Addition of VelocitiesDoppler EffectLorentz InvarianceSpace-time Interval:PHY2061 Enriched Physics 2 Lecture Notes Relativity 3 Relativity 3 Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor. Addition of Velocities Now that we know that the Galilean transformation must be modified, it’s time to revisit the topic of adding velocities. Consider two inertial frames S and S’ with a relative velocity v. x y z S x'y' z'S' v ux′ udxdtuu vucu cxxxxx′=′′=′+>′= in frame S' in a Galilean transformation, which would imply if Consider the inverse Lorentz Transformation: xxvt yyzttvxcdx dx vdtdt dt v c dx=′+′=′z=′=′+′=′+′=′+′γγγγafchafch and Take differentials:,//22 D. Acosta Page 1 8/17/2004PHY2061 Enriched Physics 2 Lecture Notes Relativity 3 Divide one by the other: where we have divided by Note that udxdtdx vdtdt v c dxudxdtvvcdxdtdtdxdtuxxx==′+′′+′=′′++′′′′′=′//221 The velocity addition formulae are: uuvvu cuuvu cuuvu cxxxyyxzzx=′++′=′+′=′+′111222///γγejej Note that even though yy zz u u u uyy z=′=′≠′≠′ and , that and z Example: Consider a spacecraft that travels at 0.8c from Earth and that launches a projectile with a relative velocity of 0.8c. What is the velocity of the projectile from Earth? Galilean: !Lorentz: ucccucccxx=+ c=>=+=<08 08 16161080 9762...... If instead of a projectile we turned on a light beam, both observers on the spacecraft and on Earth would agree that the velocity of the light beam is c, as required by Einstein’s 2nd postulate. The addition of velocity formulae tell us that nothing can exceed the speed of light. D. Acosta Page 2 8/17/2004PHY2061 Enriched Physics 2 Lecture Notes Relativity 3 Doppler Effect The Doppler effect is a change in frequency of a traveling wave when the source is moving toward or away from a receiver, like the change in pitch of a car’s engine when it travels by you. We can derive the change in “pitch” for light using what we have learned in Special Relativity. Consider a light wave traveling along the x-axis. It emits n wave crests in a time T0 in the rest frame of the emitter. crest 1 crest n λ0 vcFrame S’ Frame S ReceiverSource here Length of wavetrain = For light, frequency of light in rest frameLn cTcTnfcfcnT00000000==⇒==⇒= = =λλλλThe frequency is n crests per time T0Now consider a receiver in a different inertial frame S. Suppose that the transmitter in frame S’ is moving toward the receiver at a velocity v. Let’s compute the frequency received given that the speed of light is always a constant for all frames. L n cT vTcvnT==−⇒=−λλ length of wavetrain in frame S Distance traveled by crest 1 minus distance source moves by the time of the last crest. Now from time dilation we know that TT=γ0 λγγγ=−⇒=−=−=cvnTfcncvTffvcfnT Since 000001afaf// D. Acosta Page 3 8/17/2004PHY2061 Enriched Physics 2 Lecture Notes Relativity 3 Now we substitute in for γ: fvcvcfvc vcvcf=−−=+−−111112200/////afaf So the Doppler shift equations are: ffvcvcffvcvc=+−=−+001111//// Source and receiver approaching Source and receiver receding Thus, when source and receiver approach each other, the frequency is shifted higher. We say that the light is blue-shifted. When source and receiver recede from each other, the frequency is shifted lower. We say that the light is red-shifted. Red light has a lower frequency than blue light. Modern electronics allow us to determine frequencies very accurately, so we can measure relative velocities accurately as well using this effect. Examples include Doppler weather radar, police radar, and even the expansion of the universe! Lorentz Invariance We have seen that some quantities change from one inertial frame to another (length, time, velocity, frequency). A quantity which does not change after a Lorentz transformation is said to be Lorentz Invariant. One special invariant is the Space-time Interval: ∆∆∆∆∆∆∆sct x y ztt txx xafafafafaf222222121=−++=−=− etc. This is the generalization of Cartesian distance for 4-dimensional space-time. The same value for ∆ is obtained for any inertial frame. So although length and time separately are not invariant from one frame to another, this particular combination is. s We can prove that this is true by applying the Lorentz transformation. For example, consider a subatomic particle which decays in a time τ in its rest frame. D. Acosta Page 4 8/17/2004PHY2061 Enriched Physics 2 Lecture Notes Relativity 3 In the rest frame S’: ttxxysc12 12100′=′=′=′=′==⇒=,, ττ"∆ Now make a Lorentz transformation to another frame S moving at velocity v: xxvtyyzzttvxcxvx xvtttsc v c vccvcvcsc=′+′′==′===′+′==⇒===⇒==−= −=−−⇒=γγγτ γτγτ γτγτ γ τ γτττafchafchch as in the rest frame000111221212222 222 222 2 2222222/,,///∆∆∆∆ Some terminology: ∆∆∆sss222000>⇒=⇒<⇒ time- like light - like space- like A frame exists where 2 events occur in one place, separated by time. 2 events are separated by the speed of light. No light signal can connect the 2 events. D. Acosta Page 5
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