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By definition, a spacetime is geodesically complete if all inextendible curve are complete but is this equivalent to «if all geodesic of finite length has endpoints»?

My situation is: I have a set of null geodesics, they all have a past end point on the same surface, (they are converging because it’s a trapped surface) and thus encounter a caustic. Does geodesic completeness implies that they have a future endpoint?

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