From Eternity to Here (16 page)

Contrast the difficulty of detecting a gravitational field with, for example, the ease of detecting an electric field, which is a piece of cake. Just take your same fork and spoon, but now give the fork some positive charge, and the spoon some negative charge. In the presence of an electric field, the opposing charges would be pushed in opposite directions, so it’s pretty easy to check whether there are any electric fields in the vicinity.

The difference with gravity is that there is no such thing as a “negative gravitational charge.” Gravity is
universal
—everything responds to it in the same way. Consequently, it can’t be detected in a small region of spacetime, only in the difference between its effects on objects at different events in spacetime. Einstein elevated this observation to the status of a law of nature, the
Principle of Equivalence
: No local experiment can detect the existence of a gravitational field.

Figure 16:
The gravitational field on a planet is locally indistinguishable from the acceleration of a rocket.

I know what you’re thinking: “I have no trouble detecting gravity at all. Here I am sitting in my chair, and it’s gravity that’s keeping me from floating up into the room.” But how do you know it’s gravity? Only by looking outside and checking that you’re on the surface of the Earth. If you were in a spaceship that was accelerating, you would also be pushed down into your chair. Just as you can’t tell the difference between freely falling in interstellar space and freely falling in low-Earth orbit, you also can’t tell the difference between constant acceleration in a spaceship and sitting comfortably in a gravitational field. That’s the “equivalence” in the Principle of Equivalence: The apparent effects of the force of gravity are equivalent to those of living in an accelerating reference frame. It’s not the force of gravity that you feel when you are sitting in a chair; it’s the force of the chair pushing up on your posterior. According to general relativity, free fall is the natural, unforced state of motion, and it’s only the push from the surface of the Earth that deflects us from our appointed path.

CURVING STRAIGHT LINES

You or I, having come up with the bright idea of the Principle of Equivalence while musing over the nature of gravity, would have nodded sagely and moved on with our lives. But Einstein was smarter than that, and he appreciated what this insight really meant. If gravity isn’t detectable by doing local experiments, then it’s not really a “force” at all, in the same way that electricity or magnetism are forces. Because gravity is universal, it makes more sense to think of it as a feature of spacetime itself, rather than some force field stretching through spacetime.

In particular, realized Einstein, gravity can be thought of as a manifestation of the
curvature
of spacetime. We’ve talked quite a bit about spacetime as a generalization of space, and how the time elapsed along a trajectory is a measure of the distance traveled through spacetime. But space isn’t necessarily rigid, flat, and rectilinear; it can be warped, stretched, and deformed. Einstein says that spacetime is the same way.

It’s easiest to visualize two-dimensional space, modeled, for example, by a piece of paper. A flat piece of paper is not curved, and the reason we know that is that it obeys the principles of good old-fashioned Euclidean geometry. Two initially parallel lines, for example, never intersect, nor do they grow apart.

In contrast, consider the two-dimensional surface of a sphere. First we have to generalize the notion of a “straight line,” which on a sphere isn’t an obvious concept. In Euclidean geometry, as we were taught in high school, a straight line is the shortest distance between two points. So let’s declare an analogous definition: A “straight line” on a curved geometry is the shortest curve connecting two points, which on a sphere would be a portion of a great circle. If we take two paths on a sphere that are initially parallel, and extend them along great circles, they will eventually intersect. That proves that the principles of Euclidean geometry are no longer at work, which is one way of seeing that the geometry of a sphere is curved.

Figure 17:
Flat geometry, with parallel lines extending forever; curved geometry, where initially parallel lines eventually intersect.

Einstein proposed that four-dimensional spacetime can be curved, just like the surface of a two-dimensional sphere. The curvature need not be uniform like a sphere, the same from point to point; it can vary in magnitude and in shape from place to place. And here is the kicker: When we see a planet being “deflected by the force of gravity,” Einstein says it is really just traveling in a straight line. At least, as straight as a line can be in the curved spacetime through which the planet is moving. Following the insight that an unaccelerated trajectory yields the greatest possible time a clock could measure between two events, a straight line through spacetime is one that does its best to maximize the time on a clock, just like a straight line through space does its best to minimize the distance read by an odometer.

Let’s bring this down to Earth, in a manner of speaking. Consider a satellite in orbit, carrying a clock. And consider another clock, this one on a tower that reaches to the same altitude as the orbiting satellite. The clocks are synchronized at a moment when the satellite passes by the tower; what will they read when the satellite completes one orbit? (We can ignore the rotation of the Earth for the purposes of this utterly impractical thought experiment.) According to the viewpoint of general relativity, the orbiting clock is not accelerating; it’s in free fall, doing its best to move in a straight line through spacetime. The tower clock, meanwhile, is accelerating—it’s being prevented from freely falling by the force of the tower keeping it up. Therefore, the orbiting clock will experience more elapsed time per orbit than the tower clock—compared to the accelerated clock on the tower, the freely falling one in orbit appears to run more quickly.

Figure 18:
Time as measured on a tower will be shorter than that measured in orbit, as the former clock is on an accelerated (non-free-falling) trajectory.

There are no towers that reach to the heights of low-Earth orbit. But there are clocks down here at the surface that regularly exchange signals with clocks on satellites. That, for example, is the basic mechanism behind the Global Positioning System (GPS) that helps modern cars give driving directions in real time. Your personal GPS receiver gets signals from a number of satellites orbiting the Earth, and determines its position by comparing the time between the different signals. That calculation would quickly go astray if the gravitational time dilation due to general relativity were not taken into account; the GPS satellites experience about 38 more microseconds per day in orbit than they would on the ground. Rather than teaching your receiver equations from general relativity, the solution actually adopted is to tune the satellite clocks so that they run a little bit more slowly than they should if they were to keep correct time down here on the surface.

EINSTEIN’S MOST IMPORTANT EQUATION

The saying goes that every equation cuts your book sales in half. I’m hoping that this page is buried sufficiently deeply in the book that nobody notices before purchasing it, because I cannot resist the temptation to display another equation: the Einstein field equation for general relativity.

R
µν
- (t/2)
Rg
µν
= 8
πGT
µν
.

This is the equation that a physicist would think of if you said “Einstein’s equation”; that
E
=
mc
²
business is a minor thing, a special case of a broader principle. This one, in contrast, is a deep law of physics: It reveals how stuff in the universe causes spacetime to curve, and therefore causes gravity. Both sides of the equation are not simply numbers, but
tensors
—geometric objects that capture multiple things going on at once. (If you thought of them as 4x4 arrays of numbers, you would be pretty close to right.) The left-hand side of the equation characterizes the curvature of spacetime. The right-hand side characterizes all the various forms of stuff that make spacetime curve—energy, momentum, pressure, and so on. In one fell swoop, Einstein’s equation reveals how any particular collection of particles and fields in the universe creates a certain kind of curvature in spacetime.

According to Isaac Newton, the source of gravity was mass; heavier objects gave rise to stronger gravitational fields. In Einstein’s universe, things are more complicated. Mass gets replaced by energy, but there are also other properties that go into curving spacetime. Vacuum energy, for example, has not only energy, but also
tension
—a kind of negative pressure. A stretched string or rubber band has tension, pulling back rather than pushing out. It’s the combined effect of the energy plus the tension that causes the universe to accelerate in the presence of vacuum energy.
⁷¹

The interplay between energy and the curvature of spacetime has a dramatic consequence: In general relativity, energy is not conserved. Not every expert in the field would agree with that statement, not because there is any controversy over what the theory predicts, but because people disagree on how to define “energy” and “conserved.” In a Newtonian absolute spacetime, there is a well-defined notion of the energy of individual objects, which we can add up to get the total energy of the universe, and that energy never changes (it’s the same at every moment in time). But in general relativity, where spacetime is dynamical, energy can be injected into matter or absorbed from it by the motions of spacetime. For example, vacuum energy remains absolutely constant in density as the universe expands. So the energy per cubic centimeter is constant, while the number of cubic centimeters is increasing—the total energy goes up. In a universe dominated by radiation, in contrast, the total energy goes down, as each photon loses energy due to the cosmological redshift.

You might think we could escape the conclusion that energy is not conserved by including “the energy of the gravitational field,” but that turns out to be much harder than you might expect—there simply is no well-defined local definition of the energy in the gravitational field. (That shouldn’t be completely surprising, since the gravitational field can’t even be detected locally.) It’s easier just to bite the bullet and admit that energy is not conserved in general relativity, except in certain special circumstances.
⁷²
But it’s not as if chaos has been loosed on the world; given the curvature of spacetime, we can predict precisely how any particular source of energy will evolve.

HOLES IN SPACETIME

Black holes are probably the single most interesting dramatic prediction of general relativity. They are often portrayed as something relatively mundane: “Objects where the gravitational field is so strong that light itself cannot escape.” The reality is more interesting.