The Fabric of the Cosmos: Space, Time, and the Texture of Reality (66 page)

15. See, for example, Murray Gell-Mann,
The Quark and the Jaguar
(New York: Freeman, 1994), and Huw Price,
Time's Arrow and Archimedes' Point
(Oxford: Oxford University Press, 1996).

16. Special relativity forbids anything that has ever traveled slower than light speed from crossing the speed-of-light barrier. But if something has
always
been traveling faster than the speed of light, it is not strictly ruled out by special relativity. Hypothetical particles of this sort are called
tachyons.
Most physicists believe tachyons don't exist, but others enjoy tinkering with the possibility that they do. So far, though, largely because of the strange features that such a faster-than-light particle would have according to the equations of special relativity, no one has found any particular use for them—even hypothetically speaking. In modern studies, a theory that gives rise to tachyons is generally viewed as suffering from an instability.

17. The mathematically inclined reader should note that, at its core, special relativity claims that the laws of physics must be Lorentz invariant, that is, invariant under SO(3,1) coordinate transformations on Minkowski spacetime. The conclusion, then, is that quantum mechanics would be squared with special relativity if it could be formulated in a fully Lorentz-invariant manner. Now, relativistic quantum mechanics and relativistic quantum field theory have gone a long way toward this goal, but as yet there isn't full agreement regarding whether they have addressed the quantum measurement problem in a Lorentz-invariant framework. In relativistic quantum field theory, for example, it is straightforward to compute, in a completely Lorentz-invariant manner, the probability amplitudes and probabilities for outcomes of various experiments. But the standard treatments stop short of also describing the way in which one particular outcome or another emerges from the range of quantum possibilities—that is, what happens in the measurement process. This is a particularly important issue for entanglement, as the phenomenon hinges on the effect of what an experimenter does—the act of measuring one of the entangled particle's properties. For a more detailed discussion, see Tim Maudlin,
Quantum
Non-locality and Relativity
(Oxford: Blackwell, 2002).

18. For the mathematically inclined reader, here is the quantum mechanical calculation that makes predictions in agreement with these experiments. Assume that the axes along which the detectors measure spin are vertical and 120 degrees clockwise and counterclockwise from vertical (like noon, four o'clock, and eight o'clock on two clocks, one for each detector, that are facing each other) and consider, for argument's sake, two electrons emerging back to back and heading toward these detectors in the so-called singlet state. That is the state whose total spin is zero, ensuring that if one electron is found to be in the spin-up state, the other will be in the spin-down state, about a given axis, and vice versa. (Recall that for ease in the text, I've described the correlation between the electrons as ensuring that if one is spin-up so is the other, and if one is spin-down, so is the other; in point of fact, the correlation is one in which the spins point in opposite directions. To make contact with the main text, you can always imagine that the two detectors are calibrated oppositely, so that what one calls spin-up the other calls spin-down.) A standard result from elementary quantum mechanics shows that if the angle between the axes along which our two detectors measure the electron's spins is, then the probability that they will measure opposite spin values is cos
²
(
/2). Thus, if the detector axes are aligned (= 0), they definitely measure opposite spin values (the analog of the detectors in the main text always measuring the same value when set to the same direction), and if they are set at either +120° or -120°, the probability that they measure opposite spins is cos
²
(+120° or — 120°) =
¹
/4. Now, if the detector axes are set randomly,
¹
/3 of the time they will point in the same direction, and
²
/3 of the time they won't. Thus, over all runs, we expect to find opposite spins (
¹
/3)(1) + (
²
/3)(
¹
/4) =
¹
/2 of the time, as found by the data.

You may find it odd that the assumption of locality yields a higher spin correlation (greater than 50 percent) than what we find with standard quantum mechanics (exactly 50 percent); the long-range entanglement of quantum mechanics, you'd think, should yield a greater correlation. In fact, it does. A way to think about it is this: With only a 50 percent correlation over all measurements, quantum mechanics yields 100 percent correlation for measurements in which the left and right detector axes are chosen to point in the same direction. In the local universe of Einstein, Podolsky, and Rosen, a greater than 55 percent correlation over all measurements is required to ensure 100 percent agreement when the same axes are chosen. Roughly, then, in a local universe, a 50 percent correlation over all measurements would entail
less
than a 100 percent correlation when the same axes are chosen—i.e., less of a correlation than what we find in our nonlocal quantum universe.

19. You might think that an instantaneous collapse would, from the get-go, fall afoul of the speed limit set by light and therefore ensure a conflict with special relativity. And if probability waves were indeed like water waves, you'd have an irrefutable point. That the value of a probability wave suddenly dropped to zero over a huge expanse would be far more shocking than all of the water in the Pacific Ocean's instantaneously becoming perfectly flat and ceasing to move. But, quantum mechanics practitioners argue, probability waves are
not
like water waves. A probability wave, although it describes matter, is not a material thing itself. And, such practitioners continue, the speed-of-light barrier applies only to material objects, things whose motion can be directly seen, felt, detected. If an electron's probability wave has dropped to zero in the Andromeda galaxy, an Andromedan physicist will merely fail, with 100 percent certainty, to detect the electron. Nothing in the Andromedan's observations reveals the sudden change in the probability wave associated with the successful detection, say, of the electron in New York City. As long as the electron itself does not travel from one place to another at greater than light speed, there is no conflict with special relativity. And, as you can see, all that has happened is that the electron was found to be in New York City and not anywhere else. Its speed never even entered the discussion. So, while the instantaneous collapse of probability is a framework that comes with puzzles and problems (discussed more fully in Chapter 7), it need not necessarily imply a conflict with special relativity.

20. For a discussion of some of these proposals, see Tim Maudlin,
Quantum Nonlocalityand Relativity.

Chapter 5

1. For the mathematically inclined reader, from the equation t
_moving
= (t
_stationary
— (v/c
²
) x
_stationary
) (discussed in note 9 of Chapter 3) we find that Chewie's now-list at a given moment will contain events that observers on earth will claim happened (v/c
²
) x
_earth
earlier, where x
_earth
is Chewie's distance from earth. This assumes Chewie is moving away from earth. For motion toward earth,
v
has the opposite sign, so the earthbound observers will claim such events happened (v/c
²
)x
_earth
later. Setting v = 10 miles per hour and x
_earth
= 10
¹⁰
light-years, we find (v/c
²
) x
_earth
is about 150 years.

2. This number—and a similar number given in a few paragraphs further on describing Chewie's motion toward earth—were valid at the time of the book's publication. But as time goes by here on earth, they will be rendered slightly inaccurate.

3. The mathematically inclined reader should note that the metaphor of slicing the spacetime loaf at different angles is the usual concept of
spacetime diagrams
taught in courses on special relativity. In spacetime diagrams, all of three-dimensional space at a given moment of time, according to an observer who is considered stationary, is denoted by a horizontal line (or, in more elaborate diagrams, by a horizontal plane), while time is denoted by the vertical axis. (In our depiction, each "slice of bread"—a plane—represents all of space at one moment of time, while the axis running through the middle of the loaf, from crust to crust, is the time axis.) Spacetime diagrams provide an insightful way of illustrating the point being made about the now-slices of you and Chewie.

The light solid lines are equal time slices (now-slices) for observers at rest with respect to earth (for simplicity, we imagine that earth is not rotating or undergoing any acceleration, as these are irrelevant complications for the point being made), and the light dotted lines are equal time slices for observers moving away from earth at, say, 9.3 miles per hour. When Chewie is at rest relative to earth, the former represent his now-slices (and since you are at rest on earth throughout the story, these light solid lines always represent your now-slices), and the darkest solid line shows the now-slice containing you (the left dark dot), in earth's twenty-first century, and he (the right dark dot), both sitting still and reading. When Chewie is walking away from earth, the dotted lines represent his now-slices, and the darkest dotted line shows the now-slice containing Chewie (having just gotten up and started to walk) and John Wilkes Booth (the lower left dark dot). Note, too, that one of the subsequent dotted time slices will contain Chewie walking (if he is still around!) and you, in earth's twenty-first century, sitting still reading. Hence, a single moment for you will appear on two of Chewie's now-lists—one list of relevance before and one of relevance after he started to walk. This shows yet another way in which the simple intuitive notion of
now—
when envisioned as applying throughout space—is transformed by special relativity into a concept with highly unusual features. Furthermore, these now-lists do
not
encode causality: standard causality (note 11, Chapter 3) remains in full force. Chewie's now-lists jump because he jumps from one reference frame to another. But every observer—using a single, well-defined choice of spacetime coordinatization—will agree with every other regarding which events can affect which.

4. The expert reader will recognize that I am assuming spacetime is Minkowskian. A similar argument in other geometries will not necessarily yield the entire spacetime.

5. Albert Einstein and Michele Besso: Correspondence 1903-1955, P. Speziali, ed. (Paris: Hermann, 1972).

6. The discussion here is meant to give a qualitative sense of how an experience right
now,
together with memories that you have right
now,
forms the basis of your sense of having experienced a life in which you've lived out those memories. But, if, for example, your brain and body were somehow put into exactly the same state that they are right now, you would have the same sense of having lived the life that your memories attest to (assuming, as I do, that the basis of all experience can be found in the physical state of brain and body), even if those experiences never really happened, but were artificially imprinted into your brain state. One simplification in the discussion is the assumption that we can feel or experience things that happen at a single instant, when, in reality, processing time is required for the brain to recognize and interpret whatever stimuli it receives. While true, this is not of particular relevance to the point I'm making; it is an interesting but largely irrelevant complication arising from analyzing time in a manner directly tied to human experience. As we discussed earlier, human examples help make our discussion more grounded and visceral, but it does require us to tease out those aspects of the discussion that are more interesting from a biological as opposed to a physical perspective.

7. You might wonder how the discussion in this chapter relates to our description in Chapter 3 of objects "moving" through spacetime at the speed of light. For the mathematically disinclined reader, the rough answer is that the history of an object is represented by a curve in spacetime—a path through the spacetime loaf that highlights every place the object has been at the moment it was there (much as we see in Figure 5.1). The intuitive notion of "moving" through spacetime, then, can be expressed in "flowless" language by simply specifying this path (as opposed to imagining the path being traced out before your eyes). The "speed" associated with this path is then a measure of how long the path is (from one chosen point to another), divided by the time difference recorded on a watch carried by someone or something between the two chosen points on the path. This, again, is a conception that does not involve any time flow: you simply look at what the watch in question says at the two points of interest. It turns out that the speed found in this way, for
any
motion, is equal to the speed of light. The mathematically inclined reader will realize that the reason for this is immediate. In Minkowski spacetime the metric is ds
²
= c
²
dt
²
—dx
²
(where dx
²
is the Euclidean length dx
₁
²
+ dx
₂
²
+ dx
₃
²
), while the time carried by a clock ("proper" time) is given by d
²
= ds
²
/c
²
. So, clearly, velocity through spacetime as just defined is given mathematically by ds/d, which equals c.