Not true! The Friedmann cosmology*) is classified under the idealised cosmologies. In this particular cosmology, the universe is uniform and isotropic. For details, you may wish to consult Chapter 27 of the book Gravitation, by CW Misner, KW Thorne, and JA Wheeler (WH Freeman, San Francisco, 1973).

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*) The 'F' in FLRW refers to Friedmann.

Of course not-and the statement made about the FLRW metric is incorrect-it describes a Lorentzian manifold, not just a  three-dimensional manifold. And the spacetime geometry isn't defined just by the spatial integral of the time-time component of the energy-momentum tensor, but by all components thereof. 

The FLRW metric is written using  a particular choice of coordinates, that's all. What is of relevance isn't the metric, which transforms in a particular way under general coordinate transformations, however, but quantities that are invariant under such transformations-it can be shown that such quantities exist. All this is well known, described in all textbooks and courses on general relativity and doesn't have anything to do with the physical applications of the FLRW metric, as a particular solution of Einstein's equations. It's the other way around-since it can be shown that it is, indeed, a well-defined solution of these equations, i.e. that it does satisfy equations of motion and constraints, it makes sense to study its implications for physics, in this case, cosmology. So it would be useful to study a textbook on general relativity.

And the cosmology in question is classical, not quantum.

FRW cosmologies are based on Friedman's cosmological model, which geometrically can be explained as pseudo-hyper sphere embedded into a 5-dim flat space. If the universe expands in free fall, then no forces are experienced by a comoving observer in the same way as no forces are experienced by an observer freely falling in the Schwarzschild field. For both observers the space appears to be flat. Thus, k=0 denotes the apparent geometry for observers being in free fall in an expanding universe, while the geometry of a fictive observer being in rest is still k=1, ie a positively curves space (pseudo-hyper sphere). Have a glance at my papers on this site and also at Melia's papers.

It is a common practice among general relativity theorists to handle their theory 'à la carte':

• when they teach it they always say about the 4D manifold
• when they are forced to apply it, then they follow next two alternative paths:

See also one of my questions about application of that theory here:

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

for instance: m=m0/S (SquareRoot(1-v²/c²), when mass is in Motion, but the inverse m*S must be a slow down, while m is in Motion too. everything is in Motion.

whereby (-v)²>0, and multiplying by S (m*S) makes no difference to (/S) i.e. dividing vy S. thus: Lorentz Transformation is not consistent. More: LT is not covariant. and: when usin imaginary numbers, then S>1.

Well, now if I want to revise my question as, “Whether the space-time curvature is a physical reality or simply a mathematical caricature?”, then what will be the answer?

We don't know much, if anything at all, about things beyond the Planck scale. Consult the attached text.

http://www.symmetrymagazine.org/article/the-planck-scale

Keeping aside the Planck scale issue, we see that the space-time curvature can predict the perihelion shift of the planetary motion, while curvature of time is absent in the Friedmann cosmology. So, should we consider the sp-time curvature as a physical reality or not?

Dear Mohammad, as I was very explicit in my very first comment on this page, Friedmann's cosmology is an idealised one, that is, it is not the most general cosmology within the framework of the Einstein theory of general relativity. Why are you insisting on connecting something idealised with the question of whether or not the latter theory describes the physical reality?!

If the physical reality, whatever it may be, is describable in terms of a grand unified theory, the extant theories corresponding to the smallest and largest scales do not fall into such a unified scheme so that they must necessarily be approximations of some more fundamental theories.

Thank you Dr. Farid and Thanks to everybody.

You are welcome Mohammad.

Ps. Incidentally, leave for a moment Einstein's equations aside and consider the more transparent case of a linear partial differential equation in relation to the technique of separation of variables (think for instance of how one solves the Schrödinger equation for the hydrogen atom).

https://en.wikipedia.org/wiki/Separation_of_variables

https://en.wikipedia.org/wiki/Separation_of_variables

Gravitational red shift is caused by gravitational time dilation. Is it possible to explain this phenomenon in a FLRW framework? I will also appreciate a few references on such studies, thanks in advance.

@ Biswajoy Brahmachari - Calculation of the time dilation is achieved by the knowledge of the space-time metric, and we know the Friedmann metric. For details, you may wish to consult the following book:

Norbert Straumann, General Relativity, 2nd edition (Springer, Dordrecht, 2013).

Explicit calculation of the red-shift in the Friedmann-Lemaître Space-times is presented in Sec. 10.2.4 of this book (see also Sec. 2.6 herein). 

@ Mohammad Sharifi - You will have to excuse me, but you simply make no sense to me. I do not say this lightly, or because I enjoy interfering, but because I am clearly witnessing that you are committing academic suicide on this very page, in full view of essentially the entire world. In order to be fair to you, I have even checked your manuscript that you refer to, but that made no sense to me either; you simply do not know the literature relevant to the issues you are attacking -- yes, I even have gone through the references you cite in your manuscript.

I ask you fatherly, but very resolutely, to delete your comments on this page and stop posting comments on RG until such time that you have something serious to communicate with others. If you cannot help it, then I strongly advise you to seek assistance from a professional who can help you. Perhaps the rejection of your manuscript by various journals has given you a psychological shock. In order to recover from this shock (which is not unheard of -- strong sense of disappointment can cause major psychological shock and disorientation), you will have to use the right remedies, instead of going public and making yourself a laughing stock.

Should you delete you above above comments, I will subsequently delete this comment so that your privacy will not be in danger.

@ Mohammad Sharifi

Dear Mohammad, it is not, and has never been, my intention to interfere with your business. Nonetheless, and despite your explicit request not to write back, I feel that it is part of my social responsibility to be concerned about the conditions of my fellow human beings. Very frankly, I notice in your comments strong hints of you suffering from persecutory delusion and/or paranoid schizophrenia [1]. I do not hesitate to name these mental disorders here simply because I believe (together with many others) that these are no more serious than flu or stomach bug, and that we must get rid of the medieval notations regarding mental disorder; in the same way that we can suffer from a cold, or a heart attack, we can suffer from mental disorder -- there is of course some genetic component in mental disorder, so that some are more likely to suffer from mental disorder than others. You should therefore get in touch with a psychologist, who will certainly be able to either completely cure your problem, or at least teach you how to cope with it.

Remember that many successful scientists have suffered from some form of mental disorder. Most notably, the mathematician John Nash [1], who suffered from paranoid schizophrenia (he believed, amongst others, that he had broken a secret Russian code and therefore Russians were out to get him for he having thwarted their secret plans - in reality, he had never worked on any classified problem), and according to he himself in an interview, he had never got rid of his delusions, but had learnt how to cope with them. The film A Beautiful Mind (1998) [3], which you may or may not have seen, portrays John Nash's life.

I urge you to get yourself examined by a good psychologist, and, as I said, there is no shame in doing so. You should get out of the hard shell that you have created around yourself, rejecting outright any advice that might get you out of your unhealthy mental habits. If untreated, your problem can intensify and cause real danger to you and the people around you. You are clearly a talented person, so don't let yourself be bugged down and your life wasted by a mental condition that may be treatable by taking some tablets and performing some simple mental exercises! Ask a friend and/or close relative to help you! From time to time we all need to call on the help of others, even when we do not think that we are in need of any help. Just do that please!

[1] https://en.wikipedia.org/wiki/Persecutory_delusion

[2] https://en.wikipedia.org/wiki/John_Forbes_Nash_Jr.

[3] http://bit.ly/2vUyDiY +  http://bit.ly/2btJeri

@ Thierry De Mees

I do not know what your references are, but Halton Arp's views have been conclusively shown to be inconsistent with the later more detailed and accurate observations. For this, you may wish to consult the attached text.

https://en.wikipedia.org/wiki/Non-standard_cosmology#Redshift_periodicity_and_intrinsic_redshifts

Dear Thierry,

I do not concur with you.

Science is a human endeavour, so that it is bound to suffer here and there from human follies (e.g. journals and referees may not invariably be as unbiased as they are supposed to be), but because of the checks and balances built into its fabric, it has shown a remarkable ability towards self-correction. This is in particular the case in physics, and by extension chemistry. Social sciences sometimes, if not often, go astray, and that is because experiments in these sciences are not as straightforward as in physics, where one deals with inanimate objects which can be studied in isolation; with animate objects this is not possible, in addition to the fact that ethical considerations limit the extent to which things can be probed when the subjects are humans.

Incidentally, science as we know it is about finding a consistent explanation of as many diverse observed phenomena as possible with the minimum number of assumptions. It is not about the reality of things, whatever that may be. From this perspective, General Relativity is one of the most successful scientific theories ever devised by the human mind. It is certainly not a sacred thing, so that when new experiments prove it to be inadequate, it will be replaced, through collective effort, by a better theory. In this connection, you may or may not have seen the attached recent report (dated 9 August 2017) by Scientific American.

https://www.scientificamerican.com/article/closest-supermassive-black-hole-tests-einstein-rsquo-s-relativity/

Dear Thierry,

Thank you for your response. I have to disappoint you by saying that I do not share your views. In particular, I do not concur with the idea that people have been lying (your text is replete with the word 'lie').

Your main criticism is based on the piece in Scientific American (cited earlier by me), and your own almost entirely self-referencing manuscript; your first four references herein are elementary, and the remaining references are your own writings, which additionally have not been peer-reviewed. Please note that I am not a priori rejecting self-citation and manuscripts that have not been peer-reviewed; my main objection is calling everything and everyone as respectively lies and liars entirely on the basis of one's own works. Even if in some abstract sense you were for 100% right in your judgement, your approach falls outside the domain of scientific tradition, where scientific facts are arrived at by means of reproducible experiments and through the consensus by the scientific community. As I wrote earlier, science as we know it and as is practised in the confines of the scientific institutions the world over purses a very specific objective, which of course may not be to the liking of some individuals, such as you. This does not make traditional scientists liars or deceivers.

As for Scientific American, it is a popular journal reporting on original scientific publications. It is therefore not the appropriate source for basing one's arguments in opposition to these publications on; for such opposition, one should rely on the original relevant publications. In the case at hand, the publication by M Parsa et al.*)(The Astrophysical Journal 845:22 (2017)) is to be concentrated on (I attach the appropriate link below, as well as a link to the relevant pre-print).

If you had searched for the expert views on the above publication, you would have realised that Daniel Clery of Science Magazine has in his discussion of the above paper (dated 9 August 2017 - link attached) explicitly mentioned that Abraham Loeb of the Harvard University 'is not convinced this is enough to claim agreement with relativity because the study’s error bars are so large'. Indeed, the fitting procedure as adopted by Parsa et al. has resulted in Υ = 0.00088±0.00080, making explicit that while the observation is not inconsistent with the expected value of Υ = 0.00065, it does not prove agreement of the observations with the general relativity. Therefore, what we are observing is that there is a healthy discussion going on in the scientific community with regard to what signifies what, which is not reflected in the piece published by Scientific American. As I see things, no one is even trying to deceive.

Be it as it may, I think our perspectives on science and the way it is practised in the academia are almost, if not entirely, orthogonal with respect to each other. In contrast to you, I have no reason to believe that the science community is up to something untoward. Naturally, as a community of humans, the scientific community has also its own human-related problems, corresponding to some unscrupulous individuals in the scientific ranks who are not averse to unfounded and/or exaggerated claims, manipulation of data, perjury, etc. But this is something everyone is fully aware of, which is why publications that are deemed consequential are subjected to serious and merciless scrutiny.

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*) M Parsa et al.

https://doi.org/10.3847/1538-4357/aa7bf0

https://doi.org/10.3847/1538-4357/aa7bf0

http://www.eso.org/public/archives/announcements/pdf/ann17051a.pdf

http://www.sciencemag.org/news/2017/08/star-hurtling-towards-massive-black-hole-may-seal-deal-einstein-s-theory-general

@ Thierry De Mees

You seem to have fallen into your default position, instead of appreciating my investment of time in responding to your points, accusing me of a whole array of unholy things. You seem not even to realise that you have been insulting me by suggesting that I repeat Wikipedia! One can be a maverick, as you clearly are, but this does not justify being rude, especially to someone who has responded to you with consideration. Of course, I am not shocked by your behaviour, given the fact that you take your liberty to call all and sundry liars; for you, the entire science community appears as a giant conspiracy, intent on keeping the humanity in the dark.

End of the discussion.

@Biswajoy Brahmachari

Sir, cosmological red shift has explanation in the FLRW framework. Though it was tempting to ask, “Whether the gravitational red shift and the cosmological red shift are the same?”, but I am completely aware of not asking such question. However explicit calculations for both these cases are also available in the book “Relativistic Mechanics” by Satya Prakash (published by Pragati Prakashan).