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ModPhy1/Unit1/SpecialRelativity/RelativeView/Velocity/LorentzVelocities/ |
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Headlight Effect |
The
light radiated uniformly from a glowing object traveling close to the speed of
light looks considerably different in the rest frame. For one thing there is
shift in the spectrum of the light ahead of, behind, and to the side of the
object exactly as predicted by the relativistic transverse Doppler equation.
But there is also a focusing of the light forward exactly as predicted by
the relativistic velocity addition equations
The Headlight Effect Equation |
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q’ = angle light ray makes with x’-axis in system S’ q = angle light ray makes with x-axis in system S v = velocity of S’
along x-axis in system S |
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This
equation can be derived either from the Lorentz transformation equation (See Tipler
p. 55.) or from the above velocity transformation equation.
As
an example of the usage of this equation, consider the case where the emission
angle q’ = 90o. Then cos q’ = 0 and cos q = v/c. For an object
traveling 0.95 c, then q = 18.2 o. Since
half of light energy emitted from the object in its proper frame makes an angle
of 90 o or less, then half of the light as seen in the rest frame
makes an angle of 18.2 o or less. Therefore, the intensity of the
radiation from the object is significantly brighter in the forward direction
than to the side or backward.
Questions:
Q1. Headlight Effect
Which
of the following statements are true concerning the relativistic headlight
effect? (A) If an active galactic nucleus or a distant quasar emits two equally
intense jets of matter in opposite directions at relativistic speeds, only one
jet will be visible on a photograph. (B) A rotating neutron star sometimes
appears as a rapidly pulsating star (a pulsar) because of the relativistic
headlight effect. (C) If a passenger in a spaceship accelerating toward the
speed of light looks out the side window, he or she will see the stars moving
forward rather than backward and growing ever dimmer because most of their
energy is directed forward, away from the spaceship. (D) More than one of
these. (E) None of these.
Q2. Which of the following
statements is true concerning the relativistic headlight effect? (A) If an
active galactic nucleus or a distant quasar emits two equally intense jets of
matter in opposite directions at relativistic speeds, only one jet may be
visible on a photograph. (B) A rotating neutron star sometimes appears as a
rapidly pulsating star (a pulsar) because of the relativistic headlight effect.
(C) If a passenger in a spaceship accelerating toward the speed of light looks
out the side window, he or she will see the stars moving forward rather than
backward and growing ever dimmer because most of their energy is directed
forward, away from the spaceship. (D) More than one of these. (E) None of
these.
Problems:
P1. Photon Forward
A
beam of light is aimed at an angle of 45 o out the back window of a
spaceship traveling at 0.99 c. (So the angle between the beam and the front of
the spaceship is 135o.) (a) Use the headlight effect equation to
find the direction of the velocity of the beam of light relative to an observer
at rest? (b) Use the principle of relativity to find the magnitude of the
velocity of the beam of light in the rest frame. (c) Compare your answers to
those in part a of problem P4 on the bottom of page Lorentz Velocity
Transformation Equations.
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ModPhy1/Unit1/SpecialRelativity/RelativeView/Velocity/LorentzVelocities/