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Let $C$ be the open upper light cone in ${ℝ}^{1+n}$ with respect to the Lorentz product. The connected linear Lorentz group ${\mathrm{SO}}_{ℝ}{(1,n)}^0$ acts on $C$ and therefore diagonally on the $N$-fold product $T^N$ where $T={ℝ}^{1+n}+iC\subset {ℂ}^{1+n}.$ We prove that the extended future tube ${\mathrm{SO}}_{ℂ}(1,n)·T^N$ is a domain of holomorphy.