Why or why not?
This is closely related to another question I posited here, Does it make sense to say that objects moving at relativistic velocities appear in space-time cross-section?
What I mean is the following: Saying that moving objects appear in space-time cross-section along a space-like hypersurface (the observed 3D space view at a given moment) in the geometry of SR implies that they must undergo length contraction.
Motivation: I used a similar statement in a couple of videos I made on length contraction and the length contraction paradox and got mixed responses. My intent was in part an unorthodox approach that highlights the strong connection between length contraction and relativity of simultaneity beyond the common examples. Although the videos use strictly standard special relativity, somebody commented to me that such statements sound unprofessional and likely leave the impression of crackpottery. I posted a similar question on Reddit/r/Physics and got mostly the same reaction, which to me is a big surprise.
Are there really any conceptual objections to this that I overlooked? Or is it just a matter of the audience being unprepared for it?
Note: The original version posted on Reddit was "Moving objects appear to undergo length contraction because they are perceived in space-time cross-section". A major objection over there was that using words like "appear" and "perceive" suggest I consider length contraction some sort of optical illusion dependent on the observer. To the contrary. I never intended to debate that it is a real, measurable effect. Only meant to emphasize that length contraction appears differently in different frames, as demonstrated by measured coordinates. Rephrased here to eliminate this sort of misunderstanding.
Let me refine and clarify please:
Let us limit the question to special relativity only, for convenience and clarity. Let us also leave aside optical effects in SR, including doppler, beaming, Terrell-Penrose, etc. In other words, the question does not refer to the optical observation of length contraction by a human observer or a camera. Let us discuss only in terms of measured coordinates.
Let object A move at relativistic velocity relative to an inertial frame O. In 4D space-time (Minkowski diagram) the space view of O at any given moment of its own time is a space-like hyperplane. What O observes of A at any time is the cross-section of A's world-tube by O's corresponding constant-time hyperplane. In this cross-section A undergoes length contraction. Geometrically speaking, the length contraction is a consequence of the fact that O's constant-time hyperplanes "slice" A's world-tube at a different angle than do A's own hyperplanes of constant proper-time (corresponding to the 3D space view in its rest frame).
In view of this 4D representation, is it ok to say that, as seen in O, A undergoes length contraction because it appears in a different space-time cross-section than it does in its rest frame?