# 6: Waves

- Page ID
- 3458

- 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.

- 6.4: 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.

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

### Contributors

- Benjamin Crowell (Fullerton College). Special Relativity is copyrighted with a CC-BY-SA license.