In 1918, Emmy Noether published a paper dedicated to Felix Klein “on the occasion of the fiftieth anniversary of his doctorate.”1 Its title can be translated as “Invariant Variational Problems.”

It’s a modest title that offers no hint of the revelations within its pages. They are difficult pages to follow, even today, and must have been quite a challenge at the time, as the paper makes free use of the language and results at the cutting edge of several areas of mathematics. Even the renowned mathematician Cornelius Lanczos, certainly a modern expert in some of the techniques that Noether brought to bear to attack her problem, had to admit that “the original paper of Noether is not easy reading.”2

The difficulty is entirely due to the nature of the material and the verbal condensation demanded by the constraints of publication. The exposition itself does not belie Noether’s reputation as an elegant and clear expositor. Although she was writing for the small and highly specialized audience of her fellow research mathematicians, even those outside the treated subspecialty can glean here and in her other papers some idea of the context and aims of the work.

In this chapter, we’ll first examine the content, meaning, and importance of Noether’s theorem: the aforementioned “single most profound result in all of physics,” again to quote Frank Wilczek, who received the Nobel Prize in Physics in 2004 for his theoretical work with elementary particles.3

I then describe how Noether came to be working on a problem that led to her theorem and how she was uniquely able to make the connections that resulted in its discovery. I’ll briefly and qualitatively portray the areas of mathematics that she combined in an original way to extract new knowledge, exposing a hidden connection between two fundamental ideas in physical science. I’ll show how her result clarified much of our understanding of the previous several centuries of physics and opened a path for the physics of the future. We’ll see how, in proving her theorem, Noether also resolved a crucial issue that remained at the center of the recently published theory of general relativity. And we’ll then return briefly to the feverish months leading up to its publication and see again how she became one of its uncredited authors.

Although I can explain the intuitive and physical content of Noether’s theorem without using the customary symbology of mathematics, I cannot, without resorting to equations, explain in detail how it is proven. Consequently, you will not find here a detailed recounting of her arguments or methods, for such an explanation lies far beyond the scope of this book. For my omission of the mathematical details, I hope you can forgive me. The demonstration of a mathematical result is a mathematical argument, not translatable to nonmathematical language. This is so, even if the result (fortunately for us) has a strong intuitive content.

Noether’s theorem exposes a connection between two ideas that enjoyed a venerable status in physics and that also far transcended, in various guises, physical science: symmetries and conservation laws. To understand what the theorem says and why it opens up new avenues of thought, we first need to understand these two simple ideas.

The idea of symmetry is timeless. It was often discussed by the ancient Greeks in philosophical and mathematical contexts. It has been with us throughout our intellectual history and has always played a central role in all sciences. Symmetry, of course, exists beyond science, in philosophy and aesthetic theory in particular.

Webster’s Revised Unabridged Dictionary, in its superior 1913 incarnation, offers these characteristically elegant definitions of symmetry: “A due proportion of the several parts of a body to each other; adaptation of the form or dimensions of the several parts of a thing to each other; the union and conformity of the members of a work to the whole.” The dictionary adds a secondary definition pertaining to biology: “The law of likeness; similarity of structure; regularity in form and arrangement; orderly and similar distribution of parts, such that an animal may be divided into parts which are structurally symmetrical.” These are fine general definitions, but later I’ll offer one both simpler and more precisely attuned to the way the term is used by mathematicians and physicists.

Symmetry is everywhere in the analysis of painting and music and in psychological explanations of our sensations of beauty in the human form. It plays a descriptive and weakly explanatory role: it is supposed that symmetry pleases us because it brings a kind of logical organization to disparate elements. We are attracted to symmetry in our fellow humans because it’s a sign of health and reproductive fitness.

Symmetry is not absent even from moral philosophy, for what is the Golden Rule but an appeal to a principle of symmetry?

In biology, it’s a constant source of questions and answers, both in its presence and in its absence. Internally, the disposition of our organs departs from our external bilateral symmetry, but why is the liver on the right instead of on the left? And why, in the small minority of individuals evincing a mirror reflection of the customary internal arrangement, is it so? The shape of a helix defines a type of symmetry, and the unraveling of the helical structure of DNA opened a new epoch in the science of life.4

One of the most intriguing mysteries at the intersection of symmetry and biology has to do with the chirality, or handedness, of the amino acids and sugar molecules found in living things. We’re all familiar with the fact that, to tighten a screw, we usually have to turn it clockwise: these are right-handed screws. But we also know that we occasionally encounter a left-handed screw; we have to keep an open mind when our repairs aren’t going as planned. The reason that almost all screws are right-handed is that we’ve decided to adopt that as a standard, to make life easier. This right-hand-screw dominance in hardware is a type of asymmetry: in the absence of the chosen convention to make right-handed screws the standard, we would expect to encounter equal numbers of both possibilities. Some of the molecules of life also have a definable chirality, as they have a generally helical form, like a screw; they can be left-handed or right-handed. A departure from symmetry on the molecular level presents an enduring mystery for biologists: Why are all of life’s amino acids left-handed, while all the sugars are right-handed?5 The symmetry of the laws operating at these scales doesn’t seem to favor either chirality, so we would expect an indifference where we observe a firm exclusivity.

In physics and astronomy, symmetry has always occupied center stage. The taxonomy of crystal forms finds its explanation in the possible symmetrical arrangements of molecules. The wanderings of the electrons in those molecules, the patterns of vibration in a timpani, and the arrangements of clouds all obey symmetrical solutions of similar equations. Here, too, a conspicuous absence of symmetry occurring with no obvious, compelling explanation creates a mystery: Why do we live in a universe of matter rather than antimatter?

Most of the elementary particles of which we and our stuff are made have an antimatter twin.6 The twin is identical except that it has the opposite electrical charge. For example, the antimatter twin of the electron is called the positron. It has the same mass and other physical properties of the electron but a positive charge. We can create antimatter particles in accelerators and have even succeeded in putting these particles together to form antimatter atoms, demonstrating, in principle, that antimatter particles can combine in the same way that normal particles combine to create larger structures—structures that presumably form everything we observe in the universe, including ourselves.

Here is where the puzzle arises: the fundamental equations of physics that describe the behavior of elementary particles are indifferent to whether these particles are matter or antimatter. Yet we’re made of electrons, protons, and so on, not positrons and antiprotons. Our understanding of the Big Bang is that equal numbers of particles and antiparticles should have been created. Once again, the obstinacy of observed reality in ignoring the symmetrical perfection of our models requires explanation and provokes research. This mysterious violation of symmetry echoes the curious preference of biology for left-handed amino acids and right-handed sugars. The research is ongoing. I describe no answers to these questions, because there are no answers that explain everything or that satisfy everyone.

All of these examples reflect the roles that symmetry considerations played in science and philosophy before Noether’s theorem: symmetry was a guide to the overall patterns of reality. It served as a concise description and sometimes as an aid to constructing the solution to a problem. Its role was qualitative or semiquantitative. Symmetry told us what to expect and motivated us to explain its absence where we had expected to see it.

Until the arrival of Noether’s theorem, symmetry was not law. If it were law, there could be no exceptions. It did not govern the way the universe behaved.

Noether’s theorem established a new role for symmetry. The ancient companion and guide to humanity’s unending project to describe our universe now had new powers. The theorem explained precisely how symmetry did govern what could and could not happen wherever it was present. Noether’s theorem raised symmetry from a description of a pattern of reality to an active participant in its behavior. It promoted symmetry to the status of law.

The theorem was a signpost along the path of development of critical thought. To understand how deep its influence runs, we need first to grasp, in full generality, the concept of symmetry itself.

The generalized idea of symmetry, the particular conception of symmetry that appears in Noether’s theorem, is an extension of the familiar, visualizable examples that immediately come to mind. The word symmetry often evokes, and is illustrated by, a butterfly or an idealized face. Both images appeal to our aesthetic instincts; both are symmetrical because one-half is the mirror image of the other. Consequently, the entire object, when admired in a mirror, retains its appearance.

Simple geometrical figures embody either this symmetry or another one (or both): we can rotate them by a certain angle without changing how they appear. A circle can be rotated by any angle; a square, 90 degrees; a general rectangle, 180 degrees; and a hexagon, 60 degrees.

In each case, we have applied a transformation (a rotation or a reflection, in these examples) without causing any change. This quality is the key to understanding the generalized concept of symmetry: something has a symmetry if it is the same in every way after you apply a transformation. The type of transformation allowed defines the type of symmetry.

Once you’ve absorbed this idea, you can take it a step further: The entities embodying symmetry need not be physical objects. Nor do the transformations need to be limited to geometrical operations.

A couple of concrete examples will make this notion, which at first contact might seem forbiddingly abstract, easier to grasp. Imagine that after a period of inflation, a government decides to redenominate its currency.7 The population trades in their million-unit banknotes for one-unit banknotes; everyone’s bank account is divided by a factor of a million. Nothing in the economy changes, except that it’s more convenient to talk about money. Checks need fewer zeros. The unit of currency is simply rescaled. It takes the same number of minutes of labor to buy a loaf of bread. The transformation here is the rescaling in the unit of measurement for money; the entity that has remained the same after the transformation is the economic system. The economy is symmetrical with regard to currency rescaling. Various governments have, at times, taken advantage of this symmetry to rescale their currencies after suffering extended bouts of hyperinflation.

Another example: in the United States, we call the lobby floor of a building, the floor that you are on when you enter from the street, the first floor. We count up from number 1 (ignoring the quaint superstition that leads to omitting the number 13). In some other countries (France, for example), the first floor is one floor up from the lobby. The difference can be thought of as where to put an imaginary zeroth floor. In the United States, floor zero is one floor underground, while in France, it’s the lobby (the rez-de-chaussée). Imagine that one day France decided to come to its senses and adopt the American system. Its apartment buildings and hotels would not suddenly sink into the ground or take flight; nor would they stretch or shrink. The transformation in the location of the zero would have no effect on the reality of the structures themselves. Their heights and positions are symmetrical with regard to the change in convention.

In both these examples, we have a transformation that has no effect on reality: a symmetry, in the wider sense of the term. It is this wider sense that I intend whenever, from this point forward, I use the word symmetry.

So much for symmetry. The second big idea is that of a conservation law, a rule that demands that a conserved quantity does not change its value as time passes. This idea is even simpler than the idea of symmetry. In fact, Noether’s theorem derives much of its profundity because it reveals a hidden connection between two ideas that are so simple and that pervade much of our reality and our descriptions of it.

An example of a conservation law is the conservation of mass: the ancient concept of the invariability of the total quantity of substance. You can divide the stuff of reality into smaller bits, and you can stick the bits together, in various combinations, at will. But the total amount of stuff will never change. You cannot bring matter into existence; nor can you banish it from the world. You can even burn a piece of wood, but if you gather all the smoke, ashes, and water vapor, you will have the same mass as the log you started with. Matter can change form, but the amount of it is conserved. (Some readers may be aware that the conservation of substance is not strictly true; because of E = mc2, mass and energy can be transformed into each other. But I describe a conservation law as stated and known since antiquity; we can examine its consequences even if it fails to precisely describe the actual world. In any case, we can recover the law by replacing mass by a particular combination of mass and energy.)

Conservation principles can appeal to our intuition as deeply as symmetry. Some people are prey to an inner conviction that their luck is conserved. If something beneficial befalls them, they suffer anxiety while waiting for the other shoe to drop; the next unfortunate accident confirms their belief that every turn of chance must be balanced by one that moves their fortunes in the opposite direction. A similar superstition is known to students of probability theory as the gambler’s fallacy. It appears as an article of faith among many habitués of casinos. These people believe, for example, that a run of red numbers on a particular roulette wheel somehow makes a subsequent black number more probable than usual. Underlying this belief, which is the basis of innumerable groundless systems of betting, is an unshakable intuition that there must exist some causal force that evens out the results of chance: a law of conservation of probabilities. Of course, the gambler’s fallacy and other similar superstitions are examples of purely imaginary conservation laws, but their pervasiveness illustrates how principles of conservation are embedded in our sense of how the world should operate.

My final example is a conservation law resting on firmer ground. Something called angular momentum basically tells us how much twisting force we would have to apply to get something to stop spinning. If the spinning thing is composed of various parts, its angular momentum is higher the faster those parts are rotating, the heavier they are, and the farther they are from the center of rotation. Everyone has seen ice skaters spinning around and then spinning faster, as if by magic, as they pull in their arms. You are seeing conservation of angular momentum: as the mass of the arms is pulled in closer to the center, the skater has to spin faster to keep the angular momentum the same. It’s not magic; it’s just a physical conservation law in action.

The theorem that Noether proved in 1918 revealed a connection between symmetries and conservation laws. But this already understates its significance. For Noether’s theorem doesn’t merely show how these two ideas are connected; it proves that they are not two separate ideas but are, and have always been, the same idea.

From the earliest records of our long struggle to bring order to the universe, the ideas of symmetry and of conservation have been present in some form. Symmetry appears as an ideal of human creation and as a description of the underlying order of reality. Conservation implies the notions of constancy amid change, of something preserved amid the illusory swirls of contingency.

These two ancient principles evolved into precise statements and took on mathematical forms in the scientific age. They became tools of calculation but never lost their status as profound signifiers of an underlying structure to reality.

Until Noether’s theorem, they seemed to embody separate aspects of this underlying structure. With the precision of mathematics, the theorem shows that conservation and symmetry are the same thing.

We need, now, to make precise what “the same thing” means. I’ve been following a common practice in using the term Noether’s theorem to refer to what are in fact four theorems, all proved in the same 1918 paper. These four theorems are two mathematical statements and their converses. The difference between the two statements is rather technical, having to do with different classes of symmetries; the distinction need not concern us here. But to appreciate the full significance of Noether’s theorem, we must be clear about the meaning of converse.

A converse of a statement is the statement in reverse. For example, a statement might be “All Greek philosophers are mortal beings.” Whether you agree with that or not, let’s assume it’s true. The converse would be “All mortal beings are Greek philosophers,” which I hope you’ll agree is not true, even if the original statement is. It’s oddly common to encounter people making the mistake of assuming that a statement is the same as its converse, usually in situations where the logical flaw is not so obvious.

Noether’s 1918 paper proved that for each symmetry, there was a corresponding conservation law. The paper showed how to derive the conservation law from the symmetry. It also proved the converse: that for any conservation law, there was a corresponding symmetry. This is what I mean when I say that she proved that the two ideas were the same idea: if each implies the other, then one cannot exist without the other. They are, in every sense, the same thing, simply seen in different ways.

The symmetries dealt with in Noether’s paper are those of physical systems that admit of a certain description. But this class of systems is quite general and includes, in addition to general relativity (the field that motivated her investigation), any fundamental theory in physics. The class also includes other types of systems that lie beyond physics but can be described with similar mathematical machinery; we’ll visit some of these in Chapter 8.

I believe it is this, the existence of these theorems and their converses, that inspires the feeling of awe and profundity in many of the physicists and others who have contemplated them. If the theorems only worked in one direction, they would still be immensely important to physics and its history. But it’s the identity of symmetries and conservation laws that convinces us that we have reached a new understanding of the beauty and harmony found in nature.

Consider an example. One of the basic and intuitively obvious symmetries in physics is the time translation symmetry: the idea that we can place the zero of time anywhere convenient without affecting the predictions that physics makes about how anything will behave, evolve, and change. This idea is so obvious that most people never pause to consider it. Of course if we change, for example, where to put the year zero on our calendars, doing so has no effect on history itself, on what happened, or on how long it took to happen. This irrelevance of the placement of zero time is a symmetry, because it is a transformation that has no effect. It is the same as the previously mentioned numbering of the floors in a building and how the choice of the French or the US convention makes no difference.

Noether’s theorem demonstrates something unexpected about the symmetry of time translation: that it is equivalent to energy conservation. And because the theorem includes the converse, it’s not that time translation symmetry simply implies energy conservation, or the reverse. It’s that time translation symmetry is energy conservation.

Energy

After the dust settled at the end of 1915, when Hilbert had relinquished any claim to priority over the general theory and when he and Einstein were friends again, Hilbert turned his attention to an unresolved issue. He had tried several times to show that the general theory of relativity obeyed energy conservation, with no success. By now, scientists accepted that any reasonable theory in physics should obey energy conservation: it was obeyed by the central and fundamental theories of Newton, which explained motion, and those of Maxwell, which covered electricity and magnetism. It was the law. Just as money cannot appear or disappear from your checking account without a deposit or a withdrawal to account for it, energy should not appear or vanish from space unless its movement could be accounted for.

Here I need to be a little more precise about two forms of energy conservation. We can call these forms global and local energy conservation.

The global type is what almost everyone is familiar with on some level. It’s usually stated something like this: in an isolated system, the total amount of energy cannot change over time. When some number—whether it represents a quantity of energy or of something else, calculated from the properties of a system—does not change over time, we call that number a conserved quantity. The statement that the total energy cannot change over time is a statement of energy conservation.

Physics defines various forms of energy. The law of energy conservation means that as an isolated system evolves, some amount of one form of energy can be transformed into another form and sometimes back into the first form, and so on, but the total cannot change.

In the energy conservation law, we’re careful to stipulate an isolated system because if we allow interactions with the world outside the system, all bets are off. In general, these outside interactions—this lack of isolation—will involve some energy transfers into or out of the system, so we wouldn’t expect the system’s energy to remain unchanged.

As an example of an isolated system, take a collection of billiard balls rolling around on a table and occasionally colliding and changing direction. Of course, in real life, no system is truly isolated. Can you hear the clack of the balls colliding? Those sound waves carry energy out of the system, never to return. But part of the art of physics, and of thinking up idealized examples with which to explain physics, is to focus on the essential and to disregard small departures from an idealized picture. We try to ignore insignificant effects. The amount of energy carried away by these sound waves is pretty small, a tiny perturbation to the system. Larger is the effect of friction, which we can’t ignore and which is a bit more complicated than we’d like to include in our analysis. You know that friction is important because unless, unlike Hilbert, you’ve avoided pool halls all your life, you know that any particular ball will not keep rolling and bouncing around forever but will pretty quickly slow down and stop.

But we physicists have invented a way to completely eliminate the effects of friction and all other related, complicated phenomena: we simply say we’re talking about an ideal pool table. In this case and many others, with this one word, ideal, we can vanquish all the dragons of heat, friction, dissipation and ignore any of the other forms of energy aside from the energy of motion and other ones introduced a bit later in this chapter.

Using the assumption of an ideal system, we can begin again, this time more carefully. For an example of an isolated system, we can consider a collection of billiard balls rolling around on an ideal table and occasionally colliding and changing direction. We neglect the effects of friction and other phenomena. Here we have only one form of energy to keep track of: the kinetic energy of the system, which is the total of the kinetic energy of all the individual balls.

Kinetic energy is the modern term for the energy of motion, which we can calculate from an object’s mass and speed. This form of energy is proportional to its mass and to the square of its speed, explaining why it’s so much worse to crash your car into a tree at forty miles per hour than at twenty miles per hour.

In the ideal case, where no energy is lost to sound, heat, or any other factor outside our ideal model, if we know the system’s kinetic energy at any particular time, we know it for all times before or after that, for it can never change. That’s because energy conservation is the law, and kinetic energy is the only kind of energy in this system.

Knowing about energy conservation on the billiard table makes it far easier to solve certain types of problems, especially because all the balls (in the pool or billiard games with which I’m familiar) have the same mass, so we can just focus on the velocities. For instance, suppose I place two balls close to each other, or touching, and shoot a third ball at the spot exactly between them at a known speed. I observe that the third ball comes to a stop as the first two each fly off, and I wonder if I can calculate what the speeds of the originally stationary balls have to be.

Using a detailed force calculation and Newton’s laws of motion would be one way, but energy conservation makes it so much easier. We know from the geometrical symmetry of the problem that the first two balls must end up with equal speeds. And from the law of energy conservation, we know that the energy on the table after the collision must be the same as the energy that I imparted to the third ball. (Obviously during my interaction with the balls, the system is not isolated, but after I hit the ball and step back, the system evolves in isolation, at least according to the rules of the game.) Using just this law and knowing the expression for kinetic energy, I can calculate in my head that the speed of each of the first two balls will be the speed of the third divided by the square root of two (I certainly hope that’s correct). Conservation laws are a powerful technology for solving problems.

One more example will show how energy conservation works when we have more than one kind of energy. If there are springs, electricity, gravity, or other sources of force (other than instantaneous forces such as from ideal billiard ball collisions), then the total kinetic energy of the system is not conserved. We need to add to the energy of motion something called potential energy, which represents stored kinetic energy that is available to be returned to the masses as motion again.

With actual springs, we can see this potential energy in their compression or elongation away from their equilibrium positions. But with gravity or electricity, the idea is a bit more abstract. We see the changes of potential energy in the configuration of the system, that is, in the positions of its constituent parts.

One example is a cannonball shot straight up from the ground. Initially, the ball has a very large quantity of kinetic energy, but as it rises, the earth’s gravity slows it down until it comes to rest, for an instant, at its zenith. As it slows on its way to its zenith, the cannonball’s kinetic energy decreases, but the energy of the total combined cannonball-earth system remains the same, because the ball’s energy is stored in the form of potential energy. This potential energy is at a maximum when the ball comes to rest at its maximum height and the kinetic energy is zero. Immediately, the potential energy begins to be converted back into kinetic energy as the cannonball picks up speed on the way down. When the object reaches the ground, its kinetic energy, and therefore velocity, is the same as when it first left the barrel. (All these observations neglect the friction of the atmosphere and other sources of energy dissipation.)

As in the case of the billiard balls, it’s almost always easier to solve cannonball problems by exploiting energy conservation than by calculating forces and solving the differential equation equivalent to Newton’s second law of motion.

Energy conservation may be so familiar to us that we assume it is somehow obvious. But Newton did not know about it, and the law was not discovered, in its definitive form, for almost two hundred years after he published his laws of motion.

Since energy conservation was not yet part of science, it was not part of the wider culture, either. Shakespeare was generally aware of the scientific developments of his time.8 If the seventeenth-century worldview had included the necessity of the conservation of energy, he surely would not have written, in his Sonnet 154, “Love’s fire heats water, water cools not love,” for he would have understood that if one substance supplies energy to a second substance, then the first must lose the same quantity of energy that the second gains.9

By 1915, energy conservation had been accepted as a general law of the universe. At least, no exceptions were known.

In addition to simple classical systems such as the ones in the preceding examples, energy (and momentum) conservation worked with more complex field theories (see the previous chapter) such as electromagnetism and hydrodynamics. Energy conservation in field theories takes the form of a local energy conservation law rather than the simpler global energy conservation just described. Of course, global energy conservation still holds as well and can be derived from the local law, but the local law must be satisfied as well.

In cases such as electrodynamics, we still need a way to express the idea that energy is neither created nor destroyed. We must do so even when energy cannot be neatly assigned to individual objects, like billiard balls, flying around our system, or easily recognized as resulting from the position of a cannonball.

Local conservation laws are stricter than global ones. They require not only that the total amount of energy remain constant in time but that, in any region of space within the system, the change in energy be accounted for by the flow of energy into or out of the region. In other words, local conservation laws formalize the idea that energy cannot be created or destroyed. And this requirement applies not merely to the total, in an entire system, but to any region of the system, whether large or microscopically small.

The fact that general relativity didn’t seem to obey a local energy conservation principle was the problem that nagged at Hilbert. As far as he and Klein could see, the new theory of gravity did not respect this bedrock principle of nature.10 Apparently, energy could be illegally created and destroyed.

They naturally considered this a fatal flaw, one so profound that they wondered if they had made an error in calculation or were perhaps suffering some deep conceptual confusion. But as Hilbert repeated the calculations and as his colleagues tried to find ways around it, the problem seemed unavoidable. The situation could lead to paradoxes. When the possibility of gravitational waves was considered, it was found that an object could speed up as it lost energy by emitting gravity waves, rather than slowing down, as one would expect.

The Göttingen mathematicians were willing to accept a theory radically different from anything that had been even suggested before as a model of material reality, and the theory of general relativity certainly fit this description. Einstein’s idea turned the very structure of space and time into a player on the dynamical stage, subject to continuous change, just like the planets and other masses moving through what used to be considered the unchanging background of space. They could accept all this, as could at least part of the small fraction of the physics community capable of understanding it. But the failure of energy conservation would be a deal-breaker, at least according to what had become the received wisdom in the physical sciences. Unless this apparent defect was resolved, the theory of general relativity might be seen as a fascinating mathematical curiosity but could never be seriously considered as a description of the real world.

Hilbert was not the first person to worry about energy conservation in general relativity. Einstein himself had returned to it periodically during the development of the theory over the previous eight years or so. He had found no satisfactory solution that worked with the final, correct version of the gravitational equations.

Klein had also been communicating with Carl Runge, a well-respected mathematical physicist, about the energy issue in general relativity. Those who have studied elementary numerical methods have encountered the Runge-Kutta method; this is the same Runge. His ability to delve so deeply into Einstein’s theory so soon after its dissemination certainly indicates a high level of mathematical sophistication.

Runge thought that he had found a way to get general relativity to obey the local conservation law.11 Impressed by Runge’s concept, Klein showed it to Noether. But she immediately saw that Runge’s idea wouldn’t work, and gave Klein the bad news.

The situation in 1916, therefore, was that the status of energy conservation in general relativity was confused and uncertain.

Hilbert and Klein had been keen on getting Noether to Göttingen partly because of her expertise in invariant theory, especially aspects of it that still interested the two men. Hilbert, of course, was already deeply curious about what Einstein was doing with gravity in 1914, having tried to coax him to pay a visit to Göttingen starting in 1912. It’s unclear whether Hilbert had foreseen in particular that he himself would be studying general relativity as a practitioner and, further, that he would benefit from collaborating with Noether. It seems unlikely. There’s no sign that Hilbert put his pen to paper to work on the Einstein version of gravitation before the physicist finally came to give his lectures. As we’ve seen from Noether’s letters, Hilbert had started working on general relativity immediately after those lectures and had, from the start, brought its many mathematical puzzles to her. Once he hit the wall of energy conservation, it was clear to him that some of the methods of invariant theory might be brought to bear on the problem. He asked Noether to see if she could break through the wall.

Although Noether had no particular interest in physics, she had done her homework. She would develop a reputation in later years as a thorough scholar, intimately familiar with the literature, both historical and recent, in every field of mathematics that she explored. Her 1918 paper shows that she was keenly aware of the physical implications of its content; she knew that it had relevance both to science and to pure mathematics. She refers to its connection with general relativity throughout and mentions several special cases of her general theorems that had been unearthed in mechanics in the historical literature. She had no way of knowing that physicists and mathematicians, less conscientious in their scholarship than she, would “discover” special cases of her theorem in the future, down to the present day, and believe each time that they had found something new.12

We’re now in possession of a more general idea of symmetry: the quality that remains unchanged in an object after we apply a transformation. Similarly, a physics theory has a symmetry if all its defining equations, or their consequences, are unchanged after you apply the transformation. Take Newton’s mechanics. It consists of a few concepts and laws of motion. The second law amounts to a differential equation that we solve to get the motion of a cannonball or anything else. It contains a variable for time, a variable for each spatial dimension, and variables for the masses of the objects. It also includes the forces at play, be they gravity or anything else. The law relates these forces to the change in the velocities of the objects, where these velocities are the changes in the objects’ positions in time.

The variable of time, for example, appears in the second law only in an expression that refers to its change. The “naked” time variable does not appear. This means that we can add any constant value to the time variable or—doing the same thing—change where we define the “zero” of time, and no predictions of the second law or of Newtonian mechanics will be affected. The shift in time just drops out of the equation. In other words, if it takes you an hour to drive to work today, it will still take you an hour to drive to work after daylight saving time goes into effect and we’ve subtracted an hour from the arbitrary numbers that we use to say what time it is.

The preceding discussion means that classical mechanics is symmetric under time translation (translation is the word physicists like to use for shifting some variable). And these continuous time translations do define the type of symmetry structure to which Noether’s theorem applies. In the case of time translation symmetry, Noether’s theorem tells us that the associated conserved quantity is the total energy. As a physical system evolves, its various types of energy may each be converted into each other, but the sum of them all, the total energy, will not change with time.

Why is this a big deal? We already knew about the conservation of energy, after all. As mentioned, we “knew” about this law of nature but not about why it was true in general. The conservation of energy had a semi-empirical status. It could be proven for some systems. For example, with the cannonball shot, we can prove that the sum of the kinetic and potential energies remains constant. For other systems, scientists just assumed that energy conservation must hold because no exceptions had yet been found. That’s why it was expected to hold in general relativity as well.

Noether’s theorem showed that energy was conserved in any system with time translation symmetry. This result raised the murky idea of energy conservation to a fundamental aspect of reality, a necessary property of any universe whose basic symmetries agreed with our deepest intuitions. Because all known physical theories respected time translation symmetry, Noether’s theorem demonstrated that energy conservation was indeed a universal law. And it elevated the law from a semi-empirical observation to a mathematical truth.

Another example of a simple application of Noether’s theorem is the symmetry of spatial translations. Just as the arbitrary location where we place the zero of time should make no difference, the zero of any spatial coordinate should not matter, either. Naked spatial coordinates do not appear in the equations of motion. Another way to say this is that Newtonian mechanics is homogeneous in space. Turning the crank of Noether’s theorem shows that the symmetry of spatial homogeneity is equivalent to momentum conservation.

The conservation of momentum, a conservation law that was known before energy conservation, is implicit in Newton’s laws. It’s the conservation law that explains why rockets work and why you feel a recoil in your hand or shoulder when using firearms. The momentum of the expelled fuel from a rocket or a projectile from a firearm must be balanced by an equal and opposite momentum. A simple version of this idea is expressed in the most familiar of the laws of Newton: for every action, there is an equal and opposite reaction.

But Noether’s results showed that momentum conservation was more than a special property of Newtonian mechanics. Instead, it was a general, and necessary, feature of any spatially homogeneous theory. A theory that did not have this symmetry would imply that its results—the laws of nature—were somehow different in different places. Since this is an affront to our intuition, we naturally expect that any reasonable physics respect the symmetry of spatial homogeneity. The symmetry implications of momentum conservation exposed by Noether’s theorem elevated the conservation law to something that we would expect any theory of physics to encompass. And, as with energy conservation, momentum conservation would be a property of any universe that conformed to our deepest intuitions about the symmetries of space.

The last example of Noether’s theory of symmetry concerns the symmetry of rotations, the symmetry inherent in a circle. Noether’s procedure showed that this form of symmetry implied the conservation of angular momentum.

Rotational symmetry is also called spatial isotropy. A theory that did not respect this symmetry would admit the idea of experiments that gave a different result when repeated after rotating the laboratory. A different result doesn’t seem reasonable, so angular momentum, like the other conservation laws, could now be seen as a consequence of intuitive space-time symmetries that any rational theory would have to respect.

Note that these symmetries are symmetries inherent in the theories, as reflected in the symmetries of their equations. Clearly, if you perform an experiment twice in different locations where different conditions affect the outcome, that doesn’t mean the laws of nature have changed. Implicit in this discussion is the idea that, for example, translation in space means only shifting the location in space while holding any relevant influences constant.

Time translation, spatial translation, and rotational symmetry are just three examples of symmetries that happen to be connected, through Noether’s theorem, with familiar conservation laws. The theorem proves that any symmetry leads to a conservation law, even if it might not have a name or might be unknown. Presumably, the theorem could be used to discover new conservation laws (and was, as we’ll see in Chapter 7).

Because Noether also proved the converse of her theorems, we know not only that a specific symmetry implies a specific conservation law but also that the conservation law implies the symmetry. If you know you have a conserved quantity, then you also know that there is a symmetry in the theory, perhaps hidden, perhaps unknown. But it must be there.

As I’ll relate in Chapter 7, Noether’s theorem was slow to penetrate the consciousness of most scientists. One anecdote demonstrates something of this gradual percolation of the awareness of the theorem. In an interview, Werner Heisenberg remembered how, long before, a paper had attempted to resolve an experimental puzzle in particle physics. The data had suggested that energy was not being conserved. The paper’s authors had proposed that energy might be conserved only “statistically,” and not exactly. When Wolfgang Pauli (one of the most prominent theorists of the time) heard of this, he said, “Well, that’s too dangerous. There you try something which one shouldn’t try.”

Heisenberg goes on: “Much later, of course, the physicists recognized that the conservation laws and the symmetry properties were the same. And therefore, if you touch the energy conservation, then it means that you touch the translation in time. And that, of course, nobody would have dared to touch. But at that time, this connection was not so clear. Well, it was apparently clear to Noether, but not for the average physicist.”13

We saw in the previous chapter how, in special relativity, space and time, which had separate identities in Newton’s physics, were combined into a unified mathematical object, a four-dimensional space-time. In physics before Einstein, energy and momentum had maintained their separate identities. Each was conserved independently of the other, and Noether showed how the separate conservation laws were equivalent to separate symmetries: energy conservation with time translation and momentum conservation with space translation. But in special relativity, since space and time no longer had separate identities, Noether’s theorem suggests that energy and momentum might not have separate identities, either.

This is indeed the case. In special relativity, energy and momentum were combined into an object called the energy-momentum four vector. (It has this ungainly name because it has four components: one for the energy and three from the three components of momentum in three-dimensional space.) Therefore, the new mathematical object appearing in Noether’s paper to represent the conserved entity in general relativity was not of a completely unfamiliar kind to physicists, who had already gotten used to combining energy and momentum. In fact, when special relativity is applied to continuous fields such as electromagnetism or fluids, a related but simpler tensor already appears.14

The pre-Einstein energy and momentum conservation laws, which were two different, independent facts of nature, had been combined into one law for the conservation of relativistic energy-momentum. You can see that this had to happen, if you think about E = mc2. Mass was now another form of energy all by itself. It’s actually possible for two photons (quantum particles of light), which have no mass, to collide and become transformed into two particles with mass, an electron and a positron.15 We can’t explain the process or predict its outcome using the separate energy and momentum conservation laws.

In general relativity, the situation is considerably more complex. We can still think about something like an energy-momentum four vector, a cousin of the one that appears in special relativity. However, Einstein’s earlier theory did not demand a wholesale replacement of Newton’s law of gravity or of Euclidean geometry. In general relativity, the gravitational “force” is replaced by the properties of a curved space-time. The curvature of space-time is determined by the mass and energy-momentum contained within it. And here’s where it gets really complicated: the curvature causes things to move, which in turn changes the curvature. Also, the curvature of space-time itself contains its own contribution to the energy-momentum. This intricate interdependence is what makes the gravitational equations so hard to solve. The energy-momentum and the space-time curvature exist in a nonlinear relationship, each creating the other. In the pithy formulation of physicist and gravity expert John Wheeler, “Space-time tells matter how to move; matter tells space-time how to curve.” It took someone with the mathematical talents of an Emmy Noether to untangle this web.

Noether had broken through Hilbert’s wall. She was able to show Klein and Hilbert that they were not as confused as they might have thought. They hadn’t been able to find a local energy law, because there could not be one.

By proving its exact equivalence with the symmetry of time translation, Noether’s theorem serves as the definition of energy, a concept that, until her result, tended to attract ad hoc and imprecise descriptions.

Feza Gürsey was a Turkish mathematician and physicist who carried out fundamental research in mathematical physics on the applications of symmetry to quantum theory and general relativity.16 According to Gürsey, “Before Noether’s theorem the principle of conservation of energy was shrouded in mystery.… Noether’s simple and profound mathematical formulation did much to demystify physics.”17

Often when new entities were introduced into physics, some form of energy was defined in terms of the new entity. It was not always obvious what form this energy should take, but it needed to participate in the inviolable law of conservation and to connect correctly with other physics concepts such as force and work. So, for example, in his monograph laying out his new synthesis of electricity and magnetism, James Maxwell devoted several pages to working out just what form the energy of the electromagnetic field should take.18

It wasn’t always clear how to define the energy, because, before Noether, it wasn’t clear what exactly energy was. In the physics of today, the concept of energy is supplied by Noether’s theorem. If a physical system possesses time translation symmetry, then we say that it has an energy, and we derive the form of the energy by applying the theorem: the energy in the system is the quantity that satisfies the conservation law that Noether’s theorem shows is equivalent to the symmetry. Noether’s theorem eliminates the confusion and the ambiguity. If there is no time translation symmetry, as in general relativity, then we know that no energy can be defined.

Einstein’s special theory of relativity contains all kinds of bizarre and wonderful phenomena, such as the slowing down of clocks and the contraction of space; these things don’t exist in Newtonian mechanics. However, we know that Newton’s physics still works now just as well as it did centuries ago. Reality doesn’t start to suddenly behave differently because somebody invents a new physics. The resolution to this conflict is that the predictions of special relativity match those of Newton’s under a limiting condition: that all velocities are much smaller than the speed of light. Nobody had observed time dilation or any of the other non-Newtonian effects, because they were too small to measure. We can measure them now, of course, and they agree wonderfully with Einstein’s version of reality.

In the general theory, the relevant limiting condition is that of weak gravity. If we’re in a region far from any enormous concentration of mass or energy, the gravity will be pretty weak and space-time will be approximately flat. The geometry of the world will be very close to the Euclidean version that we learn about in high school. The angles of a triangle will add up to 180 degrees. The departure from an elliptical orbit is only observed easily with Mercury because, being the closest planet to the sun, it experiences the strongest gravity—the most warped space-time.

In finite regions of flat, or nearly flat, space-time, energy will be approximately conserved, just as it is in Newton’s universe. Since most of space-time is approximately flat, we can calculate many things as if a local energy conservation law holds, even if the equations show that, in general, it does not.

This fact was used in the first detection of gravitational waves, one of the predictions of general relativity (and one with its own stormy history). Similarly to how accelerating charges emit electromagnetic waves, accelerating masses emit waves of space-time distortions that we call gravitational waves. These are actual vibrations in the geometry of space-time that travel through the universe at the speed of light. You may remember the exciting news about the detection of gravitational waves in 2015 from the LIGO experiment.19 The experiment detected them by directly measuring space-time distortions, which caused the length of an object to oscillate. The precision of the measurements required to detect this tiny distortion in space-time is astounding. The results were widely reported in newspapers. For example, a New York Times article announced, “Gravitational Waves Detected, Confirming Einstein’s Theory.”20 Unfortunately, most headlines and articles claimed or suggested that the experiment constituted the first detection, but it was not. The finding was a historic achievement because it was the first direct detection of the waves.

However, gravitational waves were in fact first detected in 1974, when two astronomers, Russell Hulse and Joseph Taylor, discovered a pair of (probably) neutron stars zipping around each other in a fast orbit.21 The binary system is now called the Hulse-Taylor pulsar in their honor. The astronomers could detect that the orbit is slowly decaying, and successfully explained that the decay was due to the energy lost in the form of gravitational waves. The rate of decay agreed exactly with the predictions of general relativity. Hulse and Taylor received the Nobel Prize for this work about twenty years later. (The aforementioned New York Times article is a rare example of a notice in the popular press that did mention the pulsar work.)

The calculation and the verification of gravitational waves from the pulsar depended entirely on energy conservation: the energy loss in the orbit was seen to be exactly balanced by the energy carried away by the waves. But certainly, gravity and the warping of space-time were strong near the neutron stars; if they weren’t, there would be no appreciable gravitational radiation to begin with. How can we use energy conservation to verify the predictions of a theory where there is no energy conservation, and specifically in a regime where we know that the conservation law does not even hold approximately?

To understand the answer, we must first appreciate that space is vast, with enormous distances between objects. Near the orbiting neutron stars, space-time will be strongly warped, and we’ll need general relativity to describe reality. But let’s back away hundreds of light-years from these two massive objects until we reach a region far from any large masses, where gravity is weak and space-time is flat. Now let’s draw an enormous, imaginary sphere with its center near the pulsars and its perimeter at our location. Although the sphere is huge, it only contains the pulsars; this is possible because of the vastness of space.

Now keep track of all the gravitational radiation crossing the boundary of the sphere. It carries energy with it, out of the volume inside the sphere. Although in general, as Noether showed, we don’t have local energy conservation in general relativity, we have a close approximation of it in regions of flat space. Consequently, the energy crossing the spherical boundary must be balanced by the energy lost from the binary pulsar system.

Of course, nobody created any giant spheres. But the argument shows that if we can calculate the energy carried away by the gravitational waves, we can calculate the consequent decay of the orbit—and that’s how the reality of gravitational waves was first demonstrated.

(The debates over the very existence of gravitational waves lasted for decades, involving, among other incidents, Einstein’s petulant reaction to having one of his papers rejected by a referee. One productive step was taken by Richard Feynman in 1957.22 Using a simple thought experiment involving beads sliding on a rod, he convinced a crowd of physicists that these waves could exist and must carry energy.)

Noether’s theorem can make the analysis of a tricky problem straightforward. Feynman, in his (in)famous Lectures on Physics, poses a problem that he describes as a “paradox” (while making it clear that its paradoxical nature is merely apparent).23

In Feynman’s thought experiment, a metal disk, part of a simple electrical apparatus, suddenly begins to rotate, even though no obvious turning force acts on it. It’s as though the ice skater, in our earlier example, began to spin faster without any movement of the arms. In either case, angular momentum appears out of nowhere. Its conservation seems to be violated.

The resolution of the paradox is that, of course, angular momentum is still in fact conserved, as it must be. It was zero at the start, when nothing was moving, and is still zero after the disk has begun to rotate. Since the angular momentum remains zero, never changing, the conservation law is not violated. But how can this be, when the disk clearly turns and therefore obviously attains some nonzero angular momentum? The resolution is that the angular momentum of the disk is canceled by the equal and opposite angular momentum in the electric field surrounding the disk. A positive and a negative add to maintain the initially zero angular momentum. Feynman warns that this is a difficult problem, presumably because the student may not realize that the invisible force fields called electromagnetic fields can themselves carry angular momentum.

However, the student who is aware of Noether’s theorem will not only understand that electromagnetic fields have angular momentum but will also be able to use the theorem to calculate it exactly. This is another instance where the theorem not only confirms a conservation law (which holds in this case because of the symmetry of spatial isotropy) but also identifies the form that the conserved quantity takes. Here Noether’s theorem turns a conceptually confusing problem into an almost mechanical calculation.

Through its clarification of the status of conservation laws in general relativity, Noether’s theorem has important implications for cosmology, the study of the origin and evolution of the universe as a whole.

The idea of the law of conservation of energy is drilled into every physics student, usually without the context provided by Noether’s theorem. This omission is mainly due to the habit of putting off more general, symmetry-based approaches to physics until advanced classes typically at the graduate level. In advanced classical mechanics, the student may encounter Noether-style approaches, always in a watered-down form and often with no mention of Noether herself.24 Physicists who become exposed to high-energy physics necessarily learn about applications of Noether’s theorem in much greater detail. As explained in Chapter 7, the entire theoretical framework of this field is based on the theorem or special cases of it. Most of a student’s undergraduate education, and much of the coursework in graduate school, therefore pursues physics in something closer to the Newtonian style.

The result of this suboptimal educational tradition is that unless they specialize in cosmology or gravitation, many grown-up physicists believe that the conservation of energy is an inviolable law of the universe. They are surprised at the assertion that the energy of the universe as a whole is not conserved, often insisting that the assertion must be incorrect.25

The same physicists accept that the universe is expanding. Quantum mechanics can provide some insight into the consequences of that.

A photon of light has an energy inversely proportional to its wavelength. Long-wavelength photons, such as those from a microwave oven, have less energy than shorter-wavelength photons, such as those from an x-ray machine. That’s why ultraviolet light (short wavelength) from the sun can cause cancer, but cell phone radiation (long wavelength) cannot. The individual photons must have sufficient energy to damage the DNA molecules inside our cells. Saying that radio waves from a cell phone can cause cancer is similar to saying that you can knock over an elephant with a barrage of soap bubbles.

In an expanding universe consisting mainly of light, an observer sitting anywhere will observe all of its photons receding—their wavelengths will be shifted to the red (longer-wavelength) side of the visible spectrum—redshifted—just as the light from distant stars is redshifted in our actual universe, because of the Doppler effect. The Doppler effect is the same mechanism that causes a siren to have a higher pitch as an ambulance approaches you, only to shift to a lower tone as it moves away. The effect works for any wave phenomenon—light as well as sound. Since light shifted to the red has a lower energy, the total energy of this hypothetical universe must continuously decrease as it expands.

Noether’s theorem lets us prove that the total energy in an expanding universe decreases, as a direct consequence of general relativity, in agreement with the argument from photon energy. It resolves what might otherwise seem, at least to some, to be a paradox of cosmology.

In a more general sense, however, it is in fact impossible to define a quantity to fulfill the role of energy for the entire cosmos. We now know that Noether’s theorem clarifies the meaning of energy by identifying it with the conserved quantity that is equivalent with time translation symmetry. This approach replaced the older, ad hoc notions of energy and is what modern physicists turn to when a precise definition for energy is needed. But as we also now know, if the cosmos is described by general relativity, then it lacks the symmetry of time translation invariance, unlike the more intuitive Newtonian universe. If the cosmos is expanding or contracting, then the conditions are clearly not constant in time. Unlike the Newtonian universe, in Einstein’s universe, our definition of time zero makes a difference. Today’s same experiment cannot be repeated tomorrow, even in principle, because by tomorrow, today’s universe will no longer exist. If we don’t have time translation invariance, then we don’t have an energy definable through Noether’s theorem. We are led to the conclusion, still uncomfortable for some, that in our universe, no sensible total energy can be defined.26 This is even more obvious in some hypothesized universes, in which a period of expansion is followed by a contraction stage and in which we will eventually return to Big Bang–type conditions. As we approach conditions similar to the Big Bang, space-time curvature will again be extreme and there will be no region of flat space-time. In that scenario, there can be no quantity that behaves even approximately like an energy over the entire past and future history of that universe. The situation is perhaps best summed up in a widely used textbook on Einstein’s theory of gravitation: “The issue of energy in General Relativity is a rather delicate one.”27

The question of energy in general relativity is still a matter of active research. The research has expanded since 1918 in step with the expansion of our knowledge of the universe, which has grown enormously. It’s easy to forget that when Noether proved her theorem, most people thought that our Milky Way galaxy was all there was. What’s more, they presumed that the universe was static, eternally unchanging in size. In fact, when Einstein realized that his equations permitted solutions where the universe could expand or contract, he considered this result a defect and adjusted the theory to eliminate those clearly unreasonable possibilities.28 Now, beginning with Edwin Hubble’s discoveries that distant objects were receding from us, we’ve lived with an expanding universe for so long that the notion seems quite natural.

Noether’s theorem is a guide that helps us make sense of our cosmos, whether it will expand forever or one day collapse in a Big Crunch. By relating space-time symmetries to local and global conservation laws, the theorem delineates what is possible and shepherds our intuition into uncharted territories.

Emmy Noether found a solution to the energy conservation problem that had perplexed the great David Hilbert, stumped his other colleagues, and eluded Einstein. She did so by proving a set of related theorems, mathematical results of great power and generality. Although nobody realized it then, her result would eventually transcend the original context. It would profoundly change all of physics and, a hundred years later, begin to transform fields ranging from biology to economics.

We ultimately can’t explain how a mathematical genius manages to discover profound connections between things that have eluded everyone else. But in analyzing how her theorem arose from the contact between at least two widely separated areas of mathematical research that were brought together by a rare mind working at the forefront of each of them, we’ve taken a step toward an intuitive appreciation of at least one aspect of the creative process.

As briefly described in Chapter 8, one unexpected application of Noether’s theorem in recent decades has been in economics. Some of the economists who took the lead in this innovative exploitation of Noether’s mathematics also spent some time musing about how she was able to make such a breakthrough. “Noether had the ingenious insight of combining the methods of the formal calculus of variation[s] with those of Lie group theory,” they note, referring to an advanced form of calculus and part of the formal mathematics of symmetry.29 Many observers of important acts of creation in science, the arts, and technology have noticed a recurring pattern: far-reaching results often arise at the intersection of fields that have developed separately until they are brought together by an unusual individual who is intimate with all of them.

This was the case with Noether. By the time Hilbert had placed the problem of energy conservation in gravitation before her, she had become one of the world’s supreme experts in two quite different fields of mathematics. One field was related to the theory of invariants, as described in Chapter 1. This was the mathematics of symmetry in its most general form. Her greatest fame among mathematicians would eventually result from her refinement and rebuilding of related areas of mathematics, which became what we know today as abstract algebra.

The second field is the aforementioned calculus of variations (or variational calculus) and a nexus of related theories. The calculus of variations, an evolution of the calculus invented by Newton and Gottfried Wilhelm Leibniz, can calculate entire histories or paths of systems. For example, calculus can give us formulas for the speed of a baseball at any moment, but the calculus of variations can tell us its entire path as the solution to a single problem.

Perhaps it’s not too far afield at this juncture to remember what Alan Kay, the creative computer scientist and inventor of the Smalltalk programming language, said about this subject: “All creativity is an extended form of a joke. Most creativity is a transition from one context into another where things are more surprising.”30

As I mentioned earlier, Hilbert’s personal formulation of general relativity was, in a sense, more concise and elegant than Einstein’s version. Hilbert had found a way to write the gravitational field equations as a compact statement in the language of the variational calculus. But most histories of this episode fail to point out that he depended on some of Noether’s results to do so.

Noether’s Contributions to General Relativity

The “race” between Einstein and Hilbert to reach the correct gravitational field equations, recounted in the previous chapter, is usually depicted as two men scribbling mathematics in their respective solitudes and happening to reach the finish line through quite different routes. More detailed histories mention that Hilbert was happy to share his progress with Einstein, in whose fevered brain the somewhat paranoid idea of a competition germinated and festered.

There is another inaccuracy in the way the story is usually told. Although sparse, the totality of the correspondence, notes, and remembrances from this brief period makes it clear that Hilbert was not working alone. He was conferring to an extent with Felix Klein, but Klein’s own letters testify to the fact that he was more or less out of his depth. The only colleague at Göttingen, aside from Hilbert, who we are certain had complete fluency in the mathematics and methods requisite for attacking every aspect of general relativity was Noether. As historian of mathematics David Rowe points out, all of Hilbert’s and Klein’s work relating to general relativity “relied heavily on Emmy Noether’s expertise in differential invariant theory.”31

Apparently, far in advance of Noether’s formal publication of her theorem, she had worked out much of the results and shared them with Hilbert. We also know that Hilbert’s version of the gravitational field equations depended on a theorem that he discovered and that this theorem in turn depended on a version of Noether’s theorem that he was aware of before November 1915. The unavoidable implication is that Hilbert’s formulation of general relativity, which, as mentioned, employed the language of the variational calculus and was developed independently of Einstein, is the joint work of Hilbert and Noether.

Questions of credit and priority were not of any particular interest to Hilbert, who was focused on results—on seeing mathematics make progress. His attitude in this was shared completely by Noether. But perhaps his knowledge that “his” version of general relativity was not his alone contributed to his readiness to resolve the looming priority dispute with Einstein by abandoning all claims once and forever.

By this time, after all that has been written over the years about the genesis of general relativity, it should be obvious that meticulous questions of priority will never be settled. Nor do they need to be settled. Ideas are like soap bubbles, glistening with the colors of the spectrum and gliding soundlessly from brain to brain. Emmy Noether, David Hilbert, and Albert Einstein are three of the most ingenious minds that have emerged in the entire history of the human species. That they were all three together, asking the same questions at the same time and place, is a miracle of fortune. The theory of gravitation that emerged is a new way to describe reality itself, a sharp turn in the history of human thought.

Calling general relativity a new way to describe reality is not an exaggeration. The other physical theories deal with particular interactions between particular types of entities. Two electrons interact through electromagnetism but not through the strong nuclear force. Two neutrons interact through the strong force, but there is no electrical force between them. But everything—even light—interacts through gravity. General relativity has been verified from microscopic distances up to the size of the entire cosmos. It describes space-time itself, and everything—every bit of matter and energy—obeys its laws. It is literally the fabric of physical reality.

And underneath gravity, underneath the quantum theories of electromagnetism and nuclear forces, underneath any future theory yet unimagined, sits Noether’s theorem.

Hilbert’s interest in general relativity was part of an ambitious but ultimately unsuccessful effort to construct a unified theory that would encompass both gravity and the nature of matter. It was also an expression of his conviction that formal methods, with which he had had such stunning success in mathematics, were the key to putting physics on surer ground.

Einstein’s opinion of Hilbert’s attempt at a unified theory can be seen in his letter to Hermann Weyl near the end of November 1916: “Hilbert’s assumption about matter appears childish to me, in the sense of a child who does not know any of the tricks of the world outside.… Mixing the solid considerations originating from the relativity postulate with such bold, unfounded hypotheses about the structure of the electron or matter cannot be sanctioned. I gladly admit that the search for a suitable hypothesis… for the structural makeup of the electron, is one of the most important tasks of theory today. The ‘axiomatic method’ can be of little use here, though.”32

Today it would seem as if Einstein was entirely correct. Despite the unparalleled importance of Hilbert’s program for pure mathematics, a highly formalized approach to theoretical physics has rarely been useful in the discovery of new science. Physicists tend to employ a more Babylonian style of mathematics, in contrast to a classical Greek style, to use the nomenclature introduced by Feynman in a fascinating lecture (available online).33 In his talk, Feynman explains what he means by his terminology: physicists tend to use a loosely connected bundle of mathematical results, selecting formulas as they find them useful. While having some idea of how to get from one formula to another, they do not work within a formally defined axiomatic system, with all the definitions and assumptions carefully spelled out and all the derivations perfectly rigorous. This more systematic style of mathematics is what Feynman associates with the Greeks. It finds its exemplar in Euclid’s Elements, the timeless exposition and gathering of Greek geometrical knowledge.

Of course, the mathematics that Hilbert discovered (or invented, depending on your particular philosophy of mathematics) was immensely useful for all branches of physics. And it will continue to be, as long as mathematics remains the language of science. From Hilbert space to integral equations, theoretical physics would look very different without Hilbert’s contributions to the machinery. And his deep mathematical insights did often have clear implications for physics—and not only in general relativity.

For these and other reasons, Hilbert is sometimes painted as the last significant universalist by historians of mathematics. One historian describes him as having led mathematics out of the nineteenth century and into the twentieth.34

Other Symmetries

Noether’s theorem works with other symmetries besides those that have an obvious physical interpretation, such as the time translation invariance and the other symmetries just discussed.

To understand the full significance of the theorem, we need to appreciate that it applies to other types of symmetry, keeping in mind the general notion of symmetry that we now have in hand. As we’ll see in Chapters 7 and 8, its application to more abstract, mathematically defined symmetries had two crucial results. First, it allowed the theorem to become the foundation for our current theories of matter, and, second, it permitted the theorem’s application beyond the realm of the physical sciences.

The symmetries that I’m alluding to are more abstract. They do not involve space and time but are related to other properties of objects or systems. To get a taste of what some of these types of symmetries can look like, we’ll consider a familiar example involving electricity.

Have you ever noticed a bird perched on a high-voltage power line and wondered how it managed to avoid electrocution? The explanation is that only a difference in voltages produces a physical effect. The effect is an electrical current, which is the name we give to the movement of charge. All the familiar uses and abuses of electricity are the results of currents flowing through various objects: current flowing through a wire heats it up, the phenomenon that we use to build space heaters, water heaters, and stoves; a changing current passing through a wire creates a magnetic field, a phenomenon that we use to create motors and loudspeakers; sending a current through a thin wire heats it to the point that it glows, leading to the old-fashioned incandescent light bulb. In these examples, the charge that is in motion takes the form of electrons drifting through metal. In the nervous system, the charges are ions (atoms with more or fewer electrons than protons), and the ion currents contract muscles, send signals to the brain, and, in extreme cases, cause people to write books.

Even a modest electrical current passing through an animal’s heart can fatally interrupt its rhythm. Yet the bird chirps on. It can do so because all the currents just mentioned are driven by differences in voltage between two points in space. The sign on the utility pole warns of an alarmingly large number of volts, but this is the voltage difference between the wire and the ground on which we stand. If we or something like a ladder we’re holding touches the wire while our feet remain on the earth, we’re fried as this voltage difference pushes an electrical current through our bodies.

The bird has both feet on the wire and is not in contact with the ground. To be sure, there will be a tiny voltage difference between its two feet because of the nonzero resistance of the conductor, and a consequent tiny current runs through its body. But this minuscule flow of electrons doesn’t disturb the creature’s blissful ignorance of the laws of electricity.

What I’ve been calling voltage is a form of energy called the electrical potential in the field theory of electricity and magnetism. It’s similar to the gravitational potential energy that you gain each time you ascend in an elevator. And, as in the case of gravity, this electrical potential can be converted into an equal quantity of kinetic energy, or the energy of motion, as happens when the elevator fails and comes crashing down. (I’m describing an ideal, frictionless tragedy; in reality, some of the potential energy would also be converted into heat, sound, and deformation of materials.)

Here is the connection to symmetry and Noether’s theorem: the irrelevance of the value of the electrical potential at any point is a type of symmetry. We learned to see the inconsequential nature of the zero of time, or the numbering of the floors of a building, as a form of symmetry: a change or shift in an origin left everything the same. Electrical voltage is another example where the zero point, or the absolute value, of a quantity has no effect and is therefore another form of symmetry.

In the case of time, we know that the symmetry leads, through the theorem, to the law of energy conservation. It turns out that the machinery of the theorem shows that the symmetry in the electrical potential is equivalent to charge conservation. This is another basic conservation law, like mass conservation. It had been accepted for a long time as simply another rule for how the universe operates. Charge conservation means that in any system and for any collection of charged objects, the net charge (the difference between the total amount of positive charge and the total amount of negative charge) will remain constant in time forever. You cannot create or destroy charge.

Until Noether’s theorem, these two fundamental aspects to the universe, the symmetry of voltages and the conservation of charge, appeared as two separate principles. The theorem shows that the two facts about nature are in a deep sense the same fact, a hidden and unimagined connection between two principles that had seemed independent of each other.

The equivalence between electrical potential symmetry and the conservation of charge is yet another surprising and profound insight produced by Noether’s result. Without this equivalence, charge conservation, like energy conservation, would be something contingent. We observe charge conservation to be true, but it might have been otherwise. The symmetry in the electrical potential energy is not something contingent on observation but is deeply embedded in the structure of the theory. In showing that charge conservation is mathematically equivalent to potential energy symmetry, Noether’s theorem demonstrates that this crucial aspect of observed reality—that charge can be neither created nor destroyed—cannot be otherwise. That is, if Maxwell’s equations, which define the subject of electricity and magnetism, do describe reality, then this reality must include conservation of charge.

As in Newtonian mechanics, Noether’s theorem has again shone a piercing light into an old, well-established theory, illuminating it from the inside. The theorem finally reveals its hidden structure, its hidden symmetries, and its supposed contingencies as necessities and grants us insights that eluded its creator.

Up to this point, we were examining space-time symmetries: time translation, isotropy, and spatial translation. The present discussion of electrical potential considers a symmetry of a similar kind but does not involve space or time. The symmetry in the electrical potential is part of a large category of symmetries with a more mathematical flavor. Physicists call this category gauge symmetries. They are symmetries just like all the others: instances of transformations that leave everything as it was. But the transformations have now passed beyond our ancient intuitions about the structure of real-life space and time in which we live and move and into the realms of the abstractions of mathematicians and physicists. And yet intuition is not left completely behind, although our inner suspicions concerning voltage changes, and our lack of complete surprise at the oblivious resilience of our feathered friends overhead, is more of an acquired rather than an instinctual condition.

As mentioned earlier, just as Noether’s theorem shows that time translation symmetry is equivalent to energy conservation, it also shows that spatial translation symmetry is equivalent to momentum conservation.

The establishment of conservation of momentum in our laboratories today, here on earth, shows that the laws of physics are the same throughout the vast expanse of the universe (aside from complications arising from general relativity). They must be so because spatial translation symmetry means that where we decide to place the zero of our rulers cannot affect the predictions we make about the physical behavior of the cosmos or anything in it.

Various scientists and philosophers have, on occasion, suggested that we might consider an alteration of the laws of physics as we travel through space—that a different physics might hold in different parts of the universe. They point out that, after all, our science is based on observations in the corner of the cosmos where we happen to live. Why should we assume that the laws of physics are the same millions of light-years away?

Noether’s theorem tells us why. As long as momentum conservation holds, the theorem proves that the laws of physics cannot depend on where we are—spatial translation symmetry demands that “where we are” has no meaning. This is another example of the power of her mathematical result to bring clarity to questions of what is and is not possible, what we might spend time and energy musing about, and which ideas are simply nonstarters. Here, as in many other places, the theorem, a demonstration in pure mathematics, has far-reaching philosophical implications.

In his book A Beautiful Question: Finding Nature’s Deep Design, renowned physicist Frank Wilczek makes a related and intriguing physico-philosophical observation about what he calls the “uniformity of substance.”35 He points out something that many people, including many other physicists, have probably never thought about critically, because it’s something that we take for granted. It is simply that every example of a particular elementary particle, such as an electron, is exactly the same thing. Different electrons may be in different states—for example, moving with different speeds—but they all have the same properties. There’s no such thing as an electron that’s a little bigger or heavier than another under the same conditions or one that has a slightly different charge. An electron is an electron; there is no way to distinguish one from another. If you were to go to the most distant corner of the universe and do an experiment on an electron, you would get all the same results as you would on earth.

The fundamental reason for this “uniformity of substance” is that the different electrons are not really separate particles, according to modern quantum theory. They are, instead, “excitations” of a quantum field that permeates all of space. This quantum field is part of the physical description of reality; if energy and momentum conservation hold, then Noether’s theorem tells us that because of the symmetries of time and space translation, our description of physical reality must not vary in time or location. The same excitations of the quantum field must occur everywhere, and so all electrons must be identical. Through this argument, Wilczek shows how Noether’s theorem not only provides a deep explanation for the uniformity of particle properties but also assures us that these properties must be the same in all parts of the universe. This argument constrains the range of admissible cosmological theories. Whatever ideas cosmologists come up with to explain deep space or the history and structure of the universe, they should assume not only that the laws of nature are the same everywhere but also that the “stuff” that these laws act on is also identical in all parts of the cosmos.

Noether’s theorem elevates space and time themselves into active participants in reality, rather than the blank canvas on which physical things happen. The structure of space and time, and the symmetries that hold within it, give rise to the conservation laws that describe the behavior of physical reality. Something very like this is true of general relativity, which transformed our understanding of space and time into a kind of substance that gives rise to the phenomenon of gravity. But Noether’s theorem goes further, connecting all physical forces—electromagnetism, the strong nuclear force, and everything else—to a set of eternal symmetries.

Einstein and Energy Conservation After Noether’s Theorem

After some recovery time from his race to make the equations of general relativity work, Einstein returned to open questions related to the new theory. It turned out he had not forgotten about the energy conservation puzzles. Before receiving a copy of Noether’s 1918 paper in which she proved her theorem and which impressed him greatly, he was still not completely at a loss. Hilbert had been corresponding with him on the energy conservation issue, as he, Klein, and Noether were gradually coming to a more solid understanding. Or probably more likely, as Noether was working it out or already had done so, she was explaining her results to the two men.

In a letter from Einstein to Hilbert in May 1916, the physicist confesses that he’s confused about a version of an energy theorem that Hilbert had sent him. Einstein goes on to say, “It would suffice, of course, if you would charge Miss Noether with explaining this to me.”36

As much correspondence is lost, we extract whatever insight we can from what survives. Several things are clear from Einstein’s remark. It seems there was already, probably beginning soon after Einstein returned from his Göttingen lectures to his home in Berlin, established communication between him and Noether. We can surmise, from the tenor of Einstein’s comment and his other letters, that he had already become accustomed to Miss Noether in the role of his tutor, especially in regard to variational forms of the field equations, energy theorems, and related generalizations. It should not be forgotten that this period includes those months where many historians describe Einstein as working in solitude in the search for the correct field equations.

Apparently Einstein reached a better understanding of some of these issues by the end of the year. Almost exactly one year after publishing his final version of general relativity, he published a paper titled “Hamilton’s Principle and the General Theory of Relativity.” In this paper, Einstein derives the gravitational equations using the variational calculus, similarly to how Hilbert derived them in his own publication. “Hamilton’s Principle” in the title is a reference to the methods of the variational calculus. He mentions Hilbert’s derivation but points out that his own is better, because, unlike Hilbert, he makes no assumptions about the composition of matter. He also, under certain restrictions (“The energy components cannot be split into two separate parts such that one belongs to the gravitational field and the other to matter, unless one makes special assumptions”), derives a conservation law combining energy and momentum.

Einstein’s derivation of this conservation law is a very special case of Noether’s theorem that would be published later. It’s nearly impossible to avoid the conclusion that he was led there in good part by Noether’s correspondence and her tutelage, both directly and relayed through the letters of Hilbert and Klein. He mentions Noether nowhere in the paper, but we should remember that nothing on this subject had yet been published formally under her name. Nevertheless, Einstein’s parsimony in mentioning the contributions of others when publishing his works on gravitation has contributed greatly to the myth that he developed general relativity on his own. His preference for sole authorship of these papers also did nothing to counteract the erasure of Emmy Noether’s contributions to the history of physics.

Einstein seemed to be proud of this paper, sending copies of it to several colleagues. He was not aware of what would be its most enduring legacy. In this article, finally wearying of the typographical prolixity demanded by the tensor calculus, with its profusion of indices, little subscripts, and superscripts referring to coordinates, Einstein invents a clever notational innovation that makes the equations more streamlined. The scheme was universally adopted with relief and is now known as the Einstein summation convention. This innovation alone would have ensured the physicist’s immortality.

Continuing their direct correspondence, Noether sent Einstein a copy of her Noether’s theorem paper shortly before its formal publication. He reacted with another letter to Hilbert on May 24, 1918, writing with wonderment at what happened to his equations in Noether’s hands, how he had never imagined that things could be expressed with such elegance and generality. He teased the others, suggesting that they could have learned a lot from her, had they brought her aboard sooner: “Yesterday I received a very interesting paper by Ms. Noether about the generation of invariants. It impresses me that these things can be surveyed from such a general point of view. It would not have harmed the Göttingen old guard to have been sent to Miss Noether for schooling. She seems to know her trade well!”37

Apparently it took Einstein some time to absorb the implications of Noether’s paper. A note he wrote to Klein on May 28, 1918, shows that he was still struggling to fully understand energy conservation in general relativity, saying that there was still a “mathematical gap” in his argument.38 He had grappled with the issue in an earlier incarnation of the theory as well.39

One final note about the circumstances of Noether’s publication of her paper that has been the subject of this chapter: it was not published in one of the journals in which she had published her previous research, but in the Proceedings of the Royal Göttingen Academy of Science.40 The academy was where Göttingen mathematicians and other scholars liked to give talks and have discussions. Much important research had appeared, and would appear, in the pages of its Proceedings.

The Royal Academy had not permitted Noether to join or to present papers. Therefore, her paper was presented on her behalf by Felix Klein. (It was discovered recently that in the early 1930s, Hermann Weyl had tried to get Noether elected into the academy but gave up when it was clear the institution would not budge.41)

One day, Hilbert decided to punish the members of the academy for their obstinate refusal to accept his colleague, employing a withering, ironic insult. Sitting among them in their hallowed halls, he blurted out, “It is time that we begin to elect some people of real stature to this society.”

Having gotten their attention, he added, “Now, how many people of stature have we indeed elected in the past few years?”

Then he made a point of peering slowly around the room, letting his gaze fall deliberately upon a specific handful of the members there. And he answered his own question: “Zero.”

3

The Theorem