I believe this is a causality condition known as stably causal. You can read about this in Wald's "General Relativity" Chapter 8, in particular his Theorem 8.2.2.

Added

A spacetime $(M,g)$ is stably causal if there exists a continuous nowhere vanishing timelike vector $t$ such that the spacetime $(M,\tilde g)$ with $$\tilde g = g - t^\flat \otimes t^\flat$$, where $t^\flat$ is the dual one-form to $t$ relative to $g$, has no closed timelike curves.

Wald's Theorem 8.2.2 says that this is equivalent to the existence of a global time function on $M$.

Theorem 8.2.2 A spacetime $(M,g)$ is stably causal if and only if there is a differentiable function $f$ on $M$ such that its gradient is a past directed timelike vector field.

There is a whole hierarchy of causality conditions in GR. Global hyperbolicity implies stable causality and this in turn implies strong causality. Global hyperbolicity might be too strong.

Edit

The answer below is not correct. Upon further reflection, I believe that the correct causality condition is indeed global hyperbolicity and not the weaker stable causality.

I believe this is a causality condition known as stably causal. You can read about this in Wald's "General Relativity" Chapter 8, in particular his Theorem 8.2.2.

A spacetime $(M,g)$ is stably causal if there exists a continuous nowhere vanishing timelike vector $t$ such that the spacetime $(M,\tilde g)$ with $$\tilde g = g - t^\flat \otimes t^\flat,$$ where $t^\flat$ is the dual one-form to $t$ relative to $g$, has no closed timelike curves.

Wald's Theorem 8.2.2 says that this is equivalent to the existence of a global time function on $M$.

Theorem 8.2.2 A spacetime $(M,g)$ is stably causal if and only if there is a differentiable function $f$ on $M$ such that its gradient is a past directed timelike vector field.

There is a whole hierarchy of causality conditions in GR. Global hyperbolicity implies stable causality and this in turn implies strong causality. Global hyperbolicity might be too strong.

I believe this is a causality condition known as stably causal. You can read about this in Wald's "General Relativity" Chapter 8, in particular his Theorem 8.2.2.

I believe this is a causality condition known as stably causal. You can read about this in Wald's "General Relativity" Chapter 8, in particular his Theorem 8.2.2.

Added

A spacetime $(M,g)$ is stably causal if there exists a continuous nowhere vanishing timelike vector $t$ such that the spacetime $(M,\tilde g)$ with $$\tilde g = g - t^\flat \otimes t^\flat$$, where $t^\flat$ is the dual one-form to $t$ relative to $g$, has no closed timelike curves.

Wald's Theorem 8.2.2 says that this is equivalent to the existence of a global time function on $M$.

Theorem 8.2.2 A spacetime $(M,g)$ is stably causal if and only if there is a differentiable function $f$ on $M$ such that its gradient is a past directed timelike vector field.

There is a whole hierarchy of causality conditions in GR. Global hyperbolicity implies stable causality and this in turn implies strong causality. Global hyperbolicity might be too strong.