Confusion on theorem 8.1.2 and corollary in Wald's GR book In Wald's GR book theorem 8.1.2 says:

Let $(M,g_{ab})$ be an arbitray spacetime, and let $p \in M$. Then
there exists a convex normal neighborhood of $p$, i.e., an open set
$U$ such that for all $q,r \in U$ there exists a unique geodesic
$\gamma$ connecting $q$ and $r$ and staying entirely in $U$.
Furthermore, for any such $U$, $I^+(p)|_U$ consists of all points
reached by future directed timelike geodesic starting from $p$ and
contained within $U$, where $I^+(p)|_U$ denotes the chronological
future of $p$ in the spacetime $(U,g_{ab})$. In addition, $\dot I^+(p)|_U$ is generated by the future directed null geodesics in $U$
emanating from $p$.

In the next paragraph, it says if $q \in J^+(p)$, we can connect $p,q$ via causal curve $\lambda$  and cover this $\lambda$ by finitely many $U$'s in the theorem ($\lambda$ failed to be null geodesic in any such $U$), and we can deform each piece such that $\lambda$ becomes a timelike curve.
the corollary is:

If $q \in J^+(p)-I^+(p)$, then any causal curve connecting $p$ to $q$
must be a null geodesic.

My questions are:

*

*How to deform this $\lambda$ to a timelike curve by theorem? Why is the condition "$λ$ failed to be null geodesic in any such $U$" assumed? Is it a piecewise differentiable one after deformation?

*How to conclude the corollary?

 A: *

*$\lambda$ is a future directed timelike curve when the timelike vector field chosen to time-orient the manifold $M$ does not become 0 at any point on the curve:

\begin{equation}
t^{a}(s) \neq 0 \, \, , \quad \forall \, s \in \lambda
\end{equation}
This in general isn't true for every point on a causal curve, but we want all convex normal neighbourhoods between $p$ and $q$ to contain such points so that we deform $\lambda$ to include them.
As per the theorem, each convex neighbourhood contains all points on $I^{+}(p) |_{U}$ which can be reached by a future timelike geodesic starting from point $p$. Since there is a unique geodesic within each neighbourhood that stays entirely within it and it is not null, the unique geodesic for each neighbourhood $U_{i}$ that has a set of starting/ending points $(p_{i},q_{i})$ is going to be timelike and containg points from $I^{+}(p_{i}) |_{U_{i}}$. That means one can change $\lambda$ within each neighbourhood while staying within it so that $t^{a}$ never becomes 0, making the new curve future oriented timelike.
It is now clear why the initial assumption is made. If $\lambda$ contained some neighbourhood $\mathcal{U}$ where the unique geodesic that stays entirely within it is null, then you cannot always connect its starting/ending points $(\mathcal{P}, \mathcal{Q})$ with points entirely from $I^{+}(\mathcal{P}) |_{\mathcal{U}}$ without treading outside the neighbourhood. By extension, you cannot deform $\lambda$ to become timelike as a whole. Or to put it more rigorously, you could deform $\lambda$ for all points in the chronological past of $\mathcal{P}$ into a timelike curve $\ell(p)$ and all points of its chronological future into $\overline{\ell(p)}$, but $q \notin \ell(p)$ while $q \in \overline{\ell(p)}$. Hence the resulting deformed curve would not be closed.
Since the deformation is smooth, the curve is smooth itself, not piecewise differentiable.


*Again by virtue of the theorem, each null geodesic starting from a point $p_{i}$ at some neighbourhood will generate the boundary $\dot{I}^{+}(p_{i}) |_{U_{i}}$. By extension, if a point $s_{i}$ within this neighbourhood belongs to $\dot{I}^{+}(p_{i}) |_{U_{i}}$, that means it will be part of a null geodesic. The connection of each local null geodesic will complete a total null geodesic which will end up at the boundary $\dot{I}^{+}(p)$ for the complete curve. That boundary will be located on the lightcone that separates $I^{+}(p)$ from the rest of the causal future of $p$.

Now we suppose that $q \in J^{+}(p) - I^{+}(p)$. That means $q$ is part of a set that is generated by removing all points that are not causally accessible from the causal curve $\lambda$, therefore $q$ has to lie somewhere on the lightcone. Within this point's neighbourhood, $q$ not only belongs, but it is the ending point of a local null geodesic, hence the unique geodesic of that neighbourhood is null and not timelike. Using the conclusion from the previous paragraph, the condition that $\lambda$ is a closed curve, and the fact that $\lambda$ in this case cannot be deformed into a timelike curve, the total causal curve has to be a null geodesic.
