SPACETIME PHYSICS
Wherein the reader is led, once quickly (§/./),
then again more slowly, down the highways and
a few byways of Einstein's geometrodynamics—
without benefit of a good mathematical compass.
CHAPTER
GEOMETRODYNAMICS IN BRIEF
§1.1. THE PARABLE OF THE APPLE
One day in the year 1666 Newton had gone to the country,
and seeing the fall of an apple, as his niece told me, let himself
be led into a deep meditation on the cause which thus
draws every object along a line whose extension would pass
almost through the center of the Earth.
VOLTAIRE (1738)
Once upon a time a student lay in a garden under an apple tree reflecting on the
difference between Einstein's and Newton's views about gravity. He was startled
by the fall of an apple nearby. As he looked at the apple, he noticed ants beginning
to run along its surface (Figure 1.1). His curiosity aroused, he thought to investigate
the principles of navigation followed by an ant. With his magnifying glass, he noted
one track carefully, and, taking his knife, made a cut in the apple skin one mm
above the track and another cut one mm below it. He peeled off the resulting little
highway of skin and laid it out on the face of his book. The track ran as straight
as a laser beam along this highway. No more economical path could the ant have
found to cover the ten cm from start to end of that strip of skin. Any zigs and
zags or even any smooth bend in the path on its way along the apple peel from
starting point to end point would have increased its length.
"What a beautiful geodesic," the student commented.
His eye fell on two ants starting off from a common point P in slightly different
directions. Their routes happened to carry them through the region of the dimple
at the top of the apple, one on each side of it. Each ant conscientiously pursued
his geodesic. Each went as straight on his strip of appleskin as he possibly could.
Yet because of the curvature of the dimple itself, the two tracks not only crossed
but emerged in very different directions.
"What happier illustration of Einstein's geometric theory of gravity could one
possibly ask?" murmured the student. "The ants move as if they were attracted
by the apple stem. one might have believed in a Newtonian force at a distance.
Yet from nowhere does an ant get his moving orders except from the local geometry
along his track. This is surely Einstein's concept that all physics takes place by
'local action.' What a difference from Newton's 'action at a distance' view of physics!
Now I understand better what this book means."
And so saying, he opened his book and read, "Don't try to describe motion
relative to faraway objects. Physics is simple only when analyzed locally. And locally
the world line that a satellite follows [in spacetime, around the Earth] is already
as straight as any world line can be. Forget all this talk about 'deflection' and 'force
of gravitation.' I'm inside a spaceship. Or I'm floating outside and near it. Do I
feel any 'force of gravitation'? Not at all. Does the spaceship 'feel' such a force?
No. Then why talk about it? Recognize that the spaceship and I traverse a region
of spacetime free of all force. Acknowledge that the motion through that region
is already ideally straight."
The dinner bell was ringing, but still the student sat, musing to himself. "Let me
see if I can summarize Einstein's geometric theory of gravity in three ideas: (1)
locally, geodesies appear straight; (2) over more extended regions of space and time,
geodesies originally receding from each other begin to approach at a rate governed
by the curvature of spacetime, and this effect of geometry on matter is what we
mean today by that old word 'gravitation'; (3) matter in turn warps geometry. The
dimple arises in the apple because the stem is there. I think I see how to put the
whole story even more briefly: Space acts on matter, telling it how to move. In turn,
matter reacts back on space, telling it how to curve. In other words, matter here,"
he said, rising and picking up the apple by its stem, "curves space here. To produce
a curvature in space here is to force a curvature in space there," he went on, as
he watched a lingering ant busily following its geodesic a finger's breadth away from
the apple's stem. "Thus matter here influences matter there. That is Einstein's
explanation for 'gravitation.'"
Then the dinner bell was quiet, and he was gone, with book, magnifying glass—and
apple.
Space tells matter how to
move
Matter tells space how to
curve
§1.2. SPACETIME WITH AND WITHOUT COORDINATES
Now it came to me: . . . the independence of the
gravitational acceleration from the nature of the falling
substance, may be expressed as follows: In a
gravitational field (of small spatial extension) things
behave as they do in a space free of gravitation. . . . This
happened in 1908. Why were another seven years required
for the construction of the general theory of relativity?
The main reason lies in the fact that it is not so easy to
free oneself from the idea that coordinates must have an
immediate metrical meaning.
ALBERT EINSTEIN [in Schilpp (1949), pp 65-67]
Nothing is more distressing on first contact with the idea of "curved spacetime" than
the fear that every simple means of measurement has lost its power in this unfamiliar
context. one thinks of oneself as confronted with the task of measuring the shape
of a gigantic and fantastically sculptured iceberg as one stands with a meter stick
in a tossing rowboat on the surface of a heaving ocean. Were it the rowboat itself
whose shape were to be measured, the procedure would be simple enough. one
would draw it up on shore, turn it upside down, and drive tacks in lightly at strategic
points here and there on the surface. The measurement of distances from tack to