Dear Ali, This is my theory and I wrote book on universe, its called Quantum Mechanics Universe. As the name implies, the Universe is not a mechanical entity as Newton or Einstein are stating. The Universe is hollow spherical, and it rotate as well. the reason we see galaxies in different angle is, because of rotation of universe makes all galaxies rotate in opposite direction, therefore they are weightless and float in space of universe. The universe rotation is make look-a-like gravitational force, but gravity is in atom not outside of mass. https://www.academia.edu/37525574/Quantum_Mechanic_Gravity it make sense that everything must have QM foundation. best wishes

Dear Stefan Bernhard Rüster

Thank you for your reply. My point is: I want to derive a relation between the circular velocity of a certain galaxy, as a function of the distance from the center. Let's forget about dark matter for a bit. To do this in Newtonian mechanics, one equates the gravitational force to the centrifugal one. The question is: What's the equivalence of this procedure in GR?

You don't need any kind of exotic theory to derive the actual rotation curve. You simply take a spectrum of the galaxy at various positions along its major axis (usually on both sides of the center, but barring an unusual structure, one semi-major axis is just as good as the other), see what the redshift or blueshift of the major absorption lines is, then subtract the value for the redshift or blueshift of the core. The difference gives you the rotational velocity at each point on the axis, as a fraction of the speed of light, and a graph of that is the galaxy's rotation curve. (Of course when you get far from the center the stars start to spread out and you need a longer exposure to get a satisfactory spectrum, but that's just a technicality.)

The only place where you would apply Newton's Law of Gravity is when you compare the rotation curve to the distance from the center (determined by the angular distance from the center and the distance of the galaxy from us), to determine the mass between the center and a given point on the axis. General Relativity is of no use in this calculation, because it only differs from Newtonian Relativity when the gravitational force is very strong, and since galaxies are so spread out, the only place you might have a strong force is if a supermassive black hole in the center of the galaxy alters the redshift or blueshift for the center. However, that is very rare, as most supermassive black holes are not close enough to enough of the stars in the core to alter the Newtonian redshift.

In other words, it is simply a matter of measuring the rotation rate, then applying Newton's Laws to the results. General Relativity is completely irrelevant, and trying to apply it to the situation is a waste of time and effort.

Of course, the fact that almost all galaxies yield rotation rates that can only be justified by assuming that there is close to ten times as much mass in some kind of dark matter as in the visible stars means that you might want to give some thought to just what that dark matter is. There are lots of theories about that, but none of them involve General Relativity, and almost all of them (and perhaps all of them) are wrong. For a discussion of some of the theories, why the simplest ones would be so boring and hard to prove that hardly anyone is working on those ideas, and the various failures to prove that some kind of "strange" dark matter is the solution, see http://cseligman.com/text/galaxies/dark.htm (Dark Matter in Galaxies), on my website.

Using GR to model rotation curves of spiral galaxies has been explored by

Cooperstock & Tieu: https://inspirehep.net/record/728855?ln=en

but not without serious challenges: https://inspirehep.net/record/690135?ln=en

and https://inspirehep.net/record/698096?ln=en

and https://arxiv.org/abs/gr-qc/0604092

There are more recent, and apparently more successful, attempts by Rodrigues et al.: https://arxiv.org/abs/0911.4967

and https://arxiv.org/abs/1102.2188

and also applied to elliptical galaxies: https://iopscience.iop.org/article/10.1088/1475-7516/2012/09/031

There are many other technical papers using variants of, as well as modifications to, GR to fit rotation curves.

Ali, GR does not provide the flexibility to describe all types of galactic rotation curves, which is why other theories have been pursued. Other researchers have mentioned many of them above. The problem with the rotation curves led to the idea of extra matter required to explain them; and the proposal of dark matter with GR left intact. Recently, it was shown that GR is not complete in the article published in General Relativity and Gravitation Article Modified General Relativity

Dear Ali,

As a tip I would tell you that first you will start to calculate the earth-moon system with both methods and thus you make sure that you are doing it correctly

As pointed out Cooperstock & Tieu and show galactic rotation curves without Dark Matter using a GR treatment, nothing new is introduced so I am not sure why it should be challenging. The only concept one needs to understand is a GR potential. Actually the mathematics is not that difficult.

MOND certainly is not GR and has nothing to do with GR. People are literally afraid of general relativity calculations and always look for the simplest solution, in my mind to the point of simplistic. However, once understood GR is simple, and like Einstein said, "... it should be simple but no simpler."

I have spoken to Fred Cooperstock on a couple of occasions, read his papers and his book "General Relativistic Dynamics - Extending Einstein's Legacy Throughout the Universe."

Book General Relativistic Dynamics: Extending Einstein's Legacy T...

It is a simple treatment and after reading his work I cannot for the life of me understand why people still persist with Dark Matter.

My degree was in astrophysics, we were told use galactic profiles within the weak-field limit, as GR is unnecessary. When asked why, they always respond, doing a full GR profile is too difficult and when I said I wanted to do it for my thesis my supervisor relied "it gives me a stone in my stomach just thinking about it." This is why astronomers/astrophysicists try to keep the calculations simple.

Fred is a professor Emeritus and soon to retire yet his work goes largely unknown.

Rotational dynamics in GR can flatten a galaxy into a disk the same as Newtonian dynamics but the velocity of the stars are different and so too are their potentials.

Estelle, GR was constructed in a Riemannian geometry and cannot describe the energy-momentum of the gravitational field itself. Spacetime is more properly described on a time-oriented Lorentzian manifold which admits a non-vanishing vector field that, in turn, is fundamental to a tensor that does describe the energy-momentum of the gravitational field. That tensor provides the additional gravitational force in galaxies to describe dark matter; and it also describes dark energy. So while it was wise to apply the full capabilities of GR to try to describe dark matter, it cannot fully do so because it lacks the additional necessary freedom to accomplish that task. For example, the Tully-Fisher relation does not naturally come out of GR. In Modified GR, it does.

Dear Stefan Rüster,

That is the thinking, but if you read Cooperstock's paper you will see otherwise.

https://arxiv.org/pdf/1101.3224.pdf

It is about a GR potential and the effect is pronounced. Many people discount this idea without reading the work and insist on using weak-approximation profiles for calculating rotation curves.

It reminds of doing the same thing over and over again and wishing for a different result.

Guys

Getting back to first principles, Coopestock use a GR treatment that is standard in many cases, without the introduction of any new free parameters and naturally a gravitational potential arises, it is really that surprising. However, Cooperstock's treatment is not Newtonian and is based in a GR metric, regardless of the potential psi. In any case, if you look at the results of the 2010 paper you will see rotation curves, which are not tweaked, that properly normalize without the need for dark matter. The results are tantalizing, and yet people most to lineraised or weak-field approximations or even worse still to MOND, which is totally ad hoc. Here is a GR treatment that works properly.

Just because you can get results from weak-approximations of GR that are consistent with observation does not mean those weak-field methods are essentially correct, just that some information is correlated as it would be. Ptolemy's epicycles produced the same results as Copernican heliocentrism, and on the same line of though then Ptolemy's model should be correct, but we know this is not true. Just because a different method gets a similar result is not what we call robust.

In any case, GR in its standard form has passed every test, there has been no exception or deviation in any way.

https://arxiv.org/ftp/arxiv/papers/1705/1705.04397.pdf

However, people persist with weak-field limits and also MOND, which is beyond belief. I understand the linearised argument, that is seems logically consistent but I would rather start with a full GR structure than an approximation.

What is occuring now, is that we are finding galaxies without Dark Matter. Here is one such example

https://arxiv.org/pdf/1803.10237.pdf

They are using Sersic profiles, which were used in previous studies, so why no Dark Matter? All the no Dark Matter galaxies that have been found share similar traits, yet they are not the same type of galaxy. If the Dark Matter paradigm is correct we should find DM in 'all' galaxies, in all stars and indeed in everything. This is the first real crack in the DM hypothesis. Remember a dM particle has never been detected in spite of many experiments, which have all failed to find a DM particle.

It would seem relevant to review foundational physics in regards to the only real evidence of Dark Matter; namely, rotational curves. There is no other real evidence for DM, some say but what about galactic clusters, well they are part of the same methodology.

I am working on a different principle but it does employ a full GR metric but it is different to Cooperstock. In any case, we need to move away from linearised, or weak-field approximations or even MOND and work through the calculations. This is something that people say they can do, but very few actually do it.

I would like to quote Fred's paper which pretty much illustrates why a GR must be used,

" In dismissing general relativity in favour of Newtonian gravitational theory for the study of galactic dynamics, insufficient attention has been paid to the fact that the stars that compose the galaxies are essentially in motion under gravity alone (“gravitationally bound”). It had been known for many years, in fact since the time of Eddington, that the gravitationally bound problem in general relativity is an intrinsically non-linear problem even when the conditions are such that the field is weak and the motions are non-relativistic, at least in the time-dependent case. Most significantly, we have found that under these conditions, the general relativistic analysis of the problem is also non-linear for the stationary (non-time-dependent) case at hand."

Thierry De Mees I did not assert the things you say. You said, " first you pretend that Coopestock uses non-linearized GRT, while in fact he does." Yes, its non-lineaized, agreed. It is not weak-field approximate, it is non-linearised.

The curve fits are not Newtonian, I am not sure how you see it that way. If you think that is from the Bessel function, it isn't. I did not pretend DM exists, I do not believe it does! This whole treatment is non-Newtonian and according to Fred Cooperstock it isn't as well. If you could be more specific that would help.

Stefan, you state in the abstract that " the intrinsic gravitational force density, which naturally follows from the Golden Rule of Mechanics, in order to compute flat rotation curves of spiral galaxies." I have not see something like this before and at just glancing at it you do not use a metric nor the dynamical properties of gravity in fixed motion of stars but examine the centrifugal force. This does not make sense to me. Stars are in fixed gravitational dynamics are they rotate, so any centrifugal force would be restricted by momentum energy tensor. I think that has been addressed in the literature.

The key thing in Cooperstock's work is that energy within the energy-momentum tensor is localised, and so that function may be 'viewed' as linear but it is not, it is dynamically. This is where he is at odds with mainstream GR researchers, the localisation of energy suggests that energy cannot be transferred by gravitational radiation - not covered in that paper but in some of his other work. I am not sure of this aspect, but certainly it would be true in a limiting sense.

Here is the original 1992 paper about it.

Article Energy localization in general relativity: A new hypothesis

BTW Fred passed away Jan 2018.

A rotating self-gravitating mass distribution usually assumes a spheroidal shape. This result is due to Newton and MacLaurin and is of great practical importance as it applies to such diverse systems as the Earth's atmosphere and a rotating galaxy. After reading the article by Cooperstock and Tieu suggested in this thread, I think it is likely that General Theory of Relativity can be used to evaluate the similar mass distribution as well as the correct shape of the rotation curve without an ad hoc correction by adding the “black matter”.

However, the main problem lies in the internal circulation that develops in the rotating spheroidal mass distribution and in the changing in time and space release of energy that affects the mass / gravity fields and effectively the rotation rates. The authors do not address this problem, and this is not surprising since solving this task is still difficult even in the Newtonian gravity theory.

And last but not least, the question arises how the spiral arms of the galaxies are created. Is this due to the background rotation rate of the Universe?

Stefan

I know, I have that book and used it in my degree but this is the very issue, and what general relativistic dynamics will handle it better, "the gravitationally bound problem in general relativity is an intrinsically non-linear problem even when the conditions are such that the field is weak and the motions are non-relativistic, at least in the time-dependent case."

Hence you cannot use a Newtonian or linearised gravity for these calculations as their mathematically relationship is non-linear.

Janusz Pudykiewicz, thanks for your comments. Actually, what you are referring is perhaps the key issue many people have with work Cooperstock et. al. You said, " mass distribution and in the changing in time and space release of energy that affects the mass / gravity fields and effectively the rotation rates." Cooperstock published quite a bit prior to this work and its cited in these papers of their localised energy model. This is where Cooperstock departs from the mainstream, that in general relativity, energy only exists in regions of non-vanishing energy–momentum tensor. The original work goes back to 1992.

Article Energy localization in general relativity: A new hypothesis

They suggest that the energy is maintained and much of the change in spacetime of position and hence energy is conserved, for the energy is localised.

Dear Estelle,

In the second of Mitchell Struble's list of references (see July 18, above),

"Singular disk of matter in the Cooperstock-Tieu galaxy model"

[ https://arxiv.org/pdf/astro-ph/0508377.pdf ]

the author (Mikolaj Korzynski) claims that:

• "2. the model of galaxy they propose is questionable because it posseses an additional source of gravitational field in the form of a rotating flat disk at z = 0."

Regarding this additional gravitational source ("rotating flat disk"), on page 4, just below eqn (16), Korzynski writes:

• "As before, due to the presence of |z| in (14) the first equation has a distributional singularity at z = 0 rather than 0 on the right-hand side. In the following section we shall prove that this results in presence of an additional, distributional term in the stress-energy tensor T^uv."

Do you have any thoughts on this possibility, that the Cooperstock-Tieu galaxy model of 2005 [ https://arxiv.org/abs/astro-ph/0507619 ] may have included an effect from such an additional (unsuspected?) component in their model?

My reason for mentioning this is that (at first glance) such a component seems to be similar to the sort of invisible superfluid component explored by Justin Khoury (e.g. https://arxiv.org/abs/1506.07877). Khoury and others have used such disk-like components to achieve impressive matches with observation.

Note: in Khoury/Berezhiani dark-disk models, such local, superfluid components are to be distinguished from the invisible distributions of ("locally localized"?) mass-energy thought to be responsible for large scale clustering of clusters.

Some discussion in Quanta magazine:

https://www.quantamagazine.org/how-superfluid-dark-matter-mimics-an-old-idea-about-gravity-20170613/

And some critique:

https://www.quantamagazine.org/deathblow-dealt-to-dark-matter-disks-20171117/

PS: I like your idea of locally localized inputs for the relativistic treatment. Looking forward to your work on this!

Nigel

To all Readers,

Using all the interesting contributions in this thread in combination with a thought experiment with rotating self-gravitating mass distribution I have following reflections to share:

Both General Relativity (GR) and Newtonian Gravity (NG) can be used to obtain rotation curves that are consistent with observations without the need for "non-Occam" approaches such as "dark matter" or "MOND".

The above observation is not surprising since both GR and NG belong to a class of classical field theories and it is quite likely that they should converge for a certain range of parameters.

The approach presented by Stefan B. Rüster is elegant and instructive, I would also like to suggest a brief review by Hofmeister and Criss: "The Physics of the Galactic Spin" for comparison.

Deriving the rotation curves is much more complicated in the GR framework. The results discussed by Cooperstock and co-authors in a series of papers are based on a cylindrical metric and a simplified stress-energy sensor. In order to get a general solution, both simplifications should be eliminated.

One of the biggest problems with the cylindrical symmetry is that the disk, with the sharp edges stipulated by this symmetry, is gravitationally unstable due to the problem of balancing the vertical gravity component with a horizontal centrifugal force.

The solution within GR should consider a rotating relativistic fluid with a realistic distribution of mass and energy sources / sinks in the spherical coordinates. The same observation applies in the NG framework, where one can follow the theoretical model discussed by Lebovitz (1966) in "Rotating Fluid Masses".

General Relativity for billions of galaxies rotation? Where the motivation of movement of rotation of all these billions of galaxies is coming from? Why all these billions of galaxies have different rotation, and they are weightless? GR is not even working on our galaxy. The fact is the Universe is following pre-coded procedure, Maybe this article bring little better view https://www.academia.edu/38373675/Creation_of_a_Quantum_Mechanic_Universe_and_its_Rotation.doc best

Virial theorem does no longer hold in the non-Euclidean galaxy`s space-time. We derived the hyperbolic equitation of motion from the perspective of GR, exactly predicts the flat rotation curve.

Article Title: Non Dark Hyperbolic Universe

http://www.medcrave.com/articles/det/17968/Non-dark-hyperbolic-universe

The Big Bang hyperbolic universe neither needs inflation nor dark matter and dark energy: Physics Essays, Volume 26, Number 3, September 2013, pp. 422-429(8)

Article The Big Bang hyperbolic universe neither needs inflation nor...

Dear Stefan Rüst

A pressure-less flat-dust universe model is an oversimplification solution to the Einstein's Field equations. This led to a drastic failure in predicting the missing mass and the accelerating expansion of the universe. The Universe is flat if and only if DM, DE, and inflation are exist. Non of them is confirmed. We proved mathematically that the universe is hyperbolic.

Article The hyperbolic geometry of the universe and the wedding of g...

Stefan, it is very early in our exploration of the "entire universe" to declare that it is flat. More likely, it is slightly not flat. We are probably, at this stage of the exploration of space, where the "flat earth" observers were hundreds of years ago.

Dear Stefan Rüster

Results ( H0=67(Km/sec)/Mpc) of the Planck mission is in trouble.

There is a tension between the value of the Hubble constant, H0, inferred from the local distance ladder (H0=73(Km/sec)/Mpc) and the angular scale of fluctuations in the Cosmic Microwave Background CMB (H0=67(Km/sec)/Mpc). Note that H0 from CMB being model-dependent while H0 from distance ladder being model independent. So, we claim that the physics underlies the CMB is in trouble, due to the flawed presumed Dark Matter, Dark Energy and Inflation initial conditions in the early universe. See

Preprint Crisis Concerns Initial Conditions Underlying CMB Physics

Some non-homogeneities of the Hubble "constant" are probably explainable by Non-instanataneous_Big_Bang; this is a model with the mass injected from a center to the space with some finite intensity. Even if the intensity were constant, the non-homogeneity of therate of expansion must appear. A proposal of a rigorously described models is given in the paper:

Article ON A BIG-BANG-LESS MODIFICATION OF THE NEWTONIAN SUSSKIND UN...

Dear Thierry De Mees

Thank you for your comments and interesting information about the experimental results that support the idea of gravitomagnetism. However, I see that the corrections resulting from this theory are relatively small on a scale of the planetary system. I agree also that they should be included in the derivation of the rotational curves for the galaxy. It is difficult to evaluate such a model now, but I suspect that the impact will be below the threshold associated with the uncertainties associated with the stress-energy tensor (at this moment...)

There are some opinions that support further the idea suggested in your comment. According to the ESA:

“Just as a moving electrical charge creates a magnetic field, so a moving mass generates a gravitomagnetic field. According to Einstein's Theory of General Relativity, the effect is virtually negligible. However, Martin Tajmar, ARC Seibersdorf Research GmbH, Austria, and colleagues believe they have measured the effect in a laboratory”.

http://www.esa.int/Our_Activities/Preparing_for_the_Future/Discovery_and_Preparation/Towards_a_new_test_of_general_relativity/(print)

Dear @Stefan Rüster

Our hyperbolic equation of motion is completely derived from GR. Vis-viva equation is its Newtonian limit in a flat space.

"When you make use of a hyperbolic orbit a

I'm not sure that there's yet a legitimate way of doing this, as descriptions of large rotating bodies probably need to take account of gravitomagnetic effects.

Newtonian gravity seems to have gravitomagnetism-compatible equations but was never fully developed to become a gravitomagnetic theory: Einstein's GR on the other hand, requires gravitomagnetism ... but inherits a set of flat-spacetime equations from SR that aren't compatible with the concept.

Someone here on RG (I forget who) reckons that he can model galaxy rotation curves decently, nominally within GR, by using fluid dynamics. This sounds potentially credible, as that approach will presumably be leaning more heavily on statistics and field theory and ignoring GR1916's SR component.

Another problem is that we tend to assume that there's a "flat gravitational floor" between galaxies based on the assumption of a surrounding uniform distribution of matter.

More realistically, if spatial expansion is a measure of timeflow, and time runs faster in the intergalactic voids due to a slightly lower initial field density, then space should expand faster in the less dense regions, exaggerating the effect, and producing a positive-feedback system of expanding lobes of emptier space, with galaxies tending to collect as sheets between large adjacent lobes.

In this sort of unstable model, an increased rate of timeflow in regions between galaxies means that inertia within a rotating galaxy is going to be less affected by external matter than in simple Newtonian or Einsteinian calculations. Galaxies will be less strongly connected to each other, a galaxy will represent a more self-contained "island", less affected by what the outside universe is doing.

We can replicate the positive-feedback tendency to expansion by inventing dark energy, and replicate the effect of stronger internal cohesion of rotating galaxies by inventing dark matter, but hopefully both entities will be found to be redundant (and will be summarily disposed of) once we work out how to properly model the nonlinearities (and "fix" GR to work with gravitomagnetism).

@ Janusz Pudykiewicz Gravitomagnetism cannot be properly dealt with under Einstein's GR, as the velocity-dependent gravitomagnetic effect means that any relative motion of gravitational bodies is always associated with velocity-dependent deviations from flat spacetime, which breaks the "flat" Minkowski/SR description of inertial physics. And according to the principle of equivalence, all bodies with inertial mass are gravitational bodies, so it's not as if we can come up with different laws of physics for gravitational and nongravitational matter. Because there's no such thing as nongravitational matter.

As for the effect being negligible, and too small to measure, it's arguably the same effect that NASA uses to fling space probes around the solar system, and which is responsible for the geeforces pushing you back into your seat when you hit the accelerator pedal on your car.

These effects are very, very measurable. It's also worth remembering that the gm light-dragging effect for a black hole is reckoned to be 100%, so in some situations, even under Einstein's theory, the gm effect overrides everything else.

We tend to say that gravitomagnetism is vanishingly weak not because of "scientific" reasons, but

As far as I know, NASA and the ESA tend not to use GR when calculating their slingshot orbits – they fall back on using Newtonian theory with time-domain descriptions of how the fields change position over time, rather than by calculating the same momentum-exchange from a geometrical description of a moving planet's gravitomagnetic field having a gravitomagnetic distortion (curvature domain) ...

... but the fact that these space agencies depend on gravitational momentum exchange and the slingshot effect to make many of their projects viable means that the thing is hardly negligible and unimportant at solar system scales.

Dear Eric Baird

Thank you for the interesting comments. I have only one small objection to the following statement: “Gravitomagnetism cannot be properly dealt with under Einstein's GR….”.

According to Ruggiero & Tartaglia (2008): “Gravitomagnetic effects”:

“Einstein’s equations may be written in a very simple way, which leads straight to the analogy with Maxwell’s equations, if we consider the so called weak field approximation”.

The proof is provided in section 2 and equations (4) & (5) show that Einstein’s Equation can be used to explain gravitomagnetism in the framework of the weak field approximation.

I have attached a copy of the paper by Ruggiero & Tartaglia.

Dear Janusz Pudykiewicz , The problem with Einstein's GR is that the "gravitational" side of the theory requires velocity-dependent gravitomagnetic effects to exist, while the 1916 theory's need for the physics of moving bodies (including moving stars) to obey the relationships of Minkowski spacetime requires complicating v-gm effects NOT to exist.

It's a fundamental and irreconcilable conflict, and it's the reason why textbooks tend to skip over the subject of v-gm. Explanatory arguments to follow ...

Why velocity-dependent gravitomagnetism has to exist:

So a general theory requires the strength of the v-gm dragging effect to exist and to have strength of 100% for the extremal case of a moving gravitational horizon. In a continuous classical theory, the effect has to be gradual, so any mass with any gravitational distortion whatsoever must show a corresponding gravitational dragging effect.

Next: Why the effect breaks Einstein's general theory:

Why velocity-dependent gravitomagnetism breaks Einstein's 1916 theory:

Light-dragging effects alter the energy and momentum of light. If an approaching body drags light towards you, the light should arrive with increased energy and momentum, and should be blueshifted, while if a receding body drags light away from you, it should arrive at your detector with reduced energy and momentum and a redshift.

This gravitomagnetic shift looks rather like the traditional Doppler effect.

This gm shift is not included in SR's derivations.

For a black hole receding at v, where the 100% dragging effect causes the light to also change speed by v, the gm shift seems to have to be E'/E = (c-v)/c . So for a black hole, the gm shift has the same sign and magnitude as a conventional Doppler shift.

This would mean that any moving gravitational bodies must then have a motion shift law that disagrees with that given by SR, apparently as a function of their surface gravitational field intensity.

But it's a requirement of wave-compatible classical field theories and metric theories that all bodies follow precisely the same Doppler relationship regardless of their other properties. If we look at a distant galaxy consisting of stars and interstellar gas, and we change our state of motion to it along the line of sight and look at it again, then all those different light-sources need to be seen by us to change frequency by precisely the same ratio, regardless of their local gravitational properties. Having different motion-shift for strong-nd weak-gravity objects is forbidden.

So Einstein's general theory doesn't work. It requires the Doppler effect on a lone hydrogen atom to exactly agree with SR, it requires the total motion shift on a moving neutron star to diverge significantly from SR, and it also requires the motion shifts of both bodies to be precisely the same.

It's paradoxical. If we know that SR is correct for the moving atom, then we can prove that SR must also be correct for the moving star. But if the moving star has no gm effect then it has no gravitational field. So if we start with SR, we can prove that there's no such thing in our universe as a gravitational field (which is wrong).

On the other hand, if we start with a neutron star, and prove that its Doppler relationship is non-SR, then the same non-SR relationship has to hold for every object in the universe. SR is then wrong for everything. Since GR1916 lets us prove two different irreconcilable physical outcomes, the theory is pathological. It fails the test of internal consistency, and therefore isn't a valid theory.

Replacing Einstein's general theory:

It's a requirement of any theory of relativistic gravitation that a small sensor watching a strong-gravity star reports the same motion shift regardless of whether it is the star or the sensor that is said to be moving. Again, all bodies have to obey a single universal Doppler law. The "moving" sensor (or its particles) must somehow be able to mimic the same gm shift and gm dragging effect at extremely small distances as the "moving" star. The atom must somehow be able to replicate the GM shift of any moving body it could possibly encounter, up to and including a black hole.

The only way this can work if all fundamental massed particles making up stars, planets (and sensors) all have the same surface gravitational properties as a black hole.

Further, for an atom to present the same apparent gravitational differential to its surface, even if it is immersed in the larger field of a star, we must define the effective surface of the particle (w.r.t. EM emission and absorption) as being its effective curvature horizon (which can then extend or contract in response to environmental gravitational effects ).

This is the only way that all massed particles can present the same apparent gravitational properties and generate the same gravitomagnetic equations. We either have to assign all particles a distortion field of strength zero (giving SR), or we go to the opposite extreme and make all massed particles act as if they are horizon-bounded. Intermediate solutions don't seem to be workable.

That solves the problem ... and also inadvertently solves local lightspeed constancy, because all masses then drag light 100% at tiny distances ... but it means that the universal set of Doppler relationships that all matter must follow would not be the the SR relationships, but the gm shift of a black hole, E'/E=(c-v)/c

So in a general theory designed to be compatible with gravitomagnetism, the theory does not reduce to "flat SR" for inertial physics, it has to assign maximum curvature to all massed particles, and reduce to a different set of (curved-spacetime) equations that are redder than SR's by one full additional Lorentz factor.

How can one derive galaxy rotation curve using general relativity?

General Relativity is not even working on our solar system. How could we going to talk about galaxy rotation curve of billions of galaxies?

The universe is Quantum Mechanics Universe, and complete entity, all the galaxies are moving based on QM and universe's rotation. that the only way. My theory.

I just found that the topic of the flat rotation curve of galaxy is very interesting in this forum. I would like to introduce my recent paper in this topic and get your comments and feedbacks. Thanks.

Title: The Flat Rotation Curve of Spiral Galaxies may be the Result of the Perfect Combination of Fluid Mechanics and the Gravitational Mechanics

http://www.sciepub.com/IJP/abstract/10707

Dear Stefan,

Thank you for your message. The question is how accurate is the "observed mass distribution?". If the flat rotation curves cannot be obtained from Poisson equation with "the observed mass distributions ", then it is wrong. From morphology of the disc-like galaxies, they really like a rotating fluid system very similar to a hurricane, but the disc-like galaxies are special fluid systems. I will continue to work in such direction and the primary result seems matching the observation well. I will keep you posted when the result is finally out.

My word "wrong" is too strong, you have the good point. But the observation method may not be adequate enough.

Dear Stefan, nice to read your paper. You also treated the disc-like galaxies as a fluid system. The main point in my paper is that the rotating disc-like galaxies are ideal fluid systems and the flat rotation curve is the natural consequence of the such rotating fluid system, the gravitational force acts a fluid viscosity and fluid binding force, more interesting result will be found later.

Dear Stefan,

"mass distributions are not observed in disc galaxies", this is what I mentioned previously,"how accurate the observation is?". All observation is based on the optical method.The luminosity of the disc decreases dramatically with distance r, but the mass density distribution may not proportionally deceases with r like the luminosity. Decrease with exp(-r) will be much faster than decrease with (1/r), exp(-r) will not give flat rotation curve, so most likely there are substantial amount of mass (too dark, too small??) in the outer area of the disc can not be observed by available methods, but the impact of gravitational force of those unobserved mass is observed by the flat rotation curve.

Cheer!

Dear Stefan,

Definetely in GR by means of Einstein equation with covariance and mass invariance there is no way to achieve flat rotation curves. I see we have to forget mass invariance and imagine the possibility that space itself (formed by gathering matter) includes some substance with mass effect...

Dear Friends:

My new publication is just out based on the idea that the spiral galaxies are idea fluid system and I started new discussion with topic

How were the spiral arms of the galaxies formed?

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

My publication title is "New Formulas and Mechanism for the Spiral Arm Formation of Galaxies"

http://www.sciepub.com/portal/downloads?doi=10.12691/ijp-7-3-1&filename=ijp-7-3-1.pdf

Please go to my discussion site to post any comments, feedbacks and criticisms.

All are welcome. Thanks.

Dear Hongjun Pan I read your abstract regarding Spiral galaxies. The question is where all these uniform of energy of all the galaxies are coming from. You mention Black Holes (We never observed one yet), How the matter forms and where is the origination? We even do not know what is Gravity, or where the gravity is originated. Therefore we can not just say thing that is not match with any science. best.

Dear Javad, You asked a very deep question which touches the fundamental point of the nature, what is the matter and what is the gravity. Those questions are not the ones we discuss in level of the formation of the galaxies and flat rotation curves. Please read my paper to see what is addressed how is addressed. Best.

See : The Big Bang hyperbolic universe neither needs inflation nor dark matter and dark

https://ui.adsabs.harvard.edu/abs/2013PhyEs..26..422M/abstract

I am precisely developing it, I hope to finish it soon.

The only way to determine the rotation curve is by Doppler measurements of the difference in the recessional velocities on either side of the galaxy (the side rotating toward us being blue-shifted and the one moving away from us being red-shifted, relative to the average value for the galaxy). Nothing in either special or general relativity is of any use in determining the rotation curve.