Proving that the scalar product is invariant

[Someone1987]

Is there a general way of proving that the scalar product
x_ux^u = (x⁰)² - (x¹)² - (x²)² - (x³)²
is invariant under a Lorentz transformation that applies no matter the explicit form of the transformation.

 

[jdstokes]

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

 

[Fredrik]

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.