The aforementioned analogy between Minkowski spacetime and the Sphere Observer’s spacetime, holds only partly true. For clarity’s sake, I will call Minkowski's abstract 4D spacetime '3 + 1' (3 spatial dimensions + 1 of time), and the Sphere Observer's one '2 + 2' (2 spatial dimensions + 2 of time).

The analogy is only partly true because Minkowski spacetime (3 + 1) does not apply to events that are affected by gravity. (The abstract 4D spacetime of Minkowski is after all the earlier world of a spacetime without matter, e.g. intergalactic space.) The improvements in the measuring results, as described above using Pythagoras’ theorem, will therefore not apply around celestial bodies, while the Sphere Observer's 4D spacetime (2 + 2) of  applies everywhere: Both around celestial bodies and in intergalactic space, where there is no matter.

However, this difference can only be understood if you can imagine what the Sphere Observer's abstract spacetime, and similarly also the abstract underlying tissue conceived by Einstein, looks like.

 
According to the Sphere Observer, all of spacetime, i.e. all of the underlying tissue or the earlier world, consists of a conglomerate of two-dimensional spacetime shells (2 + 2), which resemble the surface of a sphere. That's because they are formed by the celestial bodies and extend themselves from it, like wrinkles in water do when a stone is thrown into it. Such a spacetime can easily be understood if you imagine those shells as being like the walls of soap bubbles, able to become infinitely large and intersecting each other without bursting. This creates the following views:

• Near celestial bodies, the invisible two-dimensional spacetime shells are arranged in an orderly fashion: they surround the body like the skirts of an onion. Through this ordering (local) gravity with its gravitational potential here prevails, as Einstein proved. At larger distances from celestial bodies, spacetime shells of various celestial bodies intersect each other in all directions of the earlier world. In its later world, gravity as we know it does not exist, because the shells are no longer arranged in an orderly fashion, meaning that there is no gravitational potential. The space-time shells, however, are still curved, which causes gravity-like phenomena, like for example, between the Sun and its planets, and the curved movements of the stars themselves. It is curved spacetime that Einstein refers to in his gravity theory. Within galaxies, spacetime is 2 + 2 anyway.

• In space where the curvature of the shells is completely negligible, so in the parts of the universe where there exists (almost) no matter, as in intergalactic space, there are no gravity or gravity-like phenomena. Therefore it is mathematically allowed to convert one dimension of time to one of space: 2 + 2 becomes 3 + 1.

According to the above, the abstract underlying spacetime of galaxies (2 + 2) can be distinguished from intergalactic spacetime (3 + 1). With the creation of the minkowski spacetime, actually a distinction is made between the timelike spacetime and the spacelike one, respectively.

Note 37 Here, intergalactic space specifically refers to the space between galaxies. Because galaxies are scattered throughout the universe with huge distances between them, intergalactic space makes up the vast majority of the universe itself. So, for example, the cross section of the Milky Way is "only" about 150,000 light years long, while the nearest galaxy, the Androma Galaxy, is at a distance of about 2.2 million light years. According to the most widely accepted theory, intergalactic space will become an exponentially increasing part of the universe in the future, as galaxies are increasingly drifting apart. Because intergalactic space is so large, the density of matter within the universe is very low and so is gravity. This allows the expansion of the universe to continue indefinitely. en.wikipedia

Continue to: 5.3.2. Minkowski spacetime nevertheless compatible with gravity?