From Eternity to Here (78 page)

192
See Neal (2006), who calls this approach “Full Non-indexical Conditioning.” “Conditioning” means that we make predictions by asking what the rest of the universe looks like when certain conditions hold (e.g., that we are an observer with certain properties); “full” means that we condition over every single piece of data we have, not only coarse features like “we are an observer”; and “non-indexical” means that we consider absolutely every instance in which the conditions are met, not just one particular instance that we label as “us.”

193
Boltzmann’s travelogue is reprinted in Cercignani (1998), 231. For more details of his life and death, see that book as well as Lindley (2001).

11. QUANTUM TIME

194
Quoted in von Baeyer (2003), 12-13.

195
This is not to say that the ancient Buddhists weren’t wise, but their wisdom was not based on the failure of classical determinism at atomic scales, nor did they anticipate modern physics in any meaningful way, other than the inevitable random similarities of word choice when talking about grand cosmic concepts. (I once heard a lecture claming that the basic ideas of primordial nucleosynthesis were prefigured in the Torah; if you stretch your definitions enough, eerie similarities are everywhere.) It is disrespectful to both ancient philosophers and modern physicists to ignore the real differences in their goals and methods in an attempt to create tangible connections out of superficial resemblances.

196
More recently, dogs have also been recruited for the cause. See Orzel (2009).

197
We’re still glossing over one technicality—the truth is actually one step more complex (as it were) than this description would have you believe, but it’s not a complication that is necessary for our present purposes. Quantum amplitudes are really
complex numbers
, which means they are combinations of two numbers: a real number, plus an imaginary number. (Imaginary numbers are what you get when you take the square root of a negative real number; so “imaginary two” is the square root of minus four, and so on.) A complex number looks like
a
+
bi
, where
a
and
b
are real numbers and “
i
” is the square root of minus one. If the amplitude associated with a certain option is
a
+
bi
, the probability it corresponds to is simply
a
²
+
b
²
, which is guaranteed to be greater than or equal to zero. You will have to trust me that this extra apparatus is extremely important to the workings of quantum mechanics—either that, or start learning some of the mathematical details of the theory. (I can think of less rewarding ways of spending your time, actually.)

198
The fact that any particular sequence of events assigns positive or negative amplitudes to the two final possibilities is an assumption we are making for the purposes of our thought experiment, not a deep feature of the rules of quantum mechanics. In any real-world problem, details of the system being considered will determine what precisely the amplitudes are, but we’re not getting our hands quite that dirty at the moment. Note also that the particular amplitudes in these examples take on the numerical values of plus or minus 0.7071—that’s the number which, when squared, gives you 0.5.

199
At a workshop attended by expert researchers in quantum mechanics in 1997, Max Tegmark took an admittedly highly unscientific poll of the participants’ favored interpretation of quantum mechanics (Tegmark, 1998). The Copenhagen interpretation came in first with thirteen votes, while the many-worlds interpretation came in second with eight. Another nine votes were scattered among other alternatives. Most interesting, eighteen votes were cast for “none of the above/undecided.” And these are the experts.

200
So what does happen if we hook up a surveillance camera but then don’t examine the tapes? It doesn’t matter whether we look at the tapes or not; the camera still counts as an observation, so there will be a chance to observe Ms. Kitty under the table. In the Copenhagen interpretation, we would say, “The camera is a classical measuring device whose influence collapses the wave function.” In the many-worlds interpretation, as we’ll see, the explanation is “the wave function of the camera becomes entangled with the wave function of the cat, so the alternative histories decohere.”

201
Many people have thought about changing the rules of quantum mechanics so that this is no longer the case; they have proposed what are called “hidden variable theories” that go beyond the standard quantum mechanical framework. In 1964, theoretical physicist John Bell proved a remarkable theorem: No local theory of hidden variables can possibly reproduce the predictions of quantum mechanics. This hasn’t stopped people from investigating nonlocal theories—ones where distant events can affect each other instantaneously. But they haven’t really caught on; the vast majority of modern physicists believe that quantum mechanics is simply correct, even if we don’t yet know how to interpret it.

2
02 There is a slightly more powerful statement we can actually make. In classical mechanics, the state is specified by both position and velocity, so you might guess that the quantum wave function assigns probabilities to every possible combination of position and velocity. But that’s not how it works. If you specify the amplitude for every possible position, you are done—you’ve completely determined the entire quantum state. So what happened to the velocity? It turns out that you can write the same wave function in terms of an amplitude for every possible velocity, completely leaving position out of the description. These are not two different states; they are just two different ways of writing exactly the same state. Indeed, there is a cookbook recipe for translating between the two choices, known in the trade as a “Fourier transform.” Given the amplitude for every possible position, you can do a Fourier transform to determine the amplitude for any possible velocity, and vice versa. In particular, if the wave function is an eigenstate, concentrated on one precise value of position (or velocity), its Fourier transform will be completely spread out over all possible velocities (or positions).

203
Einstein, Podolsky, and Rosen (1935).

204
Everett (1957). For discussion from various viewpoints, see Deutsch (1997), Albert (1992), or Ouellette (2007).

205
Note how crucial entanglement is to this story. If there were no entanglement, the outside world would still exist, but the alternatives available to Miss Kitty would be completely independent of what was going on out there. In that case, it would be perfectly okay to attribute a wave function to Miss Kitty all by herself. And thank goodness; that’s the only reason we are able to apply the formalism of quantum mechanics to individual atoms and other simple isolated systems. Not everything is entangled with everything else, or it would be impossible to say much about any particular subsystem of the world.

12. BLACK HOLES: THE ENDS OF TIME

2
06 Bekenstein (1973).

207
Hawking (1988), 104. Or, as Dennis Overbye (1991, 107) puts it: “In Cambridge Bekenstein’s breakthrough was greeted with derision. Hawking was outraged. He knew this was nonsense.”

208
For discussion of observations of stellar-mass black holes, see Casares (2007); for supermassive black holes in other galaxies, see Kormendy and Richstone (1995). The black hole at the center of our galaxy is associated with a radio source known as “Sagittarius A*”; see Reid (2008).

209
Okay, for some people the looking is even more fun.

210
Way more than that, actually. As of January 2009, Hawking’s original paper (1975) had been cited by more than 3,000 other scientific papers.

211
As of this moment, we have never detected gravitational waves directly, although indirect evidence for their existence (as inferred from the energy lost by a system of two neutron stars known as the “binary pulsar”) was enough to win the Nobel Prize for Joseph Taylor and Russell Hulse in 1993. Right now, several gravitational-wave observatories are working to discover such waves directly, perhaps from the coalescence of two black holes.

212
The area of the event horizon is proportional to the square of the mass of the black hole; in fact, if the area is
A
and the mass is
M
, we have
A
= 16π
G
²
M
²
/
c
⁴
, where
G
is Newton’s constant of gravitation and
c
is the speed of light.

213
The analogy between black hole mechanics and thermodynamics was spelled out in Bardeen, Carter, and Hawking (1973).

214
One way to think about why the surface gravity is not infinite is to take seriously the caveat “as measured by an observer very far away.” The force right near the black hole is large, but when you measure it from infinity it undergoes a gravitational redshift, just as an escaping photon would. The force is infinitely strong, but there is an infinite redshift from the point of view of a distant observer, and the effects combine to give a finite answer for the surface gravity.

215
More carefully, Bekenstein suggested that the entropy was proportional to the area of the event horizon. Hawking eventually worked out the constant of proportionality.

216
Hawking (1988), 104-5.

217
You may wonder why it seems natural to think of the electromagnetic and gravitational fields, but not the electron field or the quark field. That’s because of the difference between fermions and bosons. Fermions, like electrons and quarks, are matter particles, distinguished by the fact that they can’t pile on top of one another; bosons, like photons and gravitons, are force particles that pile on with abandon. When we observe a macroscopic, classical-looking field, that’s a combination of a huge number of boson particles. Fermions like electrons and quarks simply can’t pile up that way, so their field vibrations only ever show up as individual particles.

218
Overbye (1991), 109.

219
For reference purposes, the Planck length is equal to (
Għ
/
c
³
)
^½
, where
G
is Newton’s constant of gravitation,
ħ
is Planck’s constant from quantum mechanics, and
c
is the speed of light. (We’ve set Boltzmann’s constant equal to 1.) So the entropy can be expressed as
S
= (
c
¾
ħG
)
A
. The area of the event horizon is related to the mass
M
of the black hole by
A =
8π
G
²
M
²
. Putting it all together, the entropy is related to the mass by as
S
= (4π
Gc
³
/
ħ
)
M
²
.

220
Particles and antiparticles are all “particles,” if that makes sense. Sometimes the word
particle
is used specifically to contrast with
antiparticle
, but more often it just refers to any pointlike elementary object. Nobody would object to the sentence “the positron is a particle, and the electron is its antiparticle.”

221
“Known” is an important caveat. Cosmologists have contemplated the possibility that some unknown process, perhaps in the very early universe, might have created copious amounts of very small black holes, perhaps even related to the dark matter. If these black holes were small enough, they wouldn’t be all that dark; they’d be emitting increasing amounts of Hawking radiation, and the final explosions might even be detectable.

222
One speculative but intriguing idea is that we could
make
a black hole in a particle accelerator, and then observe it decaying through Hawking radiation. Under ordinary circumstances, that’s hopelessly unrealistic; gravity is such an incredibly weak force that we’ll never be able to build a particle accelerator powerful enough to make even a microscopic black hole. But some modern scenarios, featuring hidden dimensions of spacetime, suggest that gravity becomes much stronger than usual at short distances (see Randall, 2005). In that case, the prospect of making and observing small black holes gets upgraded from “crazy” to “speculative, but not completely crazy.” I’m sure Hawking is rooting for it to happen.

Unfortunately, the prospect of microscopic black holes has been seized on by a group of fearmongers to spin scenarios under which the Large Hadron Collider, a new particle accelerator at the CERN laboratory in Geneva, is going to destroy the world. Even if the chances are small, destroying the world is pretty bad, so we should be careful, right? But careful reviews of the possibilities (Ellis et al
.
, 2008) have concluded that there’s nothing the LHC will do that hasn’t occurred many times already elsewhere in the universe; if something disastrous were going to happen, we should have seen signs of it in other astrophysical objects. Of course, it’s always possible that everyone involved in these reviews is making some sort of unfortunate math mistake. But lots of things are possible. The next time you open a jar of tomato sauce, it’s possible that you will unleash a mutated pathogen that will wipe out all life on Earth. It’s possible that we are being watched and judged by a race of super-intelligent aliens, who will think badly of us and destroy the Earth if we allow ourselves to be cowed by frivolous lawsuits and
don’t
turn on the LHC. When possibilities become as remote as what we’re speaking about here, it’s time to take the risks and get on with our lives.