This equation doesn’t have a special name, but it will be handy to have one. I’ll call it the well-tempered equation, because it specifies the right way to tune the properties of space. Where does it come from?
It’s not immediately obvious how a universal, non-zero density can be consistent with special relativity. After all, to an observer moving at constant velocity, objects appear foreshortened in the direction of motion. It would seem, therefore, that the moving observer would see a higher density. But relativity demands that she must see the same physical laws.
The pressure that goes with density, according to the well-tempered equation, provides a loophole. The scales of the moving observer, according to the equations of special relativity, register a new density that is a mixture of the old density and the old pressure—just as, perhaps more familiarly, her clocks register time intervals that are mixtures of the old time intervals and the old space intervals. If—and only if—the old density and old pressure are related in just the way prescribed by the well-tempered equation, then values of the new density (and the new pressure) will be the same as the old values.
Another, closely related consequence of the well-tempered equation is crucial for cosmology. In an expanding universe, the density of any normal kind of matter will go down. But the density of well-tempered grid stays constant! If you’re up for a little exercise in freshman physics and algebra, here comes a pretty connection tying that constancy of density directly to Einstein’s equivalence of mass and energy. (If not, just skip the next paragraph.)
Consider a volume V of space, filled with mass density ρ. Let the volume expand by Δ V. Ordinarily, as a body expands under pressure it does work, and so loses energy. But the - sign in the well-tempered equation gives us negative pressure ρ = -p/c2. So in expanding, our well-tempered grid gains energy Δ V × p/c2. According to Einstein’s mass-energy equivalence, therefore, its mass increases by Δ V × p. And that’s just enough to fill the added volume Δ V with density ρ, allowing space to keep its density constant.
It’s possible to measure cosmic density and the pressure separately, using quite different techniques. The density affects the curvature of space, which astronomers can measure by studying the distortion such curvature causes in images of distant galaxies, or—a powerful new technique—in the cosmic microwave background radiation. Using the new technique, by 2001 several groups were able to prove that there was much more mass in the universe than could be accounted for by ordinary matter alone. About 70% of the total mass occurs appears to be very uniformly distributed, both in space and time.
The pressure affects the rate at which the universe is expanding. That rate can be measured by studying distant supernovae. Their brightness tells you how far away they are, while the redshift of their spectral lines tells you how fast they're moving away. Since the speed of light is finite, when we observe the farther-away ones we're looking at their past. So we can use supernovae to reconstruct the history of expansion. In 1998 two powerhouse teams of observers reported that the rate of expansion of the universe is increasing. This was a big surprise, because
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