Proving that the scalar product is invariant

1. Aug 29, 2008

Someone1987

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. Aug 29, 2008

jdstokes

First derive the form of a Lorentz transform in an arbitrary direction.
Then check that the scalar product is invariant.

3. Aug 30, 2008

Fredrik

Staff Emeritus
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 $x\mapsto\Lambda x:\mathbb R^4\rightarrow\mathbb R^4$ such that $\Lambda^T\eta\Lambda=\eta$ (where ^T is the transpose of the 4x4 matrix). The "scalar product" <x,y> of x and y is then defined by $\langle x,y\rangle=x^T\eta y$. If you know anything about matrices you should find it easy to prove that $\langle\Lambda x,\Lambda y\rangle=\langle x,y\rangle$ when $\Lambda$ is a Lorentz transformation.