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Δ t 0 = 2 D c . 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.

Proper Time

Proper time Δ t 0 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

Δ t = 2 s c . size 12{Δt= { {2s} over {c} } } {}

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

L = v Δ t 2 . size 12{L= { {vΔt} over {2} } } {}

Using the Pythagorean Theorem, the distance s size 12{s} {} is found to be

s = D 2 + v Δ t 2 2 . size 12{s= sqrt {D rSup { size 8{2} } + left ( { {vΔt} over {2} } right ) rSup { size 8{2} } } } {}

Substituting s size 12{s} {} into the expression for the time interval Δ t size 12{Δt} {} gives

Δ t = 2 s c = 2 D 2 + v Δ t 2 2 c . 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

( Δ t ) 2 = 4 D 2 + v 2 ( Δ t ) 2 4 c 2 = 4 D 2 c 2 + v 2 c 2 ( Δ t ) 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 Δ t 0 size 12{Δt rSub { size 8{0} } } {} , we get ( Δ t 0 ) 2 = 4 D 2 c 2 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,

( Δ t ) 2 = ( Δ t 0 ) 2 + v 2 c 2 ( Δ t ) 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 Δ t size 12{Δt} {} :

( Δ t ) 2 1 v 2 c 2 = ( Δ t 0 ) 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,

( Δ t ) 2 = ( Δ t 0 ) 2 1 v 2 c 2 . 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:

Δ t = Δ t 0 1 v 2 c 2 = γ Δ t 0 , 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

γ = 1 1 v 2 c 2 . size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}

This equation for Δ t 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 Δ t 0 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 Δ t size 12{Δt} {} , the effect is called time dilation. The Earth-bound observer sees time dilate (get longer) for a system moving relative to the 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 ( v << c size 12{v"<<"c} {} ), then v 2 c 2 is extremely small, and the elapsed times Δ t and Δ t 0 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 Δ t = γ Δ t 0 also implies that relative velocity cannot exceed the speed of light. As v size 12{v} {} approaches c size 12{c} {} , Δ t size 12{Δt} {} approaches infinity. This would imply that time in the astronaut’s frame stops at the speed of light. If v size 12{v} {} exceeded c size 12{c} {} , then we would be taking the square root of a negative number, producing an imaginary value for Δ t size 12{Δt} {} .

There is considerable experimental evidence that the equation Δ t = γ Δ t 0 is correct. One example is found in cosmic ray particles that continuously rain down on the 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 . 52 μ s size 12{1 "." "52"` "μs"} {} when it is at rest relative to the observer who measures the half-life. This is the proper time Δ t 0 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 ( Δ t size 12{Δt} {} ) varies with velocity exactly as predicted by the equation Δ t = γ Δ t 0 size 12{Δt=γΔt rSub { size 8{0} } } {} . The faster the muon moves, the longer it lives. We on the 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.

Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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