by Jed Brody
Distance = Speed × Time,
so the time interval between the emission of the light pulse and the return of the light pulse is
Time = Distance/Speed,
where Distance is 2H, twice the height of the room (figure 16), and Speed is the speed of light. An observer in the room thus finds that the time interval is 2H divided by the speed of light.
Figure 16 The total distance traveled by the light pulse is 2H, twice the height of the room.
But now imagine that the room is actually a train car, and the train is traveling at a relativistic speed, v, relative to the ground. The train car is made out of a transparent material, so someone outside the train can observe the two events. Imagine someone standing on the ground outside the train, and let’s call this person Grover (of the ground). Let’s also imagine someone named Tracy (of the train), standing in the train car.
Grover witnesses the same two events: the emission of the light pulse from the laser on the floor of the train car, and the return of the light to its source. These two events occur at different locations relative to the ground because the train moves as the light travels. In fact, to Grover, the light must travel diagonally.
As shown in figure 17, the train is in one position when the light is emitted, another position when the light reflects off the ceiling, and a third position when the light returns to its source. The total length of the two diagonal arrows is the distance traveled by the light, as seen by Grover. So the distance traveled by light, between its emission and return to source, depends on who’s watching.
Figure 17 Three positions of the train car, shown as dashed rectangles, relative to the ground: The light is emitted in the first position. The light reflects off the ceiling in the second position. The light returns to its source in the third position. The diagonal arrows represent the path of the light, and the horizontal arrow represents the motion of the train car.
Both observers agree that Time = Distance/Speed. And both observers agree that Speed is the speed of light, which is the same for everyone. But the distance is not the same. As seen by Grover, light travels a greater distance than 2H. Therefore, Grover measures a larger time interval between the two events than the train observer does. Science-fiction writers love this fact: more time elapses for people on Earth than for high-speed space travelers. This is not a technological failure of clocks, or a biological response to space travel, or any kind of flawed perspective. Time itself elapses differently for people traveling at different speeds.
Time dilation is often summarized as “moving clocks run slow.” Tracy uses a clock affixed to the floor of the train car to measure the time interval between the two events occurring there (the emission and return of the light pulse). Grover disagrees with the time interval shown by Tracy’s clock. Grover measures a larger time interval; therefore, Grover complains that Tracy’s moving clock runs slow.
It may be confusing to recognize that from Tracy’s point of view, it’s Grover who is moving. Therefore, according to Tracy, it’s Grover’s clock that runs slow! In fact, each observer complains that the other observer’s clock runs slow. We will not resolve this paradox in detail. But the resolution lies in the two other basic results of relativity: length contraction, and disagreement about chronological order. We turn to these shortly.
We may be curious about a related conundrum, the twin paradox. Suppose two twins are born on Earth. One twin hops aboard a spaceship, travels at relativistic speed, and returns to Earth to find that her twin has aged much more than she has. In fact, both twins agree on this fact when they reunite on Earth. On the one hand, this is a simple illustration of time dilation: the traveling twin’s clocks (including her physiological clock) ran slow. On the other hand, from the traveling twin’s point of view, the Earth accelerated away and then returned, while the spaceship remained still. Isn’t this a valid perspective? From a perspective at rest in the spaceship, the earthbound twin was the high-speed traveler who should have aged less.
The resolution of this paradox is that our intuition is correct: it really is the spaceship that accelerates away from Earth. If you stand on an accelerating bus, you feel the lurch that may cause you to lose your balance. The people standing on the sidewalk don’t feel the lurch of acceleration, even though, from a perspective at rest in the bus, it’s the sidewalk that accelerates away. Using accelerometers, we can prove that the bus accelerates relative to Earth; and Earth remains at rest.2 Similarly, the traveling twin experiences acceleration, but the earthbound twin does not.
There’s really quite a lot going on in the twin paradox, but we can identify the basic stages:
1.The twins start out on Earth. They’re the same age.
2.One twin hops aboard the spaceship and accelerates away from Earth. The traveling twin is accelerating, and the earthbound twin is not. The traveling twin can confirm this fact by observing an accelerometer on the spaceship.
3.The traveling twin then travels at constant, relativistic speed through space. There’s no acceleration now (the spaceship is coasting), so each twin’s perspective is valid: each twin sees that the other’s clocks are running slow. Both twins are correct; time elapses differently for different observers.
4.The spaceship decelerates to rest, and then accelerates back toward Earth for the voyage home. The earthbound twin, however, experiences neither acceleration nor deceleration.
5.The traveling twin again travels at constant, relativistic speed, this time toward Earth. Since there’s no acceleration, each twin again makes the valid observation that the other twin’s clocks run slow.
6.The traveling twin decelerates to land on Earth, while the earthbound twin remains at rest.
7.Both twins are at rest on Earth. The twin that traveled is younger because only that twin experienced acceleration and deceleration.
The bottom line is that the laws of physics are the same for all observers traveling at a constant speed (no acceleration). To make sense of what’s happening to someone who accelerates up to relativistic speeds, we can take the point of view of someone at rest (or moving at constant speed) the whole time.3
Length Contraction
Recall the experiment used to prove time dilation. Suppose that the laser happens to be directly over a railroad tie when it emits its pulse of light. Suppose too that the laser is directly over another railroad tie when the light returns to its source. Tracy and Grover now want to determine the distance between the two railroad ties. Both observers agree on two quantities: the speed of light, and the speed v at which they’re moving relative to each other.
Since Distance = Speed × Time, both observers can use this formula to find the distance between the two railroad ties. For Tracy, Speed = v, the speed at which the railroad ties are whizzing past below her. Time is the time interval between the emission of the light (when the first railroad tie is directly below) and the return of the light to its source (when the second railroad tie is directly below).
For Grover, Speed is also v, the speed at which the train is racing by. Time is also the time interval between the emission of light (when the laser is directly above the first railroad tie) and the return of the light to its source (when the laser is directly above the second railroad tie). But we found above that the two observers disagree about this time interval! Grover measures a greater time interval. Since distance is proportional to time, Grover measures a greater distance between the two railroad ties than Tracy does.
Tracy measures a shorter distance between the railroad ties. Relative to Tracy, the train tracks are moving, and the distance along the tracks is contracted, compared with Grover’s measurements. This is the contraction of the length of moving objects, along the direction of motion. Both observers agree on the height of the train because length contraction does not affect distances perpendicular to the direction of motion.
We saw that each observer complains that the other observer’s clocks run slow. Similarly, each observer complains that the other observer’s
lengths are contracted (but only along the direction of motion). Relative to Grover, the train is moving, so Grover measures that the train has a shorter length than is measured by Tracy. How can each observer claim that the other observer’s lengths are contracted? We will be able to resolve this paradox after establishing one more consequence of light’s constant speed.
The Chronological Order of Events May Depend on Who’s Observing Them
The third consequence of light’s invariant speed may be the strangest. It’s also the most significant for our discussion of quantum entanglement. Consider now a flash of light from the center of the train car. As seen by Tracy, the light simultaneously reaches the front wall and back wall of the train car (figure 18). The two events (the arrival of light at the front wall and the arrival of light at the back wall) are simultaneous. But only according to Tracy!
Figure 18 A flash of light from the center of the train car reaches both walls simultaneously—but only according to the observer on the train.
Grover sees the front wall traveling away from the light source. Similarly, the back wall is traveling toward the light source. Therefore, the light reaches the back wall first. Later, the light reaches the front wall. The two events, which were simultaneous in Tracy’s eyes, occur in sequence according to Grover. The simultaneity of two events is not a universal fact but depends on who’s observing.
We can imagine that Tracy has a stopwatch affixed to both walls of the train, and the stopwatches are designed to start counting when the pulse of light reaches them. Tracy sees both stopwatches start at the same time, so the two stopwatches are synchronized. Grover, on the other hand, sees the stopwatch on the back wall start first, so the stopwatch on the back wall shows a time in advance of the stopwatch on the front wall.
This discrepancy applies to all of Tracy’s clocks, not just stopwatches. It’s convenient to imagine that Tracy has lots of synchronized clocks along the length of the train. The further a clock toward the back of the train (to the left in figure 18), the further ahead the time that it shows, according to Grover.
This disagreement about chronology allows us to resolve the paradox about length contraction. Tracy sees that Grover’s lengths are contracted along the direction of motion, and Grover sees that Tracy’s lengths are contracted along the direction of motion. We can imagine that each observer has yardsticks aligned with the direction of motion. Each observer sees that the other observer’s yardsticks are too short—less than a yard.
Grover wants to understand how Tracy, with her contracted yardsticks, can possibly claim that Grover’s yardsticks are too short—shorter than hers. Grover sees clearly that Tracy’s yardsticks are too short—shorter than his. Grover recognizes that the length of a yardstick is the distance between the two endpoints of the yardstick. So, to measure the length of a yardstick that’s moving past you, you can record the positions of the two endpoints—but you have to be careful to record those two positions at the same time! If an arrow’s flying by you, and you record the position of front tip before you record the position of the back tip, the distance between the two positions is less than the length of the arrow.
Grover sees that Tracy’s clocks not only run slow; they disagree with one another. So when Tracy records the positions of the endpoints of Grover’s yardstick, she records the two positions at the same time—according to her own clocks. But Grover sees that Tracy’s clock toward the left shows a later time than her clock toward the right. So Grover sees that everything that happens to the left occurs too soon; when Tracy tries to perform two simultaneous measurements, she actually first performs the measurement on the left, followed by the measurement on the right. (Tracy herself observes that she performs the measurements at the same time.)
This resolves the paradox, because Grover sees the following: Making a mark on the transparent floor of her train car, Tracy identifies the point directly above the left endpoint of Grover’s yardstick. (To Tracy, the left endpoint is like the front tip of a moving arrow.) According to Grover, time elapses before Tracy marks the point directly above the right endpoint of Grover’s yardstick (which, to Tracy, is like the trailing end of the moving arrow). Thus the distance between the two marked points, according to Grover, is unfairly reduced; the two endpoints were recorded at different times. During this time interval, the train has moved relative to the yardstick, so of course the recorded distance between the two positions differs from the true length. Tracy finds that the distance between the endpoints is less than a meter—even using her own contracted (as Grover sees it) yardstick.
Tracy’s perspective is equally valid. She can explain why Grover, even with his shrunken yardsticks, claims that her yardsticks are the shrunken ones. She recognizes that Grover thinks that he’s simultaneously recording the positions of the endpoints of her yardstick, but he actually allows time to pass between the two measurements.
Again, each observer is making a mistake only from the perspective of the other observer. Both observers make equally legitimate claims about time intervals, lengths, and the order in which events occur. These properties of space and time are not universal but depend on who’s observing them.
Now consider an observer traveling to the right even more quickly than the train. To this observer, the train is receding to the left. Thus, to the speedy observer, the left wall of the train is receding from the light source, and the right wall is moving toward it. The speedy observer sees the light arrive at the right wall before it arrives at the left wall—opposite to the chronological order observed by Grover!
The length-contraction paradox was resolved, but at what price? Have new paradoxes sprung up? If the order of events depends on who’s looking, we seem to have the possibility of time travel and its attendant paradoxes. For example, consider the following two events:
1.I trip at the top of the stairs.
2.I land on my face at the bottom of the stairs.
According to my observations, Event 1 occurs before Event 2. But if you’re traveling at relativistic speed, might you see Event 2 occur first? If you see me land on my face before I even trip, could you stop Event 1 from happening, perhaps by sealing off the top of the stairway? But if you prevent Event 1 from happening, how were you able to witness Event 2, which was caused by Event 1?
Einstein showed that these paradoxes are avoided, as long as nothing travels faster than light. If nothing travels faster than light, then all observers agree that a cause occurs before all of its effects; an effect occurs after all of its causes. The order of events is different for different observers only if the events are causally unconnected.
But what of that key disclaimer, “as long as nothing travels faster than light”? Does quantum entanglement violate this condition?
The Apparent Conflict with Quantum Entanglement
Let’s return now to quantum entanglement and the observation of two entangled photons. If the measurement of one photon physically alters the other, the effect occurs instantly and with undiminished impact over any distance. On the other hand, Einstein’s relativity implies that nothing travels faster than light, whose speed is 670 million miles per hour. Suppose that I’m 670 million miles away from you. Halfway between us, a pair of entangled photons is emitted. These are the entangled photons we met in chapter 3: they are just as likely to be horizontally polarized as vertically polarized. If both photons encounter a horizontal polarizer, they will either both pass through or both be blocked.
Suppose, too, that we each decide to place a horizontal polarizer in the path of the photon heading our way. You find that your photon passes through the polarizer, and you want to convey this information to me in the fastest possible way: via radio waves, which travel at the speed of light. The radio waves hurtle through space for a full hour before they reach me. By the time your radio message arrives, it’s old news. For a whole hour, I’ve already known the result of your measurement because I too saw my photon pass through a horizontal polarizer. Have we found a way to beat relat
ivity and communicate faster than the speed of light?
Actually, the measurement of one photon cannot in any way be used to send messages via its distant twin. Both experimenters can agree to set their polarizers to the same angle, but the first photon measured is just as likely to be transmitted as blocked. The distant photon must do the same thing, but this doesn’t allow us to encode any message since the “signal” we’re sending is a random outcome over which we have no control. We can reconcile relativity with quantum entanglement by clarifying that neither mass nor messages can travel faster than light. The subtle linkage between entangled particles conveys neither mass nor messages.
Spooky action at a distance is consistent with even the strangest consequence of relativity: the chronological order of events may depend on who’s observing. You and I may agree that you measure your photon before I measure mine, but someone traveling exceptionally fast may observe, instead, that my measurement occurs first. The speedy observer sees the measurements occur in the opposite order, but the result is the same: the outcome of the first measurement is random, and the outcome of the second measurement is compelled to be the same.
6
Direct Observation Is the Only Reality?
Experiments contradict local realism, but it’s easier to reject falsehood than to establish the truth. The truth is elusive. We cannot observe photons prior to observation—so we do not have direct evidence that the observation of one photon affects another. Debate over these points has resulted in a host of interpretations of quantum mechanics.