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Common images suggest that Lorentz boosts can be interpreted as the coordinate transformation between two observers that have chosen the same basis and origin. However, I have read that one must be careful when talking about parallel axes: As far as I understand, the point is that given a Lorentz boost, the coordinate axes of one observer are not viewed as orthogonal from another observer. So does it even make sense to say that two observers have chosen the same basis?

Filippo
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1 Answers1

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I think the concern about "parallel axes" (like $x$ and $x'$) is that:

  • while these "spatial [3-]vectors" are both "colloquially parallel [in space]" in the sense of pointing in the "directions of increasing x and x' (keeping y,y' and z,z' unchanged)",
  • the corresponding spatial 4-vectors---call them $\tilde X$ and $\tilde X'$---(orthogonal to the inertial-observer 4-velocities $\tilde T$ and $\tilde T'$) are not-"parallel" in spacetime (just like, in an ordinary rotation of the $xy$-plane, the $y$- and $y'$-axes are not parallel)

I think the better spacetime-viewpoint way to describe the typical situation in a relativity boost problem (in (1+1)-spacetime) is to say that

  • "the relative motion is coplanar with the inertial observer worldlines (with their y- and z-axes orthogonal to this plane of motion)"
  • and then the "projection of $\tilde X'$ onto $\tilde X$ in the $TX$-plane"
    is equal to the "projection of $\tilde X$ onto $\tilde X'$ in the $T'X'$-plane"
robphy
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