Discussion
Started 27 April 2024
【NO.38】Doubts about General Relativity (3) - Are Space-Time Curvature and Expansion Two Different Geometrical Mechanical Properties?
Einstein field equations [1]:
Rµν - (1/2)gµνR + Λgµν = Tµν ...... (EQ.1)
where Λ is the cosmological constant, gµν is the spacetime metric, and Rµν is the Ricci tensor. EQ.1 expresses the relationship between the amount of energy-momentum (mass) and the curvature of spacetime in a region (or point) of spacetime.
The basic Friedmann equation that dominates the expansion of the universe [2]:
(a')2+K=8πGρa(t)2/3 ...... (EQ.2)
where a(t) is the Robertson-Walker scale factor, and it determines how large-scale distances in space change with time in Friedmann-Lemaître -Robertson-Walker metric:
ds2=gµνdxµdxν=dt2-a2(t)dX2 ....... EQ.3
And it is a solution of Einstein field equations. Two Space-Time properties are expressed here: curvature and expansion over time.
What causes Space-Time Curvature is local energy. What drives spacetime expansion is dark energy. ”Physics welcomes the idea that space contains energy whose gravitational effect approximates that of Einstein's cosmological constant,Λ; today the concept is termed dark energy or quintessence." [3] Dark energy is not the usual matter and radiation[2].
Our questions are:
1) Space-time is interconnected, confined by the speed of light c =Δx /Δt; the factor a(t) that determines space-time is of a kinetic nature; what makes it relevant only to time (it affects all of space in the same way as time passes) [4] and not to space?
2) Can the Einstein field equations essentially be written as two separate equations, the bending effect equation and the expansion effect equation?
3) How does Space-Time know to distinguish between energy and dark energy if Space-Time Curvature and Expansion are both different properties?
4) Can local Space-Time Curvature geometrically affect expansion if it appears to be strongly curved?
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Notes
* “How the view of space-time is unified (3)-If GR's space-time is not curved, what should it be? ”https://www.researchgate.net/post/NO17How_the_view_of_space-time_is_unified_3-If_GRs_space-time_is_not_curved_what_should_it_be
** How the View of Space-Time is Unified (4) - Is Space-Time Expansion a Space-Time Creation?
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Refererncs
[1] Grøn, Ø., & Hervik, S. (2007). Einstein's Field Equations. In Einstein's General Theory of Relativity: With Modern Applications in Cosmology (pp. 179-194). Springer New York. https://doi.org/10.1007/978-0-387-69200-5_8
[2] Weinberg, S. (2008). COSMOLOGY (Chinese ed.). Oxford University Press.
[3] Peebles, P. J. E., & Ratra, B. (2003). The cosmological constant and dark energy. Reviews of Modern Physics, 75(2), 559.
Most recent answer
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.
Popular replies (1)
M.I.M. - Musich Institute of Mathematics
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
4 Recommendations
All replies (14)
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').
M.I.M. - Musich Institute of Mathematics
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):
ISBN-978-65-00-98107-0
3 Recommendations
East China University of Science and Technology
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.
1 Recommendation
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.
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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
M.I.M. - Musich Institute of Mathematics
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:
What is missing at the moment is boldness and changing what is considered "correct", but is not correct.
3 Recommendations

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
M.I.M. - Musich Institute of Mathematics
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
3 Recommendations
M.I.M. - Musich Institute of Mathematics
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
3 Recommendations

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."
M.I.M. - Musich Institute of Mathematics
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
4 Recommendations
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.
Similar questions and discussions
【NO.37】Doubts about General Relativity (2) - Does the Energy Tensor Tµν in the Field Equations Contain the Energy-momentum of the Spacetime Field?
Chian Fan
The external spacetime field produced by an object of mass M, the Schwarzschild spacetime metric solution, is usually obtained as follows [1]:
1) Assumes a spherically symmetric spacetime metric, and is static and time invariant;
2) Assumes a vacuum conditions outside, with Tµν = 0;
3) Solve the Einstein field equation, Rµν - (1/2)gµνR=Tµν...... (EQ.1)
4) Utilize the boundary condition: the Newtonian potential ф = -GM/r, which introduces the mass M. Obtain the result:
ds2 = -(1-2GM/r)dt2 + (1-2GM/r)-1dr2 + r2dΩ2...... (EQ.2)
Overall, the Schwarzschild metric employs a priori derivation steps. The solution is unique according to Birkhoff's theorem.
Einstein does not explain why M leads to ds2, our questions are:
a) The spacetime metric is containing the energy-momentum Tspacetime , which can only originate from Tµν and is conserved. Why then must spacetime receive, store, and transmit energy-momentum by curvature* ?
b) The implication of condition 2) is that the spacetime field energy-momentum is independent of M or can be regarded as such. Comparing this to the electric field of an electron is equivalent to the fact that the energy contained in the electron's electric field is independent of the electron itself. Since Tspacetime is also bound to M, is it not part of M?
c) For complex scenarios, in the Tµν of Einstein's field equation EQ.1, should one include the spacetime energy momentum at the location? With the above Schwarzschild solution, it seems that there is none, otherwise both sides of the equation (EQ.1) become a deadly circle. So, should there be or should there not be? Does the field equation have a provision or treatment that Tµν can only contain non-spacetime energy momentum?
-----------------------------
Notes
* “How the view of space-time is unified (3)-If GR's space-time is not curved, what should it be?” https://www.researchgate.net/post/NO17How_the_view_of_space-time_is_unified_3-If_GRs_space-time_is_not_curved_what_should_it_be
** "Doubts about General Relativity (1) - Is the Geometry Interpretation of Gravity a Paradox?" https://www.researchgate.net/post/NO36_Doubts_about_General_Relativity_1-Is_the_Geometry_Interpretation_of_Gravity_a_Paradox
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References
[1] Grøn, Ø., & Hervik, S. (2007). Einstein's Field Equations. In Einstein's General Theory of Relativity: With Modern Applications in Cosmology (pp. 179-194). Springer New York. https://doi.org/10.1007/978-0-387-69200-5_8
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2024-04-26
Additional information*:
1) In his Karl Schwarzschild Memorial Lecture, Einstein summarized the many scientific contributions of his short life, stating [1], in commenting on Schwarzschild's solution, that “he was the first to succeed in accurately calculating the gravitational field of the new theory”.
(2) Einstein emphasized in his article “Foundations of General Relativity” [1], “We will make a distinction between 'gravitational field' and 'matter', and we will call everything outside the gravitational field matter. Thus the term 'matter' includes not only matter in the usual sense, but also electromagnetic fields.” ; “Gravitational fields and matter together must satisfy the law of conservation of energy (and momentum).”
(3) Einstein, in his article “Description based on the variational principle” [1], “In order to correspond to the fact of the free superposition of the independent existence of matter and gravitational fields in the field theory, we further set up (Hamilton): H=G+M
4) Einstein's choice of Riemannian spacetime as the basis for the fundamental spacetime of the universe, which I have repeatedly searched for in The Collected Papers of Albert Einstein, still leads to the conclusion that he had no arguments, even if only descriptions. In his search for a geometrical description, he emphasized that “This problem was unsolved until 1912, when I hit upon the idea that the surface theory of Karl Friedrich Gauss might be the key to this mystery. I found that Gauss' surface coordinates were very meaningful for understanding this problem.”[2] And, although many physicists also do not understand what Space-Time Curvature is all about, everyone accepted this setup. This concept of `internal curvature', which cannot be mapped to physical reality, is at least a suitable choice from a modeling point of view.
5) Einstein's initial assumptions for the field equations were also very vague, as evidenced by his use of terms such as “nine times out of ten” and “it seems”. He was hoping to obtain the gravitational field equation by analogy with the Poisson equation. Thus, the second-order derivative of the spacetime metric is assumed on the left side of the equation, and the energy-momentum density is assumed on the right side.
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* The citations therein are translated from Chinese and may differ from the original text.
[1] University, P. (1997). The Collected Papers of Albert Einstein. Volume 6: The Berlin Years: Writings, 1914-1917. In. Chinese: 湖南科学技术出版社.
[2] Einstein, A. (1982). How I created the theory of relativity(1922). Physics Today, 35(8), 45-47.
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