EInstein Revolutinary Papers

EINSTEIN'S REVOLUTIONARY PAPERS
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Although Einstein wrote five fundamental papers in 1905, only one - the article showing that light consists of discrete quantum particles - was truly revolutionary, argues John S RigdenEinstein's annus mirabilis of 1905 is rightly a cause for celebration. In less than seven months, Einstein wrote five history-making papers. He proposed the particle theory of light, developed a method to measure molecular dimensions, explained the long-puzzling Brownian motion, developed the theory of special relativity, and he finished his intellectual sprint by producing the world's most famous equation, E = mc².

New vision

The creative outpouring that Einstein exhibited in 1905 stands alone in the history of physics. After 100 years of sweeping advances in the subject since then, the content of these papers remains at the bedrock of our discipline (see Five papers that shook the world Physics World January 2005 pp16-17). But although all of Einstein's 1905 papers were fundamental, only one paper was truly revolutionary.

What makes a physics paper revolutionary? Perhaps the most important requirement is that it contains a "big idea". Next, the big idea must contradict the accepted wisdom of its time. Third, physicists capable of judging the intrinsic merit of the big idea typically reject it until they are forced to accept it. Finally, the big idea must survive and eventually become part of the woodwork of physics.

Only Einstein's March paper "On a heuristic point of view concerning the production and transformation of light" (Ann. Phys., Lpz 17 132-148) meets these criteria.

Quantum beginnings

The big idea in Einstein's March paper was his gentle suggestion that light consists of individual, discrete, localized and indivisible quantum particles. This blithely made, audacious claim contradicted a century of compelling empirical evidence, and it challenged the crowning achievement of 19th-century theoretical physics: the electromagnetic theory of light. It can be argued persuasively that Einstein's March paper was the start of quantum physics.

The quantum idea had been introduced by Max Planck in 1900; however, he did this tentatively and under duress (see "Max Planck: the reluctant revolutionary" by Helge Kragh Physics World December 2000 pp31-35). Planck's quantum had nothing whatsoever to do with the radiation he sought to explain. Rather, he divided the energies of the vibrating charged oscillators (the source of the black-body radiation) into finite energy elements so that he could find the entropy of the oscillators via Boltzmann's probabilistic approach. For Planck, the "energy elements" were not physically real, but a mathematical means to his objective. Planck was adamantly opposed to the concept of light quanta.

Einstein's path to the light quantum was not guided by experimental data: there were no data in 1905 that required light to be particulate. Einstein's starting point was the obvious contradiction between continuity and discontinuity. Physicists were pleased with their electromagnetic wave theory of light, and were intrigued by atoms and the evidence for subatomic particles. But even the cleverest among Einstein's contemporaries were not troubled by the continuity of light and the discontinuity of atoms. Einstein, however, was concerned. He recognized the fundamental problems that occur when extended light waves and point-like atoms are brought together - for example, when atoms emit or absorb light. It was this juxtaposition of light and atoms that he addressed in his March paper.

After acknowledging that the wave theory of light had "proved itself splendidly in describing purely optical phenomena", Einstein immediately points out that "optical observations apply to time averages and not to momentary values". However, he continues, observations associated with"the production [emission] or conversion [absorption] of light" are not time averages, but involve "momentary values". Einstein then writes what the science journalist Albrecht Fölsing has called the most "revolutionary" sentence written by a physicist in the 20th century.

"According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole."

Einstein's light quantum does not come from a theory that ends with quod est demonstratum. The first two sections of his March paper are tangential to his purpose, and what follows comes from Einstein's deep well of intuition; specifically, his quantum postulate emerges from an analogy between radiation and an ideal gas. Einstein derives the entropy change at constant temperature of both an ideal gas and radiation when each is compressed from a volume V₀ to a lesser volume V. Employing the Boltzmann principle S = (R/N)lnW - where S is entropy, R is the ideal-gas constant, N is Avogadro's number and W is the "relative probability" of a state - Einstein extends, by analogy, his results for a sample of an ideal gas to a sample of radiation. He concludes that "radiation...behaves thermodynamically as if it consisted of mutually independent energy quanta of magnitude Rβν/N", where β is a constant and ν is the frequency of the quanta. The ratio Rβ/N is what we now call the Planck constant, h.

Einstein's "revolutionary" paper has the strange word "heuristic" in the title. This word means that the "point of view" developed - that is, the light particle - is not in itself justified except as it guides thinking in productive ways. Therefore, at the end of his paper, Einstein demonstrated the efficacy of light quanta by applying them to three different phenomena. These were the photoelectric effect, the ionization of gases by ultraviolet light, and Stokes' rule, which says that when light of frequency n1 is converted through photoluminescence to light of frequency ν₂ then ν₂≤ν₁. But the phenomenon that demonstrated the efficacy of Einstein's light quantum most compellingly was (and is) the photoelectric effect.

In time, the photoelectric effect became a staple of physics textbooks. Teachers like it because given all the experimental data, only part of which was known in 1905, the photoelectric effect provides the basis for a simple-minded step to the hypothesis of light quanta. The pedagogical prominence given to the photoelectric effect, as well as the oft-made assumption that Planck proposed the light quantum in his earlier black-body work, have led many physicists to refer to the March paper as the "photoelectric-effect paper". The fact that Einstein won the Nobel prize for the photoelectric effect has also played a role; in truth, the photoelectric effect was a compromise solution because the Royal Swedish Academy of Sciences did not accept the quantization of light and would not recognize the theory of relativity.

All of this trivializes Einstein's only revolutionary 1905 paper. Indeed, three reputable physicists who recently debated Einstein's miracle year on Science Friday - a US radio show - did not once refer to his March paper during the one-hour programme.

Waiting for acceptance

Einstein's big idea was universally rejected by contemporary physicists; in fact, Einstein's light quantum was derisively rejected. When Max Planck, in 1913, nominated Einstein for membership of the Prussian Academy of Science in Berlin, he apologized for Einstein by saying, "That sometimes, as for instance in his hypothesis on light quanta, he may have gone overboard in his speculations should not be held against him." Moreover, Robert Millikan, whose 1916 experimental data points almost literally fell on top of the straight line predicted for the photoelectric effect by Einstein's quantum paper, could not accept a corpuscular view of light. He characterized Einstein's paper as a "bold, not to say reckless, hypothesis of an electro-magnetic light corpuscle of energy hν, which...flies in the face of thoroughly established facts of interference" (1916 Phys. Rev. 7 355-358). About the time Millikan wrote these words, Einstein wrote a letter to his friend Michele Besso and said that the existence of "the light quanta is practically certain".

In his 1922 Nobel address, Niels Bohr rejected Einstein's light particle. "The hypothesis of light-quanta", he said, "is not able to throw light on the nature of radiation." It was not until Arthur Compton's 1923 X-ray scattering experiment, which showed light bouncing off electrons like colliding billiard balls, that physicists finally accepted Einstein's idea. Bohr, however, continued to reject the light particle until mid-1925, and had even been willing to sacrifice the conservation of energy to keep the light quantum off the stage of physics.

In 1926 Einstein's light particle became the photon, named by Gilbert Lewis. Since then, the photon has become omnipresent in physics. Gone is the electromagnetic field spreading continuously through space, and in its place is a quantized field. Gone is Coulomb's force continuously filling the space between two charges; in its place are two charges exchanging localized photons. With Einstein's light particle leading the way, the other basic interactions now have their own "photons" that transmit forces through exchanges of specific field quanta. The photon is now ingrained in the woodwork of physics.

Revolutionary thoughts

In 1909, while Einstein's light quantum was being ignored by physicists, he wrote a paper entitled "On the present status of the radiation problem" (Physikalische. Zeitschrift 10 185-193). In this paper, Einstein acknowledges Planck's radiation law and writes that it "can be understood if one uses the assumption that the oscillation energy of frequency n can occur only in quanta of magnitude hn". However, Einstein continues, "it is not sufficient to assume that radiation can only be emitted and absorbed in quanta of this magnitude, i.e. that we are dealing with a property of the emitting or absorbing matter only".

Then, referring to results mentioned earlier in the paper, Einstein points out that those results are valid "if the radiation consisted of quanta of the indicated magnitude [hn]...The consequences of the theory of light quanta is, in my opinion, one of the most important tasks that the experimental physics of today must solve".

In his "autobiographical notes" of 1951 - Albert Einstein: Philosopher-Scientist (edited by Paul Schillp) - Einstein said that when Planck introduced the quantum, it "was as if the ground had been pulled out from one, with no firm foundation to be seen anywhere, upon which one could have built". In the years immediately following 1900, Einstein may well have been the only physicist to fathom the deep significance of Planck's quantum. Einstein's light particle forever changed physics by transforming Planck's quantum from a mathematical convenience to a basic physical concept. That was a revolution.

EINSTEIN'S QUESTION FOR UNIFICATION

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Einstein's quest for unification

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The last 30 years of his life were spent on a fruitless search for a unified field theory, but as John Ellis explains, Einstein put this "holy grail" of modern physics on the theoretical map.

The definitive scientific biography of Einstein, Subtle is the Lord..., which was written by Abraham Pais in 1982, delivered an unequivocal verdict on Einstein's quest for a unified field theory. Pais wrote that the time for unification had not come, and that Einstein's work "led to no results of physical interest". But a lot of water has flowed under the bridge of unification since then, allowing us to look back with perhaps more indulgence as we celebrate the centenary of Einstein's 1905 papers.

Let us briefly recall the relevant physics that was known in the 1920s, when Einstein embarked on his quest. The only known subatomic particles were the proton and the electron: the neutron and the neutrino, for example, were not predicted or discovered until the 1930s. Most "fundamental" physicists were striving to understand quantum physics - an endeavour from which Einstein stood apart. The structure of the nucleus was regarded as an interesting but secondary problem, and the unification of forces was considered, in the words of Pais, a minor issue.

For Einstein and his few unification-minded colleagues the big issue was to unify general relativity - a theory of gravity - with Maxwell's electrodynamics. Theodor Kaluza and Oskar Klein proposed starting from a 5D theory, which contained an extra "compactified" spatial dimension in addition to the three spatial and one temporal dimensions of everyday experience. Electromagnetism then emerged naturally from this extra dimension.

Perhaps more so than Pais, we now recognize these early theories as breakthroughs in unification because of their many echoes in the supergravity and string theories of the past 20 years. Einstein was an early enthusiast; as he wrote to Kaluza in April 1919, "The idea of achieving unification by means of a five-dimensional cylinder world would never have dawned on me...At first glance I like your idea enormously". Kaluza published his idea in 1921, which Einstein pursued in his first unification paper with Jacob Grommer the following year. Indeed Einstein was to return to 5D theories every few years for the rest of his life.

However, even Einstein had to admit that his unification papers were not always ground breaking. For example, after some initial confusion he recognized that the two papers he wrote in 1927 were equivalent to the work of Klein. But he might have been happy to know that some of today's particle physicists will search for Kaluza-Klein excitations using the Large Hadron Collider at CERN. Einstein had hoped to identify quantum fields with such higher components that only arose in the 5D theories.

Generalization

Another recurring theme in Einstein's quest for unification was to generalize the "metric" of relativity - the symmetric tensor that describes the curvature of space-time - so that it could also describe the electromagnetic field. He pursued many apparently blind alleys, such as asymmetric generalizations of the metric, and even postulated that there might be no tensor at all. As Einstein himself said in a letter to Klein in 1917, "this process of deepening the theory has no limits".

Unfortunately, these ideas were unsuccessful. For example, in his first unification paper in 1925 the antisymmetric part of his tensor field was not suitable for describing all the components of the electric and magnetic fields. Indeed, none of Einstein's unification attempts ever reproduced the free-field Maxwell equations. In Einstein's defence, it should be mentioned that we now recognize that other types of antisymmetric tensor fields emerge naturally from string theory. However, this type of theory had not been invented in Einstein's day.

A more basic problem with many of Einstein's proposals was that they did not include the general theory of relativity itself. However, in his final years following 1945 he returned to a theory with a fundamental tensor that was not symmetric and would include both the metric and the electromagnetic tensor, which avoided some of these problems.

No stone left unturned

It is difficult to accuse Einstein of leaving stones unturned - no matter how unpromising they might appear. For example, in the early 1940s he even toyed with the idea that nature might not be described by partial differential equations. Modern theorists can hardly be accused of excessive conservatism, but even they have not revived this startling speculation!

What is most impressive about Einstein's quest for unification was his persistent indefatigability. He tried many different ideas, and often returned to earlier theoretical haunts, such as Kaluza-Klein theories, with something new to say. However, the truth is that he was adrift from many of the most important developments in physics at the time. For instance, he was famously sceptical - if not downright hostile - towards quantum physics, and he does not seem to have followed closely the discoveries of new particles and interactions. More surprisingly, perhaps, he seems to have missed out on some of the most far-reaching new theoretical ideas of that period, which now play key roles in modern approaches to unification.

For example, Einstein recognized Hermann Weyl's seminal 1918 work on scale transformations in four dimensions, even paying it the backhanded compliment that "apart from the agreement with reality, it is at any rate a grandiose achievement of the mind". Weyl's ideas led to the discovery in the late 1920s of local phase transformations, which laid the foundations for the gauge theories of the weak and electromagnetic interactions in the 1950s and beyond. However, Einstein was never involved personally in these far-reaching developments.

He also seems to have been affected by frequent mood swings during his quest for unification. On several occasions he switched rapidly from unwarranted optimism about the prospects of a new idea to complete rejection. More alarmingly, his mood often swung in the full glare of publicity. For many years a new scientific paper by Einstein was a major public event, with hundreds of journalists hanging on the utterances of the great man. The closest present-day parallel would be Stephen Hawking and his recent comments on black holes and quantum mechanics.

Einstein's legacy

Why were Einstein's papers on unification not more successful? It is surely insufficient simply to say that only young theorists have brilliant new ideas. The many distractions of fame in his later years should also not get all the blame. Einstein himself wrote in his early years that "formal points of view...fail almost always as heuristic aids". But later he seems to have abandoned this insight in his quest for unification, and instead was seduced more by mathematical novelty than by physical intuition.

It could be, however, that Einstein was simply ahead of his time, since even if he had been following contemporary physics more closely, the information available before his death was probably insufficient to make significant progress in unification. For example, the unification of the weak and electromagnetic interactions in the 1960s required many unforeseen experimental discoveries as well as new theoretical ideas. Even now, the unification of gravity with the other interactions - which was Einstein's true dream - still eludes us.

Following Einstein, most theoretical physicists assign a central role to geometrical ideas. Most of the particle-physics community believes, for example, that string theory provides the appropriate framework for realizing Einstein's dream. Here, fascinating generalizations of Kaluza and Klein's hidden dimensions, such as "Calabi-Yau manifolds", are able to dispose of the several extra dimensions required by the theory. However, not all general relativists are convinced, and there is absolutely no experimental evidence for string theory. Are we also in danger of being seduced by formal beauty?

Although some of the unification ideas pursued by Einstein are now recognizable in developments such as string theory, this is not to say that Einstein's work actually inspired these modern unification attempts. It seems to me that the real significance of Einstein's quest for unification lies in its quixotic ambition. Einstein, more than any of his contemporaries, put unification on the theoretical map and established it as a respectable intellectual objective. Even if we do not have all the necessary theoretical tools or experimental information, unification is the "holy grail" towards which our efforts should be directed.

THE KING IS DEAD

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The king is dead. Long live the king!

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There is the Einstein who grew up, worked and died, but there is also the Einstein who became the public face of science. Robert P Crease explains the difference.

In his classic work The King's Two Bodies: A Study in Medieval Political Theology the historian Ernst Kantorowicz examined the development of the political doctrine that distinguished between a monarch's natural body and his or her political body. Whereas the monarch’s natural body is mortal – it lives, breathes, becomes ill and dies – the political body,which is the embodiment and representative of the state, is immortal. Yet somehow the two bodies comprise a single unit in making appointments, conducting wars and signing treaties. The paradox is encapsulated in the expression, "The king is dead. Long live the king!". Einstein has such a great and enduring cultural visibility that it is tempting to try to understand him in similar terms.

THE OTHE SIDE OF THE EINSTEIN

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The other side of Albert Einstein

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Einstein has attained iconic status as a scientist and humanist, but he has also been labelled a plagiarist, a philanderer and an absent father.

Einstein's standing as a scientific genius and cultural icon is second to none. His contributions to physics and his wider intellectual concerns have led to countless accolades: for example he was named "Person of the century" by Time magazine, and the greatest physicist of all time by Physics World at the turn of the millennium. But Einstein has also received his fair share of bad press in recent years.

THE POWER OF ENTANGLEMENT

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The power of entanglement

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Hating the inherent randomness of quantum mechanics, Einstein tried to show that the theory was incomplete by drawing attention to a phenomenon that we now call entanglement. As it turns out, entangled particles are the key to quantum computing.

EINSTEIN is rightly famed for his revolutionary work on relativity. But he was also one of the founders of quantum physics and in 1905 became the first physicist to apply Max Planck's quantum hypothesis to light. Einstein realized that the quantum picture can be used to describe the photoelectric effect – that only light above a certain frequency can eject electrons from the surface of a metal. Indeed, it was mainly for deriving the law of the photoelectric effect that he was awarded the 1921 Nobel Prize for Physics.

DO YOU PLAY DICE?

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Do you play dice?

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Test your knowledge of the world's greatest physicist

Note added 1 April 2005: The winner of the quiz below was Diego Castedo of the Department of Physics at the Universidad Mayor de San Andrés in La Paz, Bolivia. He got all but two of the questions right, including the fact that Einstein was not left-handed, was not a vegetarian (except in the last year of his life), and did not approve the patent for the Toblerone chocolate bar while working in the Swiss patent office; the latter story appears to be an urban myth. The correct answers are given at the bottom of this page.

Facts and figures

All the answers to the following questions appear somewhere in this issue.
1. From which university did Einstein receive his PhD?
2. How many children did Einstein have with his first wife Mileva?
3. Which two musical instruments did Einstein enjoy playing?
4. How many references did Einstein include in his first 1905 paper on special relativity?
5. What part of Einstein's body was not cremated after he died?
6. Which university currently owns Einstein's papers?

Who said that?

7. Who told Einstein to "stop telling God what to do"? (A. Niels Bohr B. Paul Dirac C. Werner Heisenberg)
8. When asked if it was true that only three people in the world understood Einstein's theory of relativity, who is reported to have said, "I'm just trying to think of who the third person might be". (A. Arthur Eddington B. Edwin Hubble C. Max Planck)
9. Who declared during a colloquium by Einstein, "You know, what Mr Einstein said is not so stupid!". (A. Paul Ehrenfest B. Wolfgang Pauli C. Erwin Schrödinger)
10. Shortly after Einstein first became known in the physics community, who said, "I only hope and wish that fame does not exert a detrimental influence on his human side". (A. His friend Michele Besso B. His sister Maja C. His first wife Mileva Maric')
11. Who declared in 1966 that Einstein "was almost wholly without sophistication and wholly without worldliness"? (A. Robert Oppenheimer B. I I Rabi C. Victor Weisskopf)
12. Who said that Einstein's work on general relativity was "one of the greatest - perhaps the greatest - achievements in the history of human thought"? (A. W H Bragg B. Ernest Rutherford C. J J Thomson)

Mix and match

About whom did Einstein say the following? Match the six quotes to the six people.
13. "He was one of the finest people I have ever known...but he really did not understand physics."
14. "[He] was as good a scholar of mechanics as he was a deplorable philosopher."
15. "She has a sparkling intelligence, but despite her passionate nature she is not attractive enough to present a danger to anyone."
16. "He is truly a man of genius...I have full confidence in his way of thinking."
17. "She is an unfriendly, humourless creature who gets nothing out of life."
18. "He was one of my dearest acquaintances, a true saint, and talented besides."
A. Niels Bohr
B. Marie Curie
C. Paul Langevin
D. Ernst Mach
E. Mileva Maric'
F. Max Planck

True or false?

19. The FBI kept a file on Einstein.
20. Einstein was left-handed.
21. Einstein was a vegetarian.
22. Einstein approved the patent for the Toblerone chocolate bar while working in the Swiss patent office.
23. Einstein won the Nobel prize for his work on special relativity.
24. Einstein worked on the Manhattan nuclear-bomb project for the Allies.

Answers
1 Einstein's PhD was from the University of Zurich.
2 His first wife Mileva had three children.
3 The two instruments he could play were the violin and the piano.
4 There were no references in Einstein's first 1905 paper on special relativity.
5 Einstein's brain was not cremated after his death.
6 His papers are currently owned by the Hebrew University of Jerusalem.
7 A (Niels Bohr)
8 A (Arthur Eddington)
9 B (Wolfgang Pauli)
10 C (Mileva Marić)
11 A (Robert Oppenheimer)
12 C (J J Thomson)
13 F (Max Planck)
14 D (Ernst Mach)
15 B (Marie Curie)
16 A (Niels Bohr)
17 E (Mileva Marić)
18 C (Paul Langevin)
19 True
20 False
21 False
22 False
23 False
24 False

STRANGE WAYS OF LIGHT AND ATOME

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Strange ways of light and atoms

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Two of Einstein's less well-known discoveries – Bose–Einstein condensation and stimulated emission – have had a huge impact on the modern world.

Einstein is best known for relativity and his other 1905 breakthroughs – explaining the photoelectric effect and his work on Brownian motion – but his ideas also underpinned the development of the laser and the creation of a new state of matter called the Bose–Einstein condensate. These discoveries, which were made in 1916 and 1924, respectively, were based on Einstein’s investigations into "bosonic" particles such as photons. Moreover, Bose–Einstein condensation was predicted to occur in one of the simplest physical systems: the ideal gas.

A SPECIAL CENTURY

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A very special centenary

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Einstein's theory of special relativity has been a cornerstone of modern physics for decades, but, as Robert Bluhm describes, physicists are still putting it to the test.

Every physics teacher recognizes the look of astonishment that appears on a student's face when they are taught special relativity. The first tenet on which the theory is built goes along with common sense: the laws of physics are the same in all inertial or nonaccelerating frames. Billiards, for example, can be played on a steady cruise ship just as well as it can be played on solid land.

It is the second tenet – that the speed of light in a vacuum is the same in all inertial frames – that causes jaws to drop. It is a bit like saying that two police officers, one standing still and the other in a fast-moving car, will both clock the same speed for a passing motorist. Clearly, this defies all common sense. It took the genius of Einstein to suspend his disbelief and explore the consequences of these two requirements.

THE SEARCH FOR GRAVITATIONAL WAVES

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The search for gravitational waves

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General relativity predicts that ripples are produced in the fabric of space–time when mass is accelerated. Detecting this phenomenon is one of the outstanding challenges in physics.

LAST AUGUST the bookmaker Ladbrokes offered the public a chance to bet on science. When the betting opened, Ladbrokes was offering odds of 500/1 that gravitational waves – a so far unconfirmed prediction of Einstein's general theory of relativity – would be detected by a laser-based experiment called LIGO before 2010. To those of us working on gravitational waves this was an opportunity not to be missed, and we quickly staked the maximum amount allowed by the bookmakers. Others did the same, and when the betting closed a few weeks later the odds had shortened to 2/1

EINSTEIN AND HIS LOVE OF MUSIC

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Einstein and his love of music

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As a keen and talented violinist, music was one of Einstein’s life-long passions. His musical tastes, however, were distinctly conservative, as Brian Foster explains.

As we celebrate the centenary of his seminal 1905 papers, it is humbling to note that Einstein was not only the outstanding scientist of the 20th century, but also a gifted and enthusiastic musician. He once said that had he not been a scientist, he would have been a musician. "Life without playing music is inconceivable for me," he declared. "I live my daydreams in music. I see my life in terms of music…I get most joy in life out of music."

LOOKING AFTER THE IMAGE OF A LEGEND

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Looking after the image of a legend

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The use of Albert Einstein’s name and image are tightly controlled. Peter Gwynne explores who owns the rights to the Einstein brand and how it is protected.

The name of Albert Einstein and the image of a moustached old man with wild hair are recognized the world over. Such is the universal appeal of the quintessential scientific genius that Einstein's image is used to sell almost anything, from T-shirts and coffee mugs to postcards and physics magazines. This will certainly be the case in 2005, as the physics community celebrates the 100th anniversary of Einstein's annus mirabilis, the year in which he published his groundbreaking papers on special relativity, Brownian motion and the photoelectric effect. However, all this activity will be carefully monitored. For the past 22 years the Hebrew University of Jerusalem (HUJ) has owned the rights to most of Einstein's words, images and personal papers. And since 1985 a US firm, the Roger Richman Agency, has acted as the exclusive licensing agent for the university.

RELATIVITY AT THE CENTENARY

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Relativity at the centenary

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Gravitational physics has become a truly experimental science as tests of the special and general theories of relativity reach new levels of precision

When I was a first-term graduate student in the late 1960s, it was said that the field of general relativity was "a theorist's paradise and an experimentalist's purgatory". There were some experiments - Irwin Shapiro, for instance, had just measured the effects of general relativity on radio waves as they passed the Sun - but the field was dominated by theory and by theorists. This seemed to reflect Einstein's own attitudes: although he had a keen insight into the workings of the physical world, he felt that the bottom line was the theory. As he once famously said, when asked how he would have reacted if an experiment had contradicted the theory, "I would have felt sorry for the dear Lord. The theory is correct".

Since that time the field has been completely transformed. Today, at the centenary of Einstein's annus mirabilis, experiment has become a central component of gravitational physics. I know of no better way to illustrate this than to cite a paper by the LIGO Scientific Collaboration that was published in Physical Review D last year (see Abbott et al. in further reading). This was one of the papers reporting results from the first science run of the Laser Interferometer Gravitational-wave Observatory (LIGO), but with 374 authors from 41 institutions in 8 countries it is reminiscent of particle physics, not general relativity.

Right on time

The breadth of current experiments - ranging from tests of classic general relativity such as the Shapiro delay and the bending of light, through space-based measurements of "frame-dragging" to searches for gravitational waves or violations of the inverse-square law - attests to the ongoing vigour of experimental gravitation. With all this data, can we still be sure that Einstein was right?

Testing the foundations

At the heart of the general theory of relativity is the equivalence principle - an idea that came to Einstein two years after he developed special relativity and led him to the dramatic conclusion that mass and gravity are intimately linked to the curvature of space-time (see figure 1 and Box 1: Special and general relativity).

Put in simple terms, the equivalence principle states that gravity and acceleration are equivalent. Embellished over the years, this idea is now called the Einstein equivalence principle and encompasses three separate principles: the weak equivalence principle, and the principles of local Lorentz and local position invariance.

The weak equivalence principle states that test bodies fall with the same acceleration independent of their internal structure or composition: in other words gravitational mass (the m in F = GMm/r², where F is the gravitational attraction between two masses a distance r apart and G is the Newtonian gravitational constant) and inertial mass (the m in F = ma, where a is the acceleration caused by any force F) are the same. There is also a strong version of the equivalence principle that goes beyond the weak version by stating that gravitational energy will fall with the same acceleration as ordinary matter and other types of energy in a gravitational field (see Box 2: Self-energy and the strong equivalence principle).

The principle of local Lorentz invariance states that the outcome of any local non-gravitational experiment carried out in a freely falling reference frame is independent of the velocity of that frame, while the principle of local position invariance holds that the outcome of any local non-gravitational experiment is also independent of where and when in the universe it is performed. In this context "local" means confined to a suitably small region of space and time, while "freely falling" means falling freely under gravity with no other forces acting.

Although Einstein used it to derive general relativity, his equivalence principle implies only that gravitation must be described by a "metric theory" - a theory in which matter responds to the geometry of space-time and nothing else. However, general relativity is not the only metric theory of gravity, and other examples include the "scalar-tensor" theory developed by Carl Brans and Robert Dicke at Princeton University in 1961, building on earlier work by Markus Fierz and Pascual Jordan.

When it comes to testing metric theories of gravity, we need to distinguish between the weak-field limit, which is valid in the solar system (see figure 2 and Box 3: Testing metric theories in the solar system), and the strong-field regime that is needed to describe regions where gravity is extremely strong, such as in the vicinity of a black hole or a neutron star. If we are being really ambitious, we might also try to describe situations where gravity is strong and quantum effects are important, such as during the Big Bang, but that is a separate story (see "Welcome to quantum gravity").

In non-metric theories matter can respond to something other than the geometry of space-time, and this can lead to violations of one or more pieces of the Einstein equivalence principle. For instance, in the string theories that seek to unify gravity with the other three forces of nature, the equivalence principle is violated because matter can respond to additional long-range fields. Searching for violations of the Einstein equivalence principle is therefore a good way to search for new physics beyond the standard metric theories of gravity.

In the balance

To test the weak equivalence principle one compares the accelerations of two bodies with different compositions in an external gravitational field. Such experiments are often called Eötvös experiments after Baron von Eötvös, the Hungarian physicist whose pioneering experiments with torsion balances provided a foundation for Einstein's ideas on general relativity.

1 Tests of the weak equivalence principle

In a torsion balance two bodies made of different materials are suspended at the ends of a rod that is supported by a fine wire or fibre. We then look for a difference in the horizontal accelerations of the two bodies as revealed by a slight rotation of the rod. The source of the horizontal gravitational force could be the Sun, a large mass in the laboratory, a nearby hill, or, as Eötvös recognized, the Earth itself. The best test of the weak equivalence principle to date has been performed by Eric Adelberger and the Eöt-Wash collaboration at the University of Washington in Seattle, who have used an advanced torsion balance to compare the accelerations of various pairs of materials toward the Earth, the Sun and the Milky Way.

A completely different test of the weak equivalence principle involves bouncing laser pulses off mirrors on the lunar surface to check if the Earth and the Moon are accelerating toward the Sun at the same rate. Lunar laser-ranging measurements actually test the strong equivalence principle because they are sensitive to both the mass and the gravitational self-energy of the Earth and the Moon. The bottom line of these experiments is that bodies fall with the same acceleration to a few parts in 10¹³ (see figure 1).

In the future, the Apache Point Observatory for Lunar Laser-ranging Operation (APOLLO) project, a joint effort by researchers from the University of Washington in Seattle and the University of California at San Diego, will use enhanced laser and telescope technology, together with a good, high-altitude site in New Mexico, to improve the lunar laser-ranging test by as much as a factor of 10 (see Williams et al. in further reading and Physics World June 2004 p9, print version only).

2 Tests of general relativity

The next major advance may occur in space, if two satellite missions are successful. MICROSCOPE, which could be launched in 2008, aims to test the weak equivalence principle to 1 part in 10¹⁵, while a later mission called the Satellite Test of the Equivalence Principle (STEP) could improve on this by a factor of 1000. These experiments will compare the acceleration of different materials moving in free-fall orbits around the Earth inside a drag-compensated spacecraft. Doing experiments in space means that the bodies are in perpetual fall, whereas Earth-based experiments at "drop towers" are over in seconds, which leads to much larger measurement errors.

Many of the techniques developed to test the weak equivalence principle have been adapted to search for possible violations of the inverse-square law of gravity at distances below 1 mm. Such violations could signal the presence of additional interactions between matter or "large" extra dimensions of space. No deviations from the inverse-square law have been found at distances between 100 μm and 10 mm, but there are enough well-motivated theoretical predictions for new effects at these distances to push experimentalists towards better sensitivities and shorter distances.

Tests with atomic clocks

The predictions of general relativity can also be tested with atomic clocks. Local position invariance requires that the internal binding energies of all atoms, and thus the time given by atomic clocks, must be independent of their location in both time and space when measured in a local freely falling frame. However, if two identical atomic clocks are placed in different gravitational potentials, they will be in different local frames and, according to the Einstein equivalence principle, they will give slightly different times.

In 1976 Robert Vessot, Martine Levine and co-workers at the Harvard Smithsonian Astrophysical Observatory and the Marshall Space Flight Center compared a hydrogen maser clock on a Scout rocket at an altitude of 10,000 km with one on the ground, and verified Einstein's 1907 prediction for this "gravitational redshift" to a few parts in 10⁴. This redshift actually has an impact on our daily lives because it must be taken into account (along with the time dilation associated with special relativity) to ensure that navigational devices that rely on the Global Positioning System (GPS) remain accurate. Relativistic effects mean that there is a 39 ms per day difference between ground-based atomic clocks and those on the GPS satellites.

Recent clock-comparison tests of local position invariance undertaken at the National Institute of Standards and Technology (NIST) in Boulder, Colorado, and the Observatory of Paris have shown that the fine-structure constant - which determines how fast the atomic clocks "tick" - is constant to 1 part in 10¹⁵ per year. The NIST team compared laser-cooled mercury ions with neutral caesium atoms over a two-year period, while the Paris team compared laser-cooled caesium and rubidium atomic fountains over five years. Plans are being developed to perform such clock comparisons in space, possibly on the International Space Station.

Atomic clocks can also be used to test the two pillars of special relativity - Lorentz symmetry and position invariance. At the centenary of special relativity, it is useful to recall that acceptance of this theory was slow in coming - Einstein's 1921 Nobel Prize was for the photoelectric effect, another of his 1905 triumphs, not for relativity. However, special relativity is now such a foundation for modern physics that it is almost blasphemy to question it, although that has not stopped a growing number of theoretical and experimental physicists searching for violations of Lorentz and/or position invariance (see "Breaking Lorentz symmetry"). In earlier times, such thinking would have been called "crackpot", but these new ideas are well rooted in attempts to find a quantum theory of gravity and, ultimately, a unified theory of the four fundamental forces of nature.

Various string theories, for instance, allow for the possibility of long-range fields that are linked to the average matter distribution of the universe. If these fields couple weakly to local matter, they could lead to effects that can be observed in experiments. In particular, we know from observations that the Earth moves through the cosmic background radiation at a speed of 350 km s^-1. With the right kind of long-range field, this motion could produce an effective interaction that has a preferred direction associated with it. If this long-range field were then to couple weakly to, say, electromagnetism, then the electromagnetic fields in atoms could be changed by an amount that depends on the orientation of the atom relative to our direction of motion through the universe.

During the late 1980s researchers at Seattle, Harvard and NIST looked for these effects by checking if atomic transition frequencies change over the course of a year as their orientation changes relative to our cosmic velocity. Exploiting the then newly developed techniques of atom trapping and cooling, the researchers found no effects down to a few parts per 10²⁶.

These "clock anisotropy" experiments are latter-day versions of the classic Michelson-Morley experiments of 1887. In the Michelson-Morley experiment the "clocks" being compared were defined by the propagation of light along each of the two perpendicular arms of an interferometer. Einstein took the null result of these experiments for granted in his 1905 paper on special relativity, although he never referred to them by name.

3 Gravity Probe B

Looking to the future, the discreteness of space-time at the Planck scale that is found in some quantum theories of gravity could also lead to effective violations of Lorentz invariance. However, a wide range of experiments, including tests of CPT (charge-parity-time) symmetry in particle-physics experiments and careful observations of gamma rays and synchrotron radiation from astrophysical sources, have ruled these out to a high-level of precision.

Does space-time do the twist?

A central prediction of general relativity is that moving matter generates a gravitational field that is analogous to the magnetic field generated by a moving charge. Thus, a rotating body produces a "gravitomagnetic" field that drags space-time around with it, and this "frame-dragging" may play an important role in the dynamics of matter spiralling into supermassive black holes in quasars and other active galaxies. Frame-dragging might also be partly responsible for the collimated relativistic jets seen in such systems.

The Gravity Probe B satellite is currently measuring this effect near the Earth. Launched on 20 April 2004, its goal is to measure the precessions of four gyroscopes relative to a telescope trained on a nearby guide star called IM Pegasi over the course of a year (until the liquid helium that is used to cool the experiment runs out). The gyroscopes are spheres that are perfect to a few parts in 10 million and are coated with a thin layer of superconducting niobium. When the spheres rotate, the superconducting films develop magnetic moments that are precisely parallel to their spin axes. This means that any precession of the spins can be measured by monitoring changes in the magnetic flux through superconducting current loops fixed in the spacecraft.

General relativity predicts that frame-dragging will lead to a precession of 41 milliarcseconds per year, and the Gravity Probe B team hopes to measure this with an accuracy of 1%. The experiment will also measure the "geodetic" precession caused by the ordinary curvature of space around the Earth. General relativity predicts a value of 6.6 arcseconds per year for this effect. Gravity Probe B has been designed so that these precessions are perpendicular to one another, and the first results from the mission are expected in early 2006 (see figure 3).

Meanwhile, last October Ignazio Ciufolini of the University of Lecce in Italy and Erricos Pavlis of the University of Maryland used techniques in which laser beams were reflected from satellites to make a measurement of frame-dragging on the orbit of a satellite. Their result agreed with general relativity, with errors at the level of 10% (see Physics World November 2004 p7).

The binary pulsar

In 1974 Russell Hulse and Joseph Taylor, then at the University of Massachusetts, discovered a binary pulsar called PSR 1913+16 that was to play a crucial role in tests of general relativity. Pulsars emit pulses of radio waves at very regular intervals and are thought to be rotating neutron stars. PSR 1913+16 was special because it was a pulsar that was in orbit around another compact object.

By carefully measuring small changes in the rate of the pulsar "clock", Hulse and Taylor were able to determine both non-relativistic and relativistic orbital parameters with extraordinary precision. In particular they were able to measure three relativistic effects: the rate of advance of the periastron (the analogue of the perihelion in a binary system); the combined effects of time-dilation and gravitational redshift on the observed rate of the pulsar; and the rate of decrease of the orbital period.

If we assume that general relativity is correct and make the reasonable assumption that both objects are neutron stars, then all three relativistic effects depend on the two unknown stellar masses. Since we have, in effect, three simultaneous equations and just two unknowns, we can determine the mass of both objects with an uncertainty of less than 0.05%, and also test the predictions of general relativity. If we assume that the orbital period of the system is decreasing due to the emission of gravitational waves, then theory and experiment agree to within 0.2%. Hulse and Taylor shared the 1993 Nobel Prize for Physics for this work.

Binary pulsars can also be used to distinguish between different theories of gravity because they have very strong internal gravity (see Stairs in further reading). Indeed, several tenths of the rest-mass energy of a neutron star is contained in the gravitational forces that hold the star together, while the orbital energy only accounts for 10^-6 of the total mass energy of the system. In the Brans-Dicke theory this internal self-gravity leads to the prediction that binary pulsars should emit both dipole and quadrupole gravitational radiation, whereas general relativity strictly forbids the dipole contribution. The emission of dipole radiation would have a characteristic effect on the orbital period of the system, but such an effect has not been seen. Several recently discovered binary-pulsar systems may allow new tests of general relativity.

Gravitational waves

One of the outstanding challenges in physics today is to detect gravitational waves, and new gravitational-wave observatories in the US, Europe and Japan hope to achieve this, possibly before the end of the decade. In addition to exploring various astrophysical phenomena, these observatories might also be able to carry out new tests of fundamental gravitational physics (see Physics World January 2005 p37, print version only).

General relativity makes three predictions about gravitational radiation that can be tested: gravitational waves have only two polarization states, whereas other theories can predict as many as six; gravitational waves travel at the speed of light, while other theories may predict different speeds; and the emission of gravitational waves acts back on the source that is emitting them in a characteristic manner.

For example, as is described above, scalar-tensor theories and general relativity make different predictions for the nature of the gravitational waves emitted by binary pulsars, and it may be possible to detect these differences. Moreover, if gravitational waves with long wavelengths travel more slowly than those with shorter wavelengths, then it might be possible to observe this behaviour - which is generally associated with massive (as opposed to massless) elementary particles - in the gravitational radiation from binary systems.

Although the collision of two compact objects to form a black hole is too complex to allow precision tests of general relativity, analysis of the gravitational waves produced in the collision will reveal information about the masses and spins of the compact objects themselves, and also about the mass and angular momentum of the final black hole. Such observations will therefore reflect dynamical, strong-field general relativity in its full glory.

Making firm predictions for this situation involves solving Einstein's equations in a regime where weak-field methods fail, and therefore requires large-scale numerical computations. This challenging task has been taken up by many "numerical relativity" groups around the world. The discovery and study of the formation of a black hole through gravitational waves would provide a stunning test of general relativity.

Relativity and beyond

Einstein's special and general theories of relativity altered the course of science. They were triumphs of the imagination and of theory, with experiment playing a secondary role. In the past four decades we have witnessed a second triumph for Einstein, with general relativity passing increasingly precise experimental tests with flying colours. But the work is not done. Tests of strong-field gravity in the vicinity of black holes and neutron stars need to be carried out. Gamma-ray, X-ray and gravitational-wave astronomy will all play a critical role in probing this largely unexplored aspect of the theory.

General relativity is now the "standard model" of gravity. But as in particle physics, there may be a world beyond the standard model. Quantum gravity, strings and branes may lead to testable effects beyond general relativity. Experimentalists will continue to search for such effects using laboratory experiments, particle accelerators, instruments in space and cosmological observations. At the centenary of relativity it could well be said that experimentalists have joined the theorists in relativistic paradise.

Box 1: Special and general relativity

When Einstein introduced the concept of "relativity" in 1905 - the notion that there is no absolute motion in the universe, only relative motion - he overthrew ideas that had been in place since the time of Newton some 200 years before. In addition to E = mc², special relativity predicted various novel effects that occurred when bodies moved at close to the speed of light: time slowed down (an effect known as time-dilation) and lengths became shorter (Fitzgerald contraction). With the general theory Einstein then went on to show that we do not reside in the flat (Euclidean) space and uniform time of everyday experience, but in curved space-time instead.

Special relativity helped us to understand the microworld of elementary particles and interactions, while general relativity revolutionized our view of the universe by predicting astrophysical phenomena as bizarre as the Big Bang, neutron stars, black holes and gravitational waves.

Big success

The theory of relativity is a single, all-encompassing theory of space-time, gravity and mechanics, although special relativity and general relativity are often viewed as being independent. Special relativity is actually an approximation to curved space-time that is valid in sufficiently small regions called "local freely falling frames", much as small regions on the surface of an apple are approximately flat, even though the overall surface is curved.

Einstein’s great insight was to realize that gravity and acceleration are equivalent in free fall, and he then went on to show that the laws of physics, such as the equations of electromagnetism, should have built-in local Lorentz and local position invariance.

In special relativity the "distance" between two points in space–time is given by the line element, ds, which is defined as ds² = -c²dt² + dx² + dy² + dz², where t is time and c is the speed of light in a vacuum. In the curved space-time of general relativity ds is defined as ds² = g_μνdx^μdx^ν, where x¹, x² and x³ are the three spatial dimensions, x⁰ = ct, and gμν, which is called the metric, is a function in space–time. The right-hand side of the equation must be summed over all values of μ and ν between 0 and 3.

General relativity provides a set of field equations that allow us to calculate the space-time metric (i.e. the amount of curvature) from a given distribution of matter - something that is not defined by the equivalence principle. Einstein’s aim was to find the simplest field equations that made this possible. The result was a set of 10 equations, symbolized by the seductively simple equation G_μν = 8πGT_μν/c⁴, where G_μν is Einstein’s curvature tensor, which can be obtained directly from g_μν and its derivatives, and T_μν is the stress-energy tensor of normal matter. Sweating the details hidden in this equation has kept generations of relativists occupied.

In the past it was customary to speak of the three classical tests proposed by Einstein: the deflection of light by a massive body; the advance of the perihelion of Mercury; and the gravitational redshift of light (although this is actually a test of the Einstein equivalence principle rather than general relativity itself). Many new tests have been developed since Einstein’s time: in 1964 Irwin Shapiro, then at the Massachusetts Institute of Technology, predicted a delay in the propagation of light past a massive body; and in 1968 Kenneth Nordtvedt Jr of Montana State University showed that theories other than general relativity do not necessarily obey the equivalence principle in certain situations. One of the most striking predictions of general relativity is the black hole: when a massive star collapses under its own gravity it can warp space-time to such an extent that nothing, not even light, can escape. There is now convincing observational evidence for these objects.

One of the outstanding problems in physics is to unify general relativity, which is our best theory of gravity, with the quantum field theories that describe the three other fundamental forces. Although this challenge defeated Einstein, it should not surprise us that all the leading candidates for a unified theory - string theory, branes and loop quantum gravity - are all fundamentally geometrical.

Box 2: Self-energy and the strong equivalence principle

Special relativity and E = mc² tell us that energy and mass are essentially the same. The mass of a proton and an electron is greater than that of a hydrogen atom because energy must be supplied to break the electromagnetic bond in the atom. The weak equivalence principle asserts that this difference will change both the gravitational mass and the inertial mass by the same amount. This means that all forms of energy at microscopic scales - electromagnetic, strong and weak - respond to gravity in the same way. But what about large bodies like the Earth and Sun, or even extreme gravitational bodies like black holes, which also have measurable gravitational binding energy? The strong equivalence principle goes beyond the weak version by stating that gravitational energy falls with the same acceleration as ordinary matter and other forms of energy in a gravitational field. Although the gravitational self-energy contained in the gravitational forces that hold the Earth together only changes its total mass energy by less than 1 part in a billion, lunar laser-ranging experiments (see main text) can achieve a precision of 1 part in 10¹³ and can therefore test the strong equivalence principle. General relativity obeys the strong equivalence principle, whereas the Brans–Dicke theory and many other alternative theories do not.

Box 3: Testing metric theories in the solar system

General relativity is one of several "metric" theories in which gravity arises from the geometry of space-time and nothing else. If we want to distinguish between different metric theories in the weak-field limit, it is customary to use a formalism that dates back to Arthur Eddington’s 1922 textbook on general relativity and was later extended by Kenneth Nordtvedt Jr and the present author. This parametrized post-Newtonian (PPN) formalism contains 10 parameters that characterize how the predictions of the different metric theories differ from those of Newtonian gravity, and therefore from each other, for various phenomena that can be measured in the solar system.

Six of these parameters are shown in the table below. For instance, γ is related to the amount of spatial curvature generated by mass and determines the size of classic relativistic effects such as the deflection of light by mass, while β is related to the degree of nonlinearity in the gravitational field. Another four parameters - ξ, α₁, α₂ and α₃ - determine if gravity itself violates a form of local position invariance or local Lorentz invariance (such as G depending on our velocity through the universe).

Table

In the PPN formalism the deflection of light and the Shapiro time delay are both proportional to (1 + γ)/2. The "1/2" corresponds to the so-called Newtonian deflection (i.e. the deflection that a body moving at the speed of light would experience according to Newtonian gravity). This result was derived over two centuries ago by Henry Cavendish, who never published it, and then discovered again by Johann von Soldner in 1803, who did publish it. The "γ/2" comes directly from the warping of space near the massive body.

The PPN parameters can have different values in the different metric theories of gravity. In general relativity, for instance, γ and β are exactly equal to one and the other eight parameters all vanish. Four decades of experiments have placed bounds on the PPN parameters that are consistent with general relativity (see figure 2).

THE 1919 ECLIPSE:A CELEBRITY IS BORN

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The 1919 eclipse: a celebrity is born

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Einstein shot to fame in 1919 when a team of astronomers led by Arthur Eddington found that the light from a distant star can be bent by the Sun, as predicted by relativity. But as Matthew Stanley explains, Eddington's expedition was partly motivated by a desire to heal the wounds between Britain and Germany after the First World War.

In the spring of 1919, while Europe was just beginning to recover from the effects of the First World War, teams of British astronomers thousands of miles from home laboured to measure a tiny effect predicted by an obscure German scientist. This scientist was Albert Einstein, and when those astronomers presented their results he would move from little-known physicist to global celebrity. How did this dramatic turn of events come to be?

EINSTEIN'S RANDOM WALK

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Einstein's random walk

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The story of Brownian motion began with experimental confusion and philosophical debate, before Einstein, in one of his least well-known contributions to physics, laid the theoretical groundwork for precision measurements to reveal the reality of atoms

Most of us probably remember hearing about Brownian motion in high school, when we are taught that pollen grains jiggle around randomly in water due the impacts of millions of invisible molecules. But how many people know about Einstein's work on Brownian motion, which allowed Jean Perrin and others to prove the physical reality of molecules and atoms?

Einstein's analysis was presented in a series of publications, including his doctoral thesis, that started in 1905 with a paper in the journal Annalen der Physik. Einstein's theory demonstrated how Brownian motion offered experimentalists the possibility to prove that molecules existed, despite the fact that molecules themselves were too small to be seen directly.

Physics in motion

Brownian motion was one of three fundamental advances that Einstein made in 1905, the others being special relativity and the idea of light quanta. Of these three great works, Einstein's analysis of Brownian motion remains the least well known. But this part of Einstein's scientific legacy was the key to a revolution that is at least as important as relativity or quantum physics. One century later, Brownian motion continues to be of immeasurable importance in modern science, from physics through biology to the latest wonders of nanotechnology. Indeed, this is reflected in citation statistics, which show that Einstein's papers on Brownian motion have been cited many more times than his publications on special relativity or the photoelectric effect.

The story of Brownian motion spans almost two centuries, its unlikely roots lying in a scientific craze that swept western Europe at the beginning of the 1800s. And it starts, surprisingly enough, not with a physicist but with a botanist.

Brown's botany

In the early 19th-century Europeans became fascinated by botany. In Britain this interest was fuelled by explorations to the corners of the growing empire, particularly Australia or "New Holland" as it was known at that time. One of the first people to get their botanical teeth into New Holland was Robert Brown, who had grown up botanizing in the Scottish hills.

After completing a medical degree at Edinburgh University and a brief period in the army, during which he spent most of his time specimen-hunting around Ireland, Brown secured a place as ship's botanist on a surveying mission to Australia in 1801. Risking attack from Napoleon's fleets, Brown spent four years exploring the Australian and Tasmanian coasts before returning to London laden with thousands of specimens of new species, his reputation as one of Europe's leading botanists already secure.

1 Random walks

But Brown was interested in more than collecting and cataloguing different species - he was also a pioneer of botany as a scientific investigation. Indeed, he is credited with the first clear description of the cell nucleus, and it was Brown that Charles Darwin came to for advice before setting out in the Beagle in 1831. In fact, the botanical craze in which Brown had played a major part laid the vital groundwork for Darwin's theory of evolution.

Brown is, of course, better known among physicists for the phenomenon of Brownian motion. In the summer of 1827 he began to make microscopic observations of suspensions of grains released from pollen sacks taken from a type of evening primrose called Clarkia pulchella. What Brown saw surprised him: the tiny grains, which were suspended in water, appeared to be in constant motion, carrying out a tireless and chaotic dance. This motion never appeared to slow or stop. Moreover, as Brown verified, it was not caused by external influences such as light or temperature. He also quickly ruled out his first idea - that the grains were somehow alive - by examining grains from inorganic minerals. So, Brown had shown that whatever it was, this incessant dance was not biology after all: it was physics.

Curiosity and paradox: Brownian motion and kinetic theory

For decades the significance of Brown's observations went almost entirely unappreciated. A few scientists returned now and then to the phenomenon, but it was seen as little more than a curiosity. In hindsight this is rather unfortunate, since Brownian motion provided a way to reconcile the paradox between two of the greatest contributions to physics at that time: thermodynamics and the kinetic theory of gases.

The laws of thermodynamics were one of the crowning achievements of physics by the middle of the 19th century. Through them a vast range of material behaviour could be understood, irrespective of particular theories of matter, simply in terms of the concepts of energy and entropy. But many scientists were not satisfied with this simple picture, and sought not just a statement but an explanation of the laws.

Chief among these were James Clerk Maxwell and Ludwig Boltzmann, who built on the 18th-century idea that matter, such as a volume of gas, is composed of many tiny particles. They showed that many of the experimental results of thermodynamics could be explained by calculating the average or statistical behaviour of such a collection of particles, in what became known as kinetic theory.

But Maxwell and Boltzmann's theory only brought into sharper focus the paradox between thermodynamics and Newtonian mechanics. Key to kinetic theory was the idea that the motion of individual particles obeyed perfectly reversible Newtonian mechanics. In other words there was no preferred direction of time. But the second law of thermodynamics expressly demanded that many processes be irreversible. Or, as Tom Stoppard puts it in his 1993 play Arcadia, you cannot "unstir" the jam from your rice pudding simply by stirring it in the opposite direction. So, if matter was made up of particles obeying perfectly reversible Newtonian equations, where did the irreversibility come from?

This violation of the second law on the scale of single particles in kinetic theory was perfectly apparent to Maxwell, but he missed the subtle link to Brownian motion that might have immediately allowed the paradox to be investigated experimentally. One clue lay in the fact that Brownian motion also apparently violated the second law, since the dance of a Brownian particle seemed to continue forever, never slowing down and never tiring. It therefore ought to be possible to extract endless work from such a particle. But such perfect conversion of heat into work was forbidden by the second law, which states that some energy must always be irreversibly lost as heat whenever work is done. And if some energy is always irretrievably lost, how can the Brownian motion continue forever?

It was not until near the end of the 19th century that scientists such as Louis Georges Gouy suggested that Brownian motion might offer a "natural laboratory" in which to directly examine how kinetic theory and thermodynamics could be reconciled. In other words they decided to turn the problem around and use Brownian motion to throw light on the great paradox of the second law.

There was, however, one problem with this natural laboratory: it was not clear which quantities needed to be measured. This was where, a few years into the 20th century, a young patent clerk called Albert Einstein came to the fore.

Atoms: philosophy, analogy or reality?

Einstein was not the kind of scientist to simply pick a problem and solve it out of idle curiosity, and this is as true of Brownian motion as it is of relativity. He had another motive for wanting to find a theory of Brownian motion, but to understand what this was we first have to consider another controversy that stemmed from kinetic theory.

Ludwig Boltzmann had championed a way out of the reversibility paradox via the statistical interpretation. He suggested that any single molecule would behave entirely in accord with reversible mechanics, but that when you put a large collection of particles together, the statistics implied irreversibility and led unavoidably to the second law. Despite its mathematical success, Boltzmann's "statistical mechanics" met with criticism. Why swap the solid ground of the laws of thermodynamics - the product of a century of careful experimental verification - for the ephemeral world of statistics and chance?

2 The reality of atoms

It seemed like a return to the chaos of the middle ages, before the time of Galileo and Newton, and it would take compelling evidence to convince people to throw this hard-won determinism away. In fact, it would take direct evidence that Boltzmann was counting something physical and real: a proof that the particles of kinetic theory really existed.

Today we take atoms for granted, but even as recently as the turn of the 20th century not everyone accepted this "discontinuous" description of matter. Even Boltzmann and Maxwell tended to sit on the fence. Boltzmann described kinetic theory as a mechanical analogy, and Maxwell never expected that his illustrative mechanisms - the pictures that helped him build mathematical theories - would be taken literally.

The so-called energeticists, such as Ernst Mach and Wilhelm Ostwald, went even further. They insisted that kinetic theory was no more than a convenient picture that should not be taken literally - certainly not, the latter argued, until you had direct evidence for the existence of atoms. Ostwald's caution was partly justified. It could be dangerous for the credibility of science to base a complete theory of matter on some hypothetical object that had never been seen - especially at a time when science was under strident philosophical attack from intellectuals, who despaired at its apparently inhumane reductionism.

But Einstein took a different view. He was one of a new generation of physicists who had grown up on a diet of Maxwell and kinetic theory, and therefore saw little reason to doubt the physical reality of atoms. Indeed, by analysing Brownian motion, Einstein set out to obtain a quantitative measure of the size of the atom so that even the most cautious sceptics would be convinced of its existence.

3 Brownian motion in action

As the great year 1905 dawned, Einstein was still an unknown physicist working in obscurity at the Bern patent office. But that year he would take the decisive theoretical step towards proving that liquids really are made of atoms. He joined the thermodynamics of liquids with statistical mechanics to obtain the first testable theory of Brownian motion, and the first chance of a direct glimpse inside the atomic world.

Quantitative predictions: Einstein and Brownian motion

In his quest for the literal truth of atoms Einstein had to accept that individual atoms could not be seen. By anyone's estimate they were simply too small and too fast. But Einstein recognized that if the predictions of statistical mechanics were correct, then any particle immersed in a "bath" of atoms must basically behave like a very large atom because it would be in thermodynamic equilibrium with the atoms in the bath. Furthermore, the equipartition of energy theorem predicted exactly how the particle's kinetic energy would depend on temperature: for each degree of freedom the average kinetic energy is k_BT/2, where k_B is Boltzmann's constant and T is the temperature of the bath.

Einstein realized that a particle with a diameter of, say, 1 μm - large enough, in other words, to be visible using a microscope - would provide a "magnifying glass" into the world of the atom. It would be like an atom you could see, and the behaviour of which you could compare directly against kinetic theory to decide once and for all whether Boltzmann's ideas agreed with reality.

Einstein predicted that, just like a molecule in solution, such a Brownian particle would diffuse according to a simple equation: D = √[(k_BT/6πηR)t], where D is the displacement (technically the root mean square displacement) of the particle, T is the temperature, η is the viscosity of the liquid, R is the size of the particle and t is time. This equation implied that large particles would diffuse more gradually than molecules, making them even easier to measure. Moreover, unlike a ballistic particle such as a billiard ball, the displacement of a Brownian particle would not increase linearly with time but with the square root of time (figure 1).

Attempts had already been made to measure the velocity of Brownian particles, but they gave a nonsensical result: the shorter the measurement time, the higher the apparent velocity. This suggested that if you could measure the velocity in an extremely short (infinitesimal) instant, you would obtain a velocity approaching infinity. But if Einstein's derivations were correct, the mystery was explained because you cannot measure the velocity of a Brownian particle simply by dividing a distance by a time. The experimenters had been measuring the wrong quantity! Thanks to Einstein's pioneering analysis, the mathematical stage was now set, and it was time for someone to get down to some serious experimenting.

The man who proved atoms are real

Jean Perrin, a physical chemist working at the Sorbonne in Paris, belonged to the same atom-believing tradition as Einstein. And it was Perrin's microscope studies of Brownian particles that confirmed Einstein's theory and sealed the reality of the discontinuous, atomic nature of matter.

These studies began in 1908, when Perrin and his team of research students embarked on an exhaustive set of experiments. Tragically, many of Perrin's team would lose their lives only a few years later in the First World War.

Their first task was to obtain a suspension of Brownian particles that were each as close as possible to being the same size, since the rate of diffusion depended on particle size, and whose size was precisely measurable. This was no mean feat for particles with a diameter of a thousandth of a millimetre. Starting with kilograms of suspended "gamboge" - a gum extract that forms spherical particles when it is dissolved in water - Perrin's team eventually managed to produce just a few grams of usable particles.

Using a microscope, Perrin showed that when these particles were dispersed in water, they formed a kind of atmosphere under gravity, since the concentration of particles decreased exponentially with height in the same way that the density of gas molecules in the Earth's atmosphere decreases. This meant that, as Einstein had predicted, the Brownian particles obeyed Boltzmann's equipartition of energy theorem just like gas molecules did (figure 2).

Perrin's group went on to measure the diffusion of the particles, confirming the square root of time law and validating Einstein's kinetic-theory approach. In further experiments over the following five years, Perrin produced a wealth of measurements that could not be contested. Soon enough even Ostwald - the arch sceptic - conceded that Einstein's theory, combined with Perrin's experiments, proved the case. It was official: atoms were real.

A fluctuating future

Science developed fast in those first decades of the 20th century. Armed with Perrin's experimental validation of statistical mechanics, there was little to stop the statistical revolution spreading into every field. Moreover, Einstein and Perrin had unknowingly paved the way for the acceptance of the inherently probabilistic quantum mechanics.

Ironically, Einstein himself never accepted the statistical interpretation of quantum mechanics. Statistics in a liquid of atoms was fine because you knew that you were counting real, physical atoms. But what did it mean to speak of the statistics of a single electron? What was "hidden" behind the electron that caused it to behave statistically? This was a question that Niels Bohr's "complementarity" simply barred you from asking, and Einstein was never satisfied with that (see p47, print version only).

The quantum revolution gained so much attention through the first half of the 20th century that it obscured the success of classical statistical mechanics. Only in recent decades has the importance of Einstein and Perrin's classical work become clearer. As physics increasingly overlaps with biology, nanotechnology and the statistics of complex phenomena, we can begin to see how understanding Brownian fluctuations is vital to everything from cell function to traffic flow, and from models of ecologies to game theory and the stock market (figure 3).

Einstein did not live long enough to appreciate the true significance of Brownian motion. In his later years, immersed in the search for a "theory of everything" through his general theory of relativity, Einstein himself dismissed his work on Brownian motion as unimportant. He was a philosopher as much as a physicist, and to him the philosophical implications of Brownian motion seemed minimal compared with those of relativity.

But if he were alive today, then perhaps he would change his mind. Since Robert Brown's first observations of Clarkia pulchella 180 years ago, scientists across many disciplines are realizing that random fluctuations are fundamentally important in many, if not most, of the phenomena around us. Without them, there would be no phase behaviour, no protein folding, no cell-membrane function and no evolution of species. And we are only beginning to realize an even deeper subtlety from the latest work on complex systems, such as molecular motors and cell membranes.

These functional biosystems must satisfy almost contradictory requirements: they must be robust to a complicated and ever-fluctuating environment, yet at the same time they must also be able to exploit the fluctuations to carry out complicated biological functions, such as the transport of vital molecules in and out of cells. Almost two centuries after Brown, this trade-off at the heart of nature is gradually becoming clearer: there is an extraordinary balance between function and fluctuation, between hard physical rules and the subtle effects of randomness.

Einstein's role in demystifying Brownian motion was pivotal in this ongoing revolution. In developing the first testable theory that linked statistical mechanics - with its invisible "atoms" and mechanical analogies - to observable reality, Einstein acted as a gateway. Through this gateway, years of confused observations could be turned into the solid results of Perrin, and from these could grow a new, proven world view with statistics at its heart.

From our more distant perspective, it is clear that the Brownian-motion papers of 1905 had just as much influence on science as did relativity or light quanta. Brownian motion was just a slower, subtler revolution: not a headlong charge, but more of a random walk into a vast and unsuspected future.

FIVE PAPERS THAT SHOOK THE WORLD

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Five papers that shook the world

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In 1905 an anonymous patent clerk in Bern rewrote the laws of physics in his spare time. Matthew Chalmers describes Einstein's miraculous year

Most physicists would be happy to make one discovery that is important enough to be taught to future generations of physics students. Only a very small number manage this in their lifetime, and even fewer make two appearances in the textbooks. But Einstein was different. In little more than eight months in 1905 he completed five papers that would change the world for ever. Spanning three quite distinct topics - relativity, the photoelectric effect and Brownian motion - Einstein overturned our view of space and time, showed that it is insufficient to describe light purely as a wave, and laid the foundations for the discovery of atoms.

Genius at work

Perhaps even more remarkably, Einstein's 1905 papers were based neither on hard experimental evidence nor sophisticated mathematics. Instead, he presented elegant arguments and conclusions based on physical intuition. "Einstein's work stands out not because it was difficult but because nobody at that time had been thinking the way he did," says Gerard 't Hooft of the University of Utrecht, who shared the 1999 Nobel Prize for Physics for his work in quantum theory. "Dirac, Fermi, Feynman and others also made multiple contributions to physics, but Einstein made the world realize, for the first time, that pure thought can change our understanding of nature."

And just in case the enormity of Einstein's achievement is in any doubt, we have to remember that he did all of this in his "spare time".

Statistical revelations

In 1905 Einstein was married with a one-year-old son and working as a patent examiner in Bern in Switzerland. His passion was physics, but he had been unable to find an academic position after graduating from the ETH in Zurich in 1900. Nevertheless, he had managed to publish five papers in the leading German journal Annalen der Physik between 1900 and 1904, and had also submitted an unsolicited thesis on molecular forces to the University of Zurich, which was rejected.

Most of these early papers were concerned with the reality of atoms and molecules, something that was far from certain at the time. But on 17 March in 1905 - three days after his 26th birthday - Einstein submitted a paper titled "A heuristic point of view concerning the production and transformation of light" to Annalen der Physik.

Einstein suggested that, from a thermodynamic perspective, light can be described as if it consists of independent quanta of energy (Ann. Phys., Lpz 17 132-148). This hypothesis, which had been tentatively proposed by Max Planck a few years earlier, directly challenged the deeply ingrained wave picture of light. However, Einstein was able to use the idea to explain certain puzzles about the way that light or other electromagnetic radiation ejected electrons from a metal via the photoelectric effect.

Maxwell's electrodynamics could not, for example, explain why the energy of the ejected photoelectrons depended only on the frequency of the incident light and not on the intensity. However, this phenomenon was easy to understand if light of a certain frequency actually consisted of discrete packets or photons all with the same energy. Einstein would go on to receive the 1921 Nobel Prize for Physics for this work, although the official citation stated that the prize was also awarded "for his services to theoretical physics".

"The arguments Einstein used in the photoelectric and subsequent radiation theory are staggering in their boldness and beauty," says Frank Wilczek, a theorist at the Massachusetts Institute of Technology who shared the 2004 Nobel Prize for Physics. "He put forward revolutionary ideas that both inspired decisive experimental work and helped launch quantum theory." Although not fully appreciated at the time, Einstein's work on the quantum nature of light was the first step towards establishing the wave-particle duality of quantum particles.

On 30 April, one month before his paper on the photoelectric effect appeared in print, Einstein completed his second 1905 paper, in which he showed how to calculate Avogadro's number and the size of molecules by studying their motion in a solution. This article was accepted as a doctoral thesis by the University of Zurich in July, and published in a slightly altered form in Annalen der Physik in January 1906. Despite often being obscured by the fame of his papers on special relativity and the photoelectric effect, Einstein's thesis on molecular dimensions became one of his most quoted works. Indeed, it was his preoccupation with statistical mechanics that formed the basis of several of his breakthroughs, including the idea that light was quantized.

After finishing a doctoral thesis, most physicists would be either celebrating or sleeping. But just 11 days later Einstein sent another paper to Annalen der Physik, this time on the subject of Brownian motion. In this paper, "On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat", Einstein combined kinetic theory and classical hydrodynamics to derive an equation that showed that the displacement of Brownian particles varies as the square root of time (Ann. Phys., Lpz 17 549-560).

This was confirmed experimentally by Jean Perrin three years later, proving once and for all that atoms do exist (see Einstein's random walk). In fact, Einstein extended his theory of Brownian motion in an additional paper that he sent to the journal on 19 December, although this was not published until February 1906.

A special discovery

Shortly after finishing his paper on Brownian motion Einstein had an idea about synchronizing clocks that were spatially separated. This led him to write a paper that landed on the desks of Annalen der Physik on 30 June, and would go on to completely overhaul our understanding of space and time. Some 30 pages long and containing no references, his fourth 1905 paper was titled "On the electrodynamics of moving bodies" (Ann. Phys., Lpz 17 891-921).

In the 200 or so years before 1905, physics had been built on Newton's laws of motion, which were known to hold equally well in stationary reference frames and in frames moving at a constant velocity in a straight line. Provided the correct "Galilean" rules were applied, one could therefore transform the laws of physics so that they did not depend on the frame of reference. However, the theory of electrodynamics developed by Maxwell in the late 19th century posed a fundamental problem to this "principle of relativity" because it suggested that electromagnetic waves always travel at the same speed.

Either electrodynamics was wrong or there had to be some kind of stationary "ether" through which the waves could propagate. Alternatively, Newton was wrong. True to style, Einstein swept away the concept of the ether (which, in any case, had not been detected experimentally) in one audacious step. He postulated that no matter how fast you are moving, light will always appear to travel at the same velocity: the speed of light is a fundamental constant of nature that cannot be exceeded.

Combined with the requirement that the laws of physics are the identical in all "inertial" (i.e. non-accelerating) frames, Einstein built a completely new theory of motion that revealed Newtonian mechanics to be an approximation that only holds at low, everyday speeds. The theory later became known as the special theory of relativity - special because it applies only to non-accelerating frames - and led to the realization that space and time are intimately linked to one another.

In order that the two postulates of special relativity are respected, strange things have to happen to space and time, which, unbeknown to Einstein, had been predicted by Lorentz and others the previous year. For instance, the length of an object becomes shorter when it travels at a constant velocity, and a moving clock runs slower than a stationary clock. Effects like these have been verified in countless experiments over the last 100 years, but in 1905 the most famous prediction of Einstein's theory was still to come.

After a short family holiday in Serbia, Einstein submitted his fifth and final paper of 1905 on 27 September. Just three pages long and titled "Does the inertia of a body depend on its energy content?", this paper presented an "afterthought" on the consequences of special relativity, which culminated in a simple equation that is now known as E = mc² (Ann. Phys., Lpz 18 639-641). This equation, which was to become the most famous in all of science, was the icing on the cake.

"The special theory of relativity, culminating in the prediction that mass and energy can be converted into one another, is one of the greatest achievements in physics - or anything else for that matter," says Wilczek. "Einstein's work on Brownian motion would have merited a sound Nobel prize, the photoelectric effect a strong Nobel prize, but special relativity and E = mc² were worth a super-strong Nobel prize."

However, while not doubting the scale of Einstein's achievements, many physicists also think that his 1905 discoveries would have eventually been made by others. "If Einstein had not lived, people would have stumbled on for a number of years, maybe a decade or so, before getting a clear conception of special relativity," says Ed Witten of the Institute for Advanced Study in Princeton.

't Hooft agrees. "The more natural course of events would have been that Einstein's 1905 discoveries were made by different people, not by one and the same person," he says. However, most think that it would have taken much longer - perhaps a few decades - for Einstein's general theory of relativity to emerge. Indeed, Wilczek points out that one consequence of general relativity being so far ahead of its time was that the subject languished for many years afterwards.

The aftermath

By the end of 1905 Einstein was starting to make a name for himself in the physics community, with Planck and Philipp Lenard - who won the Nobel prize that year - among his most famous supporters. Indeed, Planck was a member of the editorial board of Annalen der Physik at the time.

Einstein was finally given the title of Herr Doktor from the University of Zurich in January 1906, but he remained at the patent office for a further two and a half years before taking up his first academic position at Zurich. By this time his statistical interpretation of Brownian motion and his bold postulates of special relativity were becoming part of the fabric of physics, although it would take several more years for his paper on light quanta to gain wide acceptance.

1905 was undoubtedly a great year for physics, and for Einstein. "You have to go back to quasi-mythical figures like Galileo or especially Newton to find good analogues," says Wilczek. "The closest in modern times might be Dirac, who, if magnetic monopoles had been discovered, would have given Einstein some real competition!" But we should not forget that 1905 was just the beginning of Einstein's legacy. His crowning achievement - the general theory of relativity - was still to come.

Box: Elsewhere in 1905

Einstein’s annus mirabilis tends to overshadow other scientific developments that took place in 1905. So what else was going on in the year that cellophane was invented, the neon sign made its debut, and people were getting to grips with tea bags for the first time? In terms of the number of citations in physics and physical-chemistry journals since 1945, three of Einstein’s 1905 papers feature in the top five, according to Werner Marx and Manuel Cardona of the Max Planck Institute for Solid State Research in Stuttgart. Indeed, his papers on Brownian motion and special relativity take first and second place, respectively, with 1467 and 642 citations (his papers on the photoelectric effect and E = mc² are fifth and 11th). The fourth most-cited paper of 1905 was by Paul Langevin, who derived a fundamental formula in kinetic theory - clearly a popular subject at the time - while Lawrence Bragg published a paper about the energy loss of alpha particles in different media, which became the sixth most-cited paper of the year.

Hendrik Antoon Lorentz, who was influential in the development of special relativity, was elected as a fellow of the Royal Society in 1905 and published several papers, including one on the motion of electrons in metallic bodies. Nuclear physics was also a subject of intense interest at the time, with Ernest Rutherford and Frederick Soddy publishing their theory of nuclear transmutation and Bertram Boltwood demonstrating that lead is the final product of uranium decay. Further afield, Victor Goldschmidt introduced a method to reduce metallic oxides to metals, while Haldane and Priestley demonstrated the role of carbon dioxide in the regulation of breathing.

Outside the world of science, an unsuccessful revolution was beginning in Russia, Antonio Gaudi began two of his famous buildings in Barcelona, and H G Wells had written Kipps. Meanwhile, Jean-Paul Sartre and Henry Fonda were born, as was the Nobel-prize-winning physicist Emilio Segrè, who 40 years later would witness the application of E = mc² with the detonation of the first atomic bomb.