Minkowski spacetime is a Lorentzian "manifold".

I never said that worldlines are manifolds, Reality Check, I said that they are continuous, but I would not be interested in onedimensional manifolds. Threedimensional space is a manifold.Your "vacuum" ignorance again. It is persisting with world lines along with "vacuum" that suggests you think that these are the entire manifolds. You are making obvious errors about manifolds.

You are right, Newtonian spacetime with itsThink about Euclidean space in math and used in classical physics. This is a continuous space with an infinite number of points between each point. No vacuum there! Manifolds are a generalization of Euclidean space. They are mathematical. They are defined to be continuous. They do not have vacuum.standard topologymay be understood as a manifold. There is also vacuum between worldlines in Newtonian spacetime, but these points are part of the standard topology. In contrast, Lorentzian manifolds cannot be provided with a standard topology, their topology is rather a whole mess.

This is a physical reality, nothing mathematical. Space is simply not filled entirely with particle worldlines which would not leave any void. By consequence, there is void between the particles.IF03: Give your sources that say that a Lorentzian manifold must have "vacuum" or "void(s)",Rinaldo.

What are the postulates talking about? Postulate 1 talks about IRF which must be mass particles, postulate 2 talks of observers and light sources which must be mass particles. Then, postulate 2 talks about c, the speed of light in a vacuum. "in a vacuum" may not have any other sense than "without interaction with particles (or possibly: fields). We can extend this postulate to fields such like an electromagnetic field or (in GR) gravity fields which are propagating at speed of light. Laws of physics of vacuum are not mentioned, the postulates do not care about vacuum.IF04: State where the 2 postures of SR exclude ("do not apply to") "vacuum points" (presumably points in physical vacuums),Rinaldo.

Special relativity

The same is true for the whole physics which is derived from these postulates:Lorentz transformationdoes not talk about the vacuum. The Lorentz factor depends on the relative velocity between two reference frames (=particles). This is why the physics of SR do not care about the vacuum.

Lorentzian spacetimes are no manifolds, they are not continuous and differentiable, in particular not in spacelike direction.You persist with a blatant error that a Lorentzian manifold is not compatible with GR when it is easy see that it is from reading about GR. Differentiable manifolds in physics have to be continuous because we want to apply calculus and calculus needs a continuous space. A simply put example: a speed dx/dt is a limit as finite intervals in x and t tend to 0. Those intervals cannot come to a screeching halt at a finite value! GR has the additional requirement that its manifold be Lorentz invariant.

How can differential geometry be possible? The answer is simple: each of the worldlines which are parameterized by their respective proper time is perfectly continuous, differentiable and Lorentz invariant! As long as differential geometry treats worldlines of particles (including hypothetical worldlines) it works.

One counter-example: The spacelike foliation of spacetime into hypersurfaces is part of numerous theories of quantum gravity, and the corresponding application of differential geometry does not lead to any reasonable results here.

Proper time is the "time measured by a clock following a given particle". All observers agree on the proper time of the particle. Every observer can take the observed worldline of a particle (which is parameterized by the observer's coordinate time and coordinate space), and by calculation he can recover the original worldline of the particle, that is the worldline parameterized by its proper time. The resulting particle worldline will be identical for all observers.IF05b: Show that "particle worldlines are Lorentz invariant if each of them is parameterized by its respective proper time",Rinaldo.

Here is one try (there might be simpler): In order to measure an arbitrary distance from A to B in Euclidean threedimensional space, any observer in a spaceship may travel to A and then from A to B. That means, even if he does not make this travel he can at least calculate the space interval between A and B. Any observer who travels from A to B will get the same result, that means that any observer will get the same result of calculation, that means that a Euclidean threedimensional space manifold is observer-independent and Lorentz-invariant.IF05a: Show that "A threedimensional space manifold is Lorentz invariant" for example the Euclidean space used in classical physics,Rinaldo.

Note: The Euclidian space manifold used in classical physics is perfectly appropriate for modern quantum physics, and I assume that there will be also a way to express gravity in flat space (in the form of gravitational time dilation) instead spacetime curvature.

As I said, the worldlines do not form any manifold, they are solipsistic worldlines parameterized by their respective proper time, and all this within a 3D space manifold. What do you mean exactly by "definition of calculus"? See also above "How can differential geometry be possible?" in this post.IF05c: Give your definition of calculus in your "world line only manifold" and show that it works,Rinaldo.

Yes, but it has negative squares which leads to an equivalent contradiction.... look at the definition of spacetime and see it has no complex numbers!

This is not so clear, Reality Check.Spacelike spacetime intervalsare defined asbeing negative.It is an undeniable fact that physicists obscured the subject by the introduction and the maintenance of four different conventions (+,-,-,- and -,+,+,+, extraction of the root or not), this fact is not very reassuring. Has anybody reflected on this point? I studied as much literature as I could get with my limited means, but the more I read, the more I get the impression, that in 1908, people were too much absorbed by the idea of fourdimensional spacetime, and I fear that this issue has never been treated by Mainstream.