To me it becomes a geometrical effect in both ways, as 'time' and as 'distance'. It's as if the arrow, although locally invariant as measured by you, is something that 'distort' in its relations to other frames mass and motion. The next question, if this was right, would be to ask where one can set a limit for calling something 'local', and there I provisionally expect that you will find it at Planck scale. Using that as a definition of a smallest 'locality' as defined by a clock and a ruler you then have to assume that all those Plank 'points' must find time dilations as well as Lorentz contractions relative each other.

But it's somehow geometrical, all of it seems to be so to me. Even the arrow, although the arrows 'origin' (as in what I call 'time') is to be found locally as I suspect, at Planck scale. It becomes a 'inflated' universe, defined from a very small invariant 'point' as I think of it.