einstein

Light and Gravity
CHAPTER NINE
GENERAL RELATIVITY
1911–1915
After Einstein formulated his special theory of relativity in 1905, he realized that it was incomplete in at least two ways. First, it held that no
physical interaction can propagate faster than the speed of light; that conflicted with Newton’s theory of gravity, which conceived of gravity as a
force that acted instantly between distant objects. Second, it applied only to constant-velocity motion. So for the next ten years, Einstein engaged in
an interwoven effort to come up with a new field theory of gravity and to generalize his relativity theory so that it applied to accelerated motion. 1
His first major conceptual advance had come at the end of 1907, while he was writing about relativity for a science yearbook. As noted earlier, a
thought experiment about what a free-falling observer would feel led him to embrace the principle that the local effects of being accelerated and of
being in a gravitational field are indistinguishable.* A person in a closed windowless chamber who feels his feet pressed to the floor will not be able
to tell whether it’s because the chamber is in outer space being accelerated upward or because it is at rest in a gravitational field. If he pulls a
penny from his pocket and lets it go, it will fall to the floor at an accelerating speed in either case. Likewise, a person who feels she is floating in the
closed chamber will not know whether it’s because the chamber is in free fall or hovering in a gravity-free region of outer space. 2
This led Einstein to the formulation of an “equivalence principle” that would guide his quest for a theory of gravity and his attempt to generalize
relativity. “I realized that I would be able to extend or generalize the principle of relativity to apply to accelerated systems in addition to those moving
at a uniform velocity,” he later explained. “And in so doing, I expected that I would be able to resolve the problem of gravitation at the same time.”
Just as inertial mass and gravitational mass are equivalent, so too there is an equivalence, he realized, between all inertial effects, such as
resistance to acceleration, and gravitational effects, such as weight. His insight was that they are both manifestations of the same structure, which
we now sometimes call the inertio-gravitational field. 3
One consequence of this equivalence is that gravity, as Einstein had noted, should bend a light beam. That is easy to show using the chamber
thought experiment. Imagine that the chamber is being accelerated upward. A laser beam comes in through a pinhole on one wall. By the time it
reaches the opposite wall, it’s a little closer to the floor, because the chamber has shot upward. And if you could plot its trajectory across the
chamber, it would be curved because of the upward acceleration. The equivalence principle says that this effect should be the same whether the
chamber is accelerating upward or is instead resting still in a gravitational field. Thus, light should appear to bend when going through a
gravitational field.
For almost four years after positing this principle, Einstein did little with it. Instead, he focused on light quanta. But in 1911, he confessed to
Michele Besso that he was weary of worrying about quanta, and he turned his attention back to coming up with a field theory of gravity that would
help him generalize relativity. It was a task that would take him almost four more years, culminating in an eruption of genius in November 1915.
In a paper he sent to the Annalen der Physik in June 1911, “On the Influence of Gravity on the Propagation of Light,” he picked up his insight
from 1907 and gave it rigorous expression. “In a memoir published four years ago I tried to answer the question whether the propagation of light is
influenced by gravitation,” he began. “I now see that one of the most important consequences of my former treatment is capable of being tested
experimentally.” After a series of calculations, Einstein came up with a prediction for light passing through the gravitational field next to the sun: “A
ray of light going past the sun would undergo a deflection of 0.83 second of arc.”*
Once again, he was deducing a theory from grand principles and postulates, then deriving some predictions that experimenters could proceed to
test. As before, he ended his paper by calling for just such a test. “As the stars in the parts of the sky near the sun are visible during total eclipses of
the sun, this consequence of the theory may be observed. It would be a most desirable thing if astronomers would take up the question.” 4
Erwin Finlay Freundlich, a young astronomer at the Berlin University observatory, read the paper and became excited by the prospect of doing
this test. But it could not be performed until an eclipse, when starlight passing near the sun would be visible, and there would be no suitable one for
another three years.
So Freundlich proposed that he try to measure the deflection of starlight caused by the gravitational field of Jupiter. Alas, Jupiter did not prove
big enough for the task. “If only we had a truly larger planet than Jupiter!” Einstein joked to Freundlich at the end of that summer. “But nature did not
deem it her business to make the discovery of her laws easy for us.” 5
The theory that light beams could be bent led to some interesting questions. Everyday experience shows that light travels in straight lines.
Carpenters now use laser levels to mark off straight lines and construct level houses. If a light beam curves as it passes through regions of
changing gravitational fields, how can a straight line be determined?
One solution might be to liken the path of the light beam through a changing gravitational field to that of a line drawn on a sphere or on a surface
that is warped. In such cases, the shortest line between two points is curved, a geodesic like a great arc or a great circle route on our globe.
Perhaps the bending of light meant that the fabric of space, through which the light beam traveled, was curved by gravity. The shortest path through
a region of space that is curved by gravity might seem quite different from the straight lines of Euclidean geometry.
There was another clue that a new form of geometry might be needed. It became apparent to Einstein when he considered the case of a rotating
disk. As a disk whirled around, its circumference would be contracted in the direction of its motion when observed from the reference frame of a
person not rotating with it. The diameter of the circle, however, would not undergo any contraction. Thus, the ratio of the disk’s circumference to its
diameter would no longer be given by pi. Euclidean geometry wouldn’t apply to such cases.
Rotating motion is a form of acceleration, because at every moment a point on the rim is undergoing a change in direction, which means that its
velocity (a combination of speed and direction) is undergoing a change. Because non-Euclidean geometry would be necessary to describe this
type of acceleration, according to the equivalence principle, it would be needed for gravitation as well. 6
Unfortunately, as he had proved at the Zurich Polytechnic, non-Euclidean geometry was not a strong suit for Einstein. Fortunately, he had an old
friend and classmate in Zurich for whom it was.
The Math