# 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.