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Proving that the scalar product is invariant

  1. Aug 29, 2008 #1
    Is there a general way of proving that the scalar product
    xuxu = (x0)2 - (x1)2 - (x2)2 - (x3)2
    is invariant under a Lorentz transformation that applies no matter the explicit form of the transformation.
  2. jcsd
  3. Aug 29, 2008 #2
    First derive the form of a Lorentz transform in an arbitrary direction.
    Then check that the scalar product is invariant.
  4. Aug 30, 2008 #3


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    You could take the requirement that this "scalar product" is invariant (and the requirement that a Lorentz transformation is a linear transformation) as the definition of a Lorentz transformation. If you do, you don't have to prove that it's invariant.

    An alternative is to define a (homogeneous) Lorentz transformation as a linear map [itex]x\mapsto\Lambda x:\mathbb R^4\rightarrow\mathbb R^4[/itex] such that [itex]\Lambda^T\eta\Lambda=\eta[/itex] (where ^T is the transpose of the 4x4 matrix). The "scalar product" <x,y> of x and y is then defined by [itex]\langle x,y\rangle=x^T\eta y[/itex]. If you know anything about matrices you should find it easy to prove that [itex]\langle\Lambda x,\Lambda y\rangle=\langle x,y\rangle[/itex] when [itex]\Lambda[/itex] is a Lorentz transformation.
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