From Eternity to Here (71 page)

The Planck mass is about 10
^-5
grams—that would be extraordinarily heavy for a single particle, but isn’t all that much by macroscopic standards. The lightest particles that have more than zero mass are the neutrinos; we don’t know for sure what their masses are, but the lightest seem to be about 10
^-36
grams. A proton is about 10
^-24
grams, and a human being is about 10
⁵
grams. The Sun is about 10
³³
grams, a galaxy is about 10
⁴⁵
grams, and the mass within the observable universe is about 10
⁵⁶
grams.

LOGARITHMS

The logarithm function is the easiest thing in the world: It undoes the exponential function. That is, if we have some number that can be expressed in the form 10
x
—and every positive number can be—then the logarithm of that number is simply

log(10
x
) =
x
.

What could be simpler than that? Likewise, the exponential undoes the logarithm:

10
^log(
x
⁾
=
x
.

Another way of thinking about it is: If a number is a perfect power of 10 (like 10, 100, 1,000, etc.), the logarithm is simply the number of zeroes to the right of the initial 1:

log(10) = 1,

log(100) = 2,

log(1,000) = 3.

But just as for the exponential, the logarithm is actually a smooth function, as shown in Figure 89. The logarithm of 2.5 is about 0.3979, the logarithm of 25 is about 1.3979, the logarithm of 250 is about 2.3979, and so on. The only restriction is that we can’t take the logarithm of a negative number; that makes sense, because the logarithm inverts the exponential function, and we can never
get
a negative number by exponentiating. Roughly speaking, for large numbers the logarithm is simply “the number of digits in the number.”

Figure 89:
The logarithm function log(
x
). It is not defined for negative values of
x
, and as
x
approaches zero from the right the logarithm goes to minus infinity.

Just like the exponential of a sum is the product of exponentials, the logarithm has a corresponding property: The logarithm of a product is the sum of logarithms. That is:

log(
x
•
y
) = log(
x
) + log(
y
).

It’s this lovely property that makes logarithms so useful in the study of entropy. As we discuss in Chapter Eight, a physical property of entropy is that the entropy of two systems combined together is equal to the sum of the entropies of the two individual systems. But you get the number of possible states of the combined systems by multiplying the numbers of states of the two individual systems. So Boltzmann concluded that the entropy should be the logarithm of the number of states, not the number of states itself. In Chapter Nine we tell a similar story for information: Shannon wanted a measure of information for which the total information carried in two independent messages was the sum of the individual informations in each message, so he realized he also had to take the logarithm.

More informally, logarithms have the nice property that they take large numbers and whittle them down to manageable sizes. When we take the logarithm of an unwieldy number like a trillion, we get a nice number like 9. The logarithm is a monotonic function—it always increases as we increase the number we’re taking the logarithm of. So the logarithm gives a specific measure of how big a number is, but it collapses huge numbers down to a reasonable size, which is very helpful in fields like cosmology, statistical mechanics, or even economics.

One final crucial detail is that, just like exponentials, logarithms can come in different bases. The “log base
b
” of a number
x
is the number to which we would have to raise
b
in order to get
x
. That is:

log
₂
(2
^x
) =
x
,

log
₁₂
(12
x
) =
x
,

and so on. Whenever we don’t write the base explicitly, we take it to be equal to 10, because that’s how many fingers most human beings have. But scientists and mathematicians often like to make a seemingly odd choice: they use the
natural logarithm
, often written ln(
x
), in which the base is taken to be Euler’s constant:

ln(
x
) = log
e
(
x
),

e
= 2.7182818284 . . .

Euler’s constant is an irrational number, like pi or the square root of two, so its explicit form above would go on forever. At first glance that seems like a truly perverse choice to use as a base for one’s logarithms. But in fact
e
has a lot of nice properties, once you get deeper into the math; in calculus, for example, the function
e
x
is the only one (aside from the trivial function equal to zero everywhere) that is equal to its own derivative, as well as its own integral. In this book all of our logarithms have used base 10, but if you launch yourself into physics and math at a higher level, it will be natural logarithms all the way.

NOTES

PROLOGUE

1
Wikipedia contributors (2009).

2
Let’s emphasize the directions here, because they are easily confused: Entropy measures disorder, not order, and it increases with time, not decreases. We informally think “things wind down,” but the careful way of saying that is “entropy goes up.”

1. THE PAST IS PRESENT MEMORY

3
In an effort not to be too abstract, we will occasionally lapse into a kind of language that assumes the directionality of time—“time passes,” we “move into the future,” stuff like that. Strictly speaking, part of our job is to explain why that language seems so natural, as opposed to phrasings along the lines of “there is the present, and there is also the future,” which seems stilted. But it’s less stressful to occasionally give into the “tensed” way of speaking, and question the assumptions behind it more carefully later on.

4
Because the planets orbit in ellipses rather than perfect circles, their velocity around the Sun is not strictly constant, and the actual angle that the Earth describes in its orbit every time Mars completes a single revolution will depend on the time of year. These are details that are easy to take care of when we actually sit down to carefully define units of time.

5
The number of vibrations per second is fixed by the size and shape of the crystal. In a watch, the crystal is tuned to vibrate 32,768 times per second, which happens to be equal to 2 to the 15th power. That number is chosen so that it’s easy for the watch’s inner workings to divide successively by 2 to obtain a frequency of exactly once per second, appropriate for driving the second hand of a watch.

6
Alan Lightman’s imaginative novel
Einstein’s Dreams
presents a series of vignettes that explore what the world would be like if time worked very differently than it does in the real world.

7
See for example Barbour (1999) or Rovelli (2008).

8
There is a famous joke, attributed to Einstein: “When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute and it’s longer than any hour. That’s relativity.” I don’t know whether Einstein actually ever said those words. But I do know that’s not relativity.

9
Here is a possible escape clause, if we were really committed to restoring the scientific integrity of Baker’s fantasy: Perhaps time in the rest of the world didn’t completely stop, but just slowed down by a tremendous factor, and still ticked along at a sufficient rate that light could travel from the objects Arno was looking at to his eyes. Close, but no cigar. Even if that happened, the fact that the light was slowed down would lead to an enormous redshift—what looked like visible light in the ordinary world would appear to Arno as radio waves, which his poor eyes wouldn’t be able to see. Perhaps X-rays would be redshifted down to visible wavelengths, but X-ray flashlights are hard to come by. (It does, admittedly, provoke one into thinking how interesting a more realistic version of this scenario might be.)

10
Temporal
: of or pertaining to time. It’s a great word that we’ll be using frequently. Sadly, an alternative meaning is “pertaining to the present life or this world”—and we’ll be roaming very far away from that meaning.

11
As a matter of historical accuracy, while Einstein played a central role in the formulation of special relativity, it was legitimately a collaborative effort involving the work of a number of physicists and mathematicians, including George FitzGerald, Hendrik Lorentz, and Henri Poincaré. It was eventually Hermann Minkowski who took Einstein’s final theory and showed that it could be understood in terms of a four-dimensional spacetime, which is often now called “Minkowski space.” His famous 1909 quote was “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality” (Minkowski, 1909).

12
Pirsig (1974), 375.

13
Price (1996), 3.

14
Vonnegut (1969), 34. Quoted in Lebowitz (2008).

15
Augustine (1998), 235.

16
Good discussions of these issues can be found in Callender (2005), Lockwood (2005), and Davies (1995).

17
Philosophers often discuss different conceptions of time in terms laid out by J. M. E. McTaggart in his famous paper “The Unreality of Time” (1908). There, McTaggart distinguished between three different notions of time, which he labeled as different “series” (see also Lockwood, 2005). The A-series is a series of events measured relative to now, that move through time—“one year ago” doesn’t denote a fixed moment, but one that changes as time passes. The B-series is the sequence of events with permanent temporal labels, such as “October 12, 2009.” And the C-series is simply an ordered list of events—“
x
happens before
y
but after
z
”—without any time stamps at all. McTaggart argued—very roughly—that the B-series and C-series are fixed arrays, lacking the crucial element of change, and therefore insufficient to describe time. But the A-series itself is incoherent, as any specific event will be classified simultaneously as “past,” “present,” and “future,” from the point of view of different moments in time. (The moment of your birth is in the past to you now but was in the future to your parents when they first met.) Therefore, he concludes, time doesn’t exist.

If you get the feeling that this purported contradiction seems more like a problem with language than one with the nature of time, you are on the right track. To a physicist, there seems to be no contradiction between stepping outside the universe and thinking of all of spacetime at once, and admitting that from the point of view of any individual inside the universe time seems to flow.

2. THE HEAVY HAND OF ENTROPY

18
Amis (1991), 11.