Let $M$ denote a manifold admitting a Lorentzian metric $g_{ab}$. Essentially, I would like to know the "minimum domain" on which the exponential map is defined at $p\in M$. To make this concrete, let $(M,g_{ab})$ be time-orientable and let $T\in T_pM$ denote a future-directed timelike vector at $p\in M$. Let $B_T(r)$ denote the open (Euclidean) ball of radius $r$ about the origin of $T_pM$ defined with respect to $T$. Of course, there always exists $r>0$, such that the exponential map on this ball $\text{exp}_p:B_T(r)\to M$ is well defined.

**Question:** Are there known lower bounds---depending possibly, e.g., on the curvature in a neighborhood of $p$---for the minimum size of $r$? i.e., loosely speaking, for some "observer" at $p$ associated with $T$, what is the minimum (ball-shaped) domain for which the exponential map is well-defined?

*Clarification: As Igor Khavkine mentions in his answer, there exists a bound on the "injectivity radius" at $p\in M$ expressed in terms of any $r$ where the exponential map is assumed to be defined. However, I'm interested in a lower bound on $r$ itself.*

I'm most interested in the Lorentzian case, but pointers to bounds for ordinary Riemannian manifolds (with no need for the auxiliary vector $T$) may also be enlightening.