by Lee Phillips - Dec 3, 2015 2:00pm CET
It stands among the most famous theories ever created, but the general theory of relativity did not spring into being with a single, astonishing paper like the special theory of relativity in 1905. Instead, general relativity's birth was more chaotic, involving a handful of lectures, manuscripts, and more than one parent.
One hundred years ago this fall, that harrowing labor occupied almost an entire month in November 1915. When finished, Einstein finally delivered a theory perfectly formed, if not already mature, and trembling with potential. Today, the general theory retains its status as our modern theory of gravity, and its fundamental equations remain unchanged.
However, we've learned a great deal more about the back story and consequences of general relativity in the past century. In fact over time, this model of gravity, space, and time has come to be regarded by many who know it as perhaps the "most beautiful of all existing physical theories." But to fully appreciate all the complexity of general relativity—in substance and creation—you need to start before the very beginning.
Everyone knows relativity
Certainly many people are familiar with the famous theory of general relativity in the sense they're familiar with any celebrity. But what makes the theory tick isn't always so well-known. Perhaps the best approach to the general theory of relativity is by way of Isaac Newton and his theory of gravity. Newton's gravity (in concert with his laws of motion) accurately predicted the motions of the heavenly bodies for over 200 years. It was the first great unification in physics, connecting our terrestrial experience with falling apples directly to the force that binds the solar system together. Newton's work is the beginning of modern science, and the best way to begin to understand relativity is to try to understand what Einstein found unacceptable in Newton's model of the universe.
Newton explained that gravity is a force between any two objects, proportional to the product of their masses and inversely proportional to the square of the distance between them: a simple algebraic formula. This force was an instantaneous action at a distance with no medium nor mechanism behind it.
Einstein recognized several conceptual problems with the classical theory of gravity. His special theory of relativity implied that the cosmic speed limit, the velocity of light, applied to all influences, signals, and information and not merely physical particles. This is inherent in the symmetries of spacetime and the requirement that causes precede effects. But Newton's model of gravity implied that its forces turned on and off instantaneously as masses appeared and disappeared; there is nothing in the classical theory that admits a finite speed of propagation for gravity, as Maxwell's equations described the finite and definite speed of light in a vacuum.
There was also the mysterious identification of the gravitational mass with the inertial mass that appears in Newton's law of motion. This centuries-old apparent coincidence demanded an explanation.
Einstein began his critical examination of gravity as he did with his special theory—through a thought experiment. He imagined being in a windowless box, enjoying the usual experience of gravity but otherwise completely isolated from any information from the outside. After some consideration, it is clear that there is no way he would be able to determine whether he and his box were in a gravitational field, say at the surface of the Earth or in deep space far from any source of gravity, but being uniformly accelerated, say by a rocket attached to the box's bottom. There is no experiment, in practice or in principle, that would be able to distinguish the two situations (neglecting the small nonuniformity in the Earth's gravity that could be measured over a finite distance and considering the point of view of a single point in space).
Applying the maxim of William James that "a difference which makes no difference is no difference at all," Einstein elevated this observation into what he called his principle of equivalence. His insistence that a theory of gravity, and the motions it brings about, respect this principle at its core became the keystone of the general theory. The methodical working out of its consequences in mathematical form became an at times debilitating obsession over many years, culminating in the field equations of 1915 that have withstood all the challenges of the subsequent century.
One might wonder why, exactly, a theory of gravity is called a "general theory of relativity." Well, Einstein began to use this title before the theory was complete. He envisioned it as a generalization of his special theory of relativity. The special theory relates to the motion, time, and space between frames of reference that are moving at constant velocities. It showed how to relate all constant-velocity motions to each other while respecting the universally constant speed of light, and it did so by using a particular formula to transform one frame to another. Einstein assumed that he could do the same with arbitrary reference frames that might be accelerating or rotating by applying his principle of equivalence. He ultimately learned this was not quite possible—there is no general relativity of motion between accelerating frames in the sense that Einstein found in his special theory of relativity. Nevertheless, the title stuck.
What it says (relatively mathless)
To truly dig into general relativity, as in all branches of theoretical physics, it takes math. Hardcore math. The crux of the matter is the mirroring of the structure of reality in mathematical structures. And in this light, talking about a physical theory without its equations is rather like talking about music—worthwhile in some ways, but everyone can acknowledge that something is missing.
To simplify, Einstein's theory of gravity is often introduced by saying that it describes the "curvature" of spacetime. This is evocative and certainly not wrong, but the description sometimes misleads. The gravitational equations relate mass and energy to the "metric tensor," which is the mathematical object that describes this curvature. The metric tensor tells us how to measure distance at different points in space in different directions; it's like a bunch of rulers that stretch and shrink as we move around.
In the normal "flat" spacetime of Euclidean geometry, these rulers are the same everywhere: the Pythagorean theorem is always true, and the ratio of the circumference to the diameter of a circle is always π. The equations of general relativity relate this metric tensor to the distribution of matter and energy in space. A massive object actually changes the rulers in its neighborhood (including the ruler that measures time, which becomes blended with the spatial dimensions to form a unified "spacetime"). The figure here is an aid to the imagination, showing how such a curved 4d spacetime in the vicinity of a planet might be represented with a 2d projection of a 3d surface.
The planets and other masses (and massless photons, as well), rather than responding instantaneously to gravitational "forces," as in Newton's theory, follow geodesics, or shortest paths, through this curved spacetime. It is through this mechanism that the mystery of the identity of inertial and gravitational masses is resolved.
Here lies the crux of the theory's beauty. No longer is spacetime a blank canvas upon which the force vectors of gravity are drawn by a mysterious hand. Now the mass and energy in the universe create the malleable canvas of spacetime themselves, and to be in changing motion in this spacetime is the natural state (just as to be in rest, or uniform motion, was in the Newtonian universe). There is no longer a force of gravity, just spacetime and mass-energy.
As things move around due to the metric tensor; this alters the distribution of mass in the universe. This in turn changes the metric tensor, which determines how things move. This inseparability between motion and the nature of spacetime that determines it is the cause of the nonlinearity, which makes it so difficult to find exact (and, for that matter, numerical) solutions to the equations. Any physics student can use Newtonian physics to calculate the orbit of the Earth around the Sun, but similar problems in general relativity are research projects. (One class of solutions are the "singularities," or solutions with infinite densities, that are called black holes.)
This situation resembles a computationally difficult area of classical physics. Fluid dynamics is intractable because of a nonlinearity that has a similar origin: as parcels of fluid move in response to the pressure field generated by the fluid medium, that motion changes the pressure field, which in turn changes the motion, etc. Just as in general relativity, exact solutions of the Navier-Stokes equations for fluid dynamics are hard to come by, and their calculation by computer is nontrivial. As you might imagine, the equations of relativistic fluid dynamics are fairly insane.
In attempting to give concrete form to the principle of equivalence, Einstein was understandably and immediately bedeviled by the complex mathematical language that these new physical ideas seemed to demand. Like most physicists then and now, he was very familiar with multivariable calculus and differential equations as well as more elementary subjects such as Euclidean geometry. But now he found that his ideas had taken him to a place where his mathematical language was not rich enough—fortunately, he had friends who could help.
Who created general relativity?
Who is the author of Einstein's theory of general relativity? This might resemble the old joke about the New York City landmark, "Who is buried in Grant's Tomb?"—but the answer is not as self-evident.
The conventional notion is that although Einstein may have had significant help with the math, the theory is essentially his creation. His sole authorship is assumed in almost all popular, and the great majority of specialist, histories of the subject.
But you can make a good case that general relativity has three or four main authors. It was certainly not the creation of Einstein alone, working in monkish solitude: it was anything but the "theory that one man worked out a century ago with a pencil and paper," to quote a recent Discover article. This view, while not the popular one, does not disagree with recent scholarship. For example, a current paper in Nature by two historians of science who have made a close study of Einstein's notes presents a list of friends and colleagues who worked closely with him during the development of the theory. These contributions were indispensable.
The earliest of these collaborators was Marcel Grossmann, a mathematician friend of Einstein's from school who appeared as a coauthor on the first several papers on early forms of the new theory of gravity. Grossmann was well versed in the necessary mathematics of calculus in curved spaces, and he became Einstein's first tutor in the subject. My list for the primary authors of general relativity would include Einstein himself, Grossmann, Michele Besso (an engineer and another of Einstein's school friends), and the great mathematicians David Hilbert and Emmy Noether. In addition to these primary authors, there are also a handful of other researchers who made central contributions, but there is no space here to do them all justice.
The cases of Hilbert and Noether are interesting enough to dwell on. In the Spring of 1915, Hilbert invited Einstein to give some lectures at the university at Göttingen, which had become the center of mathematics in Germany and perhaps in the whole Western world. Einstein and Grossmann had published papers expounding a preliminary version of his new theory of gravity, and as Einstein was continuing to work on it Hilbert wanted to know more. He was already familiar with the exotic (to Einstein) mathematics involved (having developed further some of it himself), so as soon as he was able to achieve some understanding of the physics, Hilbert was off and running.
At first, Einstein was delighted in finding a new intellectual companion in Hilbert, someone who was able to instantly grasp the core of the problem and tackle it head-on. As he wrote in a letter near the end of November 1915, "The theory is beautiful beyond comparison. However, only one colleague has really understood it," referring to Hilbert. Einstein considered this famed mathematician a genuine "comrade of conviction" who shared his attitude about science as transcending national and ethnic boundaries. That stance might seem obvious today, but it could be considered unpatriotic in the Germany of WWI.
But this delight soon turned to resentment as a kind of race ensued, as least in Einstein's imagination, to write down the correct set of equations to describe the gravitational field. Both men understood that this was something big, and the stakes were high. Einstein carried on an intense correspondence with Hilbert and other scientists during the struggle. He became horrified that, after his years of striving, someone else might be able to hijack his work and claim credit for the complete and final theory of gravity. As he said in the same letter, "In my personal experience I have hardly come to know the wretchedness of mankind better than as a result of this theory and everything connected to it." In a note a few days after that to Michele Besso, Einstein continued. "My colleagues are acting hideously in this affair." For his part, Hilbert had already made the remark that would later become somewhat infamous: "physics is much too hard for physicists."
While author David Rowe rightfully points out we "know almost nothing about what Einstein and Hilbert talked about during the physicist's week in Göttingen," Hilbert did send off a manuscript with the correct, final field equations that comprised the fundamental content of the theory of general relativity. And he did so almost simultaneously with Einstein's public presentation of these equations. The debate over priority is still being waged by historians, but Einstein and Hilbert had forgotten their differences and moved on almost immediately. In fact, Hilbert relinquished any claim to priority and gave unqualified credit for the theory to the physicist. "Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
In the case of Emmy Noether, it is even more difficult to ascertain the exact contributions she made to general relativity. Mathematical research at the time was largely a verbal affair, with formal publication almost an afterthought, and Noether was especially fond of the conversational approach to math. Hilbert had called her to Göttingen for particular help with the immediate aftermath of the discovery of the field equations to work on the very difficult issue of the conservation of energy in general relativity. (This problem is so tricky that it was only in 1981 that Edward Witten was able to prove that the energy derived from the gravitational field equations is guaranteed to be positive.)
Further Reading
The female mathematician who changed the course of physics—but couldn't get a job
(Emmy) Noether's Theorem may be the most important theoretical result in modern physics.
Correspondence at the time from and to Einstein, including several references to a lost set of notes by Noether, make it clear that she provided critical help and tutelage during the frenzied months leading up to the final appearance of the field equations. But even more important was that her work on the energy problem led to her discovery of the far-reaching result that we now call Noether's theorem and to a mature mathematical understanding of the gravitational equations themselves.
No matter how clear Einstein's vision might have been about what the physical content of a relativistic theory of gravity should be, there was no theory until there was a set of equations that expressed those ideas and that satisfied certain mathematical and physical demands of consistency. This is why Einstein struggled for so many years to put his ideas into a form that would be worthy of the name "theory." And it's the best explanation why general relativity should rightfully be credited to a small handful of authors rather than just Einstein himself.
The matter of the correct form of the gravitational field equations aside, it is still true that the formulation of the equivalence principle, the seminal thought experiments, and therefore the initial physical impetus for general relativity was certainly Einstein's alone.
Verification and unification
World War I had left a large part of Europe in ruins. The unprecedented scale of death and destruction had caused a haze of depression and hopelessness to descend upon much of the world's population. It was with a certain gratitude and relief, therefore, that Europe's weary citizens greeted the distracting appearance of a story about the stars and about a new way of understanding the world.
This was the story that made Albert Einstein the world's most famous scientist, and a household name, overnight.
The newspaper articles were full of diagrams and dramatic descriptions of the project that successfully tested the radical, new theory. The man behind the project was Arthur Eddington, said to be the only person in the English-speaking world who understood Einstein's equations. There is a story that exists in so many variations that its truth may be lost to time, but one version goes something like this: a reporter, interviewing the now famous Eddington, mentioned that he heard there were only three people in the world who understood Einstein's theory. After Eddington was silent for a while, the reporter asked him if he had heard the question. Eddington calmly replied that he was just trying to work out who the third person could possibly be.
Eddington wasn't very popular. He had been a pacifist during the war, refusing to fight but risking his life on humanitarian missions. An atmosphere of scandal surrounded the eclipse project and Eddington's participation from the outset, as there was a strong distaste for anything of German origin in England after WWI. Few could understand why an English scientist was engaged in an elaborate undertaking in support of a German one.
The project was a pair of voyages to set up observing stations in two parts of the world in time to catch a total solar eclipse. Two stations were established for redundancy in case one encountered clouds. Measurements were to be made simply of the positions of a few stars that appeared in the sky close to the sun during the eclipse. These positions would be compared to the normal night-time star locations; the shift in apparent position would be due to the bending of the starlight as it passes through the gravitational field of the sun (as depicted in the newspaper diagram reproduced here).
This bending of the light was also predicted by Newton, but Einstein's theory of gravity predicted twice as much deflection. When all the results were examined, Einstein's theory of general relativity was declared confirmed. A new chapter in modern science had begun.
There is a popular image of Einstein as a solitary, theoretical toiler, perhaps even disdainful of experimental evidence and contemptuous of practical matters. This impression is, to be fair, reinforced by a handful of comments made by Einstein himself, some genuine, some perhaps apocryphal. He probably did say that he would have been "sorry for the Lord" had Eddington's eclipse voyages failed to confirm the predictions of his theory because "the theory is correct."
However, the letters that Einstein wrote to friends and colleagues near the end of 1915 show that he considered experimental verification of his theories crucially important. He attached his greatest hopes around this time to being able to use his developing equations to calculate the anomaly in the orbit of Mercury. When he finally got this to come out right, Einstein considered it a major piece of evidence that he had found the correct equations at last.
This anomaly is the precession in the perihelion of Mercury's orbit: it does not complete closed ellipses, as Newton's theory demands, but each time it comes around to its closest approach to the Sun (the perihelion) it's very slightly off from where it started. If we assume that Newtonian gravity is correct, Mercury's orbit requires some additional gravitational influence to explain it. For some time this explanation was sought in the existence of a new planet orbiting inside Mercury's sphere, but Einstein's calculation made the unseen planet unnecessary.
In a letter to the great physicist and teacher Arnold Sommerfeld in December of 1915, Einstein said, "The result of the perihelion motion of Mercury gives me great satisfaction. How helpful to us here is astronomy's pedantic accuracy, which I often used to ridicule secretly!" The next day, to his friend the engineer Michele Besso, Einstein shared the excitement. "The boldest dreams have now been fulfilled. [...] Mercury's perihelion motion wonderfully precise."
In the middle of February 1916, Einstein was still writing about the Mercury agreement being centrally important. To Otto Stern, he penned the following telegraphic lines, apparently in a hurry:
General relativity is now, almost exactly since we last saw each other, finally resolved. [...] Perihelion motion of Mercury explained exactly. Theory very transparent and fine. Lorentz, Ehrenfest, Planck, and Born are convinced adherents, likewise Hilbert.
General relativity today (and beyond)
Since the days of the eclipse voyages, there have been numerous additional verifications of general relativity's predictions. Gravitational lensing is now routinely observed, especially in images from the Hubble Space Telescope. The gravitational redshift—a shift in the color of light analogous to the Doppler shift that reveals the expansion of the universe, but is instead caused by gravity—has been verified. The Global Positioning System could not work if we didn't know how to make relativistic corrections to the rates of the clocks aboard the GPS satellites. A prediction of general relativity is that clocks will run more slowly on the ground, where the gravitational distortion of spacetime is larger, than in orbit.
The 1991 physics Nobel Prize was awarded for observations of the first known binary pulsar. Russell A. Hulse and Joseph H. Taylor had shown how the period of the mutually orbiting objects changed with time due to the emission of gravity waves in a way exactly as predicted by general relativity. The theory predicts that accelerating (which includes orbiting) masses will radiate waves in the form of distortions in spacetime, somewhat similarly to how charges radiate electromagnetic waves when they are accelerated. This radiation carries energy away from the object, which should change its orbital period.
Today there are efforts worldwide to try to observe gravity waves directly. The Laser Interferometer Gravitational-Wave Observatory (LIGO) consists of two interferometers, one in Hanford, Washington, and the other in Livingston, Louisiana. Each interferometer is made from two 2.5 mile long arms arranged in the shape of an L, acting as "antennae" to detect the ripples in spacetime that general relativity predicts should propagate from exploding stars, binary black holes, and other extreme objects in distant space.
An exciting recent experimental verification of general relativity was carried out by Gravity Probe B, a spacecraft that put cryogenic gyroscopes in Earth orbit. This experiment tested the predictions of frame dragging and the geodetic effect. These two effects are tiny (small fractions of a degree per year) precessions of the rotating gyroscopes. Frame dragging is caused by the effect of the rotation of the Earth, which can be thought of as a twisting of spacetime in its neighborhood, and the geodetic effect is caused by local spacetime warping due to the Earth's mass. Planning for this precision measurement (the spacecraft had a launch window of one second) began 50 years ago, the launch was in 2004, and final results were reported in 2011.
Another test of gravitational theory will occur in the near future, thanks to an expensive mistake. Last year, European GPS satellites were accidentally launched into an elliptical orbit around the Earth that makes them useless as GPS satellites, which need to orbit in circles. However, this oval trajectory takes them at times closer, and at times farther, from the Earth, which means we can use their highly accurate atomic clocks to directly test the effect of gravity on the passage of time.
General relativity, in a sense, stands apart from the rest of physics today. While quantum field theory and its elaborations have managed to unify the forces of nature into harmonious mathematical structures (structures that have been able to predict the existence of exotic new particles), gravity has so far resisted incorporation into a unified theory of nature. This is a peculiar irony, as the fundamental mathematical tool that allows physicists to predict, on the basis of nature's symmetries, the existence of new particles was created by Emmy Noether during her work on the equations of gravity.
General relativity stands apart as well because of the mathematical language it is written in. Even more starkly during the time of its creation than now, this new theory of gravity spoke an exotic dialect of geometry that was entirely unfamiliar to physicists. A current student can pass through an entire course of study in theoretical physics up to the PhD, including at least beginning courses in quantum field theory, with nothing more advanced from the mathematical tool chest than vector calculus and partial differential equations. But textbooks on general relativity, if they are cracked at all, lead the reader through several chapters introducing the intricacies of geometry and calculus on curved surfaces of multiple dimensions before talking much about physics at all.
On the scale of elementary particles, the force of gravity is negligible, but on the scale of the cosmos, it dominates the structure of the observable universe. All the forces of nature, besides gravity, and their associated particles are subsumed into mathematical structures that describe a symmetry in a multidimensional space whose dimensions describe the physical properties of the particles and forces. These abstract mathematical structures have led to predictions of new particles, such as the Higgs boson. But the spacetime symmetries of gravity seem to be of a different type, and so far they have resisted attempts to include them in a unified theory. The experimental breakthroughs described in this section provide powerful evidence that these symmetries capture the reality of spacetime and its connection with matter and energy. The challenge of future theoretical work in general relativity will be to reconcile it with our knowledge of reality at the smallest scales.
Will the next 100 years see the unification of physics and lead us to the "theory of everything?" Will some form of string theory be the answer or perhaps loop quantum gravity? Will we finally observe gravity waves or evidence of wormholes? No matter how these questions are resolved, we can be sure of only one thing: their answers will not lead to the end of science, but to further questions and an endless future of wonder and discovery. In that sense, the first 100 years of general relativity may closely parallel the next century after all.
