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Does anyone know how to show a projective transformation, i.e. a self-map of $\mathbb{RP}^n$, that takes

$S^{n-1}$={$x_0=0$}$/\sim$= $E^{n} /\sim$ (passing to the quotient in $\mathbb{RP}^n$) to itself, must be a Lorentz transformation, i.e. an element in Lorentz group $O(n,1)$?

I appreciate any hint or explanation.

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Such a map has to preserve the Hilbert metric on the open round ball and this metric is a constant multiple of the hyperbolic metric (in the Klein model). Lastly, use the hyperboloid model of the hyperbolic space.

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    Does this mean not any projective transformation preserving$S^{n-1}$ can give rise to a Lorentz transformation, only for those preserving Hilbert distances? – 2017-02-16
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    @Dai: They necessarily preserve the Hilbert metric, since projective maps of projective lines preserve the cross-ratio. – 2017-02-16
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    Oh I see Thanks! – 2017-02-16