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I would like to show that Lorentz transforms with [itex]L^{4}_4 \ge 1[/itex] preserve the sign of the time-component [itex]\Delta x^4[/itex] of the difference of two timelike separated events.

Since in the transformation law [itex]\Delta x^{4'}=L^{4'}_{\nu}\Delta x^{\nu}[/itex] we have summands whose influence on the sign must be ruled out, I cannot see how to proceed.

First I thought that for two timelike events their difference should also be timelike, so that the proof boils down to showing that the given Lorentz transforms keep the timelike vector in the upper or lower component of the cone, where it initially was. But now it seems that whether the difference is timelike or not depends on the sign of the scalar product of the vectors, which is something we don't know. So now I'm somewhat confused.

Could it actually be (somehow) enough to check it for timelike basis vector?

Any hints? Thanks

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# Before and after

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