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# 6: Waves

• 6.1: Frequency
• 6.2: Phase
We deﬁned a (Lorentz) invariant as a quantity that was unchanged under rotations and Lorentz boosts. A uniform scaling of the coordinates (t,x,y,z)→(kt,kx,ky,kz) , which is analogous to a change of units,1 is all right as long as k is nonzero. A quantity that stays the same under any diﬀeomorphism is called a scalar. Since a Lorentz transformation is a diﬀeomorphism, every scalar is a Lorentz invariant. Not every Lorentz invariant is a scalar.
• 6.3: The Frequency-Wavenumber Covector
• 6.4: Duality
To generalize this to 3+1 dimensions, we need to use the metric — a piece of machinery that we have never had to employ since the beginning of the chapter.
• 6.5: The Doppler Shift and Aberration
We generalize our previous discussion of the Doppler shift of light to 3+1 dimensions. Imagine that rain is falling vertically while you drive in a convertible with the top down. To you, the raindrops appear to be moving at some nonzero angle relative to vertical. This is referred to as aberration.
• 6.6: Phase and Group Velocity
• 6.7: Abstract Index Notation
• 6.E: Waves (Exercises)

Thumbnail: Two-dimensional representation of gravitational waves generated by two neutron stars orbiting each other. (Public Domain; NASA).