# For the future Cauchy development (or future domain of dependence), why is $D^+ (S) \subset \tilde{D}^+ (S)$?

Let $$S \subset \mathcal{M}$$ (where $$\mathcal{M}$$ is our manifold), the future Cauchy development of $$S$$ is defined as

$$D^+ (S) := \{p \in \mathcal{M} |\ \text{every past inextendible causal curve through} \ p \ \text{intersects} \ S \}$$

analogously, the same name is used for the following set

$$\tilde{D}^+ (S) := \{p \in \mathcal{M} |\ \text{every past inextendible timelike curve through} \ p \ \text{intersects} \ S \}$$

Both Minguzzi (2019) and Hawking & Ellis (1973) (in Remark 3.2 and Proposition 6.5.1 respectively) state that $$D^+ (S) \subset \tilde{D}^+ (S)$$. Why is this true? Both sources state it as a direct consequence from the former definitions with no further explanation, but I don't see it. A causal curve is either a null curve or a timelike curve. This means that for $$p \in D^+ (S)$$, there are more curves that pass through $$p$$ that intersect $$S$$ compared to the points in $$\tilde{D}^+ (S)$$, shouldn't this mean that $$\tilde{D}^+ (S) \subset D^+ (S)$$? After all, the timelike curves mentioned in the definition of $$\tilde{D}^+ (S)$$ are included in the definition of $$D^+ (S)$$.

• It all causal curves etc passing through $p$ intersect $S$, then, in particular, all timelike curves etc passing through $p$ intersect $S$. In other words, if $p\in D^+(S)$, then $p$ also belongs to the other set. That is just the thesis. Sep 1, 2021 at 15:17
• But doesn't it work the other way around too? If all timelike curves passing through $p$ intersect $S$ (thus $p \in \tilde{D}^+ (S)$), said curves are also causal by defintion, then $p$ belongs in $D^+ (S)$. I don't see where I'm wrong here, but I must be, otherwise both sets would be the same. Sep 1, 2021 at 15:22
• Consider an open spacelike disk $S$ in Minkowski spacetime. Then $\tilde{D}^+(S)$ includes all points in the future conical surface whose basis is $S$. Think of the tip of the cone in particular. These points do not belong to ${D}^+(S)$, so that the inclusion is strict in general. Sep 1, 2021 at 16:07

Suppose there exists a point $$p\in\mathcal{M}$$ for which every past inextendible timelike curve through $$p$$ intersects $$S$$, but there exist an (inextendible) null curve through $$p$$ that does not intersect $$S$$. Then $$p \in \tilde{D}^{+}(S)$$, but $$p \notin D^{+}(S)$$.
• I see. However, let's put it the other way around. Let $p \in \tilde{D}^+ (S) \Rightarrow$ every timelike curve that passes through $p$ intersects $S$. In particular, since these curves are timelike, they're also causal, doesn't this mean that $p \in D^+ (S)$? Sep 1, 2021 at 15:26
• No, because of the potential case above. There could be causal curves through $p$ that do not intersect $S$. Sep 1, 2021 at 15:28
• Indeed. I was ignoring the requirement that $\underline{\text{every}}$ causal curve through $p$ must intersect $S$ in the definition of $D^+ (S)$. Thank you for your explanation. Sep 1, 2021 at 15:37