Steve Nadis is a contributing editor to *Discover* magazine and a contributing writer to *Quanta*.

Shing-Tung Yau is a mathematics professor at Tsinghua University and professor emeritus at Harvard University. The recipient of the Fields Medal, National Medal of Science, and a MacArthur Fellowship.

Below, co-authors Steve and Shing-Tung share five key insights from their new book, *The Gravity of Math: How Geometry Rules the Universe*. Listen to the audio version—read by Steve—in the Next Big Idea App.

### 1. Gravity is geometry.

Just to be clear, that’s not an idea that Yau and I came up with. It’s something Albert Einstein figured out more than 100 years ago, and it probably is not what he expected to find when he set out in 1905 to create a new theory of gravity. The prevailing theory of gravity, devised in the mid-to-late 1600s by Isaac Newton, breaks down when objects are moving very fast (at speeds close to that of light) and when gravitational fields are very strong. Einstein sought a law that was not subject to those same limitations.

Although Newton’s law works well in ordinary, non-extreme situations, the picture it paints is all wrong, according to Einstein. Newton regarded gravity as an attractive force between two or more massive objects. Einstein envisioned something very different. A massive object like the sun, he said, bends the fabric of space and time around it—just as a bowling ball would deform a stretched rubber sheet when placed in the middle. The deformation caused by the sun, for example, keeps smaller objects—like the planets in our solar system–within the sun’s gravitational grip.

In other words, a massive object causes a surface to bend or curve. And the surface that does the curving in Einstein’s theory is the combination of space and time called “spacetime.” The effect we call gravity stems directly from the curvature of spacetime. But the shape and curvature of any surface relate closely to its geometry. Which is why one can say that gravity is, in essence, geometry.

### 2. Although Einstein was surely a genius, he was not a “lone genius” when it came to developing his theory of gravity.

That theory grew increasingly mathematical as Einstein became more deeply enmeshed in this decade-long endeavor, and mathematics was not his strong suit. He had little interest in the subject as a student and young scientist, and he consequently paid little attention to it.

But when he realized that his gravitational theory would depend heavily on curved spaces, involving a kind of geometry he was unfamiliar with, he called on a friend for help—a former college classmate named Marcel Grossmann, who specialized in geometry. Grossmann introduced Einstein to the work of Bernhard Riemann, who had invented in the 1850s a new branch of geometry that focused on higher-dimensional curved spaces. That turned out to be just what Einstein needed.

“When he realized that his gravitational theory would depend heavily on curved spaces, involving a kind of geometry he was unfamiliar with, he called on a friend for help.”

In 1913, Einstein and Grossmann published a paper that presented the first draft of the equations of gravity—also known as the field equations of general relativity. Einstein wrote the physics part; Grossmann wrote the mathematics part. That paper almost got things right, but it had some fatal flaws that took Einstein another two years to sort out. In those years, he got help from other leading mathematicians, including Tullio Levi-Civita and David Hilbert. Although Einstein did not go it entirely alone, his accomplishment was still extraordinary.

### 3. Einstein produced a revolutionary theory of gravity, but he was conservative when it came to the implications of that theory.

The equations of gravity that Einstein published in November 1915 had just a few terms and could fit onto a single line. They looked deceptively simple, but they were actually quite complicated. He was not sure that an exact solution to them could ever be found. As things turned out, he did not have to wait long.

About a month later, Einstein received the first of two letters from Karl Schwarzschild, an astrophysicist who was then fighting for the German cause in World War I. Schwarzschild had found solutions to Einstein’s equations that described the geometry of spacetime both inside and outside a spherical star. Schwarzschild showed that if the star’s interior contained enough mass packed into a small enough radius, the pressure and density at the star’s center would approach infinity. In other words, he had described what 50 years later would be called a black hole.

Although the strange star conjured up by Schwarzschild sprung directly from Einstein’s equations, Einstein did not believe that such an object could actually exist. “If the result were real, it would be a true disaster,” he claimed. “There should be a law of nature to prevent a star from behaving in such an absurd way,” his colleague Arthur Eddington maintained. But to the contrary, all evidence suggests that some stars do behave in such an absurd way.

### 4. Einstein originated the concept of gravitational waves, yet he doubted their physical reality.

In 1916, drawing upon his theory of general relativity, Einstein predicted that massive objects accelerating through spacetime would cause ripples to form. Gravitational waves are analogous to the waves generated by a powerboat speeding across a once-calm pond, except that these waves would travel in every direction. It was a prescient notion that took a century to confirm. Einstein yet again doubted the strength of his theory.

In 1936, twenty years after his original prediction, Einstein was prepared to speak at Princeton University on the “Nonexistence of Gravitational Waves.” However, he softened his stance when a colleague pointed out an error in the paper that Einstein had planned to base his talk on. So, instead, he adopted a more moderate position in his lecture, saying, “If you ask me whether there are gravitational waves or not, I must answer that I don’t know. But it is a highly interesting problem.”

“An international study revealed that there may be a permanent gravitational wave background reverberating throughout the universe.”

That turned out to be a much better way of phrasing things. In 2015, the first gravitational waves were detected at the LIGO Observatory. Since then, there have been roughly 100 other detections associated with collisions between two black holes, between two neutron stars, or even the collision of a black hole and a neutron star. Just last year, an international study revealed that there may be a permanent gravitational wave background reverberating throughout the universe.

It appears that Einstein got things right the first time.

### 5. What Einstein called his “greatest blunder,” the cosmological constant, is now known to dominate our universe.

In 1917, Einstein had a realization that might be called an epiphany. His gravitational theory did not just apply to the motions of planets in our solar system or to the motions of stars in our galaxy. It could be applied to the behavior of the universe as a whole.

Einstein recognized that the universe could expand or contract unless he added a term to his equations called the cosmological constant, a calming influence that would keep things just as they were—consistent with the view that he and others then shared about a tranquil, unchanging universe.

But Einstein never liked the term because it messed up his otherwise elegant equations. “I am unable to believe that such an ugly thing should be realized in nature,” he said. It turns out that he was right to insert the term, though he did so for the wrong reasons.

In 1998, it was discovered that the universe is expanding at an accelerated rate, spurred on by something called dark energy. The cosmological constant that Einstein fretted over for so long is now believed to be dark energy. Dark energy is an invisible energy field that permeates all space and is thought to comprise 70 percent of the universe’s total mass and energy. That percentage, moreover, is steadily growing.

Mathematical development of general relativity did not end in 1915. New academic papers on this subject come out practically every day, probing the nature of black holes and investigating the beginning of our universe, among other topics that theorists like to think about. Mathematical relativity is still a rich, active, and evolving field. May the force of gravity be with you.

*To listen to the audio version read by co-author Steve Nadis, download the Next Big Idea App today:*