From Eternity to Here

by Sean Carroll

Our simple estimate of what the vacuum energy should be comes out to about 10105 joules per cubic centimeter. That’s a lot of vacuum energy. What we actually observe is about 10-15 joules per cubic centimeter. So our estimate is larger than the experimental value by a factor of 10120—a 1 followed by 120 zeroes. Not something we can attribute to experimental error.

The fact that the vacuum energy is so much smaller than it should be is a serious problem: the “cosmological constant problem.” But there is also another problem: the “coincidence problem.” Remember that vacuum energy maintains a constant density (amount of energy per cubic centimeter) as the universe expands, while the density of matter dilutes away. Today, they aren’t all that different: Matter makes up about 25 percent of the energy of the universe, while vacuum energy makes up the other 75 percent. But they are changing appreciably with respect to each other, as the matter density dilutes away with the expansion and the vacuum energy does not. At the time of recombination, for example, the energy density in matter was a billion times larger than that in vacuum energy. So the fact that they are somewhat comparable today, uniquely in the history of the universe, seems like a remarkable coincidence indeed. Nobody knows why.

Either the universe is accelerating under the gentle influence of vacuum energy, or something even more dramatic and mysterious is going on.

THE DEEPEST FUTURE

As far as we can tell, the density of vacuum energy is unchanging as the universe expands. (It could be changing very slowly, and we just haven’t been able to measure the changes yet—that...

The universe will continue to expand, cool off, and become increasingly dilute. Distant galaxies will accelerate away from us, becoming more and more redshifted as they go. Eventually they will fade from view, as the time between photons that could possibly reach us becomes longer and longer. The entirety of the observable universe will just be our local group of gravitationally bound galaxies. Galaxies don’t last forever. The stars in them burn their nuclear fuel and die. Out of the remnant gas and dust more stars can form, but a point of diminishing returns is reached, after which all of the stars in the galaxy are dead. We are left with white dwarfs (stars that once burned, and ran out of fuel), brown dwarfs (stars that never burned in the first place), and neutron stars (stars that used to be white dwarfs but collapsed further under the pull of gravity). These objects may or may not be stable in their own right; our best current theoretical guess is that the protons and neutrons that make them up aren’t perfectly stable themselves but will eventually decay into lighter particles. If that’s true (and admittedly, we’re not sure), the various forms of dead stars will eventually dissipate into a thin gas of particles that disperse into the void. It won’t be quick; a reasonable estimate is 1040 years from now. For comparison, the current universe is about 1010 years old.

Besides stars, there are also black holes. Most large galaxies, including our own, have giant black holes at the center. In a galaxy the size of the Milky Way, with about 100 billion stars, the black hole might be a few million times as massive as the Sun—big compared to any individual star, but still small compared to the galaxy as a whole. But it will continue to grow, sweeping up whatever unfortunate stars happen to fall into it. Ultimately, however, all of the stars will have been used up. At that point, the black hole itself begins to evaporate into elementary particles.

Due once again to quantum fluctuations, a black hole can’t help but gradually radiate out into the space around it, slowly losing energy in the process. If we wait long enough—and now we’re talking 10100 years or so—even the supermassive black holes at the centers of galaxies will evaporate away to nothing.

In the very far future, the universe becomes once again a very simple place: It will be completely empty, as empty as space can be.

THE ENTROPY OF THE UNIVERSE

1. In the early universe, before structure forms, gravity has little effect on the entropy. The universe is similar to a box full of gas, and we can use the conventional formulas of thermodynamics to calculate its entropy. The total entropy within the space corresponding to our observable universe turns out to be about 1088 at early times.

2. By the time we reach our current stage of evolution, gravity has become very important. In this regime we don’t have an ironclad formula, but we can make a good estimate of the total entropy just by adding up the contributions from black holes (which carry an enormous amount of entropy). A single supermassive black hole has an entropy of order 1090, and there are approximately 1011 such black holes in the observable universe; our total entropy today is therefore something like 10101.

3. But there is a long way to go. If we took all of the matter in the observable universe and collected it into a single black hole, it would have an entropy of 10120. That can be thought of as the maximum possible entropy obtainable by rearranging the matt...

Our challenge is to explain this history. In particular, why was the early entropy, 1088, so much lower than the ...

1. The Big Bang was truly the beginning of the universe, the moment when time began. That may be because the true laws of physics allow spacetime to have a boundary, or because what we call “time” is just an approximation, and that approximation ceases to be valid near the Big Bang. In either case, the universe began in a low-entropy state, for reasons over and above the dynamical laws of nature—we need a new, independent principle to explain the initial state.

2. There is no such thing as an initial state, because time is eternal. In this case, we are imagining that the Big Bang isn’t the beginning of the entire universe, although it’s obviously an important event in the history of our local region. Somehow our observable patch of spacetime must fit into a bigger picture. And the way it fits must explain why the entropy was small at one end of time, without imposing any special conditions on the larger framework.

As to which of these is the correct description of the real world, the only answe...

PART TWO

TIME IN EINSTEIN’S UNIVERSE

4

TIME IS PERSONAL

LOST IN SPACE

Had we been very good record keepers about the amount and duration of our acceleration, we could possibly calculate the distance we had traveled; but just by doing local experiments, there doesn’t seem to be any way to distinguish one location from another. Likewise, we can’t seem to distinguish one velocity from another.

We can tell whether we are spinning or not spinning; but if we fire the appropriate guidance rockets (or manipulate some onboard gyroscopes) to stop whatever spin we gave the ship, there is no local experiment we can do that would reveal the angle by which the ship had rotated.

Naturally, there are names for the symmetries we have uncovered. Changing one’s location in space is known as a “translation”; changing one’s orientation in space is known as a “rotation”; and changing one’s velocity through space is known as a “boost.” In the context of special relativity, the collection of rotations and boosts are known as “Lorentz transformations,” while the entire set including translations are known as “Poincaré transformations.”

THE KEY TO RELATIVITY

Indeed, the entire content of special relativity boils down to these two principles:

No local experiment can distinguish between observers moving at constant velocities.

The speed of light is the same to a...

Indeed, “light” is not all that important in this game. What’s important is that there exists some unique preferred velocity through spacetime. It just so happens that light moves at that speed when it’s traveling through empty space—but the existence of a speed limit is what matters, not that light is able to go that fast.

SPACETIME

Extraneous motion decreases the time elapsed between two events in spacetime, whereas it increases the distance traveled between two points in space.

STAYING INSIDE YOUR LIGHT CONE

EINSTEIN’S MOST FAMOUS EQUATION

E=mc2 isn’t just about atomic bombs; it’s a profound feature of the dynamics of energy all around us.

5

TIME IS FLEXIBLE

Newtonian mechanics implied that the relative velocity of two objects moving past each other was simply the sum of their two velocities; Maxwellian electromagnetism implied that the speed of light was an exception to this rule. Special relativity managed to bring the two theories together into a single whole, by providing a framework for mechanics in which the speed of light did play a special role, but which reduced to Newton’s model when particles were moving slowly.

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.

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

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 th...

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.

EINSTEIN’S MOST IMPORTANT EQUATION

The interplay between energy and the curvature of spacetime has a dramatic consequence: In general relativity, energy is not conserved.

HOLES IN SPACETIME

WHITE HOLES: BLACK HOLES RUN BACKWARD