Let $\eta$ be a bi-linear form on $\mathbb{R}^4$ given by $$\eta\left(x,\,y\right)=x_0y_0-x_1y_1-x_2y_2-x_3y_3$$ The Poincare group is defined as the group of isometries of $\eta$, that is, the group of all maps $\Lambda:\mathbb{R}^4\to\mathbb{R}^4$ such that for all $(x,\,y)\in\left(\mathbb{R}^4\right)^2$, $$ \eta\left(\Lambda\left(x-y\right),\,\Lambda\left(x-y\right)\right)=\eta\left(x-y,\,x-y\right)$$

Then one usually says that these comprise affine transformations, for example, in the Wikipedia article on the Poincare group: "The Poincaré group itself is the minimal subgroup of the affine group".

My question is: is there a mathematical theorem that says, just given the data I specified so far (or more data?), that $\Lambda$ has to be affine (and if so, what is that theorem and its proof?), or is it part of the physical input which stipulates that lines must be sent to lines?

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    $\begingroup$ For the specific reformulation in terms of isometries and affine transformations, see Qmechanic's answer $\endgroup$ – ACuriousMind Dec 21 '16 at 1:43
  • $\begingroup$ @ACuriousMind, Thanks for the link, it helped me. I am still not sure why the Jacobian of an arbitrary transformation has to be also a Lorentz transformation (a condition which appears prominently in some of the answers). One explanation I could give myself is we assume that the transformation is real analytic, plug in an expansion into the isometry condition and then employ the constraint order by order, thereby getting the constraint on the first order, that is, on the Jacobian. But why should this transformation be real analytic? $\endgroup$ – PPR Dec 21 '16 at 9:11
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    $\begingroup$ You need to think more carefully, on a general background, what the metric acts on. It does not act on spacetime points, but on tangent vectors (although you may identify those in the case of Minkowski space), and tangent vectors transform by the Jacobian. $\endgroup$ – ACuriousMind Dec 21 '16 at 16:56