einstein

the observed mass,” Einstein told Schwarzschild. “It can be put this way. If I allow all things to vanish, then according to Newton the Galilean inertial
space remains; following my interpretation, however, nothing remains.” 5
The issue of inertia got Einstein into a debate with one of the great astronomers of the time, Willem de Sitter of Leiden. Throughout 1916,
Einstein struggled to preserve the relativity of inertia and Mach’s principle by using all sorts of constructs, including assuming various “border
conditions” such as distant masses along the fringes of space that were, by necessity, unable to be observed. As de Sitter noted, that in itself would
have been anathema to Mach, who railed against postulating things that could not possibly be observed. 6
By February 1917, Einstein had come up with a new approach. “I have completely abandoned my views, rightly contested by you,” he wrote de
Sitter. “I am curious to hear what you will have to say about the somewhat crazy idea I am considering now.” 7 It was an idea that initially struck him
as so wacky that he told his friend Paul Ehrenfest in Leiden, “It exposes me to the danger of being confined to a madhouse.” He jokingly asked
Ehrenfest for assurances, before he came to visit, that there were no such asylums in Leiden. 8
His new idea was published that month in what became yet another seminal Einstein paper, “Cosmological Considerations in the General
Theory of Relativity.” 9 On the surface, it did indeed seem to be based on a crazy notion: space has no borders because gravity bends it back on
itself.
Einstein began by noting that an absolutely infinite universe filled with stars and other objects was not plausible. There would be an infinite
amount of gravity tugging at every point and an infinite amount of light shining from every direction. On the other hand, a finite universe floating at
some random location in space was inconceivable as well. Among other things, what would keep the stars and energy from flying off, escaping,
and depleting the universe?
So he developed a third option: a finite universe, but one without boundaries. The masses in the universe caused space to curve, and over the
expanse of the universe they caused space (indeed, the whole four-dimensional fabric of spacetime) to curve completely in on itself. The system is
closed and finite, but there is no end or edge to it.
One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a twodimensional
universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has
no meaning to them.
Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them)
gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein
put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these
flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our
planet heading straight off over the seas would eventually return from the other horizon.
The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction
they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting
from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was
like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it. 10
By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite
system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean
geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well.
In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on
itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,”
the physicist Max Born has declared. 11
Yes, but what is outside this curved universe? What’s on the other side of the curve? That’s not merely an unanswerable question, it’s a
meaningless one, just as it would be meaningless for a flatlander to ask what’s outside her surface. One could speculate, imaginatively or
mathematically, about what things are like in a fourth spatial dimension, but other than in science fiction it is not very meaningful to ask what’s in a
realm that exists outside of the three spatial dimensions of our curved universe. 12
This concept of the cosmos that Einstein derived from his general theory of relativity was elegant and magical. But there seemed to be one hitch,
a flaw that needed to be fixed or fudged. His theory indicated that the universe would have to be either expanding or contracting, not staying static.
According to his field equations, a static universe was impossible because the gravitational forces would pull all the matter together.
This did not accord with what most astronomers thought they had observed. As far as they knew, the universe consisted only of our Milky Way
galaxy, and it all seemed pretty stable and static. The stars appeared to be meandering gently, but not receding rapidly as part of an expanding
universe. Other galaxies, such as Andromeda, were merely unexplained blurs in the sky. (A few Americans working at the Lowell Observatory in
Arizona had noticed that the spectra of some mysterious spiral nebulae were shifted to the red end of the spectrum, but scientists had not yet
determined that these were distant galaxies all speeding away from our own.)
When the conventional wisdom of physics seemed to conflict with an elegant theory of his, Einstein was inclined to question that wisdom rather
than his theory, often to have his stubbornness rewarded. In this case, his gravitational field equations seemed to imply—indeed, screamed out—
that the conventional thinking about a stable universe was wrong and should be tossed aside, just as Newton’s concept of absolute time was. 13
Instead, this time he made what he called a “slight modification” to his theory. To keep the matter in the universe from imploding, Einstein added
a “repulsive” force: a little addition to his general relativity equations to counterbalance gravity in the overall scheme.
In his revised equations, this modification was signified by the Greek letter lambda, λ, which he used to multiply his metric tensor g μν in a way that
produced a stable, static universe. In his 1917 paper, he was almost apologetic: “We admittedly had to introduce an extension of the field
equations that is not justified by our actual knowledge of gravitation.”
He dubbed the new element the “cosmological term” or the “cosmological constant” (kosmologische Glied was the phrase he used). Later,*
when it was discovered that the universe was in fact expanding, Einstein would call it his “biggest blunder.” But even today, in light of evidence that
the expansion of the universe is accelerating, it is considered a useful concept, indeed a necessary one after all. 14
During five months in 1905, Einstein had upended physics by conceiving light quanta, special relativity, and statistical methods for showing the
existence of atoms. Now he had just completed a more prolonged creative slog, from the fall of 1915 to the spring of 1917, which Dennis Overbye
has called “arguably the most prodigious effort of sustained brilliance on the part of one man in the history of physics.” His first burst of creativity as
a patent clerk had appeared to involve remarkably little anguish. But this later one was an arduous and intense effort, one that left him exhausted
and wracked with stomach pains. 15
During this period he generalized relativity, found the field equations for gravity, found a physical explanation for light quanta, hinted at how the