einstein

When Einstein moved back to Zurich from Prague in July 1912, one of the first things he did was call on his friend Marcel Grossmann, who had
taken the notes Einstein used when he skipped math classes at the Zurich Polytechnic. Einstein had gotten a 4.25 out of 6 in his two geometry
courses at the Polytechnic. Grossmann, on the other hand, had scored a perfect 6 in both of his geometry courses, had written his dissertation on
non-Euclidean geometry, published seven papers on that topic, and was now the chairman of the math department. 7
“Grossmann, you’ve got to help me or I will go crazy,” Einstein said. He explained that he needed a mathematical system that would express—
and perhaps even help him discover—the laws that governed the gravitational field. “Instantly, he was all afire,” Einstein recalled of Grossmann’s
response. 8
Until then, Einstein’s scientific success had been based on his special talent for sniffing out the underlying physical principles of nature. He had
left to others the task, which to him seemed less exalted, of finding the best mathematical expressions of those principles, as his Zurich colleague
Minkowski had done for special relativity.
But by 1912, Einstein had come to appreciate that math could be a tool for discovering—and not merely describing—nature’s laws. Math was
nature’s playbook. “The central idea of general relativity is that gravity arises from the curvature of spacetime,” says physicist James Hartle. “Gravity
is geometry.” 9
“I am now working exclusively on the gravitation problem and I believe that, with the help of a mathematician friend here, I will overcome all
difficulties,” Einstein wrote to the physicist Arnold Sommerfeld. “I have gained enormous respect for mathematics, whose more subtle parts I
considered until now, in my ignorance, as pure luxury!” 10
Grossmann went home to think about the question. After consulting the literature, he came back to Einstein and recommended the non-Euclidean
geometry that had been devised by Bernhard Riemann. 11
Riemann (1826–1866) was a child prodigy who invented a perpetual calendar at age 14 as a gift for his parents and went on to study in the great
math center of Göttingen, Germany, under Carl Friedrich Gauss, who had been pioneering the geometry of curved surfaces. This was the topic
Gauss assigned to Riemann for a thesis, and the result would transform not only geometry but physics.
Euclidean geometry describes flat surfaces. But it does not hold true on curved surfaces. For example, the sum of the angles of a triangle on a
flat page is 180°. But look at the globe and picture a triangle formed by the equator as the base, the line of longitude running from the equator to the
North Pole through London (longitude 0°) as one side, and the line of longitude running from the equator to the North Pole through New Orleans
(longitude 90°) as the third side. If you look at this on a globe, you will see that all three angles of this triangle are right angles, which of course is
impossible in the flat world of Euclid.
Gauss and others had developed different types of geometry that could describe the surface of spheres and other curved surfaces. Riemann
took things even further: he developed a way to describe a surface no matter how its geometry changed, even if it varied from spherical to flat to
hyperbolic from one point to the next. He also went beyond dealing with the curvature of just two-dimensional surfaces and, building on the work of
Gauss, explored the various ways that math could describe the curvature of three-dimensional and even four-dimensional space.
That is a challenging concept. We can visualize a curved line or surface, but it is hard to imagine what curved three-dimensional space would be
like, much less a curved four dimensions. But for mathematicians, extending the concept of curvature into different dimensions is easy, or at least
doable. This involves using the concept of the metric, which specifies how to calculate the distance between two points in space.
On a flat surface with just the normal x and y coordinates, any high school algebra student, with the help of old Pythagoras, can calculate the
distance between points. But imagine a flat map (of the world, for example) that represents locations on what is actually a curved globe. Things get
stretched out near the poles, and measurement gets more complex. Calculating the actual distance between two points on the map in Greenland is
different from doing so for points near the equator. Riemann worked out ways to determine mathematically the distance between points in space no
matter how arbitrarily it curved and contorted. 12
To do so he used something called a tensor. In Euclidean geometry, a vector is a quantity (such as of velocity or force) that has both a magnitude
and a direction and thus needs more than a single simple number to describe it. In non-Euclidean geometry, where space is curved, we need
something more generalized—sort of a vector on steroids—in order to incorporate, in a mathematically orderly way, more components. These are
called tensors.
A metric tensor is a mathematical tool that tells us how to calculate the distance between points in a given space. For two-dimensional maps, a
metric tensor has three components. For three-dimensional space, it has six independent components. And once you get to that glorious fourdimensional
entity known as spacetime, the metric tensor needs ten independent components.*
Riemann helped to develop this concept of the metric tensor, which was denoted as g mn and pronounced gee-mu-nu. It had sixteen
components, ten of them independent of one another, that could be used to define and describe a distance in curved four-dimensional
spacetime. 13
The useful thing about Riemann’s tensor, as well as other tensors that Einstein and Grossmann adopted from the Italian mathematicians
Gregorio Ricci-Curbastro and Tullio Levi-Civita, is that they are generally covariant. This was an important concept for Einstein as he tried to
generalize a theory of relativity. It meant that the relationships between their components remained the same even when there were arbitrary
changes or rotations in the space and time coordinate system. In other words, the information encoded in these tensors could go through a variety
of transformations based on a changing frame of reference, but the basic laws governing the relationship of the components to each other
remained the same. 14
Einstein’s goal as he pursued his general theory of relativity was to find the mathematical equations describing two complementary processes:
1. How a gravitational field acts on matter, telling it how to move.
2. And in turn, how matter generates gravitational fields in space-time, telling it how to curve.
His head-snapping insight was that gravity could be defined as the curvature of spacetime, and thus it could be represented by a metric tensor.
For more than three years he would fitfully search for the right equations to accomplish his mission. 15
Years later, when his younger son, Eduard, asked why he was so famous, Einstein replied by using a simple image to describe his great insight
that gravity was the curving of the fabric of spacetime. “When a blind beetle crawls over the surface of a curved branch, it doesn’t notice that the
track it has covered is indeed curved,” he said. “I was lucky enough to notice what the beetle didn’t notice.” 16
The Zurich Notebook, 1912