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How is the relativistic Doppler effect different from the classical Doppler effect?

The relativistic Doppler effect takes into account the special relativity concept of time dilation and also does not require a medium of propagation to be used as a point of reference (light does not require a medium for propagation).

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Section summary

  • With classical velocity addition, velocities add like regular numbers in one-dimensional motion: u=v+u size 12{ ital "u=v+u" rSup { size 8{'} } } {} , where v size 12{v} {} is the velocity between two observers, u size 12{u} {} is the velocity of an object relative to one observer, and u size 12{u rSup { size 8{'} } } {} is the velocity relative to the other observer.
  • Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:
    u= v+u 1 + v u c 2
  • An observer of electromagnetic radiation sees relativistic Doppler effects    if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation
    λ obs s 1 + u c 1 u c size 12{λ rSub { size 8{"obs"} } ital "=λ" rSub { size 8{s} } sqrt { { {1+ { {u} over {c} } } over {1 - { {u} over {c} } } } } } {}
    λ obs size 12{λ rSub { size 8{"obs"} } } {} is the observed wavelength, λ s size 12{λ rSub { size 8{s} } } {} is the source wavelength, and u size 12{u} {} is the relative velocity of the source to the observer.

Conceptual questions

Explain the meaning of the terms “red shift” and “blue shift” as they relate to the relativistic Doppler effect.

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What happens to the relativistic Doppler effect when relative velocity is zero? Is this the expected result?

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Is the relativistic Doppler effect consistent with the classical Doppler effect in the respect that λ obs size 12{λ rSub { size 8{"obs"} } } {} is larger for motion away?

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All galaxies farther away than about 50 × 10 6 ly size 12{"50" times "10" rSup { size 8{6} } `"ly"} {} exhibit a red shift in their emitted light that is proportional to distance, with those farther and farther away having progressively greater red shifts. What does this imply, assuming that the only source of red shift is relative motion? (Hint: At these large distances, it is space itself that is expanding, but the effect on light is the same.)

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Problems&Exercises

Suppose a spaceship heading straight towards the Earth at 0 . 750 c size 12{0 "." "750"c} {} can shoot a canister at 0 . 500 c size 12{0 "." "750"c} {} relative to the ship. (a) What is the velocity of the canister relative to the Earth, if it is shot directly at the Earth? (b) If it is shot directly away from the Earth?

(a) 0 . 909 c size 12{0 "." "909"c} {}

(b) 0 . 400 c size 12{0 "." "400"c} {}

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Repeat the previous problem with the ship heading directly away from the Earth.

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If a spaceship is approaching the Earth at 0.100 c and a message capsule is sent toward it at 0.100 c relative to the Earth, what is the speed of the capsule relative to the ship?

0 . 198 c size 12{0 "." "198"c} {}

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(a) Suppose the speed of light were only 3000 m/s . A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1000 m/s . What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

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If a galaxy moving away from the Earth has a speed of 1000 km/s size 12{1,"000"" m/s"} {} and emits 656 nm size 12{"656"" nm"} {} light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?

a) 658 nm size 12{"658""nm"} {}

b) red

c) v / c = 9 . 92 × 10 5 size 12{v/ ital "c="9 "." "92" times "10" rSup { size 8{ - 5} } } {} (negligible)

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A space probe speeding towards the nearest star moves at 0 . 250 c size 12{0 "." "250"c} {} and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on the Earth?

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If two spaceships are heading directly towards each other at 0 . 800 c size 12{0 "." "800"c} {} , at what speed must a canister be shot from the first ship to approach the other at 0 . 999 c size 12{0 "." "999"c} {} as seen by the second ship?

0 . 991 c size 12{0 "." "991"c} {}

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Two planets are on a collision course, heading directly towards each other at 0 . 250 c size 12{0 "." "250"c} {} . A spaceship sent from one planet approaches the second at 0 . 750 c size 12{0 "." "750"c} {} as seen by the second planet. What is the velocity of the ship relative to the first planet?

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When a missile is shot from one spaceship towards another, it leaves the first at 0 . 950 c size 12{0 "." "950"c} {} and approaches the other at 0 . 750 c size 12{0 "." "750"c} {} . What is the relative velocity of the two ships?

0 . 696 c size 12{0 "." "696"c} {}

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What is the relative velocity of two spaceships if one fires a missile at the other at 0.750 c and the other observes it to approach at 0.950 c ?

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Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?

0 . 01324 c size 12{0 "." "01324" c} {}

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A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning echo has a frequency 15.0 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the effect is small.

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Prove that for any relative velocity v size 12{v} {} between two observers, a beam of light sent from one to the other will approach at speed c size 12{c} {} (provided that v size 12{v} {} is less than c size 12{c} {} , of course).

u = c , so

u = v+u 1 + ( vu ′/ c 2 ) = v+c 1 + ( vc / c 2 ) = v+c 1 + ( v / c ) = c ( v+c ) c+v = c size 12{alignl { stack { ital "u=" { { ital "v+u'"} over {1+ \( ital "vu""'/"c rSup { size 8{2} } \) } } = { { ital "v+c"} over {1+ \( ital "vc"/c rSup { size 8{2} } \) } } = { { ital "v+c"} over {1+ \( v/c \) } } {} #{ {c \( ital "v+c" \) } over { ital "c+v"} } = {underline {c}} {} } } } {}

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Show that for any relative velocity v size 12{v} {} between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that v size 12{v} {} is less than c size 12{c} {} , of course).

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(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy 12 . 0 × 10 9 ly size 12{"12" "." 0 times "10" rSup { size 8{9} } " ly"} {} ly away is receding from us at 0. 0.900 c , at what velocity relative to us must we send an exploratory probe to approach the other galaxy at 0.990 c , as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)

a) 0 . 99947 c

b) 1 . 2064 × 10 11 y size 12{1 "." "2064" times "10" rSup { size 8{"11"} } " y"} {}

c) 1 . 2058 × 10 11 y size 12{1 "." "2058" times "10" rSup { size 8{"11"} } " y"} {} (all to sufficient digits to show effects)

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Practice Key Terms 3

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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