# On “the geometry of free fall and light propagation” paper by Ehlers

In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), I reach to an axiom which says:

There exists a collection of triplets $(U,P,P^{\prime})$ where $U\subset M$, $P, P^{\prime}\in \mathscr{P}$ such that the system of maps $x_{PP^\prime}|_U$ is a smooth atlas for $M$.

In the axiom above $\mathscr{P}$ is a subset of $M$ called whose members are world lines of particles, and $x_{PP^\prime}:M\to \mathbb{R}^4$ is a map that sends every $e\in M$ to $(u,v,u^{\prime},v^{\prime})\in\mathbb{R}^4$ which is called radar coordinate.(following figure)

Afterwards, it is concluded according to the axiom that the dimension of the tangent space of $M$ at a point equals the dimension of $M$.

Question:

I know the mathematical proof of $\dim T_p M=\dim M$, but I cannot understand how one concludes it from that axiom. Can anyone help me?

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These axioms make M into a diff. manifold. And that is true for every manifold. –  MBN Aug 19 '13 at 12:45