The opening statement is wrong. Gravitational force is described, quantitatively, by the curvature of spacetime, through Einstein's equations. The matter described by the Standard Model just doesn't curve spacetime enough for this to be noticeable. And this is obvious from Einstein's equations, whose RHS contain the energy-momentum tensor of matter. The numerical value of the RHS is equal to the ratio between the typical energy of matter and the Planck energy; at the LHC, this ratio is approximately 10 TeV/1019 GeV=104 GeV/1019 GeV=10-15. So the spacetime metric differs from the Minkowski metric to that extent, which is negligible.

Stam Nicolis "Gravitational force is described, quantitatively, by the curvature of spacetime, through Einstein's equations."

Yes, exactly that is the point. Gravitation is described but no causal reason for its occurrence is presented. Gravitation also is described quantitatively with reference to potential energy. But this also is no causal reason why gravitational forces occur.

But field energy dependency on object movement is a comprehensible reason for a force on an object.

The opening statement is correct because curvature of spacetime does not provide a causal description. It only provides an observational description.

It's the symmetries that are the ``cause'' and these must be discovered-and, for gravitation-to a large part-have been discovered.

It's been understood since 1918 and the work of Emmy Noether, that the symmetries come first and determine the properties of the dynamics. That this isn't as widely known as it ought to be is sociology, not physics. Many people remain stuck before 1918, apparently.

So gravitation is the expression of invariance of spacetime under diffeomorphisms. One shouldn't confuse history of physics with physics. How it was understood that gravitation is described by the invariance of spacetime geometry under diffeomorphisms is distinct from the fact that it is and working out the consequences thereof.

Curvature of spacetime captures only part of the effects of this symmetry-there's much more.

What isn't fully understood, for example, is, whether there are fields, beyond the metric tensor, that define the spacetime geometry. The most general expression thereof is supergravity, but experiments that can probe its consequences haven't been done.

So any alternative theory of gravity has to start from a different assumption about the symmetries of spacetime and work out the consequences.

The Standard Model, indeed, does make a statement about gravity, since its symmetry group is the direct product of the Poincaré group and the internal symmetry group. So the spacetime geometry is assumed to be flat-which means that the symmetry group is that of global Lorentz transformations, rotations and translations. This group is a limit of the group of diffeomorphisms.

Stam Nicolis "It's the symmetries that are the ``cause'' and these must be discovered-and, for gravitation-to a large part-have been discovered."

Yes, I fully agree to this statement. But I do not agree to your conclusions.

Symmetries are the foundation of conservation laws. The conservation laws lead to causal relations about conserved entities.

In the case we are considering, the conserved entity is energy. The force law based on the energy content in force field overlays, uses the conservation of mechanical energy "force times path-way" combined with the "energy content of the gravitational field".

This is a genuine causal relation because a conserved entity is involved. Relations derived by relativity theory or potential energy are in this sense not causal because no conserved entity is involved.

I even claim that only those relations can be causal, which are based on conserved entities. All other relations, which are not based on conserved entities, are only observational or empirical. Those relations are like alleged proofs based on "as we all know" or "as experience has shown".

"So the spacetime geometry is assumed to be flat-which means that the symmetry group is that of global Lorentz transformations, rotations and translations. This group is a limit of the group of diffeomorphisms."

I don't think that there is a difference between "flat" and "locally flat". What you consider as being a consequence of a flat space also applies to manifolds. This means that all conservation laws also would apply to our space if this space has the structure of a 3-sphere.

The symmetry group that describes gravitational phenomena is the group of diffromorphisms, of which the time translations-the global invariance under which describes the conservation of energy-is a subgroup.

Symmetries do much more than imply conservation laws-they imply the dynamics. The only equations, that are invariant under diffeomorphisms, that contain up to two derivatives and involve only the metric tensor are Einstein's equations and the only way that matter can contribute in them is through its energy-momentum tensor. If additional fields are assumed to describe the spacetime geometry, the equations can be found in a similar way-the most general are those of supergravity, where other properties of matter couple to the additional fields.

Newtonian gravity is recovered in the non-relativistic limit.

All this is now known and understood.

Stam Nicolis "The symmetry group that describes gravitational phenomena is the group of diffeomorphisms, of which the time translation is the global invariance under which describes the conservation of energy-is a subgroup."

Yes according to Emmy Noether this is the base of energy conservation.

"Symmetries do much more than imply conservation laws..."

Yes, but what has this to do with the ultimate reason for the gravitational force?

By the way, it is not my idea, that the gravitational field contains an energy density.

see:

[1811.10602] The Mass of the Gravitational Field - arXiv.org

Charles. T. Sebens

Once more: Gravitational force means nothing more and nothing less than the property that spacetime geometry is described by Einstein's equations and that matter moves along the geodesics of the spacetime manifold-if it is, also, assumed that the geometry is defined by the metric tensor; additional fields contribute in a corresponding way.

``Gravitational field'' is just another term for the field strength corresponding to the ``metric tensor'' (which can be identified, in the simplest case, with the curvature of spacetime). That it can't describe the propagation of massive excitations is a consequence of invariance under diffeormorphisms. It is possible to define fields, whose 1-particle excitations do have non-zero mass and spin 2, these, however, don't describe the geometry of spacetime, because their symmetries aren't appropriate-another instance of symmetries implying dynamics. They describe matter moving in spacetime.

The paper referred to is, simply, wrong for that reason.

Stam Nicolis "Gravitational force means nothing more and nothing less than the property that spacetime geometry is described by Einstein's equations and that matter moves along the geodesics of the spacetime manifold.."

Yes, this is a nice view and suggests that with the Einstein equations and the metric tensor and all the differential geometry gravity can be described well.

But it does not answer why this is the case. There is no causal reason given.

The metric explanation does not differentiate between an instantaneous interaction over light years and a local force.

Is there an instantaneous action over a distance? What provides the energy needed for gravitational acceleration? Is it the metric?

The best high sophisticated description how gravity works only is an eyewash without providing the cause for the gravitational force. Naming the cause requires that an entity is involved, which is subject of a conservation law.

Wrong-the invariance under diffeomorphisms means that the propagation isn't instantaneous.

In fact all of the properties of the gravitational force between objects are the consequence of imposing invariance of the equations of motion under differomorphisms.

All these are now standard exercises in any textbook on general relativity.

Symmetries don't only imply conservation laws, they imply the expression of interactions. Imposing invariance under diffeomorphisms leads to the geodesic equation for the motion of a particle and to the interaction between two particles.

Stam Nicolis

All what you are telling about symmetries, diffeophormy, and so on is correct, but it does not provide information about the cause of the gravitational force.

What entity provides the energy needed to gravitationally accelerate an object towards the centre of our galaxy?

How do you answer that question with metric concepts, diffeomorphy, or Einstein's field equations?

Please don't misunderstand me, I am not saying you are wrong. But I have some questions, which cannot be answered with the information you are presenting or at least you have presented up to now.

The cause of the gravitational force is mass. The quantum of gravity is mass and masses are quantum at all energy scales. Quantum mass theory computes the mass value of each fundamental particle. In three dimensions, quantum mass theory maintains a minimum of three generations of fundamental particles. In four dimensional spacetime, quantum mass theory accesses an infinite number of parallel universes in the local neighborhood of each spacetime point, requiring new terms such as threading. Quantum mass theory originates from the Einstein-hole problem in all dimensions.

George Soli "The cause of the gravitational force is mass."

How does mass cause the gravitational force?

The entity that provides the ``energy'' needed to accelerate an object is the curvature of spacetime-it's the fact that time translations aren't globally defined that implies that there's a force. That's what the geodesic equation expresses. In fact the presence of any force can be described, equally well, by the property that the object in question moves in a spacetime that's curved-and the absence of any force by the property that the object in question moves in a spacetime that's flat.

What mass can do is curve spacetime (since it contributes to the energy-momentum tensor, that appears on the RHS of Einstein's equations)-but, in our universe, ordinary masses, below that of the Earth, produce negligible spacetime curvature.

“…Standard Theory admits that no plausible explanation for the gravitational force has yet been found. Therefor gravity remains a myth.…..”

- that isn’t, of course, correct. In “Standard Theory” [earlier “Standard particle Model”, what was much more reliable] there are only 3 fundamental Nature Weak, EM, and Strong, forces not because that the ST admits that “no plausible explanation for the gravitational force has yet been found”, that is because of that in mainstream physics the GR is standard theory, and it completely and without any doubts, explains Gravity;

- where is no [in this case - gravitational] field, and so in the ST there is no corresponding particle this would be Gravity Force mediator – in contrast to the 3 bosons for the other Forces in the ST.

In the GR it is postulated that observed gravitational effects/interactions in gravitationally coupled systems are caused at interactions in systems “mass-spacetime-mass”, where some “mass” for/by some really by no means grounded/explained, i.e. completely transcendent, in the GR reason/way “curves Matter’s spacetime”, and this curved spacetime forces, again for/by some really by no means grounded/explained, i.e. completely transcendent, in the GR reason/way, forces other “mass” to move “along geodesics”, say, curved by Sun spacetime forces Earth rotate around.

Really, as that rigorously scientifically proven in the Shevchenko-Tokarevsky’s informational physical model, which is based on the philosophical 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

https://arxiv.org/abs/1004.3712

- where the fundamental phenomena/notions, first of all in this case “Matter”, “Consciousness”, “Space”, “Time”, “Energy”, “Information”, which were/are fundamentally completely transcendent/uncertain/irrational in mainstream philosophy and sciences, including physics, and including the GR author, are rigorously scientifically defined,

- everything in Matter fundamentally cannot, and so doesn’t, impact on/affect Matter’s space/time/spacetime, and the space/time/spacetime fundamentally cannot, and so doesn’t, impact on/affect everything in Matter.

Really Gravity is, of course, nothing else than the “fourth” fundamental Nature force. More about what is Gravity [and similar Electric Force] see the Shevchenko-Tokarevsky’s 2007 initial Planck scale model of these Forces in

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces

To read SS post on page 1, 7 days ago, in

https://www.researchgate.net/post/Can_kinetic_and_gravitational_potential_energy_could_both_be_zero_and_can_gravitational_potential_energy_be_equal_to_kinetic_energy it is useful as well.

Cheers

Stam Nicolis "The entity that provides the ``energy'' needed to accelerate an object is the curvature of spacetime-it's the fact that time translations aren't globally defined that implies that there's a force."

Ok, the curvature of spacetime is the entity, which provides the energy needed to accelarate an object by a gravitational force.

But this means that the curvature of spacetime intrinsically contains a distributed energy, an energy density, which gets partially transferred to the accelerated object. It equalizes the curvature of spacetime to a scalar field with an energy density. That is just the view of the energy density contained in the gravitational field. It does not make a difference if we consider the gravitational field as something which contains a gravitational energy density or as something which embodies a space curvature, which contains a kind of tension energy density.

The important point is that the energy, which finally gets transformed to kinetic energy of a falling object is not yet already contained as potential energy inside the object.

What's unclear in your statement is the relation to time translations and to the force. It even seems that you are not actually thinking about an energy density which literally is present in a space curvature. Did I revert what you wanted to say?

If the invariance under time translations isn't or can't be globally defined, then there's a force present. There's nothing unclear about this.

Once more, it suffices to write the Lagrangian for a particle in a curved spacetime, deduce the geodesic equation, which is the equation of motion and, also, deduce the Hamiltonian. And recover, in the non-relativistic limit, Newton's law of motion.

Sergey Shevchenko

Sergey, I am sorry, but your contribution with a hint on "information as absolute" is not related to the energy density of gravitational fields.

Your trial to describe such a relation is much too far fetched and incomprehensible.

Stam Nicolis "Once more, it suffices to write the Lagrangian for a particle in a curved spacetime, deduce the geodesic equation, which is the equation of motion and, also, deduce the Hamiltonian."

A particle in a curved spacetime does not explain gravity. In this view, modifying the particle position does not modify the energy density content of the gravitational field or of spacetime.

You may consider the Lagrangian view as being sufficient and recommend to do not ask further. But in fact, this does not explain the source of the energy needed for gravitational acceleration.

For the umpteenth time: A spactime metric that's not the flat metric explains (a) gravity and (b) in fact ANY force (either derived from a potential, or a force that's not-the latter defines a ``flux background'') acting on a particle. It suffices to actually write down the equation of motion for the particle, that's the geodesic equation in that metric. The metric in turn is a solution to Einstein's equations. In the limit in which the motion of the particle doesn't influence the evolution of the metric, the equations decouple.

The ``source of energy'' is nothing more or less than the property that, on a curved spacetime (once more NOT space), time translation isn't a global symmetry. The energy-momentum tensor is ``covariantly'' not ``globally'' conserved in curved spacetime; it's globally conserved in flat spacetime, that's why there's no gravitational force in flat spacetime.

"How does mass cause the gravitational force?".

In general relativity, if spacetime is locally flat, then non-gravitational force causes time dilation that is universally observable. For a general understanding and cosmological implications see [1]. In locally flat spacetime curvature, in standard general relativity, everybody, or all observers, observe time dilation if you are not in freefall. If you are in freefall accelerating towards a mass, you are not time dilating as seen by all observers. All accelerations, except gravitational freefall accelerations, cause universally observable time dilation. So, all non-gravitational force cause time dilation, or non-gravitational force is time dilation. Force and time dilation are not separable because they are universally observable.

[1] Fulvio Melia, “The Origin of Rest-mass Energy” July 2021, The European Physical Journal C 81(8) https://link.springer.com/article/10.1140/epjc/s10052-021-09506-w

Stam Nicolis

Umpteenth repetitions do not increase reliability. Curved spacetime describes gravity, but does not explain it.

"The ``source of energy'' is nothing more or less than the property that, on a curved spacetime (once more NOT space), time translation isn't a global symmetry"

A property cannot be the source of energy. Energy is something which can be given away. If it has been given away, then less of it is available. You cannot give away a part of a property.

But why are you circling around in all this spacetime terminology, without a chance to approach the energy concept. It is well known that concerning energy, relativity theory has an issue.

But before we focus on linguistic subtleties let us have a look on field theoretic aspects:

Article Self-Consistent, Self-Coupled Scalar Gravity

Wolfgang,

“…Sergey, I am sorry, but your contribution with a hint on "information as absolute" is not related to the energy density of gravitational fields.

Your trial to describe such a relation is much too far fetched and incomprehensible…..”

- the SS&VT “The Information as Absolute” conception directly relate to any really fundamental problem in physics, including the problem “what is a fundamental Nature force”, including what is “what is a fundamental Nature Gravity force”, because, again, only in this conception the fundamental phenomena/notions, first of all in this case “Matter”, “Consciousness”, “Space”, “Time”, “Energy”, “Information”, are rigorously scientifically defined,

- whereas all these phenomena/notions were/are fundamentally completely transcendent/uncertain/irrational in mainstream philosophy and sciences, including physics, and, say, including the GR author,

- and for the posters in this thread.

So mainstream physicists, really having no any scientific understanding about what are the phenomena/notions above, in this case first of all what are Matter”, “Space”, “Time”, “Energy”, completely logically inevitably compose really only transcendent theories, postulating in the theories fantastic/mystic properties, write really transcendent claims, etc.;

in this [Gravity] case, first of all considering purely fantastic interactions in systems “mass-spacetime-mass”, despite that in the conception it is rigorously proven that such interactions fundamentally cannot and so don’t exist; etc. – and despite that rather evidently be unable to understand even for themselves – why their postulates could be correct.

Again, the uniquely now really rigorously scientifically grounded, and so practically for sure correct, model of Gravity [and similar Electric Force] Forces is the SS&VT 2007 initial Planck scale models of these Forces, the links see the SS post on page 2;

- where, including, it is rigorously scientifically shown that the at least fundamental Gravity, Electric, and Nuclear [more see the 2023 initial model of last Force in https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force ] Forces fields fundamentally don’t contain/carry energy, and only provided this fact at these Forces interactions the energy conservation law really acts.

More see the SS posts above and links in the posts, here only repeat: any really scientific elaboration of any really fundamental scientific problem is possible only basing on the SS&VT conception above, and so every scientist, who really wants to develop really scientific something, should study and understand the conception.

Cheers

Gravity is the consequence of a curved metric-while in the Newtonian approximation the starting point is Newton's law of motion, in general relativity the starting point is the spacetime metric, which satisfies Einstein's equations and leads to Newton's law of gravitation between two point-like sources in the Newtonian limit. That's the bottom line; the details are the topic of courses in general relativity.

It's not possible to deduce general relativity from Newtonian mechanics; it is possible to deduce Newtonian mechanics from general relativity. The error that's being systematically made is trying to deduce general relativity from Newtonian mchanics.

No, neither make any sense in GR, they're avatars of the limiting case. Potential energy only makes sense in the non-relativistic, Newtonian, approximation and the energy-momentum tensor of the gravitational field itself vanishes identically, by the fact that the metric satisfies Einstein's equations.

Wolfgang Konle Especially the core equations to verify (or falsify) your claims are important. For the record and clarification - from an iSpace theoretical view I fully agree with you, that gravity is a (seemingly) attractive only electromagnetic rooted quantized force:

Force-quantum-of-gravitation = (alphaG)^2 * PlanckForce

with alphaG = (ElectronMass/PlackMass)^2

Conference Paper iSpace - Deriving G from α, e, R∞, μ0 and Quantum of Gravita...

Conference Paper iSpace - Exact Symbolic Equations for Important Physical Con...

Preprint Hubble constant H0 is derived from Newtonian gravitational c...

Preprint iSpace - Quantization of Time in iSpace-IQ Unit-System by 1/...

Article New novel physical constants metric and fine structure const...

Stefan Bernhard Rüster "…gravitational potential energy is gravitational binding energy while energy of the gravitational field is stored as space-time curvature."

Let us look at the following thought experiment:

Given is a plate capacitor with a large plate area A and a small plate distance d. But d is in range of cm, which excludes microscopic effects. The electric field between the plates is homogenous. It has exactly the same strength close to the plates as in the middle of the plates. If we modify the distance between the plates, we only modify the volume, which is filled with the electric field. The electric energy stored in the capacitor is exactly proportional to the volume, filled with the electric field.

Now we reflect if it would be possible that charge carriers in the plates could contain the electric energy stored in the capacitor as potential energy or as binding energy. This would require at least a theoretical possibility that the plate distance could be noticed by the charge carriers in the plates. But in this thought experiment there is not the faintest causal relation between the plate distance and the charge carriers. The absolute homogeneity of the electric field prohibits that any information about the plate distance reaches the charge carriers. Consequently it is obvious that the energy content of the capacitor is contained in the electrical field and not in the potential energy of charge carriers.

Now we switch to a gravitational capacitor geometry. The two plates attract each other gravitationally. But now the volume between the plates is field free. The gravitational field is present in the volume outside of the two plates. If we now increase the distance between the plates, we put mechanical energy W=F*∆d into the system. In this case we increase the field free volume by A*∆d by providing mechanical energy. But of course the sum of field free and field filled volume remains the same if we move the plates. This means by adding the energy W we reduce the volume filled with the field energy density E. We get W=-A*∆d*E. Because W is positive, the energy density E is negative.

Again we now consider an eventually possible causal relationship between the plate distance and the potential energy of the mass carriers in the plate. But we also see in this case that due to the homogeneity of the field, a causal relation is excluded. Consequently in the gravitational capacitor geometry like in the electric capacitor, the system energy is contained in the field and not in a potential energy of mass carriers.

The ultimate reason of this fact is because in the capacitor plate geometry the mass carriers have no possibility to notice the modification of the energy content of the system. In scenarios with inhomogeneous fields a theoretical possibility exists, how the mass carriers could have noticed energy modifications. But even this requires capabilities which cannot be derived from physical laws.

Consequently we must abandon the idea that something like potential energy or binding energy could be present in addition to field energy. The binding energy contained in molecules, liquids, or solid state objects is field energy. It also should be obvious that binding energy and field energy cannot be both present. It is an either or selection, clearly decided in favour of field energy because potential energy has a causality issue.

Dear Stefan Bernhard Rüster

Thank you for the link with the nice explanation of the capacitor. Unfortunately it does not contain any relevant new information. It simply equates field energy and potential energy. But if field energy is the same as potential energy then it is also obvious that only one form of this energy exists. Another fallacy of this school level consideration of the capacitor is that causality is not at all an issue.

Considering two objects attracting each other in free space, we easily see that having "potential energy" and "field energy" and "kinetic energy" leads to a contradiction. Or can you illustrate the interaction of those three forms of energy under energy conservation without a contradiction?

Christian G. Wolf "Especially the core equations to verify (or falsify) your claims are important. For the record and clarification - from an iSpace theoretical view I fully agree with you, that gravity is a (seemingly) attractive only electromagnetic rooted quantized force"

Thank you for this far reaching contribution, which demonstrates big thinking.

Relations between natural constants can reveal new fundamental concepts of nature. But unfortunately we do not know the exact values of natural constants. Therefor we can probe random arithmetic combinations until we find those, matching within the accuracy limits. Considering a considerably large number of constants, allowing each constant to appear in a certain power and considering all combinations, which match in the physical units, leaves a huge variety of possibilities. It is a feasible task for a computer to do all those evaluations. However if an arbitrary new constant is involved then any possible relation to a possible (hidden) concept of nature is diluted.

I do not intend to allege that this kind of method has led to your finding. A great method would be to understand the concept of nature, which is behind the revealed relation.

Please provide a confirmation that Fg is not such an arbitrary new constant which would dilute the relation between your finding and new fundamental concepts of nature. I would very much appreciate if that what you are describing, leads to new concepts to understand nature.

My hope is that we can find a way to determine the upper and lower limit of the energy density of the cosmic field, which prohibits an absolute negative value of the energy density of gravitational fields.

Wolfgang Konle Wolfgang, thank you for your encouraging reply! However - fortunately we do know exact values for constants of nature (most but all) - and thanks to extreme first order multiplicative simplicity of iSpace theory we do know then in iSpace-SI and new iSpace-IQ unit systems both of which are exact (zero error bar by design of changed distance definition based integer geometry) yet still fully MKSA/SI lab compatible and also lossless convertible by each other simply by the help of knowing the quantum of time to be 1/6961 iSpaceSecond (exactly). Please (all) take the time to at least a bit thoroughly going thru above given RG links to my papers before stating „impossible“ - it can be done … as it has been!

Dear Stefan Bernhard Rüster "Potential energy is converted in kinetic energy, while energy of the gravitational field in matter-free space-time remains constant."

Let us consider two objects of mass M and m separated and then combined. The difference of field energy between the two states is -2GmM/r0. Integrating over -g²/(8πG) from r0 to infinity leads to that result. This is just the same amount as you are assuming for the difference of the potential energy.

But nothing except the gravitational field distribution and the kinetic energy is modified between these two states.

“…Dear Wolfgang Konle that, what you described in your last comment is not the field energy (energy of the gravitational field) but the gravitational potential energy, because gravitational potential energy is the amount of energy that needs to be used to separate the particles of a bound gravitational system to infinity, ….”

- yeah, that is so, however that

“….while the energy of the gravitational field in the Newtonian theory of gravitation is the amount of energy that is stored in the gravitational field surrounding the masses. Physically, these are two completely different kinds of energies.….”

- really is incorrect in at least two points: in the Newtonian theory of gravitation there is no energy that is stored in the gravitational field surrounding the masses, Newton didn’t claimed that; and further that practically wasn’t in mainstream physics, because that the GR was claimed standard completely perfect theory; though yeah, the stored in the electromagnetic field surrounding electric charges energy is postulated in the other theories – mainstream classical electrodynamics and QED;

- and – though that would be indeed two completely different kinds of energies [including in Electric/Electromagnetic Force fields], really there cannot be fundamentally any stored in the any, including Gravity, Force field energy, that , as that is rigorously shown in the Shevchenko-Tokarevsky’s 2007 initial Planck scale models of Gravity and Electric Forces, see in

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces ,

- and 2023 model of Nuclear Force, see in https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force– is prohibited by energy conservation law.

Correspondingly that

“…In GR, the energy of the gravitational field is completely different from that as it stores the energy of the gravitational field as curvature of space-time. This has absolutely nothing to do with gravitational potential energy. Additionally, the values of the gravitational potential energy and the energy of the gravitational field do not have the same value in GR.…..”

- also is incorrect in again at least two points:

- first of all, as that is rigorously scientifically proven in the SS&VT philosophical 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception [the link see SS post page 2, 4 days ago now] there fundamentally cannot be any “curvature of space-time”, Matter’s spacetime is fundamentally absolute, fundamentally continuous, fundamentally flat, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct), and

- even that would be as that is completely arbitrarily without any rational physical grounds postulated in the GR, again that would violate the energy conservation law.

Cheers

Stefan Bernhard Rüster,

Firstly a note to

“…Moreover, space-time of course is curved.…..”

- that is simply fundamentally wrong, Matter’s spacetime is, as that is rigorously scientifically shown in the SS&VT informational physical model, fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct),

- and – that is rigorously scientifically proven in the model – spacetime the fundamentally cannot be impacted/transformed by anything in Matter; including be “curved”, and fundamentally cannot impact on anything in Matter, say, to force Earth to rotate around Sun. It is fundamentally nothing else than logical possibility for Matter to exist and to change, for what is quite enough to be the fundamentally infinite by definition “empty container”.

That is rather numerously written in SS posts, including in this thread, but seems sometimes is too complex to understand; so here let consider firstly more simple case – the SR, where yet in 1908 firstly “relativistic properties and effects”,

- i.e. some transformations of space/time/spacetime, i.e. “space contraction” and “time dilation” were postulated – completely equally with the GR postulates without any scientific grounds

- that a moving inertial reference frame or even a moving particle for/by completely mystic reason and way “contract space” and “dilate time”, and these contracted space and dilated time again for/by completely mystic reason and way really contract real lengths of real moving bodies, and slow tick rates of real clocks – while the last both are really observed experimentally.

Really in these experimentally observed effects there is nothing mystic, what happens in this case really – more see SS&VT whole informational physical model paper https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics , here briefly

– yeah, if some concrete material body is impacted by some material force and after corresponding acceleration moves with some speed V, say, along X-axis in 3D space, the body’s length in the 3D space, where most [all in classical mechanics] of interactions in Matter happen, is really contracted because that the body rotates at the acceleration in the (X, cτ) plane of Matter’s absolute utmost universal 4D space with metrics (cτ,X,Y,Z), so that its 3D projection is contracted in Lorentz factor, the intrinsic processes really slow down in this factor,

- and, at that, the front end of the body becomes be lower in the cτ-axis than the back end on the Voigt-Lorentz/“relativity of simultaneity” factor VL/c2. Since in physics the absolute motion in the utmost universal [5]4D spacetime with metrics (cτ,X,Y,Z, ct), isn’t observed till now because of the really extremely mighty Galileo-Poincaré relativity principle action, really in everyday practice, including physics, the really spatial cτ-dimension is used as the time dimension, the body’s front end becomes be “younger” than the back end.

Nothing at that happens with space/time/spacetime, the “relativistic properties and effects” appeared in the SR only because of that Minkowski completely illusory, and having for that no any scientific grounds, postulated that letters x,y,z,t in Lorentz transformations relate to all/every point in [infinite] spacetime, while these letters really relate only to the points that at given time moment the body occupies.

In the GR the SR “properties and effects” above remain, while additional really mystic – i.e. by no means scientifically grounded – “ space-time “curvature” is nothing else than another postulated illusion; in both cases fundamentally nothing happens in the Matter’s spacetime when some bodies are in it.

The post is rather long already, so now

Cheers,

Sergey

Sergey Shevchenko

Sorry, but still you gave no comprehensible explanation about what you are proposing instead of an energy content in force fields.

The energy content depends on the separation of the objects which cause the field. This dependency leads to the force on the objects.

What is your alternate proposal to explain the force on objects, which contribute to a force field.

Stefan Bernhard Rüster [and Wolfgang Konle],

Now let to continue

“…Sergey Shevchenko it is well known that the energy density of the gravitational field is given by epsilon = -1/(8 pi G) ( grad Phi )2 in analogy to electrodynamics. This kind of energy is stored in the field….”

- yeah, that is well known in mainstream physics, both – mainstream fundamental Nature EM [really in the mainstream only Electric] and Gravity [not the GR] forces fields contain some “energy”; and forces, which act on concrete charged bodies in fields, are caused by the fields “grad Phi”; though, since now the Gravity theory isn’t developed on sufficient level, while in the GR that looks as too problematic at all, since, say, on an event horizon the “space-time curvature” is infinite, so “grad Phi” is infinite also, and energy of the “gravitational field”[which doesn’t exist in the GR, though] is infinite,

- so further in this case let to look at much more developed rationally classical electrodynamics

- where electromagnetic fields contain energy; the energy density W=(E2+H2)/8π,i.e. W~1/r4, and so corresponding whole energy integral from some r0 to infinity is well limited. [So, say classical ED quite rationally considers the case when the particle’s “electron” mass has completely EM origin and really is the mass the Electric field energy/c2].

So the field energy can be – and is - as that is postulted in classical ED - radiated infinite time inside the infinite space volume, be at that well limited by the intergral above value. Howewer that can be rational only at statics. If a charge moves, it radiates energy not in the stable space volume above, radiating, say, if an electron charge has “electromagnetic radius” ~λ/α, λ is electron’s Compton length, α is fine structure constant, in a space volume between this radius, and say, 1mm radius, practically all energy in the integral, and so if motion is rather long, the radiated energy would be enough to create a lot of electrons and their “electromagnetic masses”.

In QED electrically charged particles also constantly and always radiate “virtual photons”, which, though are “virtual” in QED nonetheless reallyimpact on other particles, transmitting to them quite non-virtual energy/momentum, i.e. in QED electric charge also radiates practically in infinite time the equally as in the classics mystic energy.

Again, what really happens in coupled by some fundamental Nature forces systems of having corresponding Forces charges particles, bodies, etc., is quite rigorously explained in this [energy] case for at least Gravity, Electric, and Nuclear, Forces [Gravity Force fundamentally is some “ordinary” Nature force, and by no means that is what is in the GR] in the Shevchenko-Tokarevsky’s 2007 initial Planck scale Gravity and Electric Forces models in

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces ;

- and 2023 initial model of Nuclear [that binds nucleons in nuclei] Force in https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force.

In the models it is rigorously shown that all these Forces act in Matter by the same scheme, where the Forces mediators – and the Forces fields that are flows of the mediators, which are radiated by a particle’s corresponding Force charge, and hit into other “irradiated” particle, mediators don’t contain/carry energy. That are some logical marks/signals that when, hit into the logical algorithm “particle”, that constantly runs, triggers in this algorithm releasing of its own energy/momentum so, that the particle, say, if is free, is accelerated; at statics along the line between particles direction. Again, in a Force field irradiated particle/body, etc., moves as a human swims in water – spending own energy.

I.e. in every case in Matter the potential energy, U,is the current own energy of interacting particles, which is equal or lesser than the sum of intrinsic energies of particles that they have if are free at rest in the absolute 3D space of Matter’s absolute spacetime: U

Dear Sergey Shevchenko "If a charge moves, it radiates energy..."

Yes, this is the basis. But why does it radiate?

It radiates because its field contains energy and because its field is co-moving with the charge. The radiation has an aperiodic component. After the acceleration has ended and after the aperiodic component of the radiation has passed, the field is co-moving with the new velocity of the charge.

The impact of the aperiodic radiation part is exactly equal to the space and time difference between the field configuration at the old charge location and velocity and the new charge location and velocity.

This relation exactly defines the aperiodic radiation part. The periodic radiation part then only fulfils the wave equation in combination with the aperiodic part.

Without an energy content of the static field, the aperiodic radiation part would be zero. The consideration of the aperiodic radiation part is a new aspect of theoretical physics.

QED applies to extremely high energies. We should not consider it here.

Dear Wolfgang Konle,

“…[SS quote]..If a charge moves, it radiates energy..[end quote] ."

Yes, this is the basis. But why does it radiate?…”

- as that is pointed in the SS posts above in the mainstream classical ED an electric charge for/by some mystic reason/way constantly and always radiates energy yet at statics, and so in the mainstream every charge’s Coulomb field has “energy density” W – see in the SS posts CGSE equation above, if H=0; what, again, can be only if the charge is source of infinite energy; though that isn’t too evident in this case since the integral energy value is limited,

- while the integral is defined on infinite space, “infinity” is rather vague thing, and so, say, some “electron’s electromagnetic mass” looks as rather reasonable; and is in all mainstream textbooks.

However this radiation fundamentally continues when a charge moves, including with a small speed, when additional momentum/energy that the field has after an impact on the charged body that accelerated the charge up to a speed V [in classic ED Poynting vector] is much lesser than the “Coulomb” energy;

- thus, as that is pointed in the SS post above, if a moving charge passes some distance, the radiated energy volume becomes be well larger than it is at statics, and so in this case there can be rather independently radiated a lot of “electron’s electromagnetic masses”, etc.

As to

“…[a charge] radiates because its field contains energy and because its field is co-moving with the charge…”.

- in the mainstream ED, and that is, if we don’t say about really mystic “the E/EM field energy density”, quite rational - that happens really, that is another thing that, as that is rigorously shown in the SS&VT Gravity, Electric and Nuclear, Forces models [see links in the SS posts above] the radiated by corresponding Forces’ charges fields fundamentally don’t contain/carry energy,

- a charge’s field by no means is “co-moving with the charge”, at motion a charge in every space point radiates “new field”, while just radiated on earlier trajectory points field impacts on other charges – see in ED textbooks “retarded potentials”, “Liénard–Wiechert potentials”, etc. [and SS models]; and as to

“…The radiation has an aperiodic component. After the acceleration has ended and after the aperiodic component of the radiation has passed, the field is co-moving with the new velocity of the charge. …..”, etc.

- sorry, but I comment as a rule only mainstream physics, and don’t comment some alternative approaches.

More see the SS posts above and links in the posts.

Cheers,

Sergey Shevchenko "a charge’s field by no means is “co-moving with the charge”, at motion a charge in every space point radiates “new field”, while just radiated on earlier trajectory points field impacts on other charges – see in ED textbooks “retarded potentials”, “Liénard–Wiechert potentials”, etc. [and SS models]"

Sorry, but the field around a charge contains energy. Therefor the charge cannot renew the field without losing energy.

You see that the energy content of force fields often is in contradiction to some standard assumptions.

Wolfgang Konle,

“…Sorry, but the field around a charge contains energy. Therefor the charge cannot renew the field without losing energy. You see that the energy content of force fields often is in contradiction to some standard assumptions.……”

- again, that the Forces fields around charges contain energy; and that the charge constantly and always renew the fields without losing energy, are “standard assumptions” in mainstream physics soon 200 years already; and in this case the energy content of force fields by no means is in contradiction to the standard assumptions.

That is another thing that for any normal human all in the above can be only for some completely mystic reasons and by some completely mystic ways.

Cheers,

Sergey Shevchenko

Mystic contradictions are not a base for serious science.

Force fields either contain energy or not. If they contain energy they cannot be renewed because every renewal would cost energy. If they don't contain energy, they could be constantly renewed, but this alleged renewal would mean just that nothing happens.

Wolfgang Konle

- what is the fundamental Nature Gravity force, how it acts in Matter, and how the energy exchange between coupled by Gravity Force particles – and so further bodies – happens it is quite clearly explained in the SS posts in the thread already, which, of course, are intended not for only you, but for other readers, who really want to understand what exists and happens in Matter in this case. The explanation is quite clear, but not trivial and non-standard, so it is necessary to spend some time on thinking at attempts to understand what is written, but you again practically immediately in fact repeat the claims, which are already commented in the SS posts.

So for other readers more see also the SS posts at least on page 5, here a brief comment to

“…..Mystic contradictions are not a base for serious science. Force fields either contain energy or not. If they contain energy they cannot be renewed because every renewal would cost energy. If they don't contain energy, they could be constantly renewed, but this alleged renewal would mean just that nothing happens.…..”

- again, Gravity field fundamentally is constantly and always renewed/radiated by every particle, however that fundamentally doesn’t cost energy, since radiated by the particle field fundamentally doesn’t contain energy.

However that fundamentally doesn’t mean “just that nothing happens”, the Gravity Force mediators, that compose gravitational fields, cause releasing in “irradiated” particles releasing energy portions of the irradiated particle own energies, and the irradiated particle so is impacted by corresponding 3D space momentum/force, and, say, irradiated particle moves, at statics toward the “radiating” particle having corresponding kinetic energy/momentum.

Just so in the case when this released energy is dissipated from a system of bodies, say, when the bodies compose one body, this body’s mass, M, is lesser on the negative gravitational mass defect, ΔM, and corresponding energy, E= ΔMc2, than the sum of mass/energies when the bodies were free. Say, Earth negative gravitational mass defect is ~ 2.5×1015 kg, corresponding energy U ~2.5×1032 J; more see, e.g. https://en.wikipedia.org/wiki/Gravitational_binding_energy, though all that is in textbooks.

And that, as that is rigorously scientifically shown in the SS&VT models of Gravity, Electric, and Nuclear Forces [the links see SS posts on page 5, 6 and 3 days ago now] on the Planck scale happens by the same one scheme, so coupled electric particles/bodies, nuclei, quite for/by the same reason and way have negative mass defects.

Cheers,

Sergey Shevchenko "However that fundamentally doesn’t mean “just that nothing happens”, the Gravity Force mediators, that compose gravitational fields, cause releasing in “irradiated” particles releasing energy portions of the irradiated particle own energies, and the irradiated particle so is impacted by corresponding 3D space momentum/force, and, say, irradiated particle moves, at statics toward the “radiating” particle having corresponding kinetic energy/momentum."

What particle flow are you assuming? What entity is source and what entity is sink of the flow. What are the particles transporting? How does the balance of the transported entity work? What does "irradiated particle" and "radiating particle" mean in that context?

Sorry but this particle story is neither comprehensible nor convincing.

Wolfgang Konle,

- you again instead of to read a next SS post and to think some time aimed at to attempt to understand what you read, write again post seems without thinking:

“…What particle flow are you assuming? What entity is source and what entity is sink of the flow. What are the particles transporting? How does the balance of the transported entity work? What does "irradiated particle" and "radiating particle" mean in that context?….”

- in the SS post above it is quite clearly written, that every having charges of some fundamental Nature forces particle constantly and always radiates the Forces’ mediators, which compose the Forces’ “fields”, and so, say, if there are two particles that are interacting by some Force – in this case by Gravity Force, every of the particles is radiating the Force mediators, and simultaneously, is irradiating by the other particle’s Force mediators.

At that all particles in Matter contain/carry energy E=mc2, while the mediators aren’t particles, and so don’t contain/carry energy; so particles billions of years radiate fields, however are stable having always the energy above.

For the readers, who really want to understand what are fundamental Nature forces and how the Forces, Including Gravity, act, more see the SS posts above in the thread an SS&VT papers that are linked in the posts.

Cheers

Sergey Shevchenko "At that all particles in Matter contain/carry energy E=mc2, while the mediators aren’t particles, and so don’t contain/carry energy; so particles billions of years radiate fields, however are stable having always the energy above."

Sorry but your story about particles which radiate force mediators and which are irradiated by force mediators radiated by other particles is neither proven nor comprehensible.

How can it be made plausible that such a cloud of particles and force mediators would lead to an exactly defined attractive force between masses? What is the movement pattern of the particles? How do force mediators work. What exactly is their impact.

You are insisting on your request to read your publications instead of doubting your theories. But the problem is that your publications only contain those unproven stories about that particle model of gravitation. What you have written there, for you seems to appear completely clear and look like the only possibility to explain gravity.

But for others it is mysterious how those particles can exert an attractive force. It is even more enigmatic, what kind of entity a force mediator is. Is that force mediator something like the gravitational field? But then, why do we need particles?

To summarize, re-reading your theories does not help because the foundations of your theory are illogical.

Stefan Bernhard Rüster "Physically, these are two completely different kinds of energies. However, in Newtonian theory of gravitation these two physically different kinds of energies have the same value."

By partial integration over the potential energy of the complete mass in the potential generated by that mass we can proof that the field energy is just this potential energy.

We get the complete energy by integrating 0.5 * "mass density" multiplied with the potential over all space.

Replacing the "mass density" by the expression of the potential in the Poisson equation (mass density = ∇²ɸ/(4πG))

We then get from W=∫ ɸ∇²ɸ/(8πG) after partial integration W=-∫(∇ɸ)²/(8πG)).

The residual integral vanishes because the potential is zero in spatial infinity.

The Poisson equation shows, that these energies are physically the same.

I think gravitation may be understood only together with inertia.

My intuition is saying me that inertia is possible for desription only in absolute space. Provably for that reason we rarely hear about inertia from relativists.

Stefan Bernhard Rüster "However in case the energy density of the gravitational field would appear as a source on the right hand side in the Poisson equation this whole derivation becomes irrelevant. Moreover, these two kinds of energies are physically not the same as I explained many times."

But now you amaze me. If you take intrinsic gravity into account in the Poisson equation and then plug that back into the potential energy formula, then partial integration provides an additional term for energy density. This can be continued iteratively. Other terms quickly become tiny but mathematically extremely complicated. However, this does not make anything irrelevant. Only the expression for the energy density of the gravitational field is corrected.

It cannot be said that the field energy and the potential energy are different. In the formula expression with partial integration, these two manifestations of energy are equated with each other. To regard them as physically different would be a completely new interpretation of the meaning of the mathematical equal sign in physics.

Stefan Bernhard Rüster "However, I don't think that the iterative method will lead to success, because for each gravitational system the energy density of the gravitational field in matter-free space-time must be equal to the total energy density, the latter of which is simply a parameter that is a different constant for different gravitational systems."

The gravitational fields are not restricted to matter free space. The fact, that the field itsself corresponds to a certain matter density underlines this aspect. Just this fact, that the field energy density has a matter equivalent, leads to such a self reflection, expressed by the iterative method.

Therefor the iterative (or recursive) method, which extends the partial integration, is the only way to derive a consistent expression for the energy density of the gravitational field.

The total energy contained in every local assembly of massive objects is determined by the poisson equation, which relates the potential to the mass density. It is a fact, that the potential is just the intrinsic potential of the mass assembly, and this exactly determines the parameter you mention.

You should not try to implement an additional arbitrary freedom in the energy relation with a selectable parameter. The poisson equation is unambigous.

In the SS posts above in the thread it is well clearly it is explained that, as that is completely rationally scientifically, and so with a well non-zero probability really rigorously scientifically, shown in the Shevchenko-Tokarevsky’s 2007 initial Planck scale Gravity and Electric Forces models in

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces ;

[and in 2023 initial model of Nuclear [that binds nucleons in nuclei] Force in https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force , where it is rigorously scientifically shown that this Force acts by the same scheme that Gravity and Electric Forces act]:

- [with fundamentally 100% probability] Gravity is fundamentally nothing else than some fundamental Nature force – as that also Weak, Electric, and Strong/Nuclear Forces are, and

- the Forces act as having corresponding Forces charges particles [at that all particles have Gravity charges “gravitational masses”] radiate the Forces mediators, flows of which compose what in mainstream physics is called “Forces fields”; and if a mediator hits into other particle that has the same Force charge, the other particle obtain 3D space directed momentum/on the particle the Forces’ force acts, and so, say, if the “irradiated” particle is free, it moves in the 3D space having some kinetic energy, Ek.

At that the mediators aren’t particles, and so [at least at statics] don’t contain/carry any energies/momentums – so the fields don’t contain energy – in fundamental contrast to the mainstream classical fields theories and QFTs , where the classical Forces fields and QFT flows of “virtual” Forces’ mediators contain some really mystic energy, which are constantly and always radiated by particles/charges, nonetheless by some mystic way the particles are stable billions of years; and

- another fundamental contrast is in that while in the SS&VT models above the mediators are real, in the classics the fields exist for/by completely transcendent mystic reasons and ways; in QFT “virtual photons/gravitons/gluons/mesons” are quite frankly evidently transcendent mystic physical material entities.

So for those readers that really want to understand what exists and happens in this case more see the SS linked above papers and posts above, where, including, the questions in

“…..How can it be made plausible that such a cloud of particles and force mediators would lead to an exactly defined attractive force between masses? What is the movement pattern of the particles? How do force mediators work. What exactly is their impact.….”

- are rigorously scientifically answered; here only a comment to the last discussion, say, to

“…It cannot be said that the field energy and the potential energy are different. In the formula expression with partial integration, these two manifestations of energy are equated with each other. To regard them as physically different would be a completely new interpretation of the meaning of the mathematical equal sign in physics.……”

- what really is in full accordance with classic mystic “fields” in mainstream physics theories.

However really the “field energy” and the “potential energy” are, of course, different, though at that it is necessary to understand that the wording “potential energy” has a sense only if there exist a system of at least two particles/bodies/etc. that compose a coupled by a Force system;

- and in this case “potential energy” is just potential energy of a system’s components. If one of component mass, M, is much more than the other’s mass, m, that is practically potential energy of m, which at motion of m in M’s field is transformed into kinetic energy, etc.

- while the, again really mystic in the mainstream, and really non-existent, “field energy” in the mainstream exists always, including when there exist only one particle/body/etc. , and so there exist no any “potential energy”.

Etc. – such strange things in mainstream physics are numerous, though.

Cheers

Stefan Bernhard Rüster “The Poisson equation is just an approximation. It is not the ultimate truth. In matter-free space-time, the energy density of the gravitational field must be constant.”

If we add the mass equivalent of the field energy to the Poisson equation, we significantly increase the accuracy. If we apply the iterative correction to the Poisson equation, we get from the partial Integration, we can achieve an arbitrary high accuracy. I do not see any effect, which prevents to get an exact solution.

In matter free space the energy density of the gravitational field is proportional to the inverse fourth power of the distance to the field source. But the mass equivalent of the field energy density leads to a slight modification in the case of weak fields.

Why do you think that this would rule out my conclusion about an exact solution? Do you see a further reason additional to the mass equivalent of field energy, which could make an approximation out of the Poisson equation?

Sergey Shevchenko

At least you are trying to explain those ideas about gravitons, force mediators, radiation, and irradiation. But sorry this trial did not help. Finally you are not substantially detailing your explanations and again you post vague references to alleged further declarations in your publications.

Either it is possible for you to directly and definitively present your ideas in a comprehensible fashion, or we finally must refute them.

It is unacceptable to be obliged gathering suitable arguments from your publications, which would support your ideas.

Stefan Bernhard Rüster "the Poisson equation is obtained in the limit of weak gravitational fields from Einstein's field equations. This is why the Poisson equation is just an approximation"

Are you claiming that an extension of the Poisson equation, which consistently considers the mass equivalent of the gravitational field energy, does not make the Poisson equation in the gravitational case to an exact equation?

Why is the poisson equation in the electrostatic case an exact equation? Or are you caliming as well that the Poisson equation in the electrostatic case also only is an approximation?

In fact, Siméon Denis Poisson had no idea about Einstein's field equations when he derived this equation. The application of this equation in physics is by far not restricted to gravitation.

Dear Stefan Bernhard Rüster "An approximation is an approximation and no exact expression. We need an exact expression and no approximation."

For what do we need an exact expression? By adding correction terms in the partial integration equation we can improve the accuracy up to any level we would need.

The key point is that this partial integration formula reveals that the potential energy of all mass in the gravitational field of that mass, is contained as field energy density intrinsically in the gravitational field. Increasing the accuracy of available numerical descriptions of that energy density does not change something in principle.

In fact, there is absolutely no need for an exact expression. We can use the approximation for any purpose, for which we would use the exact expression.

An alternative possibility for the existence of a cosmic field that compensates for the negative energy density of gravitational fields does not seem to have occurred to anyone until now.

Therefore, we should all assume that such a field exists and can interact with our world.

The most important interaction is that gravitational fields must therefore be understood as an indentation of the cosmic field. Any tiny or arbitrarily strong gravitational field is nothing more than a weakening of the cosmic field.

We can compare these weakenings to low-pressure systems that influence our weather patterns.

The movement of low-pressure systems causes air currents. Therefore, the movement of indentations of the cosmic field also causes a flow of the cosmic field. This means that even an unaccelerated movement is slowed down, even if only minimally, by the cosmic field.

The mechanism that ensures that mechanical energy is transferred to the cosmic field now has cosmological consequences.

The energetically extremely active processes in the interior of stars ensure that the cosmic field is noticeably strengthened. Since an amplification is the opposite of an indentation, this amplification is also associated with a gravitational effect.

We no longer need MOND theory or dark matter theory to explain gravitational anomalies in galaxies. The gravitational interaction with the cosmic field is quite sufficient to explain.

Dear Stefan Bernhard Rüster "what you call correction terms are in fact no correction terms"

The terms, I am proposing to add, take into account the mass equivalent of the field energy in the poisson equation.

Why do you think that this does not improve the accuracy of the Poisson equation? Do you expect another fallacy as the missing mass equivalent?

Matter free space does not exist if we consider the mass equivalent of field energy. Matter free space would be field free space.

Wolfgang Konle : The most important interaction is that gravitational fields must therefore be understood as an indentation of the cosmic field.

You as usual mix up two things. The basic gravitational force is generated by each proton in this universe. (As I did show in SOP some years ago) The change of this force is what Einstein (GR) and others tried to describe.

So it looks like you mean the later.

The other fundamental problem is the believe that e.g. GR physically is a complete model, what holds just for its math part, but not for its physical mapping, that fails. So you must distinguish between abstract gravitation and real physical gravitation.

Real gravitation fields are not divergence free because celestial bodies are not infinitely rigid. The treatment of tidal (divergence) forces is cumbersome and leads to non 1/r total potential energies that shine up as heat (photons) or change in rotation speed.

As you also learned in the mean time E mc2! what means you cannot directly convert mass into photonic/field energy as already Poincaré found and Einstein misunderstood. So there is a difference in mass like action (Poincaré ) and direct mass action = gravitation (Einstein).

So for me the true prime question is whether G is a real constant or whether it depends on the mass structure! If G const then you can forget the "simple" models.

Stefan Bernhard Rüster "One can deduce the Newtonian Poisson equation from Einstein's field equations but not vice versa."

Yes, one can deduce the Newtonian Poisson equation, without considering the mass equivalent of the gravitational field, from Einstein's field equations. We can do this, because Einstein's field equations also do not consider this mass equivalent.

But this means that Einstein's field equations are an approximation as well. In fact they are, because they also do not consider the mass equivalent of the field energy.

Therefor beeing in agreement with Einstein's field equations, is not something like the ultimative goal.

What I simply want is not being conformal with Einstein's field equations. I simply want to proof that the field energy concept is the only real concept, and that potential energy is virtual. Using the partial integration for that proof, does not at all lead to an inconsistency.

Dear Stefan Bernhard Rüster

Now you are astonishing me again. In the work you cited here, you have added something to the theory of general relativity. Just with your complements the energy density of the gravitational field has been taken into account. The original theory did not consider the mass equivalent of the field energy.

Just this extension of the theory has been my motivation for a deeper look into your work.

Jürg Wyttenbach

You obviously missed the proof about the field energy density contained in gravitational fields. The dependency of the field energy content on the distance between objects is the cause for the gravitational force between the objects.

The proof presented 6 days ago is as follows:

By partial integration over the potential energy of the complete mass in the potential generated by that mass we can proof that the field energy is just this potential energy.

We get the complete energy by integrating 0.5 * "mass density" multiplied with the potential over all space.

Replacing the "mass density" by the expression of the potential in the Poisson equation (mass density = ∇²ɸ/(4πG))

We then get from W=∫ ɸ∇²ɸ/(8πG) after partial integration W=-∫(∇ɸ)²/(8πG)).

The residual integral vanishes because the potential is zero in spatial infinity.

The Poisson equation shows, that these energies are physically the same.

Wolfgang Konle : The proof presented 6 days ago is as follows:

By partial integration over the potential energy of the complete mass in the potential generated by that mass we can proof...

This is exactly what I meant: You cannot distinguish between physics and math. I agree that on paper you can construct a self consistent formula system for GR.

But this is not real physics. In general the GR field - in real world - is not divergence free. Energy (from tidal = divergence forces) leaks out of your field energy density. This is a process that over billions of years dominates.

Photons escape your reference frame!!!

You also ignore that "G" itself needs a source! And this source is exactly known since about 3 years now.

Jürg Wyttenbach ""G" itself needs a source"

What is the source of "G"? G only is a constant which determines the relation between the gravitational acceleration and the energy content of the gravitational field.

Stefan Bernhard Rüster "the Einstein tensor, which is already contained in Einstein's field equations, represents the energy density of the gravitational field"

This is a very important claim, which leads us to the essential point of our discussion.

We must differentiate between the spatial impact of gravitational energy density and the gravitational energy density itsself.

The gravitational energy density is given by the relation E=-g²/(8πG). The relation between field strength and energy density is determined by fundamental natural laws (conservation of energy and momentum) which are applicable to all force fields. Therefor this relation is beyond our discussion and not subject of any correction or modification.

The Einstein tensor is on the level of the 1/r² law like the Poisson equation. What we are discussing here exclusively is the geometrical aspect. The 1/r² law is violated if we consider the mass equivalent of the field energy. This violation concerns the Poisson equation and the Einstein tensor.

Therefor the Einstein tensor does not represent the energy density of the gravitational field. It embodies the geometrical properties of the gravitational field in space and time. The field energy is exclusively determined by the quadratic dependency on the field strength.

Stefan Bernhard Rüster "You now know that this (E=-g²/(8πG)) is only an approximation and cannot be the exact value of a precise theory."

No, this is the exact relation. The quadratic dependency of field strength and field energy is a basic natural law.

(Only the 1/r² law is not exact in the case of gravity.)

Wolfgang Konle : G only is a constant which determines the relation between the gravitational acceleration and the energy content of the gravitational field.

Why did you study physics? Just to tell us jokes?

Every constant in a physical model needs a physical origin! So how do you get G from a proton?

Also needless to tell you that a force constant (G) cannot be uniquely derived from the energy.

Stefan Bernhard Rüster : Additionally, this tensor has the unit of measurement of an energy density required to represent a tensor of any kind of energy...

This is your statement and truly is the origin of the failure of most classic models. Physics theorists have forgotten to define, what the basic form of energy is. In my view the photon is the best candidate but most equations use the classic Poincaré EM - action coupling equivalence.

But as soon as you understand that the conversion of energy cannot be described by a Hamilton function - this is just envelope of all possible transformations - then you must start to explain how you want to convert mass energy into (your total) field energy. The same for other forms like orbit energy. Fact is that complex rotor systems only partially convert energy never totally.

So this does raise the key question: Has energy a kind of invariant kernel that cannot be transformed?

This ads to the fact that real gravitational systems always leak energy in the form of photons.

One answer to this is to understand which forms of energy do = influence "G" and which do just change "g" or only act following "g".

Jürg Wyttenbach ""G" itself needs a source"..."Every constant in a physical model needs a physical origin!"

Sorry, but a physical origin is something like an explanation and a source physically provides something.

Are we doing hairsplitting here or do we have fundamentally different perceptions about the meaning of physical terms?

Wolfgang Konle : No hair splitting. What causes "G" ?? You cannot derive "G" from an energy. Any possible solution of a differential equation of the form

0 =x'' +x' +x + k delivers an other "G". So if you always measure about the same "G" how can it depend on the field energy??

G is not a mathematical constant! It's a force constant with units attached.

Further it is not at all clear whether "G" is constant at all.

Jürg Wyttenbach "So if you always measure about the same "G" how can it depend on the field energy??"

No, it is the other way round. The gravitational field energy density depends on the field strength squared and on G.

"Further it is not at all clear whether "G" is constant at all."

G is one of the fundamental natural constants. Being constant is the essence of constants. By doubting this, you are doubting anything.

The question of constants of nature has been once and for all clarified by iSpace theory. Constants are exact integer geometric based and hence constant (unless one claims any integer or irrational number can change…).

Christian G. Wolf

Do you have any proof for this integer Space theory?

Wolfgang Konle Wolfgang, dear all, if you (one) accept "resonable beyond any doubt" as a mathematical proof - then yes. Of course I am aware that experiment only should and will have last word. The Quantum of time to be exactly 1/6961 iSpaceSecond (in iSpace-SI unit system) is what will is and will be the smoking gun in the sende of more than convicing enought - but honestly to read and understand about 100+ lines is everyones own task!

In the increasing order of time it takes to immediately grasp this is factual:

Method iSpace - Quick check of α and Φ0 from Markoulakis & Antonidakis

Preprint Values of the Fundamental Physical Constants - Comparions of...

Preprint Hubble constant H0 is derived from Newtonian gravitational c...

Preprint iSpace - Quantization of Time in iSpace-IQ Unit-System by 1/...

Article New novel physical constants metric and fine structure const...

Conference Paper iSpace - Deriving G from α, e, R∞, μ0 and Quantum of Gravita...

Conference Paper iSpace - Exact Symbolic Equations for Important Physical Con...

As well as every other preprint as published on my RG home. Multiplication is (or should be) really not that difficult to comprehend. The base thoughts of integer geometric and possibly even fractal model behind all this are not neccessary at all to read, grasp and understand the equations numerical outcome, exact (zero error bar per design of the iSpace theoretical model), especiall and more so for the problem unsolved 100 years by CODATA, namely exact self referential inter-constant relations (for all constants!).

Wolfgang Konle : G is one of the fundamental natural constants....

This is an assumption only. Only experiments can prove it. But so far experiments tell that G is not constant.

I also have to remind you that real physics can nothing prove by measuring (e.g. field) energy as real experiments only can measure forces or count wave numbers. So there is no way you can prove that G depends on field energy etc...

Understanding logic is the base of math and even more of real = experimental physics.

Christian G. Wolf

Sorry, but I think, products of natural constants are only matching incidentally. We cannot derive laws, understanding of relations, or other useful findings from those matches. Another point is that because all natural constants only have a limited accuracy, those matches are not precise, and therefor they cannot be confirmed conclusively.

For me, this searching for relations, which undoubtedly is quite wide spread, is reading tea leaves.

Jürg Wyttenbach "But so far experiments tell that G is not constant."

What experiments tell us that G is not constant?

Jürg Wyttenbach "So this does raise the key question: Has energy a kind of invariant kernel that cannot be transformed?"

Yes, energy has something like an invariant kernel. Energy conservation, based on a principal time symmetry in our world, shows this. E=mc² also shows that energy is the base of everything substantial.

Because energy is conserved and because gravitational fields contain energy, G must be absolutely constant.

Denying this would mean to question the deepest intrinsic building blocks of physics.

Wolfgang Konle Wolfgang, the whole scientific world has more than 100 years waited - no worshiped - for a solution of what constants of nature like FSC are based on and how their are related., Now that we have the answer - simple multiplicative iSpace intrger geometry - you and many old school mainstream arrested thinkers do not like it and seemingly sole for the purpose it did not come with all the massive pseudo-clever apparatus of integral and differential mathematics. In other words - anyone can do physics from now on - and you all seem to be sort of out of business. Understandable that you don’t want to even consider iSpace to be the solution, but in the end Occsm will teach you all.

Christian G. Wolf "In other words - anyone can do physics from now on - and you all seem to be sort of out of business. Understandable that you don’t want to even consider iSpace to be the solution, but in the end Occsm will teach you all."

Sorry, the problem is not about liking iSpace or not, no it is that iSpace relations do not help to solve any problem.

No natural laws can be derived from those alleged relations based on integer numbers. Even if all the derived formulae about multiplicative relations between natural constants would be fact, then "so what?". What is the benefit?

In a certain sense I can understand the enthusiasm about having found a new approach to natural sciences. But unfortunately this is a chimera, closely related to a group of other chimera, containing "consciousness research", "finding irrational numbers in nature", and even "looking for a proof of the existence of god with science methods".

All those chimera only waste our time.

Stefan Bernhard Rüster

Here are some new thoughts about the question if the Einstein tensor, the energy momentum tensor, represents all substantial aspects of matter and all forms of energy. Indeed, I think the Einstein tensor has been designed that way. My claim that the Einstein tensor would be on the same level as the 1/r² law is not correct. It is of the level of the density term in the Poisson equation. Einstein’s field equations then are an extension of the Poisson equation to the time dimension.

According to its definition the Einstein tensor could include the static field energy density. But the field energy density and the other components of the Einstein tensor are related. The relation is expressed by the space curvature, which is subject of Einstein’s field equations. Assuming that the field energy density is included in the Einstein tensor, then compromises the field equations.

I think the problem is that the wonderful differential geometric approach of Einstein has got the nimbus of infallibility. In your work, you substantially addressed the fundamental problem of the energy density in static gravitational fields in Einstein’s context, but you could not resolve all aspects.

The original Einstein field equations are inconsistent. But the fundamental problem is that they do not become exact by adding a single term, which considers the field energy density.

Wolfgang Konle Wolfgang, that justifies just one final reply (no answer required) from my side: regarding „Sorry, the problem is not about liking iSpace or not, no it is that iSpace relations do not help to solve any problem.“ How could be solving (assuming iSpace to be the solution for a moment) the really epic search in last century for the true reason behind the physically omnipräsent FSC, whose inverse value has to be placed on every bathroom mirror (don’t have the exact quote of Feynman to present here) not help to solve any problem? The only real problem is, that it’s simple exact geometric result equations directly hint to the proper spacetime structure, in case of FSC it’s e.g. 2 Pi times Planck-resistance over Quantum-Hall resistance (that simple!) and in case of iSpace derivation of G it shows with multiplicative FSC^10 term (!!) clearly and directly hinting to the 10D= 3D+6D+1D (sub)space-time geometry behind all this. Its a real pity so many on RG love to write instant replies of outdated beliefs based on their (often own …) favourite models and thinking instead of downloading and actually thorough studying something before writing some solid (and of course also very valid) different point of view. I hope everyone is aware on RG we can see who just barks back and who downloaded, read, understood, thought about and then answered.