Strong causality in Choquet-Bruhat's "GR and the Einstein Equations"

Question

I'm having trouble in understanding Choquet-Bruhat's definition of a strongly causal spacetime ("GR and the Einstein Equations", OUP, sec. XII.10). Here she defines a strongly causal spacetime as a time-oriented Lorentz manifold $(M,g)$ such that

for any $x\in M$ and any neighbourhood $\Omega$ of $x$ there is a neighbourhood $U\subseteq \Omega$ such that $I_{x}^{+}\cap U$ is connected

Here $I_{x}^{+}$ is as usual the chronological future of $x$.

Now, aside from the fact that $U$ could be interpreted either as a neighbourhood of $x$ or as a generic neighbourhood, i.e. a generic open set, not necessarily containing $x$ (I'm leaning towards the first possibility, but different definitions of the strong causality condition make me wonder whether this is in fact so), it seems to me that even the Minkowski torus has this property, and as a Minkowski torus possesses closed causal curves, this cannot be possible. By Minkowski torus I mean the set $[-1,+1]\times [-1,+1]$ with metric $g=-dt^2+dx^2$ quotiented on its sides as in the usual construction for the one-torus, and orientation given by $\partial/\partial t$.

The proof of the validity of the property for the torus goes as follows. Consider the point $p=(0,0)$ and any other point $q=(t_{q},x_{q})$ with $|x_{q}|<1$. Construct a piecewise smooth curve by joining the timelike future-directed geodesic that goes from $p$ to $q'=(1,x_{q})$ and the timelike future-directed geodesic that goes from $q''=(-1,x_{q})$ to $q$. Modulo identifications that come from the quotient, $q'=q''$, and the curve is well defined and timelike. As for $q=(t_{q},±1)$, construct a piecewise-smooth curve by first going from $p$ to the side $t=1$ by means of a timelike future-directed geodesic, then going from the $t=-1$ side to the $x=1$ (or $x=-1$) side again through a timelike future-directed geodesic, then, if needed, going vertically up the $x=1$ (or $x=-1$) side. It is easy to see that by adjusting the inclination of the geodesics and modulo the identifications given by the quotient, any point on the $x=\pm 1$ side can be reached by means of a timelike future-directed piecewise smooth curve. This shows that $I_{p}^{+}$ is equal to the whole torus: in the Minkowski torus, every point is in the chronological future of $p$ (of course one can go from $p$ to $p$ too, by following a vertical future-directed closed geodesic). Hence, as $a\,\partial /\partial t + b\, \partial/\partial x$ with $a$ and $b$ arbitrary constants is Killing for the metric, the Minkowski torus is $\Bbb{R}^{2}$-homogeneous and the same must be true for any $p$ in the torus. It follows that $I_{p}^{+}\cap V=V$ for any $p$ and any subset $V$ in the torus, so that for any $p$ and any $\Omega$ the connected component of $\Omega$ containing $x$ is the connected neighbourhood we are looking for. I can't see why this proof should fail, but as I'm not an expert in the field there could well be something I'm missing.

I can't understand what is going wrong. I even though that there could be an error in the book (as I found many before). Elsewhere, I found definitions of strongly causal spacetimes in very different terms and I understood most of them (e.g. Penrose's definition in terms of causally convex neighbourhoods). Does anybody know how Choquet-Bruhat's definition is equivalent to the more common ones?

Answer 1

The definition I read in the book is: A spacetime is called strongly causal if for each point $x$ and each neighbourhood $\Omega$ of $x$ there is a neighbourhood $U_x\subset \Omega$ such that $I^+(x)\cap U_x$ is a connected subset, i.e. $I^+(x)\cap U=I^+_{x,U}$.

Anyway, this is not the definition of strong causality. The definition of strong causality is: for each point $x$ and each neighbourhood $\Omega$ of $x$ there is a neighbourhood $U_x\subset \Omega$ such that $\gamma^{-1}(U_x)$ is connected (possibly empty) where $\gamma$ is any causal curve. So it is related to connectedness but of a different object.

Also the equation she writes at the end of the definition is related to the future distinction property rather than to strong causality, where $I^+_{x,U}$ has to be understood as the chronological future of $x$ in the spacetime $(U,g\vert_U)$.