Orthochronous Lorentz Transformations

[vertices]

How can I convince myself of the following statement:

If x²<0, there exists L orthochronous Lorentz tranformation such that:

Lx = -x

My concern is this:

If for example, we take x^µ=(1,0,0,0), then Lx in component form is:

Λ^µ_βx^β=Λ^µ₀x⁰
=(Λ⁰₀, Λ¹₀, Λ²₀, Λ³₀).

By definition, if it is an orthochronous LT, we must have Λ⁰₀≥1 so it is clear that the statement above is false for this example (as the RHS in the case of our example vector, can't be positive).

Can anyone think of an example x^µ that would actually satisfy the condition Lx = -x?

 

[Fredrik]

You're asking the first question in a way that strongly suggests that the statement is supposed to be true for any x with x²<0. (And the fact that you tried x=(1,0,0,0) must mean that you're using a -++++ metric). But Λ^µ_βx^β=(Λ⁰₀x⁰,0,0,0) proves that the statement is false.

Are you really looking for a specific choice of Λ and x such that Λx=-x? It looks impossible to me, but I haven't proved it. I suggest that you try playing around with the explict formula for Λ in 1+1 dimensions:

$$\Lambda=\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}$$

$$\gamma=\frac{1}{\sqrt{1-v^2}}$$

(This is in units such that c=1).

Hm...or maybe not. Maybe it won't work in 1+1 dimensions but will work in 2+1 and 3+1 because then you will have the opportunity to flip the sign of two spatial components with a rotation.