There is no "space-time" because this is no "Relativity of Simultaneity" --- distant (and together) events are simultaneous in both stationary and moving frames. And so there is an "Absolute character to time" (t = t') and an "Absolute character to breadth" (x = x').

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Space-Time Curvature arises from the fundamental forces of Physics and not from gravity, as the latter arises from the former. Expansion arises from the inertia generated by fundamental forces. Therefore, both have mechanical properties as a result of their intrinsic characteristics.

See my book (It's in my book):

https://archive.org/details/mus-mathematimus-hyperelliptical-geometry-202403311048/

ISBN-978-65-00-98107-0

Wolfgang Konle has provided opinions elsewhere that are relevant to this thread. It is copied here especially for reference.

Wolfgang Konle：

"Are Space-Time Curvature and Expansion Two Different Geometrical Mechanical Properties?"

If we consider Einstein's field equations with the cosmological constant in matter-free space-time, we get an interesting answer to your question:

Gij + Λgij = 0, where Gij = Rij - 1/2 gij R. Now contract both sides in Rij - 1/2 gij R + Λgij = 0 by using the inverse metric tensor gij to get gij Rij - 1/2 gij gij R + Λgij gij = 0. With the well-known results gijgij = 4 and R = gij Rij you then obtain R - 2 R + 4 Λ = 0, which gives R = 4 Λ, and means nothing else than a constant scalar curvature R of space-time in matter-free space-time.

This means that the absolute value of an energy density, let us call it a dark energy density, is related to a constant spacetime curvature.

If we then consider the field equations with matter, we find that matter leads to a gradient in spacetime curvature.

But concerning expansion or shrinking of the universe, the field equations do not tell us something by default. We explicitly must assume an expansion of the universe in the field equation to get information about the properties of an expanding universe.

This short analysis shows that you are addressing a relevant cosmological question. The hint, we get from Einstein's field equations, is that there are indeed two different geometrical mechanic properties. One property is that a constant homogenous energy density is related to a constant homogenous scalar space curvature and that a mass is related to a gradient in the space curvature.

But the field equations cannot answer the question if the universe is expanding or not. They only can give us some information about what else is happening if the universe is expanding.

Chian Fan:

Thank you for providing a very professional answer, which clarifies the important spacetime issue.

Regarding the concept of spacetime curvature, some Chinese books call it intrinsic curvature, which is somewhat similar to intrinsic spin. This intrinsic curvature undoubtedly exists mathematically, but there may be an obstacle in understanding how to manifest it physically. A two-dimensional sphere of constant curvature is easy to understand, but a three-dimensional global space of constant curvature is hard to map. I don't know if you have any insights on this point.

Best Regards, Chian Fan

Wolfgang Konle:

"A two-dimensional sphere of constant curvature is easy to understand, but a three-dimensional global space of constant curvature is hard to map."

A three-dimensional global space of constant curvature 1/R² is known as 3-sphere or S³. It has a volume of 2π²R³ but no surface. Mathematically we call such spaces "manifolds". Locally they are Euclidean. Geodesics in this space are circles of radius R. Any two points, except antipodes, in that space are connected by a specific geodesic circle arc. Antipodes are located on opposite sides of a geodesic circle. You find all properties of the S³ meticulously described in textbooks about differential geometry.

Since the expansion of the Universe is accelerating, why isn't it equivalent to gravity according to the founding tenet of general relativity? It seems the tail (cosmological constant) might be wagging the dog (general relativity). Large scale repulsion could only appear attractive at the smaller scales, as the material of a sponge is among the voids; the expanded voids are the principal structural elements, not the material among the voids.

The idea that space expands originate from the hypothesis that our present universe originate from a point like energy concentration (the big-bang hypothesis).

Curved spacetime originates from Einstein’s hypothesis that gravity isn’t a force field but is caused by the geometry of space itself.

If we measure the non-Doppler red shift of a cluster of galaxies we don’t measure the expansion of the volume of space that the galaxies occupy. The explanation is that the galaxies are “tied” together by gravity. “Solitary” galaxies seem to drift apart.

There is only 1 conclusion possible: the expansion of space and the curvature of space are not the same “spacetime property”.

With kind regards, Sydney

Sydney Ernest Grimm - Your allegations are valid and corroborate my research. There is a need to update calculations based on the new M.U.S. (new pi) and the application of Hyperelliptical Geometry, included in my book.

My book is available free of charge to all researchers:

https://archive.org/details/mus-mathematimus-hyperelliptical-geometry-202403311048

What is missing at the moment is boldness and changing what is considered "correct", but is not correct.

Dear Stenio Musich

My previous comment about the supposed properties of spacetime is not my research (I am not an astronomer). It was published a couple of years ago.

Actually, if the non-Doppler red shift of the light of distant galaxies has nothing to do with the expansion of spacetime because it has another causation, then the consequence seems to be that the clustering of galaxies is an ongoing process of energy concentration.

You can find some 150 "frontier" papers at https://qiss.fr about quantum gravity (and quantum information theory). But in spite of all the efforts there is still no breakthrough. No matter the participation of quite a number of scientific institutions and universities.

Your link above needs the download of a 1312 MB PDF file. I am sorry for you but I am afraid that most people don't want to download your file.

With kind regards, Sydney

Dear Sydney Ernest Grimm

I liked your observations.

Regarding my book, I published my research in full as a way of advancing the research, because if I were to publish it with all the details I think, I think it will take another 40 years and 18,000 pages takes up a lot of space. I published it to help other researchers in the field of mathematics and physics, and it is also useful for an astronomer.

Regards

Stenio

Dear Sydney

The book is free, the only cost is to download it, if it doesn't work, delete it. What the Bible says, what you receive freely, give freely. It wasn't that cheap, I researched for 16 years, I had work and costs, but all for the benefit of science.

Regards

Stenio

Dear Stenio Musich

It is not because of the costs, it is because of time. ArXiv.org expected the upload of about 6 - 7 million scientific papers between 2023 en 2027. Papers about physics, astronomy, mathematics, quantum computing. And arXiv.org is not the only reposterory. So there are far, far more "important" papers than a person can ever read. And the personal research project(s) are the most important thing...

To study someones research costs a lot of time. So if you want to get attention for your ideas/concepts, don't offer the whole research. You better "advertise" the essence. So people can make a personal decision because of their interest/expertise and perhaps download everything.

With kind regards, Sydney

P.S. Suppose that Einstein lives now. And the Home feed of ResearchGate shows me his new papers about relative time and the origin of gravity. I know for sure that I will not download his papers. Probably I will think something like: "Be wise, don't spoil your time, these ideas lack reliable foundations."

Dear Sydney Ernest Grimm

Thank you for the suggestions and tips, but I published for the love of science and my publication alone is enough for me.

I thought you were a very nice person, I like people who speak the truth and objectively. Congratulations.

I stopped researching due to lack of time and money, because without support the research would take too long and therefore, I published all the research for other researchers, if they want, they can continue with the research.

Best Kind Regards

Stenio

https://www.researchgate.net/post/One_can_see_that_Special_Relativity_and_General_Relativity_are_in_error_from_the_below_wiki_figure2

Preprint Yet further Animadversions upon the wiki figure purporting t...

Newton suggested that -if- there were a substance that could be made into a perpetual motion machine (ie. the "eternal supercurrent in a ring," for example) that is : a superfluid,---- then this substance would not suffer from the effects of gravity, and we would therefore expect it to "weigh nothing."

I say that the superfluid-part, of Liquid Helium cooled below the transition temperature, "weighs nothing."

That's why a beaker of superfluid Liquid Helium "empties itself" Cf.

https://www.youtube.com/watch?v=6WHhrFcH3b0