by Don Lincoln, Senior Scientist, Fermi National Accelerator Laboratory; Adjunct Professor of Physics, University of Notre Dame

This November marks the centennial of Albert Einstein’s theory of general relativity. This theory was the crowning achievement of Einstein’s extraordinary scientific life. It taught us that space itself is malleable, bending and stretching under the influence of matter and energy. His ideas revolutionized humanity’s vision of the universe and added such mind-blowing concepts as black holes and wormholes to our imagination.

Einstein’s theory of general relativity describes a broad range of phenomena, from nearly the moment of creation to the end of time, and even a journey spiraling from the deepest space down into a ravenous black hole, passing through the point of no return of the event horizon, down, down, down, to nearly the center, where the singularity lurks.

Deep into a quantum world

If you were reading that last paragraph carefully, you’ll note that I used the word “nearly” twice. And that wasn’t an accident. Einstein’s theory has been brilliantly demonstrated at large size scales. It deftly explains the behavior of orbiting binary pulsars and the orbit of Mercury. It is a crucial component of the GPS system that helps many of us navigate in our cars every day.

But the beginning of the universe and the region near the center of a black hole are very different worlds — quantum worlds. The size scales involved in those environments are subatomic. And that’s where the trouble starts.

Einstein’s heyday coincided with the birth of quantum mechanics, and the stories of his debates with physicist Niels Bohr over the theory’s counter intuitive and probabilistic predictions are legendary. “God does not play dice with the universe,” he is famously reported to have said.

However, regardless of his disdain for the theory of quantum mechanics, Einstein was well aware of the need to understand the quantum realm. And, in his quest to understand and explain general relativity, he sought to understand how of gravity performed in his epic theory when it was applied to the world of the super small. The result can be summarized in three words: It failed badly.

Bridging the quantum world to relativity

Einstein spent the rest of his life, without success, pursuing ways to integrate his theory of general relativity with quantum mechanics. While it is tempting to describe the history of this attempt, the effort is of interest primarily to historians. After all, he didn’t succeed, nor did anyone in the decades that followed. [Einstein’s Biggest Triumph: A Century of General Relativity (Op-Ed)]

Instead, it is more interesting to get a sense of the fundamental problems associated with wedding these two pivotal theories of the early 20th century. The initial issue was a systemic one: General relativity uses a set of differential equations that describe what mathematicians call a smooth and differentiable space. In layman’s terms, this means that the mathematics of general relativity is smooth, without any sharp edges. 

In contrast, quantum mechanics describes a quantized world, e.g. a world in which matter comes in discrete chunks. This means that there is an object here, but not there. Sharp edges abound.

The water analogy

In order to clarify these different mathematical formulations, one need think a bit more deeply than usual about a very familiar substance we know quite well: liquid water. Without knowing it, you already hold two different ideas about water that illustrate the tension between differential equations and discrete mathematics. 

For example, when you think of the familiar experience of running your hand through water, you think of water as a continuous substance. The water near your hand is similar to the water a foot away. That distant water might be hotter or colder or moving at a different speed, but the essence of water is the same. As you consider different volumes of water that get closer and closer to your hand, your experience is the same. Even if you think about two volumes of water separated by just a millimeter or half a millimeter, the space between them consists of more water. In fact, the mathematics of fluid flow and turbulence assumes that there is no smallest, indivisible bit of water. Between any two arbitrarily-close distances, there will be water. The mathematics that describes this situation is differential equations. Digging down to its very essence, you find that differential equations assume that there is no smallest distance.

But you also know that this isn’t true. You know about water molecules. If you consider distances smaller than about three angstroms (the size of a water molecule), everything changes. You can’t get smaller than that, because when you probe even smaller distances, water is no longer a sensible concept. At that point, you’re beginning to probe the empty space inside atoms, in which electrons swirl around a small and dense nucleus. In fact, quantum mechanics is built around the idea that there are smallest objects and discrete distances and energies. This is the reason that a heated gas emits light at specific wavelengths: the electrons orbit at specific energies, with no orbits between the prescribed few. 

Thus a proper quantum theory of water has to take into account the fact that there are individual molecules. There is a smallest distance for which the idea of “water” has any meaning.

Thus, at the very core, the mathematics of the two theories (e.g. the differential equations of general relativity and the discrete mathematics of quantum mechanics) are fundamentally at odds.

Can the theories merge?

This is not, in and of itself, an insurmountable difficulty. After all, parts of quantum mechanics are well described by differential equations. But a related problem is that when one tries to merge the two theories, infinities abound; and when an infinity arises in a calculation, this is a red flag that you have somehow done something wrong. 

As an example, suppose you treat an electron as a classical object with no size and calculate how much energy it takes to bring two electrons together. If you did that, you’d find that the energy is infinite. And infinite to a mathematician is a serious business. That’s more energy than all of the energy emitted by all of the stars in the visible universe. While that energy is mind-boggling in its scale, it isn’t infinite. Imagining the energy of the entire universe concentrated in a single point is just unbelievable, and infinite energy is much more than that. 

Therefore, infinities in real calculations are a clear sign that you’ve pushed your model beyond the realm of applicability and you need to start looking to find some new physical principles that you’ve overlooked in your simplified model.

In the modern day, scientists have tried to solve the same conundrum that so flummoxed Einstein. And the reason is simple: The goal of science is to explain all of physical reality, from the smallest possible objects to the grand vista of the cosmos. 

The hope is to show that all matter originates from a small number of building blocks (perhaps only one) and a single underlying force from which the forces we currently recognize originates. Of the four known fundamental forces of nature, we have been able to devise quantum theories of three: electromagnetism, the strong nuclear force, and the weak nuclear forces. However, a quantum theory of gravity has eluded us. 

General relativity is no doubt an important advance, but until we can devise a quantum theory of gravity, there is no hope of devising a unified theory of everything. While there is no consensus in the scientific community on the right direction in which to proceed, there have been some ideas that have had limited success. 

Superstring theory

The best-known theory that can describe gravity in the micro world is called super string theory. In this theory, the smallest known particles should not be thought of as little balls, but rather tiny strings, kind of like an incredibly small stick of uncooked spaghetti or a micro-miniature Hula-Hoop. The basic idea is that these tiny strings (which are smaller compared to a proton than a proton is compared to you) vibrate, and each vibration presents a different fundamental particle.

Employing a musical metaphor, an electron might be an A-sharp, while a photon could be a D-flat. In the same way that a single violin string can have many overtones, the vibrations of a single super string can be different particles. The beauty of super string theory is that it allows for one of the vibrations to be a graviton, which is a particle that has never been discovered but is thought to be the particle that causes gravity. 

It should be noted that super string theory is not generally accepted, and indeed, some in the scientific community don’t even consider it to be a scientific theory at all. The reason is that, in order for a theory to be scientific, it must be able to be tested, and have the potential to be proven wrong. However, the very small scale of these theoretical strings makes it difficult to imagine any tests that could be done in the foreseeable future. And, some say, if you can’t realistically do a test, it isn’t science.

Personally, I think that is an extreme opinion, as one can imagine doing such a test when technology advances. But that time will be far in the future.

Another idea for explaining quantum gravity is called “loop quantum gravity.” This theory actually quantizes space-time itself. In other words, this model says that there is a smallest bit of space and a shortest time. This provocative idea suggests, among other things, that the speed of light might be different for different wavelengths. However, this effect, if it exists, is small and requires that light travel for great distances before such differences could be observed. Toward that end, scientists are looking at gamma-ray bursts, explosions so bright that they can be seen across billions of light-years — an example of the cosmic helping scientists study the microscopic.

The simple fact is that we don’t yet have a good and generally accepted theory of quantum gravity. The question is simply just too difficult, for now. The micro world of the quantum and the macro world of gravity have long resisted a life of wedded bliss and, at least for the moment, they continue to resist. However, scientists continue to find the linkage that blends the two. In the meantime, a theory of quantum gravity remains one of the most ambitious goals of modern science — the hope that we will one day fulfill Einstein’s unfinished dream.

Read Lincoln’s other essays from this three-part series in Einstein’s Biggest Triumph: A Century of General Relativity (Op-Ed) and Einstein’s Biggest Triumph: A Century of General Relativity (Op-Ed).