I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any), and what physical consequences do we observe from them?
The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.
Euclidean geometry is characterized by Euclid's five postulates being true
This is true, but what characterizes non-Euclidean geometry are deviations from the parallel postulate, specifically.
Minkowski space with the flat Minkowski metric has submanifolds that have non-Euclidean geometry (hyperbolic geometry). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $\pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $\pi$ radians, since the separation between parallel lines increases or decreases, respectively.
EDIT: After thoughtful feedback in the comments, I tried to make it clear that the fifth postulate is violated on the submanifolds of a Lorentzian manifold, whereas the entire Minkowski spacetime is indeed flat (and affine meaning Euclid/Playfair parallelism holds on the Lorentz manifold).
Minkowski spacetime violates Euclid's First Postulate, when expressed in a form like the Playfair postulate. Physically, this violation tells us something about the Causal Structure in spacetime.
The first postulate can be expressed like the "dual" version of the fifth postulate:
"Given a point, and a line not through that point, there exists no point on that line which cannot be joined (by an “ordinary” line) to the given point."
So, for special relativity, I write it like:
"Given an event, and a worldline not experiencing that event, there exists infinitely many events on that worldline which are not “timelike-related” to the given event."
For Galilean relativity,
"Given an event, and a worldline not experiencing that event, there exists one event on that worldline which is not “timelike-related” to the given event."
(I think the Parallel Postulate (suitably expressed, e.g. Playfair) is okay for Galilean and Minkowski spacetimes. I think it's what allows us to draw parallel lines... like the ends of a meterstick on a spacetime diagram.)
This viewpoint is what I physically interpreted from
I.M. Yaglom's "A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity"
See Figures 188a-c on page 220, which I inserted below.
(From Yaglom, p220)
In accordance with the requirement of congruence of any two lines in the plane, it is natural to define the "Minkowskian plane" as the points of the (ordinary) plane and the lines of one kind—say, the first. In this connection we note that in Euclidean geometry every point on a line a can be joined by a line to a point A not on a (Fig. 188a); in Galilean geometry, each line a contains a unique point, "parallel" to A, which cannot be joined by an (ordinary) line to A (Fig. 188b); in Minkowskian geometry, there are infinitely many points on a line a of the first kind which cannot be joined to A by means of lines of the first kind (Fig. 188c). On the other hand, all three geometries share the property that through any point A not on a there passes a unique line not intersecting a (a unique line parallel to a).