Asif I believe there are two distinct questions lurking here. 

The easy one is why gravity is non linear in the sense that it is not a perfect inverse square acceleration law which would produce Keplerian orbital forms.  The electrostatic law is really not linear at all, it is inverse square on radius.  What is usually meant when one says it is linear is that its effect can be calculated with superposition.  This is a kind of linearity I suppose.  The force of N charges is N times the force of one charge.  I will call this superpositional instead of linear.  (a word I made up just now)

But gravity is not a force law.  What we have is time and space being affected by the proximity of matter.  If the weak field is superpositional, then in a strong field when other affects become noticeable it cannot remain superpositional.  The fact that gravity is not superpositional and is universal in its effect should probably be taken as evidence that gravity is not based on a virtual boson field like electromagnetics.

The second and more subtle question is with regard to inertia.  It is well known for 100 years that inertia can be expressed approximately as the sum of 1/R potentials with respect to all other masses in the universe.  Such a sum is a linear function of Mi/Ri.  However, it is not exact because GR in the weak field limit preserves the Newtonian force law rather than the Newtonian potential law.  In the paper I sent you earlier I show that it cannot in the limit enforce both and must choose.

If GR had made the other choice, to enforce Newtonian potential, we'd hardly notice any difference in its predictions, in fact none that I'm aware of.  The difference is extremely slight and all the solar system effects would be indistinguishable.  But while gravity would still be nonlinear, in your sense, inertia would be superpositional.  Instead of black holes we'd have what I call black dots, which in some ways are much more terrifying, though I wouldn't care to fall into either.

Photons don't carry electric charge themselves-gravitons do carry energy and momentum. So photons don't have self-interactions, gravitons do.

Robert I appreciate new term “superpositional”. Mathematically linear relations are superpositional, I don’t know reverse is true or not in mathematics. This reminds me Kirchhoff’s laws in circuit theory, which are superpositional in nature. These are linear V I (superpositional) laws and not applicable to non linear devices. However linear models of nonlinear devices are made, which then allows application of Kirchhoff’s laws.

The electrostatic fields are superpositional i.e. the resultant field at any point is vector sum of fields due to all sources around. In this sense the electrostatic (ES) fields are linear. To my knowledge, there is no maximum limit on field strength for ES field to be linear. Nuclear ES fields (particularly with high atomic number) are very strong and we simply take the sum of ES fields of protons i.e. ES is superpositional even in this limit.

The Newtonian gravity is superpositional and does not put a maximum limit on field strength for it to be linear. However it is inevitable to include non linear effects when fields are stronger. The example is perihelion of Mercury; which cannot be explained with Newtonian gravity.

The GR has the answer. However as you mentioned @“ But gravity is not a force law. What we have is time and space being affected by the proximity of matter.” i.e. GR is theory of Space-time curvature and is not real interactional force theory. In this sense it is futile to compare it with other physical theories as mentioned by Robert J. Low @” GR is a physical theory, and electrostatic interactions are physical phenomena.” Now should we accept the “reality” that gravity in principle cannot be integrated with other forces!

Again @ Robert “The fact that gravity is not superpositional and is universal in its effect should probably be taken as evidence that gravity is not based on a virtual boson field like electromagnetics.” I believe that gravity is due to boson field of spin 2. Since it is spin 2 field it cannot be linearly added and is not superpositional like spin 1 electromagnetic field.

Unfortunately I don't know enough about the implications of a spin 2 field to comment intelligently.  I do think at some point one has to give up making new particles to explain things, and have the interacting objects represent themselves.  With gravity, bosons themselves have gravity.  So they have to make 2nd order bosons to do the interacting, aka gravitons.  These gravitons themselves interact as Stam pointed out, so you have to make gravit2ons.  But the gravit2ons also interact, so you have to make something else.  It reminds me of a Dr. Seuss children's book about cleaning up messes and the alphabet.  Each letter pulled out of a hat is a cleaner, but ultimately it just spreads the mess around and requires a new cleaner.  Finally Z comes out of the hat and cleans everything up.  It appears Seuss was optimistic though.  In fairness to the man, he didn't claim to be a particle physicist.  We have Z bosons, but they certainly weren't the final word.

In my 2011 inertia paper I had everything representing itself - no intermediaries.  It requires a teeny tiny little change in the Schwarzschild coefficient, nothing measurable within the solar system (just the thing to preserve the potential law I mentioned above), but physicists are actually in no mood to change it.  Thus some of the more heated discussions on some of my threads.

It is interesting to realize the nonlinearity of GR seems to allow an absolute measurement of the scale of things.  Which is very un-relativistic.

Robert. I cautiously used term “spin 2 field” not “graviton”. Although theories are there but we have not been able to detect a single graviton (this by the way is quantum gravity topic) and gravity waves to date. We have the field theories (without the mention of quantization) with spin properties, such as circular polarized EM waves (we need quantization to make it a photon- spin 1). Higher spin field theories are there but beyond spin 2, theories become more complex. So far nothing could replace GR, in spite of its altogether different approach. Pl see Gravity probe B mission of NASA (http://www.nasa.gov/mission_pages/gpb/). “The experiment launched in 2004, used four ultra-precise gyroscopes to measure the hypothesized geodetic effect, the warping of space and time around a gravitational body, and frame-dragging, the amount a spinning object pulls space and time with it as it rotates. GP-B determined both effects with unprecedented precision by pointing at a single star, IM Pegasi, while in a polar orbit around Earth.”

By the way the frame dragging is inertial effect (and appears as small correction term in GR). I call this effect the energy-momentum dynamism. The source of gravity in GR theory is Energy-momentum stress tensor which unlike other theories does not generate field (e.g. charge source generates E field) but curvature in space. I believe some day we will be able to find gravitational potential based gravity theory.

Robert J. Low : Maxwellian electrodynamics is nice theory, which gave birth to Special theory of relativity; and first invariance-covariance principle; which is core of modern Physics theories including QM . GR doesn’t fit to this paradigm. There are many (approximate) methods to make it look like linear point by point or making linear approximations in week field limit. This fact is another bottleneck for unification of gravity with other forces.

Robert I missed to respond to your note on graviton -graviton interaction and further interactions. To my understanding it is similar to multiple Fourier components of non linear wave-shape. However you have to decide to include reasonable no. of components, which say contain 90% energy.  The same may be applicable to multiple interaction of non linear gravitons ? However it will be very difficult to go beyond 2nd interaction level.  Dr. Seuss book's example is interesting; We have reached to Z bosons, and there are no further alphabets  for Bosons, newly discovered.

I had long discussions with you on your inertia paper. To me it seems realistic idea and an alternate  explanation.   

Asif, yes my inertia paper intends to present an alternative quantum approach to energy-boson fields.  That old paper does not derive curvature, but if I can get all the i's dotted and t's crossed on the new "three steps to curvature" paper, then essentially it will retroactively apply to the inertia paper.

There is very obvious reason that Electric field is a tensor of first rank only while the gravity is tensorial field of more than one rank.

We are perhaps on the brink of detecting the gravitational waves predicted by GR. But this is a long way from detecting gravitons, nothing in GR even predicts them.

If the recoil of the motion of electrically charged matter on the electromagnetic field is NOT neglected electrodynamics in the presence of charged matter is a highly non-linear theory that exhibits relatively poorly understood phenomena (e.g., instabilities, such as run-away solutions, etc. that have been studied for more than a century). Many people expect that the theory is mathematically meaningful only after it is quantized, i.e., in the form of quantum electrodynamics (QED), (provided an ultraviolet cutoff is introduced at an enormous energy scale). The resulting theory is the most precise theory of Physics.

Now, there are many interesting, concrete questions of physics and engineering that can be answered within certain approximations to full-fledged electrodynamics; e.g., by neglecting the recoil of matter on the electromagnetic field. This is fortunate - if it were otherwise radios, radars, television, X-ray scanners, etc. would never have been invented!

I should add that, in the absence of matter, electrodynamics is a linear theory.

General Relativity is non-linear even in the absence of matter (for reasons described, heuristically and roughly, by Stam Nicolis. Obviously, this makes it harder to deal with. However, for people who like sophisticated mathematics and hard work, there are now wonderful mathematical tools to analyze GR even in the presence of matter - great progress has been made during the past thirty years.

To conclude, I would like to emphasize that ALL theories of Physics that describe interesting situations where interactions are at work and events can be observed are NON-LINEAR!

And, by the way, Professor Roos apparently likes to answer questions that have not been asked. But so be it!

I should have written:

"Now, there are many interesting, concrete questions of physics and engineering that can be answered within certain approximations to full-fledged electrodynamics; e.g., by neglecting the effects of the electromagnetic field on the motion of matter."

In some of the comments to the question debated here I have read that Maxwell theory is linear. Nothing could be farther from the truth! It is linear ONLY in the absence of (charged) matter or if the recoil of the electromagnetic field on the motion of matter is neglected. (All realistic equations of motion of matter are NON-LINEAR.)

Robert Low is right :  the non-linearity comes from the constitutive equations; but their precise forme are dependent on the  structure of the coupling of the Maxwell field with the matter field (whose unknown structure is presumably the source of the non-linearity).

To say that "the structure of the coupling of the Maxwell field with the matter field is unknown" is a little too pessimistic, after eighty years of QED, quantum optics and condensed matter physics! What some people may perceive as a problem is that there do not appear to exist any consistent CLASSICAL models of matter interacting with the electromagnetic field. The situation is much better when quantum-mechanical matter is studied. One may then be able to derive "constitutive relations", and, on a somewhat heuristic level, this has, of course been done for many models of matter.

The fundamental difference between Maxwell's theory and GR is that the electromagnetic field is not its own source, while the metric field is, which implies that Einstein's field equations are non-linear even in the absence of matter.

This answer to this question is clear from the point of view of the classical limit of gauge theory. The fundamental interactions (separating electromagnetic and weak) are the electromagnetic, weak, strong and gravitational. Each of these may be viewed as arising from making corresponding symmetries - U(1), SU(2), SU(3), Lorentz - local. Of these symmetry groups, only U(1) is a commuting group (U(1) transformations act equivalently in any order). This results in there being no quadratic term in the EM field strength. Each of the other symmetries involves nontrivial commutators, making them nonlinear.

In the quantum theory, QED, the situation changes. A pair of photons may scatter off one another, for example, via virtual pair creation, making the effective interaction of the photons nonlinear.

I'd recommend Sidney Coleman's 1961 article, ``Classical electron theory from a modern  standpoint'',http://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf, for a wonderful presentation of the issues pertaining to runaway solutions, among other issues.

I think there was some confusion here because in the discussion it should have been stated clearly what is a linear function of what. Secondly, we are talking of mathematical models of physics; if we add all the dirty side effects nothing is linear anymore. In Maxwell's theory, the em fields are linear functions of the charged sources and currents that are around, but if you take into account that these sources back react, then the combined equations become non-linear.

Only in this sense, the question posed is a meaningful one: if we keep the sources and currents fixed, then our mathematical models say that the em fields are linear but the gravity fields are not.

In mathematical terms, this can be explained by the fact that the local gauge group in electromagnetism is Abelian (i.e. the effect of two consecutive gauge transformations does not depend on the order) while in gravity it is non-Abelian (the effect of two consecutive curved coordinate transformations does depend on the order). Physically, this means that gravity carries energy and momentum (although this depends on the curved coordinates chosen), so gravity generates gravity, while em fields are electrically neutral.

All of this did not require the consideration of quantum mechanics. In ordinary quantum mechanics, what I say above is still valid. But now, even the vacuum has vacuum fluctuations of charged particles and they cause non-linearities in light when you include the back reaction of the vacuum.

Actually, photons DO have self-interactions, albeit tiny ones that can be neglected most of the time. (In perturbation theory, these self-interactions can be understood as being created by pairs of oppositely charged virtual particles.) It is conceivable that, with very powerful lasers, self-interactions of electromagnetic radiation will become observable.

Such interactions are due to quantum effects, however; whereas the self-interactions of (macroscopic superpositions of) gravitons are due to classical effects.

As Gerard said (and Stam implied) gravity generates gravity, But gravitons are hypothetical particles which need both the classical gravitational field and quantum mechanics for their description. Such a theoretical description is lacking, no astrophysical effect is known which needs it, thus it is premature to speculate about the interactions of gravitons.

When you speak of gravitons you already think quantum-mechanically!

The statement that "gravity generates gravity" is meaningless. Gerard said something more precise: Theories with a non-abelian gauge principle (such as QCD and General Relativity) have the feature that the fundamental gauge field has non-linear self-interactions.

Wikipedia is not an argument, Manuel.

I am getting tired of finding all the nonsense ResearchGate sends to my mail box. Why don't more people address the questions that are asked - rather than make us waste our time with remarks that don't have anything to do with those questions or, worse, are just nonsense.

Maxwell's theory is linear and it is it's drawback.

Einstein's General Relativity is non linear, but it is linear over a huge range of fields. In fact, if you follow the math for the observered black hole merger, you can see that the gravtitational wave energy was still in a fairly linear regime when its energy density was six orders of magnitude higher than the maximum energy density in the Schwinger Limit for electromagnetism. 

Gravity is the strongest force when measured as the maximum energy density, power transmitted, etc.

https://gravityphysics.com/2016/02/11/gw150914-enormous-gravitational-wave-energy/

First I very much agree with Juerg-Froehlich answer. I also agree with his calls(I have called the same in regard to the question of the radiation origin of matter) for the website to differentiate between comments and answers and label these correctly(according to the wish of the contributor of course) so that it becomes easier to follow the good answers of many that get lost by so many political rather than scientific answers/comments metaphorically speaking. Relying on the recommendations alone is not sufficient as you can have good answers that are not well understood resulting in small support.

I have a background in fluid mechanics and EM and the way I see it is that GR should be linear and people should abandon the pseudo 4 coordinates and stick to 3 space and 1 time. This doesn't lessen from the theory of course.

Interactions in the presence of matter can then be added like viscosity and turbulence models of fluids, and similar closer models in nonlinear Schrodinger, EM itself in optics, and in plasma electrodynamics and others. More progress in space research can be made this way I feel.

Since empty space is linear, all equations in their origin must be linear. The reason interactions cause non- linearity is the existence of two coordinate systems at the same time.

Wave equations are hyperbolic and represent propagation only.

If we decide to use Maxwell equations in empty space to represent matter and radiation, we have to realise that matter requires elliptic equation that are 'non-propagating'. It is here where extra (closure) models are needed to govern the exchange of energy and the change of phase between the two.

Jürg Martin Fröhlich is right. There should be a clear distinction between 2 or 3 categories: 1) Questions (open to anyone), 2) Answers (only for professional scientists), 3) Reactions and opinions (open to anyone, but regularly cleansed by an editor).

But the present site is an old one, we can’t change too much in it. I would be in favour of an edited site.

Dear Professor Gerard t Hooft , perhaps the catch is in who deserves to be called "professional".

Jose Gaite Gerard t Hooft No thank you. Leave this only free site. We've had enough of you 'professionals' and your monopoly.

The downvoting option offered in the past for each answer, should be enough if accompanied with a threshold (of bad scoring, negative ) set for each question. People scoring below such level are prevented to add posts in that specific thread.

It's interesting that a paper by Sidney Coleman, https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf on how to treat, in a consistent way, the radiation of an electric charge in classical electrodynamics isn't as well known as it ought to be.

The similarity between the Coulomb potential and the Newtonian potential is more fortuitous than fundamental, as a study of the symmetries of the two interactions, at a deeper level, shows. That's why it doesn't make sense according significance to their apparent similarity.

Stefano Quattrini, Alireza Jamali , Jose Gaite ,

Exactly, like Jorg Frohlich I would limit my responses on a site where professionals give answers to interested lay persons. This distinction should not be based on upvotes or downvotes, but on university connections. This site demonstrates clearly why we need such a splitting. I am sorry, but this is how the world works. I know what comments I’ll get. If that will be more than I can bear (and I can bear more than most of my colleagues) then I’ll simply quit.

Gerard t Hooft If you are here to answer 'laymen' you have definitely misunderstood the whole idea of ResearchGate. You've come to the wrong place to preach and teach your academic wisdom. For that purpose other sites like Reddit and Quora exist.

for very strong electric/magnetic fields, maxwell theory is replaced by a nonlinear theory, the Euler-Heisenberg theory. hence, maxwell theory is not sufficient for the whole spectrum of electromagnetic phenomena, but a nonlinear Heisenberg-Euler lagrangian is needed.

Since I raised the point, let me add something. I dare say that I have read on this site rather unprofessional comments, but I do not know how to define "professionalism". A dictionary says: "the qualities connected with trained and skilled people". Scientific training is necessary, but in which subject?

"University connections" is an objectionable criterion (in addition to being somewhat indefinite). ArXiv has an endorsement system, but I am afraid that that is not suitable here.

Dear Dr. Mohammad Asif,

You write: "The perihelion of orbit of Mercury in the gravitational field of the Sun is an example of non linearity of gravitational interaction".

Non linearity of what? gravitational interaction?

It is quite possible to construct a linear theory of gravitation (field equations) for which the relativistic Mercury perihelion precession is satisfied trivially.

So though this observation is consistent with general relativity GR, this observation does not require GR.

If you refer to nonlinearity of the solutions and not of the theory then It is also pertinent to remark that a non-linear gravity theory does not necessarily imply non-linear solutions to the field equations

For instance, in Eddington co-ordinates even the exact Schwarzschild metric is perfectly linear.

I left a proper comentary on your question to the future.

Bestwishes

J.F. Pascual-Sánchez

Mohammad Asif Gerard t Hooft Alireza Jamali Jose Gaite Here's is the story. After publishing his theory of Special relativity, Einstein was looking to include gravity within his special relativity. While he was working on that, in 1911 Max Abraham put forward a theory in which he included gravity within special relativity. But with a catch, Abraham made a mistake by concluding that to include gravity within special relativity speed of light must be a variable. Einstein didn't like that and rejected his theory saying a variable speed of light would destroy Lorentz invariance. Same thing happened with Nordstrom theory of gravity. To maintain the constancy of speed of light and to include gravity within relativity, Einstein needed a mathematical trick which was to curve spacetime. This resulted in a non linear theory different from Maxwell's theory. Now the question is whether curving spacetime is the only option we have to include gravity with relativity and keeping speed of light constant? There are also alternatives to general relativity like scalar theories of gravity but with not much success. The reason for all this lies at the heart of classical mechanics which poses a hindrance. Even Maxwell's equation, being a classical theory, suffers from that problem. For long physicists were unable to quantize it untill feynmann came along but that's a different story. Btw, Gerard is an expert in ironing out this problem with renormalization technique.

Rudra Pandey So what? I knew your `story'. What is the point of tagging me in your exalting of `Gerard'? I have no interest in hearing these old stories.

• Alireza Jamali ok dude. My mistake. Won't tag you again. Pheww.

A small comment about the constancy of the speed of light which seems to come up often in our present discussion.

Many seem to think that by saying that; the ether is superfluous, Einstein meant that it doesn't exist. I think we should interpret that statement as not needed not doesn't exist. All that is needed is the constancy of the speed of light for everything to work out in his case.

Now, in my opinion, the constancy of the speed of light can be a proof of the existence of a medium not the other way round. Take sound speed in a medium. It is constant and is independent of the source or the receiver motion.. because it is determined by the properties of the medium in c^2=K/rho, where c is the speed of sound, K some elasticity constant of the medium, and rho is density. Thus if you send a sound pulse it will move in the same speed regardless of you sending it from a moving car or receive it in a moving car.. all that can change is the frequency being shifted one way or the other- Doppler shift.

The speed of sound will only change if the whole medium moves as in sound in a moving water pipe for example. As the speed of light doesn't change according to all experiments, we should conclude that ether(whatever is that) never moves. It is interesting to note that E=mc^2 can be written as c^2=E/m and if you divide top and bottom by V the volume that contained the two we get; c^2=energy density (pressure)/matter density.. This is the same as the sound speed in matter. This idea is not originally mine but the person who wrote it was asking; would I be considered a genius if I wrote Einstein formula like that!!.

I then asked myself why then the ether it trying to imitate matter in this behaviour when it doesn't have matter and doesn't know what is matter. The answer seems to be is that matter is imitating empty space not the other way round... I realised this when I learned that 99.999..% of matter is empty space.. so ..no wonder.

Riadh Al Rabeh wrote: “in my opinion, the constancy of the speed of light can be a proof of the existence of a medium not the other way round.”

In the frame of the “theory of informatons”, it is assumed that “g-information”, respectively “e-information” is the substance of gravitational and electromagnetic fields. The elementary quantity of g-/e-information is carried by a mass and energy less granular entity, called an “informaton”. Informatons are emitted by material objects at a rate proportional to the mass of their emitter. Relative to an inertial reference frame they rush away with the speed of light and they carry information about the position, the velocity and - where appropriate - the electrical charge of their emitter.

The “theory of informatons” - where informatons are the constituent elements of the media hat mediate in the gravitational and in the electromagnetic interactions (the gravitational, respectively the electromagnetic fields) – perfectly explains the description of gravity by gravitoelectromagnetism (GEM - the Maxwell-Heaviside equations) and the classical description of electromagnetism (EM – Maxwell’s equations). In that context a gravitional wave as well as an electromagnetic wave (light) are understood as “trains of informatons” that – relative to an IRF - move with the speed of light.

Article INFORMATION AS THE SUBSTANCE OF GRAVITATIONAL AND ELECTROMAG...

The confidence (arrogance?) of some people who understand little and berate experts is truly something marvelous. I don't know why ResearchGate suggested I read this, but I did get a laugh.

Stefan Bernhard Rüster At a glance it looks like you just decide the Einstien tensor is the the energy-momentum of gravity, which means in the absence of a cosmological constant it is exactly equal to the matter stress-energy tensor. Which is obviously nonsense. If this made sense, what incentive would they/we have to "deny" it? I'd love to see gravitational energy understood, but there are so many immediate problems with your idea and it's clear you have no respect for those who actually do understand things better than you. What's more likely, that your idea is wrong or that the world's experts conspired together against you?

Stephen McCormick wrote: “I don't know why ResearchGate suggested I read this”

GR is a brilliant theory about the gravitational phenomena and interactions and experts in GR are brilliant theoretical physicists. But GR is not the ultimate theory of gravity and experts in GR are not possessors of the absolute truth.

The sensitivities of some of these experts cannot be a reason to introduce on Research Gate censorship with the intend to silence those who propose consistent alternative gravitational theories.

Stefan Bernhard Rüster no, I only need to look at a little to see it's obviously not true, and given how you've responded to others I certainly don't owe you my time. It isn't any kind of argument at all to insist that all people either agree with you, or do not understand things -- you seem to completely rule out any possibility that anyone might understand things better than you. This isn't how science, or even general human interaction, works.

Everyone is free to open his mouth, but every now and then I like to close my ears.

"Everyone is free to open his mouth, but every now and then I like to close my ears."

ResearchGate.net does not put requirements to the level of skill of the participants of discussions. So, you must expect all kinds of contributions and many of them are emotional or intuitive. I learn a lot from the reactions of discussion partners and not because they show sufficient scientific skill. If the subject concerns the structure or behavior of physical reality or about the interpretation that humans give to these subjects, then the input of unskilled participants can be as valuable as the input of a skilled scientist. I am convinced that in its essence the structure and behavior of physical reality are very simple. It must be possible to explain these essentials to both skilled and unskilled intelligent humans. Applied exact science is very successful and reliable because it applies precise descriptions and does not require precise explanations. In contrast, theoretical exact science is supposed to supply explanations and is not very successful in that task. None of the existing theories starts at sufficiently low levels such that that level can be used as a trustworthy foundation of the theory.

It is possible to start at a very simple level and construct an alternative theory that explains much of our living environment. This is shown in "How to create a universe" ; https://vixra.org/abs/2108.0004

Following Feynman et al, gravity cannot be fully described but by means of non-linear field equations, which is the case of Einstein's field equations. This non-linearity is due to the affine connections as pseudotensors of third rank.

To linearize gravity, one can apply the simplified perturbation theory with gauge invariance to the metric tensor and EFEs become linear partial differential equations of second order.

This is not the case of Maxwell's differential equations,which are linear of first order.

More details can be read in "Feynman Lectures on Gravitation."

On ``Why...?'' questions: https://www.youtube.com/watch?v=Q1lL-hXO27Q

Without an agreement on what's assumed known, discussion is pointless and just generates replies and reads.

The first part of the question, ``Why is GR highly non-linear'' doesn't have anything to do with the second part ``while electrostatic interactions...'', since the Newtonian and Coulomb potentials are not identical-charges can be shown to have both signs, masses, however, can only have one sign and this is a profound difference-so the answer to the first part is independent of the second part. The comparison is meaningless.

Incidentally, ``highly non-linear'' doesn't mean anything-the distinction is between ``linear'' and ``non-linear'' (and what's meant by the qualifiers refers to the equations of motion).

The answer to the first part was given by 't Hooft some replies ago and has to do with the properties of GR, in the generic case.

(It should be mentioned that, in certain special cases, Einstein's equations are, in fact, linear, in the sense that they are integrable, cf. Article Thirty years of studies of integrable reductions of Einstein...

Most of the comments here-and elsewhere-are, simply, due to not having studied GR; it might be useful to actually study it, for instance here: Article Lecture Notes on General Relativity

or here: https://webspace.science.uu.nl/~hooft101/lectures/genrel_2013.pdf

After that, it's possible to understand that any modification of GR builds upon it, doesn't replace it (since GR is defined by the property of invariance under general coordinate transformations; the only question is, whether the metric is the only field that defines the spacetime geometry-and it's possible to show that the most general formulation, involving fields with spins up to 2, is known, namely, supergravity. Generalizations that involve fields with spins greater than 2 are, however, also, studied, but a complete formulation is not, yet, known). The presence of fields beyond the metric, that define the spacetime geometry, and can't be understood as matter fields, by a redefinition, can be probed by searching for violations of the equivalence principle, since the additional fields would be sensitive to the particle content of matter, not just its energy-momentum tensor and this has been used to put bounds on certain, though not all, generalizations of GR.

Why is GR highly non linear while electrostatic interactions are linear even in the comparatively stronger electric force limit?

As a matter of fact , amongst all the various inconsistencies that there exist in GR theory , this is one of the most conspicuous ones. However , there ARE ways and means to remedy it .

In general, a theory of physics could have defects of all sorts. I divide them in 3 categories:

A) There could be internal logical inconsistencies in its mathematical construction;

B) There could be conceptual difficulties in the physical interpretation of a theory,

and C) It may well be that the theory does not describe the physical world in the way it claims to.

It would be helpful if those who criticise GR would disclose which of these three kinds of deficiencies they talk about.

In quantum mechanics, for instance, A and C are perfect, but in B, many researchers have complaints; I nyself have serious objections to what is often said.

The Standard Model of the subatomic particles is good but not perfect in all three manners: it has internal mathematical shortcomings, it shares its interpretation problems with quantum mechanics, and there are indications that it does not agree perfectly with all observations. Of course, it does not describe gravitational forces. Remarkably, if not perfect, the Standard Model works amazingly well.

In the latest arguments raised in the discussions here, it seems that some people think that GR is (A) internally inconsistent, others think it to be (B) wrongly interpreted, and (C) some think that GR does not apply to the real world.

My response to that: (A) and (B) are perfect. We know exactly what GR says and how to calculate anything we want to know. As for (C), GR does not include quantum effects in a satisfactory manner, so that it can at best be a (very good) approximation. In my opinion people who think there are fallacies of type A or B should find themselves a good text book to learn how the theory really works. As for C, the underlying principle, invariance under general coordinate transformations is difficult to drop since it would make the gravitational force uncalculable; I personally think nothing can replace it. Yet QM would force a re-thinking of these principles, probably replacing them by something better. The universe is almost perfectly flat in the spacial direction and this could serve as a starting point, but most investigators, myself included, are not smart enough to do this right.

Outside any popular theory: Photons of an electric field are not attracted to the source. The energy or mass of a time varying gravitational field (like all mass) is attracted to the source of the field. Static gravity by definition is not moving things and therefore changing so it does not have to transfer (energy or mass). The case of back hole the only way gravity's mass is by quantum tunneling (having a wave function longer that the hole). Note gravity wave have clocked at speed of light.

In at lest that way gravity differs from electro-magnetic fields. Also although the potential of static electric filed is liner, the intensity (area density of energy proportional to area density of the photons) of a time varying field has a inverse square relation with distance. Like wise for weak time varying gravitational waves the area density of energy (area density of gravitons) has a inverse square law. But for things approaching the time varying gravity of black holes things get more non-linear.

Gerard t Hooft I don't think GR is any of those 1. Internally inconsistent 2.wrongly interpreted 3. It does not apply to the real world. It's none of those, it's a mathematically sound theory, no doubt about it. It is more of a unnecessary theory, it is not required. GR is the only theory which is background dependent, out of 4 forces it's the only description of force which requires curvature of spacetime. It predicts precession of mercury, bending of light etc. but that can be achieved by alternative theories of gravity too.GR solves many problems but it has created many other problems. Gravitoelectromagnetism can be a good alternative to GR if we can remove the problems associated with it. We can then put gravity on equal footing with GR.

Misgivings about GR are to my mind misplaced; discarding GR would put us in the scientific Stone Age < 1900. Reconciling it with QM is a challenge, even if difficult.

If cosm. const. would not be constant, on what should it depend? GR with variable Lambda would me a monstrosity even when compared to GR with fixed nonzero Lambda.

Wait for some time, you'll have to throw it away.

Theories don't get ``thrown away'' unless they're found to be mathematically inconsistent. (If they are mathematically consistent, but seem not to describe experiments, they provide the framework for describing the backgrounds to experiments.) And what matters isn't that they're mathematically inconsistent, but what the mathematically consistent formulation is. For every mathematical statement has assumptions, so what's relevant is what the assumptions are, that lead to the alleged inconsistencies.

GR is mathematically consistent (though not mathematically complete), that's been known for many years (cf. here: Article Initial Value Problem in General Relativity

The cosmological constant is as much a part of GR as Newton's constant is. Once upon a time this wasn't understood; now it is.

Even were the accelerating expansion of the Universe not fully described just by a positive value of the cosmological constant-which, for the moment seems to be the case- this would, simply, mean that there are additional fields, beyond the metric, that determine the spacetime geometry. Such fields are known and the most general way to take them into account, if their spin doesn't exceed 2, is in supergravity. And, even in that case, a cosmological constant is an inevitable feature (in four dimensions). It's possible to probe the existence of such fields by deviations of the equivalence principle-that's how it's possible to state that the only significant contribution to the accelerating expansion, i.e. dark energy, is the cosmological constant.

The statement that GR doesn't describe quantum effects is, as usual, the result of a confusion: GR is the classical limit of some quantum theory. One must, first, construct the quantum theory and then show that it possesses a classical limit and that this limit is GR. It's not possible to deduce the quantum theory from its classical limit. That's unfortunately a common error.

Stam Nicolis

Mathematic consistency alone is not criterium for acceptable physics . Mathematical consistency has to , then , be in harmony with a variety of experimentations .

Respectfully

REZA

The CC can not be constant. Its actual value consistent with the flat and slow changed Universe, there are no reasons why it must be the same at the Big Bang moment.

Gravitation is a complicated activity and the electric field differs from the gravitation field. Please read "How to create a universe" ; https://vixra.org/abs/2108.0004

Dear Gerard t Hooft, I readily agree with you. The CC becomes exactly constant = ZERO in  vacuum as a result of the novel matter-antimatter (OM/AM) gravity "charges" symmetry - (antigravity), predicted in my periodic 3D-waveguides 4D-multiverse (PWM)-concept (see [1, 2] and citations therein).

[1] Gribov I.A. 2019 Periodic waveguided multiverse as a source of dark matter and dark energy. J. Phys.: Conf. Ser. 1147 012083. http://iopscience.iop.org/article/10.1088/1742-6596/1147/1/012083/pdf

[2] Gribov I.A. 2021 Singularity-free stable black holes, holding the baryon conservation law in the periodic waveguided multiverse J. Phys.: Conf. Ser. 1787 012040

https://iopscience.iop.org/article/10.1088/1742-6596/1787/1/012040/pdf

The 4D-PWM-concept creates totally novel - pure multiversally united physics (the co-emergent 3D SR/QM/EP/GR, etc) in the periodic 4D-chain of the intrinsically physically identical 3D-waveguides with the co-emergen ~3D OM/AM/DM/DAM... universes / antiuniverses with the multiversally expanded PWM-GR [2], where our visible universe is a mixture of equally presented (OM+DM) clusters and (AM+DAM) anticlusters with ~70% of repulsive gravity (as the DE-nature) and 30% attractive one in this weightless = (spatially flat) mixture (now on the large ~10-20 Mpc scale). Notably, the PWM-concept has some experimentally testable predictions [1,2], including OM/AM antigravity and this tremendous prediction will be verified (as I strongly believe) oder rejected very soon at CERN in the antihydrogen-gravity test. Waiting for your responce,

Sincerely, Dr. Iourii Gribov

It oughtn’t be unknown that, due to the fact that time translation invariance is a local, not a global, symmetry, focusing on the conservation of energy in curved spacetime is an exercise in futility. This doesn’t have anything to do with the mathematical consistency of GR, however (it may have had some historical interest, but how to deal with local symmetries is, now, part of standard courses in field theory and homework problems). So any statement that, because energy and momentum aren't conserved globally, might be a cause for claiming GR is inconsistent-or incomplete-is, just, wrong.

Tensors aren't observables in curved spacetime, because they carry spacetime indices (they're not observables already in flat spacetime for the same reason). They're building blocks of gauge-invariant quantities, i.e. quantities that are invariant under diffeomorphisms. That's why it's not useful focusing on them and not completing the calculations, that belong to private notebooks, not on discussion fora.

It might be useful to peruse Preprint Mathematical General Relativity

Stam Nicolis

I agreed with much of what you say but not all. There is nothing wrong with a properly defined energy concept in gravity. Sometimes people want to define energy in a completely covariant way and that is not possible (you could use the einstein tensor, but then the total energy vanishes, which makes it not very useful). But you can define energy as soon as you use an arbitrarily chosen background metric. That metric could be completely flat Minkowski spacetime, or a completely smooth, almost flat cosmological configuration. The actual spacetime metric, as dictated by Einstein's equations including matter, is then treated as a perturbation with respect to this background. You can then use time translation invariance of the background (if it exists!) to define energy momentum tensors. If the background is not time translation invariant the energy is not conserved, but you can easily compute what happens. The background delivers (or absorbs) energy to your system. It’s all in the equations, you shouldn’t be scared of using them ...

@Gerard 't Hooft I agree, everything is in the equations, I just wanted to point out that non-invariant quantities are building blocks. Choosing a background is a necessary procedure, but nothing can depend on the background chosen, isn't that so (just like nothing can depend on the choice of gauge)? So how can the statement that a background delivers or absorbs energy be made meaningful? Something more is needed, right? If a background metric is fixed, then, by construction, only its isometries are relevant and the conservation laws are defined with respect to it, if I'm not mistaken. That, of course, is possible. However, to avoid confusion with the issue of ``background (in)dependence'' I would just prefer to use another term (``reference metric''?).

Stam Nicolis ,

that is not at all as such, unless you consider conservation of energy only the one which is prescribed by SR in flat space-time. Conservation of energy is a very important topic relevant to the "exotic matter" it is strongly related to the role of a background...

One author who points out the importance of energy conservation, introducing the line element field is Gary Nash in his theory.

mathematical consistency doesn't have much to do with respecting physical constraints...

Because a strong gravitational field converts a centrifugal force into an electrostatic force of attraction. See here,

https://www.researchgate.net/publication/354656658_Centrifugal_Force_in_the_Schwarzschild_Field

Stefano Quattrini Conservation of energy means time translation invariance, so it's only defined in flat spacetime, for sure. In curved spacetime, where time translation invariance is local, this means that conservation of energy can only be defined, with respect to a reference metric, that has certain mathematical properties and then perturbations with respect to this metric can be described as transfer of energy. Quantities that can be changed by applying symmetry transformations aren't useful beyond using them for constructing quantities that don't change under such transformations.

``Physical constraints'' is a term that doesn't mean anything, unless there's a mathematical way of defining it, so only the mathematical definition matters. It's necessary to be able to test that the words are consistent with the equations. So there can't be any difference between the two.

Einstein's equations take the cosmological constant (and Newton's constant) as input, not output. They define an initial value problem whatever the value of the cosmological constant, in particular, the value zero. They, also, have solutions for non-zero values of whatever sign, so it's not true that there's a mathematically well-defined way of selecting the solutions with non-zero cosmological constant from those with zero. The solutions don't cease to exist. Experiment only can indicate what are the values of the cosmological constant and of Newton's constant that describe solutions, that lead to agreement between the calculations and the measurements (that have been designed on the basis of Einstein's equations; extensions with additional fields can be tested this way).

Stam Nicolis, Indeed, ‘reference metric’ would be a good word to use here. In many practical calculations with gravity, you want to define the concept of energy and the price is to refer to a reference metric. By preference this reference metric has a timelike Killing vector. Then energy is conserved. If you want to talk about energy in a moving elevator, or in a tidal energy generator, these might be useful concepts. Of course, there, relativity is not a big issue, so you can use your common sense to find out what happens. But now we can add that GR does not affect the laws of energy conservation, if introduced with sufficient care.

The Cosm.Const. does not change the fundamental principles, if done right; it could be regarded as the existence of a special kind of scalar matter field with non vanishing (ubiquitous) vacuum value and negligible kinetic term. So I would add this to the matter side of the equations. I’m not impressed by this very tiny modification.

This is an excellent discussion of GR and its structure. What is missing in the discussion is a fundamental understanding of which spacetime describes reality. I think all relativists and particle physicists would agree that a manifold with a Lorentzian metric is the best description of spacetime that we have. In a Lorentzian spacetime, GR and and equivalence principle nicely blend with SR during free-fall. However, this leads to the problem of the non-localization of gravitational energy-momentum in GR. In fact, this problem is so well known, some relativists claim that grav. e-m does not exist, and others claim that the nonlinearity in the Einstein tensor automatically takes care of the fact that gravity gravitates. The second claim can be shown to be the same as the first by writing the Lorentzian metric as a background metric plus a tensor representing a small perturbation around the background. The Ricci tensor can then be expanded to second order in the perturbed metric. An energy-momentum tensor that depends only on the background gravitational field and contains terms quadratic in the perturbed metric and its first two covariant derivatives is obtained. However, it is not invariant when the connection coefficients vanish as in free-fall. A connection invariant local expression for gravitational energy-momentum does not exist in GR.

What is poorly understood by almost all physicists is that a Lorentzian spacetime does not exist without a line element field. That follows from a theorem that states a non-compact paracompact manifold admits a Lorentzian metric if and only if it admits a line element field. Modified General Relativity is then constructed from the line element field and Einstein's original postulate of a total energy-momentum tensor that includes the energy-momentum of the gravitational field, phi_ab, which is introduced by a decomposition of symmetric tensors; the Orthogonal Decomposition Theorem. phi_ab is connection invariant because it is constructed from the Lie derivative of both the Lorentzian metric and a product of unit line element covectors. Local gravitational e-m exists in MGR.

In MGR, the cosmological constant is dynamically replaced by the trace of phi_ab. Local e-m conservation is given by the vanishing divergence of the total e-m tensor and not the matter e-m tensor. MGR completes GR.