The Fabric of the Cosmos: Space, Time, and the Texture of Reality (68 page)

11. You might wonder, then, why water ever turns into ice, since that results in the H
₂
O molecules becoming more ordered, that is, attaining lower, not higher, entropy. Well, the rough answer is that when liquid water turns into solid ice, it gives off energy to the environment (the opposite of what happens when ice melts, when it takes in energy from the environment), and that raises the environmental entropy. At low enough ambient temperatures, that is, below 0 degrees Celsius, the increase in environmental entropy exceeds the decrease in the water's entropy, so freezing becomes entropically favored. That's why ice forms in the cold of winter. Similarly, when ice cubes form in your refrigerator's freezer, their entropy goes down but the refrigerator itself pumps heat into the environment, and if that is taken account of, there is a total net increase of entropy. The more precise answer, for the mathematically inclined reader, is that spontaneous phenomena of the sort we're discussing are governed by what is known as
free energy.
Intuitively, free energy is that part of a system's energy that can be harnessed to do work. Mathematically, free energy, F, is defined by F = U — TS, where U stands for total energy, T stands for temperature, and S stands for entropy. A system will undergo a spontaneous change if that results in a decrease of its free energy. At low temperatures, the drop in U associated with liquid water turning into solid ice outweighs the decrease in S (outweighs the increase in —
TS
), and so will occur. At high temperatures (above 0 degrees Celsius), though, the change of ice to liquid water or gaseous steam is entropically favored (the increase in S outweighs changes to U) and so will occur.

12. For an early discussion of how a straightforward application of entropic reasoning would lead us to conclude that memories and historical records are not trustworthy accounts of the past, see C. F. von Weizsäcker in The Unity of Nature (New York: Farrar, Straus, and Giroux, 1980), 138-46, (originally published in
Annalen der Physik
36 (1939). For an excellent recent discussion, see David Albert in
Time and Chance
(Cambridge, Mass.: Harvard University Press, 2000).

13. In fact, since the laws of physics don't distinguish between forward and backward in time, the explanation of having fully formed ice cubes a half hour earlier, at 10 p.m., would be
precisely
as absurd—entropically speaking—as predicting that by a half hour later, by 11:00 p.m., the little chunks of ice would have grown into fully formed ice cubes. To the contrary, the explanation of having liquid water at 10 p.m. that slowly forms small chunks of ice by 10:30 p.m. is
precisely
as sensible as predicting that by 11:00 p.m. the little chunks of ice will melt into liquid water, something that is familiar and totally expected. This latter explanation, from the perspective of the observation at 10:30 p.m., is perfectly temporally symmetric and, moreover, agrees with our subsequent observations.

14. The particularly careful reader might think that I've prejudiced the discussion with the phrase "early on" since that injects a temporal asymmetry. What I mean, in more precise language, is that we will need special conditions to prevail on (at least) one end of the temporal dimension. As will become clear, the special conditions amount to a low entropy boundary condition and I will call the "past" a direction in which this condition is satisfied.

15. The idea that time's arrow requires a low-entropy past has a long history, going back to Boltzmann and others; it was discussed in some detail in Hans Reichenbach,
The
Direction of Time (Mineola, N.Y.: Dover Publications, 1984), and was championed in a particularly interesting quantitative way in Roger Penrose,
The Emperor's New Mind
(New York: Oxford University Press, 1989), pp. 317ff.

16. Recall that our discussion in this chapter does not take account of quantum mechanics. As Stephen Hawking showed in the 1970s, when quantum effects are considered, black holes do allow a certain amount of radiation to seep out, but this does not affect their being the highest-entropy objects in the cosmos.

17. A natural question is how we know that there isn't some future constraint that also has an impact on entropy. The bottom line is that we don't, and some physicists have even suggested experiments to detect the possible influence that such a future constraint might have on things that we can observe today. For an interesting article discussing the possibility of future and past constraints on entropy, see Murray Gell-Mann and James Hartle, "Time Symmetry and Asymmetry in Quantum Mechanics and Quantum Cosmology," in Physical Origins of Time Asymmetry, J. J. Halliwell, J. Pérez-Mercader, W. H. Zurek, eds. (Cambridge, Eng.: Cambridge University Press, 1996), as well as other papers in Parts 4 and 5 of that collection.

18. Throughout this chapter, we've spoken of
the
arrow of time, referring to the apparent fact that there is an asymmetry along the time axis (any observer's time axis) of spacetime: a huge variety of sequences of events is arrayed in one order along the time axis, but the reverse ordering of such events seldom, if ever, occurs. Over the years, physicists and philosophers have divided these sequences of events into subcategories whose temporal asymmetries might, in principle, be subject to logically independent explanations. For example, heat flows from hot objects to cooler ones, but not from cool objects to hot ones; electromagnetic waves emanate outward from sources like stars and lightbulbs, but seem never to converge inward on such sources; the universe appears to be uniformly expanding, and not contracting; and we remember the past and not the future (these are called the thermodynamic, electromagnetic, cosmological, and psychological arrows of time, respectively). All of these are time-asymmetric phenomena, but they might, in principle, acquire their time asymmetry from completely different physical principles. My view, one that many share (but others don't), is that except possibly for the cosmological arrow, these temporally asymmetric phenomena are not fundamentally different, and ultimately are subject to the same explanation—the one we've described in this chapter. For example, why does electromagnetic radiation travel in expanding outward waves but not contracting inward waves, even though both are perfectly good solutions to Maxwell's equations of electromagnetism? Well, because our universe has low-entropy, coherent, ordered sources for such outward waves—stars and lightbulbs, to name two—and the existence of these ordered sources derives from the even more ordered environment at the universe's inception, as discussed in the main text. The psychological arrow of time is harder to address since there is so much about the microphysical basis of human thought that we've yet to understand. But much progress has been made in understanding the arrow of time when it comes to computers—undertaking, completing, and then producing a record of a computation is a basic computational sequence whose entropic properties are well understood (as developed by Charles Bennett, Rolf Landauer, and others) and fit squarely within the second law of thermodynamics. Thus, if human thought can be likened to computational processes, a similar thermodynamic explanation may apply. Notice, too, that the asymmetry associated with the fact that the universe is expanding and not contracting is related to, but logically distinct from, the arrow of time we've been exploring. If the universe's expansion were to slow down, stop, and then turn into a contraction, the arrow of time would still point in the same direction. Physical processes (eggs breaking, people aging, and so on) would still happen in the usual direction, even though the universe's expansion had reversed.

19. For the mathematically inclined reader, notice that when we make this kind of probabilistic statement we are assuming a particular probability measure: the one that is uniform over all microstates compatible with what we see right
now.
There are, of course, other measures that we could invoke. For example, David Albert in
Time and Chance
has advocated using a probability measure that is uniform over all microstates compatible with what we see
now
and what he calls
the past hypothesis—
the apparent fact that the universe began in a low-entropy state. Using this measure, we eliminate consideration of all but those histories that are compatible with the low-entropy past attested to by our memories, records, and cosmological theories. In this way of thinking, there is no probabilistic puzzle about a universe with low entropy; it began that way, by assumption, with probability 1. There is still the same huge puzzle of
why
it began that way, even if it isn't phrased in a probabilistic context.

20. You might be tempted to argue that the known universe had low entropy early on simply because it was much smaller in size than it is today, and hence—like a book with fewer pages—allowed for far fewer rearrangements of its constituents. But, by itself, this doesn't do the trick. Even a small universe can have huge entropy. For example, one possible (although unlikely) fate for our universe is that the current expansion will one day halt, reverse, and the universe will implode, ending in the so-called big crunch. Calculations show that even though the size of the universe would decrease during the implosion phase, entropy would continue to rise, which demonstrates that small size does not ensure low entropy. In Chapter 11, though, we will see that the universe's small initial size does play a role in our current, best explanation of the low entropy beginning.

Chapter 7

1. It is well known that the equations of classical physics cannot be solved exactly if you are studying the motion of three or more mutually interacting bodies. So, even in classical physics, any actual prediction about the motion of a large set of particles will necessarily be approximate. The point, though, is that there is no fundamental limit to how good this approximation can be. If the world were governed by classical physics, then with ever more powerful computers, and ever more precise initial data about positions and velocities, we would get ever closer to the exact answer.

2. At the end of Chapter 4, I noted that the results of Bell, Aspect, and others do not rule out the possibility that particles always have definite positions and velocities, even if we can't ever determine such features simultaneously. Moreover, Bohm's version of quantum mechanics explicitly realizes this possibility. Thus, although the widely held view that an electron doesn't have a position until measured is a standard feature of the conventional approach to quantum mechanics, it is, strictly speaking, too strong as a blanket statement. Bear in mind, though, that in Bohm's approach, as we will discuss later in this chapter, particles are "accompanied" by probability waves; that is, Bohm's theory always invokes particles
and
waves, whereas the standard approach envisions a complementarity that can roughly be summarized as particles
or
waves. Thus, the conclusion we're after— that the quantum mechanical description of the past would be thoroughly incomplete if we spoke exclusively about a particle's having passed through a unique point in space at each definite moment in time (what we
would
do in classical physics)—is true nevertheless. In the conventional approach to quantum mechanics, we must also include the wealth of other locations that a particle could have occupied at any given moment, while in Bohm's approach we must also include the "pilot" wave, an object that is also spread throughout a wealth of other locations. (The expert reader should note that the pilot wave is just the wavefunction of conventional quantum mechanics, although its incarnation in Bohm's theory is rather different.) To avoid endless qualifications, the discussion that follows will be from the perspective of conventional quantum mechanics (the approach most widely used), leaving remarks on Bohm's and other approaches to the last part of the chapter.

3. For a mathematical but highly pedagogical account see R. P. Feynman and A. R. Hibbs,
Quantum Mechanics and Path Integrals
(Burr Ridge, Ill.: McGraw-Hill Higher Education, 1965).

4. You might be tempted to invoke the discussion of Chapter 3, in which we learned that at light speed time slows to a halt, to argue that from the photon's perspective all moments are the same moment, so the photon "knows" how the detector switch is set when it passes the beam-splitter. However, these experiments can be carried out with other particle species, such as electrons, that travel slower than light, and the results are unchanged. Thus, this perspective does not illuminate the essential physics.

5. The experimental setup discussed, as well as the actual confirming experimental results, comes from Y. Kim, R. Yu, S. Kulik, Y. Shih, M. Scully, Phys. Rev. Lett, vol. 84, no. 1, pp. 1-5.

6. Quantum mechanics can also be based on an equivalent equation presented in a different form (known as matrix mechanics) by Werner Heisenberg in 1925. For the mathematically inclined reader, Schrödinger's equation is:
H
(
x,t
) =
i
(
d
(
x,t
)/
dt
), where
H
stands for the Hamiltonian, stands for the wavefunction, and is Planck's constant.

7. The expert reader will note that I am suppressing one subtle point here. Namely, we would have to take the complex conjugate of the particle's wavefunction to ensure that it solves the time-reversed version of Schrödinger's equation. That is, the T operation described in endnote 2 of Chapter 6 takes a wavefunction (
x,t
) and maps it to *(
x
,—
t
). This has no significant impact on the discussion in the text.