To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by
v=L0Δt.v=L0Δt. size 12{v= { {L rSub { size 8{0} } } over {Δt} } } {}
(3)The time relative to the Earth-bound observer is ΔtΔt size 12{Δt} {}, since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by
v=LΔt0.v=LΔt0. size 12{v= { {L rSub { size 8{0} } } over {Δt} } } {}
(4)The moving observer travels with the muon and therefore observes the proper time Δt0Δt0 size 12{Δt rSub { size 8{0} } } {}. The two velocities are identical; thus,
L0Δt=LΔt0.L0Δt=LΔt0. size 12{ { {L rSub { size 8{0} } } over {Δt} } = { {L} over {Δt rSub { size 8{0} } } } } {}
(5)We know that Δt=γΔt0Δt=γΔt0 size 12{Δt=γΔt rSub { size 8{0} } } {}. Substituting this equation into the relationship above gives
L=L0γ.L=L0γ. size 12{L= { {L rSub { size 8{0} } } over {γ} } } {}
(6)Substituting for γγ size 12{γ} {} gives an equation relating the distances measured by different observers.
Length contraction LL size 12{L} {} is the shortening of the measured length of an object moving relative to the observer’s frame.
L=L01−v2c2.L=L01−v2c2. size 12{L - L rSub { size 8{0} } sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } {}
(7) If we measure the length of anything moving relative to our frame, we find its length LL size 12{L} {} to be smaller than the proper length L0L0 size 12{L rSub { size 8{0} } } {} that would be measured if the object were stationary. For example, in the muon’s reference frame, the distance between the points where it was produced and where it decayed is shorter. Those points are fixed relative to the Earth but moving relative to the muon. Clouds and other objects are also contracted along the direction of motion in the muon’s reference frame.
Suppose an astronaut, such as the twin discussed in Simultaneity and Time Dilation, travels so fast that γ=30.00γ=30.00 size 12{γ="30" "." "00"} {}. (a) She travels from the Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an Earth-bound observer. How far apart are the Earth and Alpha Centauri as measured by the astronaut? (b) In terms of cc size 12{c} {}, what is her velocity relative to the Earth? You may neglect the motion of the Earth relative to the Sun. (See Figure 3.)
Strategy
First note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance L0L0 size 12{L rSub { size 8{0} } } {}, because it is measured by an Earth-bound observer to whom both stars are (approximately) stationary. To the astronaut, the Earth and the Alpha Centauri are moving by at the same velocity, and so the distance between them is the contracted length LL size 12{L} {}. In part (b), we are given γγ size 12{γ} {}, and so we can find vv size 12{v} {} by rearranging the definition of γγ size 12{γ} {} to express vv size 12{v} {} in terms of cc size 12{c} {}.
Solution for (a)
- Identify the knowns. L0−4.300 lyL0−4.300 ly; γ=30.00γ=30.00
- Identify the unknown. LL size 12{L} {}
- Choose the appropriate equation. L=L0γL=L0γ size 12{L= { {L rSub { size 8{0} } } over {γ} } } {}
- Rearrange the equation to solve for the unknown.
L
=
L
0
γ
=
4.300 ly
30.00
=
0.1433 ly
L
=
L
0
γ
=
4.300 ly
30.00
=
0.1433 ly
alignl { stack {
size 12{L= { {L rSub { size 8{0} } } over {γ} } } {} #
= { {4 "." "300"" ly"} over {"30" "." "00"} } {} #
=0 "." "1433"" ly" {}
} } {}
(8)
Solution for (b)
- Identify the known. γ=30.00γ=30.00 size 12{γ="30" "." "00"} {}
- Identify the unknown. vv size 12{v} {} in terms of cc size 12{c} {}
- Choose the appropriate equation. γ=11−v2c2γ=11−v2c2 size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}
- Rearrange the equation to solve for the unknown.
γ
=
1
1
−
v
2
c
2
30
.
00
=
1
1
−
v
2
c
2
γ
=
1
1
−
v
2
c
2
30
.
00
=
1
1
−
v
2
c
2
alignl { stack {
size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {} #
"30" "." "00"= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } {}
} } {}
(9)Squaring both sides of the equation and rearranging terms gives
900
.
0
=
1
1
−
v
2
c
2
900
.
0
=
1
1
−
v
2
c
2
size 12{"900" "." 0= { {1} over {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } {}
(10)so that
1
−
v
2
c
2
=
1
900
.
0
1
−
v
2
c
2
=
1
900
.
0
size 12{1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } = { {1} over {"900" "." 0} } } {}
(11)and
v2c2=1−1900.0=0.99888....v2c2=1−1900.0=0.99888.... size 12{ { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } =1 - { {1} over {"900" "." 0} } =0 "." "99888" "." "." "." } {}
(12)Taking the square root, we find
vc=0.99944,vc=0.99944, size 12{ { {v} over {c} } =0 "." "99944"} {}
(13)which is rearranged to produce a value for the velocity
v=0.9994c.v=0.9994c. size 12{ ital "v="0 "." "9994"c} {}
(14)
Discussion
First, remember that you should not round off calculations until the final result is obtained, or you could get erroneous results. This is especially true for special relativity calculations, where the differences might only be revealed after several decimal places. The relativistic effect is large here (γ=30.00γ=30.00 size 12{ ital "γ=""30" "." "00"} {}), and we see that vv size 12{v} {} is approaching (not equaling) the speed of light. Since the distance as measured by the astronaut is so much smaller, the astronaut can travel it in much less time in her frame.
People could be sent very large distances (thousands or even millions of light years) and age only a few years on the way if they traveled at extremely high velocities. But, like emigrants of centuries past, they would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on the Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies than classical physics predicts would be needed to achieve such high velocities. This will be discussed in Relatavistic Energy.
Why don’t we notice length contraction in everyday life? The distance to the grocery shop does not seem to depend on whether we are moving or not. Examining the equation L=L01−v2c2L=L01−v2c2 size 12{L=L rSub { size 8{0} } sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } {}, we see that at low velocities (v<<cv<<c size 12{v"<<"c} {}) the lengths are nearly equal, the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle, like an electron, traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer. (See Figure 4.) As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3 km long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and the Earth are all moving by and are length contracted. The relativistic effect is so great than the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, since the beam does not have to be as precisely aimed to get down a short pipe as it would down one 3 km long. This, again, is an experimental verification of the Special Theory of Relativity.
