Einstein: His Life and Universe

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 four-dimensional 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

Beginning in that summer of 1912, Einstein struggled to develop gravitational field equations using tensors along the lines developed by Riemann, Ricci, and others. His first round of fitful efforts are preserved in a scratchpad notebook. Over the years, this revealing “Zurich Notebook” has been dissected and analyzed by a team of scholars including Jurgen Renn, John D. Norton, Tilman Sauer, Michel Janssen, and John Stachel.¹⁷

In it Einstein pursued a two-fisted approach. On the one hand, he engaged in what was called a “physical strategy,” in which he tried to build the correct equations from a set of requirements dictated by his feel for the physics. At the same time, he pursued a “mathematical strategy,” in which he tried to deduce the correct equations from the more formal math requirements using the tensor analysis that Gross-mann and others recommended.

Einstein’s “physical strategy” began with his mission to generalize the principle of relativity so that it applied to observers who were accelerating or moving in an arbitrary manner. Any gravitational field equation he devised would have to meet the following physical requirements:

• It must revert to Newtonian theory in the special case of weak and static gravitational fields. In other words, under certain normal conditions, his theory would describe Newton’s familiar laws of gravitation and motion.

• It should preserve the laws of classical physics, most notably the conservation of energy and momentum.

• It should satisfy the principle of equivalence, which holds that observations made by an observer who is uniformly accelerating would be equivalent to those made by an observer standing in a comparable gravitational field.

Einstein’s “mathematical strategy,” on the other hand, focused on using generic mathematical knowledge about the metric tensor to find a gravitational field equation that was generally (or at least broadly) covariant.

The process worked both ways: Einstein would examine equations that were abstracted from his physical requirements to check their covariance properties, and he would examine equations that sprang from elegant mathematical formulations to see if they met the requirements of his physics. “On page after page of the notebook, he approached the problem from either side, here writing expressions suggested by the physical requirements of the Newtonian limit and energy-momentum conservation, there writing expressions naturally suggested by the generally covariant quantities supplied by the mathematics of Ricci and Levi-Civita,” says John Norton.¹⁸