How can I convince myself of the following statement:(adsbygoogle = window.adsbygoogle || []).push({});

If x^{2}<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}_{β}x^{β}=Λ^{µ}_{0}x^{0}

=(Λ^{0}_{0}, Λ^{1}_{0}, Λ^{2}_{0}, Λ^{3}_{0}).

By definition, if it is anorthochronousLT, we must have Λ^{0}_{0}≥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?

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# Orthochronous Lorentz Transformations

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