# Learning Objectives

### Learning Objectives

By the end of this section, you will be able to do the following:

• Describe simultaneity
• Describe time dilation
• Calculate $γγ$
• Compare proper time and the observer’s measured time
Figure 11.4 Elapsed time for a foot race is the same for all observers, but at relativistic speeds, elapsed time depends on the relative motion of the observer and the event that is observed. (Jason Edward Scott Bain, Flickr)

Do time intervals depend on who observes them? Intuitively, we expect the time for a process, such as the elapsed time for a foot race, to be the same for all observers. Our experience has been that disagreements over elapsed time have to do with the accuracy of measuring time. When we carefully consider just how time is measured, however, we will find that elapsed time depends on the relative motion of an observer with respect to the process being measured.

# Simultaneity

### Simultaneity

Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to use the arrival of light from the event, such as observing a light turning green to start a drag race. The timing will be more accurate if some sort of electronic detection is used, avoiding human reaction times and other complications.

Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps. (See Figure 11.5.) Two flash lamps with observer A midway between them are on a rail car that moves to the right relative to observer B. In the frame of reference of observer B, the light flashes are emitted just as A passes B, so that both A and B are equidistant from the lamps when the light is emitted. Observer B measures the time interval between the arrival of the light flashes. According to postulate 2, the speed of light is not affected by the motion of the lamps relative to B. Therefore, light travels equal distances to him at equal speeds. Thus, observer B measures the flashes to be simultaneous.

Figure 11.5 Observer B measures the elapsed time between the arrival of light flashes as described in the text. Observer A moves with the lamps on a rail car. Observer B perceives that the light flashes occurred simultaneously. Observer A perceives that the light on the right flashes before the light on the left.

Now consider what observer B sees happen to observer A. Observer B perceives light from the right reaching observer A before light from the left, because she has moved toward that flash lamp, lessening the distance the light must travel and reducing the time it takes to get to her. Light travels at speed $cc size 12{c} {}$ relative to both observers, but observer B remains equidistant between the points where the flashes were emitted, while A gets closer to the emission point on the right. From observer B’s point of view, then, there is a time interval between the arrival of the flashes to observer A. Observer B measures the flashes to arrive simultaneously relative to him but not relative to A.

Now consider what observer A sees happening. She sees the light from the right arriving before the right to the left. Since both lamps are the same distance from her in her reference frame, from her perspective, the right flash occurred before the left flash. Here a relative velocity between observers affects whether two events are observed to be simultaneous. Simultaneity is not absolute.

This illustrates the power of clear thinking. We might have guessed incorrectly that if light is emitted simultaneously, then two observers halfway between the sources would see the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was brilliant at this type of thought experiment, or Gedankenexperiment in German. He very carefully considered how an observation is made and disregarded what might seem obvious. The validity of thought experiments, of course, is determined by actual observation. The genius of Einstein is evidenced by the fact that experiments have repeatedly confirmed his theory of relativity.

In summary—two events are defined to be simultaneous if an observer measures them as occurring at the same time, such as by receiving light from the events. Two events are not necessarily simultaneous to all observers.

# Time Dilation

### Time Dilation

The consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect.

### Time dilation

Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer.

Suppose, for example, an astronaut measures the time it takes for light to cross her ship, bounce off a mirror, and return. (See Figure 11.6.) How does the elapsed time the astronaut measures compare with the elapsed time measured for the same event by a person on Earth? Asking this question, which is another thought experiment, produces a profound result. We find that the elapsed time for a process depends on who is measuring it. In this case, the time measured by the astronaut is smaller than the time measured by the Earth-bound observer. The passage of time is different for the observers because the distance the light travels in the astronaut’s frame is smaller than in the Earth-bound frame. Light travels at the same speed in each frame, and so it will take longer to travel the greater distance in the Earth-bound frame.

Figure 11.6 (a) An astronaut measures the time $Δt0Δt0$ for light to cross her ship using an electronic timer. Light travels a distance $2D2D$ in the astronaut’s frame. (b) A person on Earth sees the light follow longer path $2s2s$ and take a longer time $ΔtΔt$. (c) These triangles are used to find the relationship between the two distances $2D2D$ and $2s2s$.

To quantitatively verify that time depends on the observer, consider the paths followed by light as seen by each observer. (See Figure 11.6(c).) The astronaut sees the light travel straight across and back for a total distance of $2D2D size 12{2D} {}$, twice the width of her ship. The Earth-bound observer sees the light travel a total distance $2s2s size 12{2s} {}$. Since the ship is moving at speed $vv size 12{v} {}$ to the right relative to Earth, light moving to the right hits the mirror in this frame. Light travels at a speed $cc size 12{c} {}$ in both frames, and because time is the distance divided by speed, the time measured by the astronaut is

11.1 $Δt0=2Dc.Δt0=2Dc. size 12{Δt rSub { size 8{0} } = { {2D} over {c} } } {}$

This time has a separate name to distinguish it from the time measured by the Earth-bound observer.

For GPS navigation to work properly, satellites have to take into account the effects of both special relativity and general relativity. GPS satellites move at speeds of a few miles per second, and although these speeds are just tiny fractions of the speed of light, the accuracy of timing that is needed to pinpoint a position requires that we account for the effects of special relativity--that is, the slower motion of satellite time relative to an observer on Earth. Additionally, GPS satellites are in orbit roughly ten thousand miles above Earth, where the gravitational force is weaker. From the theory of general relativity, the weaker gravitational force means that time on the satellite is ticking faster. If these two relativistic effects were not accounted for, GPS units would lose their accuracy in a matter of minutes.

### Proper Time

Proper time $Δt0;Δt0; size 12{Δt rSub { size 8{0} } } {}$ is the time measured by an observer at rest relative to the event being observed.

In the case of the astronaut observe the reflecting light, the astronaut measures proper time. The time measured by the Earth-bound observer is

11.2 $Δt=2sc.Δt=2sc. size 12{Δt= { {2s} over {c} } } {}$

To find the relationship between $Δt0Δt0 size 12{Δt rSub { size 8{0} } } {}$ and $Δt,Δt, size 12{Δt} {}$ consider the triangles formed by $DD size 12{D} {}$ and $s.s. size 12{s} {}$ (See Figure 11.6(c).) The third side of these similar triangles is $LL size 12{L} {}$, the distance the astronaut moves as the light goes across her ship. In the frame of the Earth-bound observer,

11.3 $L=vΔt2.L=vΔt2. size 12{L= { {vΔt} over {2} } } {}$

Using the Pythagorean theorem, the distance $ss size 12{s} {}$ is found to be

11.4 $s=D2+vΔt22.s=D2+vΔt22. size 12{s= sqrt {D rSup { size 8{2} } + left ( { {vΔt} over {2} } right ) rSup { size 8{2} } } } {}$

Substituting $ss size 12{s} {}$ into the expression for the time interval $ΔtΔt size 12{Δt} {}$ gives

11.5 $Δt=2sc=2D2+vΔt22c.Δt=2sc=2D2+vΔt22c. size 12{Δt= { {2s} over {c} } = { {2 sqrt {D rSup { size 8{2} } + left ( { {vΔt} over {2} } right ) rSup { size 8{2} } } } over {c} } } {}$

We square this equation, which yields

11.6 $(Δt)2=4D2+v2(Δt)24c2=4D2c2+v2c2(Δt)2.(Δt)2=4D2+v2(Δt)24c2=4D2c2+v2c2(Δt)2. size 12{ $$Δt$$ rSup { size 8{2} } = { {4 left [D rSup { size 8{2} } + { {v rSup { size 8{2} } $$Δt$$ rSup { size 8{2} } } over {4} } right ]} over {c rSup { size 8{2} } } } = { {4D rSup { size 8{2} } } over {c rSup { size 8{2} } } } + { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } $$Δt$$ rSup { size 8{2} } } {}$

Note that if we square the first expression we had for $Δt0Δt0 size 12{Δt rSub { size 8{0} } } {}$, we get $(Δt0)2=4D2c2(Δt0)2=4D2c2 size 12{ $$Δt rSub { size 8{0} }$$ rSup { size 8{2} } = { {4D rSup { size 8{2} } } over {c rSup { size 8{2} } } } } {}$. This term appears in the preceding equation, giving us a means to relate the two time intervals. Thus,

11.7 $(Δt)2=(Δt0)2+v2c2(Δt)2.(Δt)2=(Δt0)2+v2c2(Δt)2. size 12{ $$Δt$$ rSup { size 8{2} } = $$Δt rSub { size 8{0} }$$ rSup { size 8{2} } + { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } $$Δt$$ rSup { size 8{2} } } {}$

Gathering terms, we solve for $Δt.Δt. size 12{Δt} {}$

11.8 $(Δt)21−v2c2=(Δt0)2.(Δt)21−v2c2=(Δt0)2. size 12{ $$Δt$$ rSup { size 8{2} } left (1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } right )= $$Δt rSub { size 8{0} }$$ rSup { size 8{2} } } {}$

Thus,

11.9 $(Δt)2=(Δt0)21−v2c2.(Δt)2=(Δt0)21−v2c2. size 12{ $$Δt$$ rSup { size 8{2} } = { { $$Δt rSub { size 8{0} }$$ rSup { size 8{2} } } over {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } {}$

Taking the square root yields an important relationship between elapsed times:

11.10 $Δt=Δt01−v2c2=γΔt0,Δt=Δt01−v2c2=γΔt0, size 12{Δt= { {Δt rSub { size 8{0} } } over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } =γΔt rSub { size 8{0} } } {}$

where

11.11 $γ=11−v2c2.γ=11−v2c2. size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}$

This equation for $ΔtΔt size 12{Δt} {}$ is truly remarkable. First, as contended, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. Proper time $Δt0Δt0 size 12{Δt rSub { size 8{0} } } {}$ measured by an observer, like the astronaut moving with the apparatus, is smaller than time measured by other observers. Since those other observers measure a longer time $Δt,Δt, size 12{Δt}{}$ the effect is called time dilation. The Earth-bound observer sees time dilate (get longer) for a system moving relative to Earth. Alternatively, according to the Earth-bound observer, time slows in the moving frame, since less time passes there. All clocks moving relative to an observer, including biological clocks such as aging, are observed to run slow compared with a clock stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light ($vcvc size 12{v"$), then $v2c2v2c2$ is extremely small, and the elapsed times $ΔtΔt$ and $Δt0Δt0 size 12{Δt rSub { size 8{0} } } {}$ are nearly equal. At low velocities, modern relativity approaches classical physics—our everyday experiences have very small relativistic effects.

The equation $Δt=γΔt0Δt=γΔt0$ also implies that relative velocity cannot exceed the speed of light. As $vv size 12{v} {}$ approaches $c,c, size 12{c}{}$$ΔtΔt size 12{Δt}{}$ approaches infinity. This would imply that time in the astronaut’s frame stops at the speed of light. If $vv size 12{v} {}$ exceeded $c,c, size 12{c}{}$ then we would be taking the square root of a negative number, producing an imaginary value for $Δt.Δt. size 12{Δt} {}$

There is considerable experimental evidence that the equation $Δt=γΔt0Δt=γΔt0$ is correct. One example is found in cosmic ray particles that continuously rain down on Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons. The half-life (amount of time for half of a material to decay) of a muon is $1.52μs1.52μs size 12{1 "." "52" "μs"} {}$ when it is at rest relative to the observer who measures the half-life. This is the proper time $Δt0.Δt0. size 12{Δt rSub { size 8{0} } }{}$ Muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an Earth-bound observer ($ΔtΔt size 12{Δt}{}$) varies with velocity exactly as predicted by the equation $Δt=γΔt0.Δt=γΔt0. size 12{Δt=γΔt rSub { size 8{0} } } {}$ The faster the muon moves, the longer it lives. We on Earth see the muon’s half-life time dilated—as viewed from our frame, the muon decays more slowly than it does when at rest relative to us.

### Example 11.1Calculating $ΔtΔt$ for a Relativistic Event: How Long Does a Speedy Muon Live?

Suppose a cosmic ray colliding with a nucleus in Earth’s upper atmosphere produces a muon that has a velocity $v=0.950c.v=0.950c. size 12{v=0 "." "950"c}{}$ The muon then travels at constant velocity and lives $1.52μs1.52μs size 12{1 "." "52" ital "μs"} {}$ as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal clock.) How long does the muon live as measured by an Earth-bound observer? (See Figure 11.7.)

Figure 11.7 A muon in Earth’s atmosphere lives longer as measured by an Earth-bound observer than measured by the muon’s internal clock.

Strategy

A clock moving with the system being measured observes the proper time, so the time we are given is $Δt0=1.52μs.Δt0=1.52μs.$ The Earth-bound observer measures $ΔtΔt$ as given by the equation $Δt=γΔt0.Δt=γΔt0.$ Since we know the velocity, the calculation is straightforward.

Solution

1) Identify the knowns. $v=0.950cv=0.950c$, $Δt0=1.52μsΔt0=1.52μs$

2) Identify the unknown. $ΔtΔt$

3) Choose the appropriate equation.

Use,

11.12 $Δt=γΔt0,Δt=γΔt0,$

where

11.13 $γ=11−v2c2.γ=11−v2c2.$

4) Plug the knowns into the equation.

First find $γ.γ.$

11.14 $γ=11−v2c2=11−(0.950c)2c2=11−(0.950)2=3.20γ=11−v2c2=11−(0.950c)2c2=11−(0.950)2=3.20$

Use the calculated value of $γγ size 12{γ} {}$ to determine $Δt.Δt.$

11.15 $Δt=γΔt0=(3.20)(1.52μ s)=4.87μ sΔt=γΔt0=(3.20)(1.52μ s)=4.87μ s$

Discussion

One implication of this example is that since $γ=3.20γ=3.20 size 12{γ=3 "." "20"} {}$ at $95.0 percent95.0 percent size 12{"95" "." 0%} {}$ of the speed of light ($v=0.950cv=0.950c size 12{v=0 "." "950"c} {}$), the relativistic effects are significant. The two time intervals differ by this factor of 3.20, where classically they would be the same. Something moving at $0.950c0.950c size 12{0 "." "950"c} {}$ is said to be highly relativistic.

Another implication of the preceding example is that everything an astronaut does when moving at $95.0 percent95.0 percent size 12{"95" "." 0%} {}$ of the speed of light relative to Earth takes 3.20 times longer when observed from Earth. Does the astronaut sense this? Only if she looks outside her spaceship. All methods of measuring time in her frame will be affected by the same factor of 3.20. This includes her wristwatch, heart rate, cell metabolism rate, nerve impulse rate, and so on. She will have no way of telling, since all of her clocks will agree with one another because their relative velocities are zero. Motion is relative, not absolute. But what if she does look out the window?

### Real-World Connections

It may seem that special relativity has little effect on your life, but it is probably more important than you realize. One of the most common effects is through the Global Positioning System (GPS). Emergency vehicles, package delivery services, electronic maps, and communications devices are just a few of the common uses of GPS, and the GPS system could not work without taking into account relativistic effects. GPS satellites rely on precise time measurements to communicate. The signals travel at relativistic speeds. Without corrections for time dilation, the satellites could not communicate, and the GPS system would fail within minutes.

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to Earth would age less than her Earth-bound twin. Imagine the astronaut moving at such a velocity that null as in Figure 11.8. A trip that takes 2 years in her frame would take 60 years in her Earth-bound twin’s frame. Suppose the astronaut traveled 1 year to another star system. She briefly explored the area, and then traveled 1 year back. If the astronaut was 40 years old when she left, she would be 42 upon her return. Everything on Earth, however, would have aged 60 years. Her twin, if still alive, would be 100 years old.

The situation would seem different to the astronaut. Because motion is relative, the spaceship would seem to be stationary and Earth would appear to move. This is the sensation you have when flying in a jet. If the astronaut looks out the window of the spaceship, she will see time slow down on Earth by a factor of null To her, the Earth-bound sister will have aged only 2/30 (1/15) of a year, while she aged 2 years. The two sisters cannot both be correct.

Figure 11.8 The twin paradox asks why the traveling twin ages less than the Earth-bound twin. That is the prediction we obtain if we consider the Earth-bound twin’s frame. In the astronaut’s frame, however, Earth is moving and time runs slower there. Who is correct?

As with all paradoxes, the premise is faulty and leads to contradictory conclusions. In fact, the astronaut’s motion is significantly different from that of the Earth-bound twin. The astronaut accelerates to a high velocity and then decelerates to view the star system. To return to Earth, she again accelerates and decelerates. The Earth-bound twin does not experience these accelerations. So the situation is not symmetric, and it is not correct to claim that the astronaut will observe the same effects as her Earth-bound twin. If you use special relativity to examine the twin paradox, you must keep in mind that the theory is expressly based on inertial frames, which by definition are not accelerated or rotating. Einstein developed general relativity to deal with accelerated frames and with gravity, a prime source of acceleration. You can also use general relativity to address the twin paradox and, according to general relativity, the astronaut will age less. Some important conceptual aspects of general relativity are discussed in General Relativity and Quantum Gravity of this course.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around Earth on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, since gravity and accelerations were involved as well as relative motion.