Time Dilation​
​

Jesse Heidhorn

Introduction

In everyday life, velocities of objects add like vectors. This means that the velocity of an object that you observe depends not only on the object’s motion, but on your own motion as well.

Vector Addition

◼
  • V
    AC
    =
    V
    AB
    +
    V
    BC
    • Graphics[Arrow[{{0,0},{2,0}},{0,1}]]​​Style["+",40,Bold]​​Graphics[Arrow[{{2,0},{0,0}},{0,1}]]​​Style["=",40,Bold]​​Graphics[Arrow[{{2,0},{6,0}}]]
    +
    =
    If you are sitting in a moving car and then throw a football straight up, you will see it as traveling this path:
    ListLinePlot[Table[{6.2,n},{n,0,5}],PlotRange{{0,6.4},{0,5.2}}]​​
    0
    1
    2
    3
    4
    5
    6
    0
    1
    2
    3
    4
    5
    But a person standing on the sidewalk will see the ball travel a path like this:
    Plot[-12
    2
    (x-3)
    +5,{x,0,6.5}]
    The ball gets velocity added in the horizontal direction for the sidewalk observer, making the overall velocity of the ball greater.

    The Speed of Light

    Light Clock in a Car

    Even if our light beam is in a car, it will have the same velocity. However, we now notice that the sidewalk observer sees it traveling a greater distance.
    Since Distance = (Rate) * (Time), and rate is always the same, then for a larger distance, time must be larger as well. This is time dilation.
    Graphics[​​{EdgeForm[Thick],White,Triangle[{{0,0},{1,1},{2,0}}],​​Style[Line[{{1,0},{1,1}}],Thick,Red],​​Style[Line[{{-.2,0},{.2,0}}],Thickness[.035],Black],​​Style[Line[{{.8,1},{1.2,1}}],Thickness[.035],Black],​​Style[Line[{{1.8,0},{2.2,0}}],Thickness[.035],Black],​​Rotate[Text[Style["L",15,Bold,Blue],{.95,.4}],90Degree],​​Rotate[Text[Style["D",15,Bold,Blue],{.4,.5}],48Degree],​​Rotate[Text[Style["D",15,Bold,Blue],{1.6,.5}],-48Degree],​​Text[Style["(velocity of car)*(t')",15,Bold,Blue],{1,-.1}],​​Text[Style["
    2
    D
    =
    2
    L
    +(1/4)​(v*t'
    2
    )
    ",20,Bold,Black],{1,-.5}]}]
    ​
    In one up-and-down bounce, the observer in the car sees light traveling a path 2*L, but an observer on the sidewalk sees it traveling a path 2*D.
    ​​Style["1. 2(L)=(c)*(t)",Bold,20]​​Style["2. 2(D)=(c)*(t')",Bold,20]​​Style"3. D=
    2
    L
    +
    1
    4
    2
    v
    2
    (
    ′
    t
    )
    ",Bold,20​​Style["4. Plug in and solve:",Bold,20]​​Style"t'=​
    1
    1-
    2
    v
    2
    c
    ​t ,  γ=
    1
    1-
    2
    v
    2
    c
    ",Bold,20
    We see how the blue light traces out a longer path but is traveling with the same speed, so it should take the light in the blue one longer to bounce up and down once. One “tick” is larger in that frame.
    FURTHER EXPLORATIONS
    ​
    AUTHORSHIP
    Jesse Heidhorn