Determining 4-vector character of a 4-tuple

  1. Suppose you're given a 4 tuple and told that its scalar product with any 4-vector is a lorentz scalar. How do I show that this implies the 4-tuple is a 4-vector?

    Thanks
     
  2. jcsd
  3. Fredrik

    Fredrik 10,470
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    It can't be done unless you're told that its scalar product with any 4-vector is a scalar.

    If the given 4-tuple is x and the (arbitrary) 4-vector is y,

    [tex]x_\mu y^\mu=x'_\mu y'^\nu=x'_\mu\Lambda^\mu{}_\nu y^\nu[/tex]

    [tex]x_\mu=\Lambda^\mu{}_\nu x'_\mu[/tex]

    Now do some raising and lowering of indices and apply a Lorentz transformation to solve for x', and you're done. This post should help with the notation.
     
  4. dextercioby

    dextercioby 12,314
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    Just apply the definitions on the <scalar> product. Denoting by F the matrix the 4-tuple (index down) uses to transform under a Lorentz group element, you'll end with a matrix equation

    [tex] \mathbb{F} \Lambda = \mbox{1}_{4\times 4} [/tex].

    Since [itex] \Lambda [/itex] is invertible, the conclusion follows easily.
     
    Last edited: Feb 18, 2010
  5. Fredrik

    Fredrik 10,470
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    Here's how I would do it in matrix notation:

    [tex]x^T\eta y=x'^T\eta y'=x'^T\eta\Lambda y[/tex]

    [tex]x^T\eta=x'^T\eta\Lambda[/tex]

    [tex]\eta x=\Lambda^T\eta x'[/tex]

    [tex]x'=\eta^{-1}(\Lambda^T)^{-1}\eta x=\eta^{-1}(\eta\Lambda\eta^{-1})\eta x=\Lambda x[/tex]

    The fact that [tex](\Lambda^T)^{-1}=\eta\Lambda\eta^{-1}[/tex] follows from the definition of a Lorentz transformation, [tex]\Lambda^T\eta\Lambda=\eta[/tex]. Just multiply both sides with [tex]\eta^{-1}[/tex] from the right.

    I suggest that you be a bit more careful with the terminology. A 4-tuple can't ever be 4-vector. In order to define a 4-vector you must specify a 4-tuple for each coordinate system. It's the assignment of 4-tuples to coordinate systems that defines a 4-vector, not a single 4-tuple. The assignment is of course usually done by specifying the 4-tuple that you want to associate with a specific coordinate system, and then explicitly stating that the 4-tuples associated with all the other coordinate systems are given by the tensor transformation rule.
     
    Last edited: Feb 18, 2010
  6. Yes, that was sloppy of me. The 4-tuple I had in mind only has to be a continuous function of the coordinate transformation. But I wasn't thinking of it being confined by the tensor transformation rule.
     
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