From Eternity to Here (70 page)

That’s okay. We find ourselves, not as a central player in the life of the cosmos, but as a tiny epiphenomenon, flourishing for a brief moment as we ride a wave of increasing entropy from the Big Bang to the quiet emptiness of the future universe. Purpose and meaning are not to be found in the laws of nature, or in the plans of any external agent who made things that way; it is our job to create them. One of those purposes—among many—stems from our urge to explain the world around us the best we can. If our lives are brief and undirected, at least we can take pride in our mutual courage as we struggle to understand things much greater than ourselves.

NEXT STEPS

It’s surprisingly hard to think clearly about time. We’re all familiar with it, but the problem might be that we’re
too
familiar. We’re so used to the arrow of time that it’s hard to conceptualize time without the arrow. We are led, unprotesting, to temporal chauvinism, prejudicing explanations of our current state in terms of the past over those in terms of the future. Even highly trained professional cosmologists are not immune.

Despite all the ink that has been spilled and all the noise generated by discussions about the nature of time, I would argue that it’s been discussed too little, rather than too much. But people seem to be catching on. The intertwined subjects of time, entropy, information, and complexity bring together an astonishing variety of intellectual disciplines: physics, mathematics, biology, psychology, computer science, the arts. It’s about time that we took time seriously, and faced its challenges head-on.

Within physics, that’s starting to happen. For much of the twentieth century, the field of cosmology was a bit of a backwater; there were many ideas, and little data to distinguish between them. An era of precision cosmology, driven by large-scale surveys enabled by new technologies, has changed all that; unanticipated wonders have been revealed, from the acceleration of the universe to the snapshot of early times provided by the cosmic microwave background.
³⁰³
Now it is the turn for ideas to catch up to the reality. We have interesting suggestions from inflation, from quantum cosmology, and from string theory, to how the universe might have begun and what might have come before. Our task is to develop these promising ideas into honest theories, which can be compared with experiment and reconciled with the rest of physics.

Predicting the future isn’t easy. (Curse the absence of a low-entropy future boundary condition!) But the pieces are assembled for science to take dramatic steps toward answering the ancient questions we have about the past and the future. It’s time we understood our place within eternity.

APPENDIX: MATH

Lloyd: You mean, not good like one out of a hundred?
Mary: I’ d say more like one out of a million.
[pause]
Lloyd: So you’re telling me there’s a chance.

—Jim Carrey and Lauren Holly, Dumb and Dumber

In the main text I bravely included a handful of equations—a couple by Einstein, and a few expressions for entropy in different contexts. An equation is a powerful, talismanic object, conveying a tremendous amount of information in an extraordinarily compact notation. It can be very useful to look at an equation and understand its implications as a rigorous expression of some feature of the natural world.

But, let’s face it—equations can be scary. This appendix is a very quick introduction to exponentials and logarithms, the key mathematical ideas used in describing entropy at a quantitative level. Nothing here is truly necessary to comprehending the rest of the book; just bravely keep going whenever the word
logarithm
appears in the main text.

EXPONENTIALS

These two operations—exponentials and logarithms—are exactly as easy or difficult to understand as each other. Indeed, they are opposites; one operation undoes the other one. If we start with a number, take its exponential, and then take the logarithm of the result, we get back the original number we started with. Nevertheless, we tend to come across exponentials more often in our everyday lives, so they seem a bit less intimidating. Let’s start there.

Exponentials just take one number, called the
base
, and raise it to the power of another number. By which we simply mean: Multiply the base by itself, a number of times given by the power. The base is written as an ordinary number, and the power is written as a superscript. Some simple examples:

2
²
= 2 • 2 = 4,

2
⁵
= 2 • 2 • 2 • 2 • 2 = 32,

4
³
= 4 • 4 • 4 = 64.

(We use a dot to stand for multiplication, rather than the × symbol, because that’s too easy to confuse with the letter
x
.) One of the most convenient cases is where we take the base to be 10; in that case, the power simply becomes the number of zeroes to the right of the one.

10
¹
= 10,

10
²
= 100,

10
⁹
= 1,000,000,000,

10
²¹
= 1,000,000,000,000,000,000,000.

That’s the idea of exponentiation. When we speak more specifically about the exponential
function
, what we have in mind is fixing a particular base and letting the power to which we raise it be a variable quantity. If we denote the base by
a
and the power by
x
, we have

a
x
= a • a • a • a • a • a ... • a, x
times.

This definition, unfortunately, can give you the impression that the exponential function makes sense only when the power
x
is a positive integer. How can you multiply a number by itself minus-two times, or 3.7 times? Here you will have to have faith that the magic of mathematics allows us to define the exponential for
any
value of
x.
The result is a smooth function that is very small when
x
is a negative number, and rises very rapidly when
x
becomes positive, as shown in Figure 88.

Figure 88:
The exponential function 10
x
. Note that it goes up very fast, so that it becomes impractical to plot it for large values of
x
.

There are a couple of things to keep in mind about the exponential function. The exponential of 0 is always equal to 1, for any base, and the exponential of 1 is equal to the base itself. When the base is 10, we have:

10
⁰
= 1,

10
¹
= 10.

If we take the exponential of a negative number, it’s just the reciprocal of the exponential of the corresponding positive number:

10
^-1
= 1/10
¹
= 0.1,

10
^-3
= 1/10
³
= 0.001.

These facts are specific examples of a more general set of properties obeyed by the exponential function. One of these properties is of paramount importance: If we
multiply
two numbers that are the same base raised to different powers, that’s equal to what we would get by
adding
the two powers and raising the base to that result. That is:

10
x
•
10
y
= 10
⁽
x
⁺
y
⁾

Said the other way around, the exponential of a sum is the product of the two exponentials.
³⁰⁴

BIG NUMBERS

It’s not hard to see why the exponential function is useful: The numbers we are dealing with are sometimes very large indeed, and the exponential takes a medium-sized number and creates a very big number from it. As we discuss in Chapter Thirteen, the number of distinct states needed to describe possible configurations of our comoving patch of universe is approximately

10
¹⁰¹²⁰

That number is just so enormously, unimaginably huge that it would be hard to know how to even begin describing it if we didn’t have recourse to exponentiation.

Let’s consider some other big numbers to appreciate just how giant this one is. One billion is 10
⁹
, while one trillion is 10
¹²
; these have become all too familiar terms in discussions of economics and government spending. The number of particles within our observable universe is about 10
⁸⁸
, which was also the entropy at early times. Now that we have black holes, the entropy of the observable universe is something like 10
¹⁰¹
, whereas it conceivably could have been as high as 10
¹²⁰
. (That same 10
¹²⁰
is also the ratio of the predicted vacuum energy density to the observed density.)

For comparison’s sake, the entropy of a macroscopic object like a cup of coffee is about 10
²⁵
. That’s related to Avogadro’s Number, 6.02 • 10
²³
, which is approximately the number of atoms in a gram of hydrogen. The number of grains of sand in all the Earth’s beaches is about 10
²⁰
. The number of stars in a typical galaxy is about 10
¹¹
, and the number of galaxies in the observable universe is also about 10
¹¹
, so the number of stars in the observable universe is about 10
²²
—a bit larger than the number of grains of sand on Earth.

The basic units that physicists use are time, length, and mass, or combinations thereof. The shortest interesting time is the Planck time, about 10
^-43
seconds. I nflation is conjectured to have lasted for about 10
^-30
seconds or less, although that number is extremely uncertain. The universe created helium out of protons and neutrons about 100 seconds after the Big Bang, and it became transparent at the time of recombination, 380,000 years (10
¹³
seconds) after that. (One year is about 3 • 10
⁷
seconds.) The observable universe now is 14 billion years old, about 4 • 10
¹⁷
seconds. In another 10
¹⁰⁰
years or so, all the black holes will have mostly evaporated away, leaving a cold and empty universe.

The shortest length is the Planck length, about 10
^-33
centimeters. The size of a proton is about 10
^-13
centimeters, and the size of a human being is about 10
²
centimeters. (That’s a pretty short human being, but we’re only being very rough here.) The distance from the Earth to the Sun is about 10
¹³
centimeters; the distance to the nearest star is about 10
¹⁸
centimeters, and the size of the observable universe is about 10
²⁸
centimeters.