Instead, the covariant observation is that null geodesics always reach the asymptotic boundary of AdS, while timelike ones never do. Likewise, by definition any curve (of any signature) that reaches the asymptotic boundary does so in infinite proper time/distance/affine parameter. For instance, you say that null geodesics reach infinity in finite coordinate time, but for a given null geodesic I could of course choose a time coordinate that diverges when the null geodesic reaches infinity, making that geodesic reach the boundary in infinite coordinate time.
coordinate-dependent) quantity, while the latter is physical but uninteresting in the context of the asymptotic boundary.
proper time/affine parameter is a bit of a red herring, since the former is a purely gauge (i.e. Clearly these intuitions conflict with each other. But on the other hand, the fact that null geodesics can reach spatial infinity in finite coordinate time suggests that it's "easier" to reach spatial infinity in AdS space than in Minkowski space. The fact that all timelike geodesics are bounded away from spatial infinity suggests to me that it's "harder" to reach spatial infinity in AdS space than in Minkowski space.
My intuition is that an accelerated timelike trajectory could reach spatial infinity, but only in both infinite proper time and infinite coordinate time.īut I'm not confident that I'm correct, because I don't have good intuition for the boundary behavior of AdS space. Intuitively, it seems to me that the case of accelerated timelike trajectories is intermediate between the case of timelike geodesics and the case of null geodesics, because an accelerated timelike trajectory can "escape from the origin better than" any timelike geodesic by accelerating against the AdS gravitational attraction, but it can't "escape from the origin as well as" a null geodesic because it can never hit the speed of light. non-geodesic) timelike trajectories? Can they reach spatial infinity in finite coordinate time? Finite proper time? In not, can they come arbitrarily close, unlike timelike geodesics? I understand why in anti-de Sitter (AdS) spacetime, null geodesics can reach spatial infinity in finite coordinate time, while timelike geodesics cannot reach spatial infinity at all, not even in infinite coordinate time (in the sense that any timelike geodesic's radius is bounded).
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