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UF PHY 3101 - Relativity3

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PHY3101 Modern Physics Lecture Notes Relativity 3 D. Acosta Page 1 8/25/00 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. udxdtu u vu c u cxx xx x′=′′=′+>′= in frame S' in a Galilean transformation, which would imply if Consider the inverse Lorentz Transformation: x x vt y y z zt t vx cdx dx vdtdt dt v c dx=′+′=′=′=′+′=′+′= ′ + ′γγγγafchafc h and Take differentials:,//22 x y z S x' y' z' S' v ux′PHY3101 Modern Physics Lecture Notes Relativity 3 D. Acosta Page 2 8/25/00 Divide one by the other: where we have divided by Note that udxdtdx vdtdt v c dxudxdtvv cdxdtdtdxdtuxxx= =′+′′ + ′=′′++′′′′′=′//221 The velocity addition formulae are: Note that even though y y z z u u u uy y z z= ′ = ′ ≠′≠′ and , that and 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: uccccucc cxx=+=>=+= <080816161080 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. uu vvu cuuvu cuuvu cxxxyyxzzx=′++′=′+′=′+′111222///γγe je jPHY3101 Modern Physics Lecture Notes Relativity 3 D. Acosta Page 3 8/25/00 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. Now 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 vTc vnT==−⇒ =−λλ length of wavetrain in frame S Now from time dilation we know that TT=γ 0 λ γγγ=−⇒ =−=−=cvnTfcnc v Tffv cf n T Since 0000 01afaf// crest 1 crest n λ0 Length of wavetrain = For light, frequency of light in rest frameLncTcTnf cfc nT0 0 00000 0==⇒ ==⇒ = = =λλλλFrame S’ c The frequency is n crests per time T0 Distance traveled by crest 1 minus distance source moves by the time of the last crest. Source here Receiver Frame S vPHY3101 Modern Physics Lecture Notes Relativity 3 D. Acosta Page 4 8/25/00 Now we substitute in for γ: fv cvcfv c v cvcf=−−=+ −−111 112 20 0/// //afaf So the Doppler shift equations are: 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: ∆ ∆ ∆ ∆ ∆∆∆s c t x y zt t tx x xa fafafafaf2 2 2222 121= − + += −= − etc. This is the generalization of Cartesian distance for 4-dimensional space-time. The same value for ∆s is obtained for any inertial frame. So although length and time separately are not invariant from one frame to another, this particular combination is. 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. f fv cv cf fv cv c=+−=−+001111//// Source and receiver approaching Source and receiver recedingPHY3101 Modern Physics Lecture Notes Relativity 3 D. Acosta Page 5 8/25/00 In the rest frame S’: t t x x ysc1 2 1 2 10 0′=′=′=′=′= =⇒=, , ττL∆ Now make a Lorentz transformation to another frame S moving at velocity v: x x vt y y z zt t vx cx v x x vt t ts c v c v ccv cv csc=′+′′==′=== ′ + ′= = ⇒ == = ⇒ == − = −=−−⇒=γγγ τ γ τγτ γτγ τ γ τ γ τττafc ha fc hc h as in the rest frame00011122 12 122 2 2 2 2 2 2 2 2 2 22 22 22 2/,,///∆∆∆∆ 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


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