity (the laws of transformation formulas to relate results by different observers in space-time) and general relativity (a framework to describe space-time independently of the arbitrariness of coordinates).

As in these examples, the personal fate and success of an individual can be inspirational; stories of people, in lively contrast to the impersonal results of science, can motivate (or sometimes deter). However, in the inevitable selection of stories to tell, there lies a danger. The importance and influence of those named may seem exaggerated; those unnamed one cannot see. Like ants exploring a jungle to scavenge for food, scientists start at a firm base and spread out to reach new territory. To stay secure, one should not lose ties to others; ties may sometimes be thin, but are strengthened by success: more are attracted to the same spot, to help or envy. Or ties die down when a source is depleted, most scavengers moving elsewhere. It is not a genius ant which finds new food and brings home most of it, but one that is hard-working and keeps its eyes open, to be ready at the right moment and place. It is difficult to do justice to the many contributors to science; referring to the usual stories about those already standing in bright light only makes things worse by further dimming all others. There is not much more annoying than having justice attempted in an unjust way.

What is more, focusing on the bright side ignores, downplays or denies the darkness that can befall scientific developments, like all human activities. It goes without saying that scientists sometimes make mistakes, errors which may be worth pointing out, not to slander but because one can learn much even from mistakes. The best scientists have not hesitated to admit mistakes; some even highlighted them to warn or teach others. A significant number of important theories, including general relativity and quantum mechanics, started with versions that were, in strict terms, wrong. And yet, they were of great influence; the “correct” theories we now work with would not have come to existence without their predecessors. Initial errors were results of some kind of incompleteness which could rather easily be overcome once the groundwork was set down. Nobody should be ashamed of such errors, and no one should fault anybody for them. Only mediocre individuals need be afraid of talking about their mistakes, a true scientist can rest assured in the validity of Theodore Roosevelt’s ode to the hero: “It is not the critic who counts: not the man who points out how the strong man stumbles or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood, who strives valiantly, who errs and comes up short again and again … who spends himself in a worthy cause; who, at the best, knows, in the end, the triumph of high achievement, and who at the worst, if he fails, at least he fails while daring greatly.” For physicists trying to translate the book of nature, it holds that6) “Translation, almost by definition, is imperfect; there is always ‘room for improvement,’ and it is only too easy for the late-comer to assume the beau rôle.”

Darkness in science can reach personal levels. Scientists must sometimes face attacks from outside, hindering progress.7) Although science does occasionally follow the mythical idea of a siblinghood with unselfish interest in understanding nature, many moves, including those by some prominent scientists, can only be understood as attempts to keep competition in check. Science, much like politics and so much else, is to a large degree powerplay. Not only has modern science with its large collaborations and often high stakes (not to speak of tight job and funding markets) made acquaintance with the role of power; examples can be found in the days when it was pushed only by a few. Galileo Galilei, for one, attacked his contemporary, and great ally in heliocentric pursuits, Johannes Kepler, with whom he otherwise corresponded in a friendly manner, on the theory of the tides, developed in differing ways by both scientists. (Kepler’s version turned out to be closer to the correct explanation.) In retrospect, history seems to be made by the powerful, but in a more Tolstoyan view, progress is driven by the often unseen masses, by the many contributions from individuals.

Some scientists, of course, deserve more credit than others. When contributions made by someone are especially important and attract many followers, even more so when new contributions are made by the same person several times in a lifetime, the person, not the contributions, is often placed in the foreground. However, even if a person is granted multiple discoveries, personal acclaim may not always be justified. Power gained from the first success becomes a self-enhancing tool to facilitate more success. More funding and colleagues can be attracted with more power, and new influential work is a consequence gained more easily than by someone starting from scratch. Power is attracted by the powerful; only during revolutions does transfer of power happen in the opposite way. (Einstein, for instance, joined the powerful after his first works, but his role is special: he continued to stay outside the mainstream, preferring to follow his own ideas as far as he could without the need to buttress old successes. He did not follow and he did not force. One might even go as far as saying that this was his greatest achievement.) Science is often seen as open to diverse opinions, but it is much less susceptible to revolutions or true democracy. The powerful stand in bright light, while others may campaign in vain. In addition to the self-enhancement caused, there is, sadly, a form of corruption in science. Publications, citations and the credit one receives are the currency of science. Power increases one’s visibility, and can be used to gain more credit by self-enrichment. Sometimes, the research field occupied by a powerful scientist appears as a fiefdom, as a piece owned by the scientist. Opposing opinions are suppressed, and credit is given to close followers or within a small citation circle of scientists primarily citing one another’s work. In the end, the competition stifled, it is not always the correct view that emerges.

When examples of scientists are given, one should see them not as owners, in general, not even as originators of what they contributed. The admirable way of doing science is not to conquer, but to explore; of being creative, not to sire, but to conceive. The new arises not in brief ecstatic breakthrough moments, but by long, painful nurturing. In the main story lines of science, discoveries have been made by long contemplations of existing knowledge and of what was dissatisfying or inconsistent about it. Often with luck and hard work, one converged to a better version. A successful scientist does not own contributions any more than a construction worker owns the house he helps to build. If a piece of the truth is owned by someone, or by a group of people, it ceases to be true.8)

Mathematics The hallmark of science is its principle that, in trying to understand nature, the ultimate judgment about the correctness of our ideas can only come from nature herself We observe phenomena, form ideas about their origin and build theories about their workings, then perform experiments to test our theories, comparing theoretical predictions with observations. Science begins and ends with nature and our role in it.

In the intermediate stages of this process, one sees another virtue of science: mathematics used to organize observational data, to model phenomena, and to derive new predictions. The true and unique virtues of science can be found in this interplay of noting nature and minding mathematics, the two incorruptible pillars in the world of the mind. Without mathematics, an understanding of the universe as profound as we have it today would not have been possible. Mathematics in science holds a role both of power and of artistic