Дорогой Preston Guynn ,

You need to study. long and hard.

Good luck!

Dear Stefan Bernhard Rüster ,

All this is true in general for the natural sciences. The transition from theory to hypotheses, to one’s own humanitarian methods without exact laws. For example, the editor of a prestigious hydrodynamics journal said 30-40 years ago, “hydrodynamics is a humanities science.” The absence of theory gives rise to hypotheses.

Einstein's theory of space deals with a new physical object, Riemannian space. Einstein expressed this clearly in his 1920 work "Ather und Relativitdtstheorie. "

Riemann space, like any physical object, has a mass density and a momentum density. More precisely, it has an energy-momentum-tension tensor. Einstein came close to this value in 1915; in 2021 I proved the tensor nature of this quantity and showed the asymptotic equality of the energy density of this field energy density tensor in the Newtonian approximation.

Preprint New version of General Relativity and the Structure of "Black Holes"

this is the latest version of work 2021

Article Exact equation of the gravitational field based on the Einst...

Stefan Bernhard Rüster ,

Вы правильно заметили: "you wrote in your article: "The fact that a tensor can be zeroed with the choice of coordinates can only mean that it is associated with a gravitational field, since only the gravitational field can be zeroed by transforming coordinates.""

I meant something slightly different. tensor Bik=0 when choosing g=1, including in the inertial reference frame. It was just a guess that the Bik tensor is the energy-momentum tensor of the field. In the last article I found a more powerful argument B00→-1/c^4(𝛛𝛗/𝛛x)^2 in the Newtonian approximation.

Thank you

Stefan Bernhard Rüster

"local inertial frame" what is it and how to form it?

Preprint Consistent definition of the local equivalence principle

the gravitational field does not exist in the inertial coordinate system. This is a fact not related to any transformations. Moreover, there are no transformations with non-zero curvature into an inertial system, since scalar curvature is INVARIANT.

Your Valery

I can think about Penrose singularity theorems, positive energy theorem of Witten, Milgrom's MOND, Brans–Dicke theory..

Dear Rüster,

1. No, not dark matter, dark energy or gravitational singularities.

2. Another metric theory of gravity could be a better model for gravitational phenomena.

3. Same as 2.

4. Yes, since there is no time translation invariance according to Noethers theorems.

Manuel Urueña Palomo ,

The energy of the gravitational field is negative. This has been proven by approaching the Newtonian limit.

Singularities in general relativity are a defect of the Einstein equation. Einstein proposed to rid general relativity of singularities. I did it.

Preprint New version of General Relativity and the Structure of "Black Holes"

From the equivalence principle at the root of general relativity, gravity and acceleration should be indistinguishable. Then why is the observed acceleration of the Universe an exception, and not equivalent to gravity? In answer is a series of articles about reseating Newton's gravity in accelerated expanding space, so that the cosmological constant is implicit. https://www.linkedin.com/pulse/gravity-redux-warren-frisina/?trackingId=NLvYVsQbQ4iJI0fk7dgKzw%3D%3D

Stefan Bernhard Rüster ,

gik=Jli Jmk ηlm.

From this equality it follows that gik is an inertial frame. The reference system with such a transformation (LL-2 § 83. Curvilinear coordinates) does not change. Such transformations change ONLY the coordinate system, but not the reference system.

For example, a Cartesian coordinate system cannot be converted into a spherical one in this way.

Preprint Are arbitrary transformations in general relativity arbitrar...

Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, 2nd Edition, Dover, New York, 2000.

g'ik=Jli Jmk glm.

Linear transformation preserving invariants.

https://en.wikipedia.org/wiki/Invariant_(physics)

Warren Frisina ,

"Then why is the observed acceleration of the Universe an exception, and not equivalent to gravity?"

the acceleration of the universe is just a hypothesis. Not a fact.

Preprint Cosmological Redshift of the Spectrum of Galaxies in a Stati...

Dear Stefan Bernhard Rüster

You say "This metric tensor gik becomes the Minkowski metric tensor ηlm when a coordinate transformation to a local inertial frame is performed by using gik=Jli Jmk ηlm."

There is no such conversion, don't even try to write it in the form

gik=Jli Jmk ηlm

Give it a try.

Kind regards

Valery

Dear Stefan Bernhard Rüster ,

if you claim that it is possible to transform the Minkowski metric into the Schwarzschild metric, find the Jacobian matrices and write the transformation. I did this to convert Cartesian coordinates to spherical ones. BUT this transformation is non-linear. and it is impossible to lure him into Linear Transformation! Please read carefully.

Preprint Are arbitrary transformations in general relativity arbitrar...

g'ik=Jli Jmk glm is a linear transformation! There is a useful science called “linear algebra”; it puts things in order in linear transformations of matrices and in the head. This is the little that remains in my head after 50 years as a mathematician.

Dear Valery Borisovich Morozov

Consider the following natural law:

The force on an object, which contributes to a force field is given by the dependency of the energy content of the field overlay on the distance between the object and the field source.

This law applies to all kinds of force fields including the gravitational field.

Mathematically the dependency is expressed by the directional derivative of the energy content in respect to the distance.

This natural law describes a plausible reason for all forces in our world. It indeed is a shame that this law is not omnipresent in the theory of gravity.

Dear Stefan Bernhard Rüster ,

"In a local inertial frame, which is the origin of the free-falling reference frame, there you have glm = ηlm. A little bit besides the origin of the free-falling reference frame the metric tensor glm starts to differ from the Minkowski metric tensor ηlm."

In any arbitrarily small region of a point in Riemannian space, the space remains Riemannian. In this case, the value of the Riemann tensor remains constant.

True, space in a small region becomes LOOKING LIKE affine space. But this is always Riemann space. This is exactly what Landau-Lifshitz and Möller wanted to say. But they were good physicists, but bad mathematicians.

Preprint Consistent definition of the local equivalence principle

Dear Wolfgang Konle ,

I can guess what you are trying to say. In fact, gravity is eliminated in a completely different way from electrodynamics. In fact, this is a theory of space because, as it turned out, space is curvilinear; bodies move with acceleration, which is given by the curvature of space. The field energy density is not set as simply as we would like. So Einstein did not find this quantity. And the conservation law does not exist in Einstein’s theory. This has led to the fact that the laws of general relativity do not work for large fields. Human imagination knows no boundaries, so in place of the incomprehensible, certain fantasies have arisen, and these are what most astrophysicists are now studying. I managed to find the energy density of the gravitational field and this immediately corrected everything.

Invented objects have disappeared, the theory works in large fields. "Black holes" turned out to be potential pits.

I wish you success!

Dear Stefan Bernhard Rüster ,

And what are you claiming? That the transformation is nonlinear? there is a definition of linear transformations;

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

if you are so sure that it is possible to convert the Minkowski metric to the Schwarzschild metric? Then convert. Take your time.

Find the Jacobian matrix and calculate gik=Jli Jmk ηlm

However, my example should convince you. No unnecessary calculations. Nonlinear coordinate transformations are unacceptable, only linear transformations of the form Ai=Jki A'k or Aik=Jil Jkm Alm.

The existence of the Jacobian matrix does not mean that there is a linear transformation gik=Jli Jmk ηlm. You didn't give an example of such a transformation transformation for between Cartesian and Schwarzschild coordinates! and you will never bring it. This is impossible!

No words needed, you can calculate. For your example, do what I did.

I don't need to prove anything. Everything has been proven. You're the one trying to persuade me. There is no need, I trust only strict evidence, not words.

Valery Borisovich Morozov

All what you are showing with the Christoffel symbols and related differential geometrical calculations is correct and not subject of a discussion.

The only point is that this consideration does not imply an impact on the geometry. The geometry is considered to be absolutely static and unfolds its impact without any retroaction. In reality gravitational fields get modified if they do work. This important fact is missing in the geometric considerations.

Stefan Bernhard Rüster

"The Christoffel symbols Gamma^a_{bc} are no tensors. Also the contracted Christoffel symbol Gamma^a_{ba} is no tensor, means that it is no vector."

A contracted tensor of course is no tensor because it only has components of a vector. It is well known that Christoffel symbols do not transform like tensors. But that contracted Christoffel symbols can't be vectors is new. Can you proof that?

But my point is different, because restricting the discussion to differential geometry is not according to the physical facts of gravity.

Wolfgang Konle ; "The geometry is considered to be absolutely static and unfolds its impact without any retroaction. "

This is the geometry of space-time. I'm just looking at static solutions. But in my energy density, there is a term that depends on time. It is also contained in the corresponding part of the Ricci tensor.

Stefan Bernhard Rüster ..." https://physics.stackexchange.com/questions/620602/jacobi-matrix-between-cartesian-and-schwarzschild-coordinates"

This is not a tensor transformation of coordinates introduced by Einstein look in Fock's book § 37. You have not proven anything. There is no transformation; the fact that the Jacobian matrix exists does not mean anything.

Here is an example of the Jacobian matrix existing, but when trying to transform according to Einstein. nothing works out.

Stefan Bernhard Rüster incorrect, it is not the determinant that is transformed, but the Ricci tensor.

Wolfgang Konle , Correctly the Christoffel symbol Γ_ik^m is not a tensor, but I am considering a convolution Γ_ik^k.

Stefan Bernhard Rüster ,

a mathematician is always looking for an error in a proof. This is the shortest way. If there is no error, you will see it. I once attended a seminar by the great I.M. Gelfand.

Stefan Bernhard Rüster

We know:

Tensorial properties like the the Riemannian curvature tensor express a space property which is independent from the selected coordinate system.

The Christoffel symbols however depend on the selected coordinate system.

You have shown, that in general also contracted Christoffel symbol values depend on the selected coordinate system. But this does not proof that in any case contracted Christoffel symbol values would depend on the selected coordinate system. As an example you cannot exclude that Christoffel symbols lead to constant values.

"the appearance of the second term, which in general is not vanishing, is responsible that Gamma^a_{ba} is no tensor. A tensor behaves like"

The theorem is correct. Nobody found the error.

Wolfgang Konle "Tensorial properties like the the Riemannian curvature tensor express a space property which is independent from the selected coordinate system."

The Riemann tensor is not an invariant. Scalar curvature is an invariant. invariant is the interval between points. Simplification of the Christoffel symbol is an invariant.

Elementary

Valery Borisovich Morozov "The Riemann tensor is not an invariant."

The Riemann tensor describes a physical space property. This means at any space position, it defines values which describe in respect to any orientation, what happens to a vector, parallel transported around a closed path. The Riemann tensor expresses the relative difference between the original vector and the parallel transported vector after the round trip. It describes the vector difference for any original unit vector, any transport start direction, and any orientation of the plane in which the transport path is closed.

But this parallel transport deviation is a space property, which is independent from the coordinate system we are using to describe the space curvature.

Wolfgang Konle ,

Right. It can be said more simply. The reference system does not change during coordinate transformation. It's strange, but I've never seen anything like this.

Stefan Bernhard Rüster ,

"The contracted Christoffel symbols vanish in the Cartesian coordinate system, while in the polar coordinate system"

there is no transformation Christoffel symbols by the transformation from Cartesian to polar coordinates. PROVEN!

Preprint Are arbitrary transformations in general relativity arbitrar...

Stefan Bernhard Rüster ,

I remind you that the coordinate transformation of a tensor is a linear transformation according to Einstein’s rule. And nothing else!

https://de.wikipedia.org/wiki/Tensor

Dear Stefan Bernhard Rüster ,

All this is correct. And I always knew this well. With the exception of Eq. (85.15). Didn't pay attention. It was you who showed it to me, for which I am grateful to you. This helped to find a good version of the proof of the tensor character of the simplified Christoffel symbol.

Preprint New version of General Relativity and the Structure of "Black Holes"

THANKS

Dear Valery Borisovich Morozov

Your preprint puts some momentum into the frozen theory about gravitation. However, like the frozen parts of the theory, it also ignores the energy content of the gravitational field. What makes things worse is that this energy content appears to be negative, and an absolute negative energy density cannot exist.

The energy density W of the gravitational field must then be given by

W=W0-g²/(8πG).

W0 is the energy density of an omnipresent background field, which prevents the gravitational field energy density to become negative. But the existence of this background field sets an upper limit for the gravitational field strength g. This upper limit falsifies everything we thought to know about black holes.

Therefor if some real momentum shall be put into the frozen gravitation theory, the energy density aspects of force fields must be considered.

Stefan Bernhard Rüster "I see absolutely no reason why the energy density of the gravitational field must not be negative."

The reason is in the relation between radiation and the field energy content. Contradictory effects would emerge if via radiation an accelerated mass adapts its gravitational field to its new velocity. A negative amount of energy would be necessary in the aperiodic part of the radiation to modify the field according to the new state of the particle movement. But acceleration of a mass exclusively involves positive energy contributions.

If the absolute amount of the energy contained in the gravitational field would be negative, then a mass had no chance to adapt via radiation its gravitational field to its new velocity.

The only mechanism for this field adaption, we can imagine, is radiation. Something else would imply a completely new aspect in physics.

The energy transfer from acceleration to radiation involves a force, which reduces the acceleration. But if the field energy would be absolutely negative, the reaction force of the radiation would increase the acceleration.

Every particle then would be subject of a run-away effect.

Keep in mind that all these aspects only emerge from the energy content of force fields. Without this energy content the aperiodic radiation component would be zero and a negative energy density also would't be an issue.

Stefan Bernhard Rüster

The energy of the gravitational field has a mass equvalent. If the field source gets accelerated, the field also must be accelerated. Otherwise the source would get a decentralized position in respect to the field.

This field acceleration needs radiatiation. More precise the adaptation is done by the aperiodic part of radiation. A small amount of kinetic energy is necessary to accelerate the mass equivalent of the field energy accordingly, if the field source gets accelerated.

With other words, only the rest mass of the field is conserved. The adaptation of the relativistic mass to the new velocity needs some energy.

But with an absolute negative mass equivalent the necessary energy transfer via radiation would not work.

Stefan Bernhard Rüster "what you state are good arguments for other kind of field energies"

Indeed, electric fields and gravitational fields are different. But what makes you think that the methods, how field sources keep the field shape according to their movent, could be different? We know electromagnetic waves and gravitational waves. This makes it more probable that there is a common aperiodic radiation mechanism, which adapts all kinds of force fields to the accelerated movement of the field source.

In fact such a radiative field repair mechanism must exist in all cases. Otherwise a field source would lose its field if it gets accelerated.

Therefor other kind of field energy definitively cannot matter. The mechanism of radiative energy transfer must be the same. But this mechanism only can work with positive field energies. Therefor the field energy density W of the gravitational field must be given as W=W0-g²/(8πG).

Cosmology then will have to handle W0.

Dear Stefan Bernhard Rüster "This means, that in case a mass moves, then at the old place, the total energy density is just the energy density of the gravitational field while at the new place by motion the total energy density equals the energy density of mass plus the energy density of the gravitational field. And so on."

It seems you see some iterative mechanism in the movement of a field source and its field. But because a field is something real with an energy density it simply moves with its source. There is nothing which relates the old and the new position of field and field source. Field and field source are an inseparable unit which moves as a unit.

Only the acceleration of the field source stresses the connection between field and source. The only possibility to communicate between source and field is radiation. Therefor radiation is involved in the stress between field and source, if the source gets accelerated.

In the case of a constant velocity, and considered in the rest system of the field source, there is no dynamic at all. Everything is perfectly at rest.

Dear Stefan Bernhard Rüster

Any mass causes a depression in the omnipresent total energy density. The depression is equal to -g²/(8πG). The movement of the depression includes some movement of the surrounding field, which also includes a certain diffusion of kinetic energy within that medium.

Gravitational waves become periodic and aperiodic oscillations of the omnipresent medium. The propagation velocity is given by the square root of pressure divided by density, which is the speed of light.

We can assume that this medium is capable to support an internal movement in all possible directions without a conflict. This medium is capable to absorb kinetic energy indistinguishable from energy density.

We can compare that medium to an ideal gas without collisions between gas particles. Instead of particles, the medium consists of a continuum without dissipative effects and without a temperature.

The properties of that medium lead to a totally different cosmology, compared to the currently assumed patchwork. The consequences of the existence of this medium challenge the current cosmology in an embarrassing way. This is the reason why the majority of the scientists refutes to even discuss this approach.

The thread question “What has been done in the theory of gravity over the past 100 years? The answer is NOTHING”

- looks as rather questionable, and would be more correct if would be as “What has been done in the theory of gravity over the past more 300 years – up to 2007 year? The answer is NOTHING”.

Cheers

Stefan Bernhard Rüster, Valery Borisovich Morozov

Something fundamental (1) from Stefan and (2) from Joel Franklin (Reed College) to be discussed:

(1)

------------------------

In Newton’s theory of gravitation, the energy density of the gravitational field amounts to

𝜀gf (r) = − [∇Φ(r)]28𝜋𝐺 . (10)However, in the Poisson equation,ΔΦ(r) = 4𝜋𝐺 𝜚(r) , (11)

there does not appear the energy density of the gravitational field, but only the mass distribution on the right-hand side as a source of gravitation.

(4) (PDF) Energy Conservation in General Relativity and Flat Rotation Curves of Spiral Galaxies. Available from: https://www.researchgate.net/publication/374337609_Energy_Conservation_in_General_Relativity_and_Flat_Rotation_Curves_of_Spiral_Galaxies [accessed Oct 02 2023].

------------------------

Please compare this with an according statement from the article

“J. Franklin, Self-consistent, self-coupled scalar gravity.”

(2)

------------------------

“The work required to build a configuration of mass with mass density ρ(r) and associated potential Φ(r) is given by W = Integral all space ρ(r) Φ(r) dτ. (5)

Using (1), we can replace ρ(r) with 1 /(4 π G) ∇²Φ(r) and integrate by parts, assuming that Φ(r) vanishes at spatial infinity (the boundary of our integration domain) to get W = − Integral all space 1/( 8 π G)) ∇Φ(r) · ∇Φ(r) dτ. (6)

The integrand is an energy density that we interpret as the energy density associated with

Φ(r): E = − 1/(8 π G) ∇Φ(r) · ∇Φ(r).”

------------------------

This basic mathematic relation (derived by partial integration) between potential energy and field energy in a mass centric geometry is of fundamental importance.

The relation should be discussed in detail to finally clarify the eventualities about a duplicate existence of potential energy and field energy.

My understanding is that the partial integration relation proofs the either or relation.

Dear Stefan Bernhard Rüster

I recommend you to look at the formulae (1), (5), (6) in the screen shots above, which proof that the complete potential energy of all mass located in the potential, spanned by that mass, is contained in the field.

The proof is contained in the work of Joel Franklin (Reed College) titled "Self-Consistent, Self-Coupled Scalar Gravity"

The citation from that article proofs the equivalence of potential energy and field energy. The poisson equation equates the total energy contained by all the mass positioned in the potential, generated by that mass, to the field energy. With other words, the field contains the complete potential energy.

This fact is mathematically proven by partial integration over the potential energy of all mass in all space. Please consider the cited formulae.

Wolfgang

That c depends on the medium will be questioned in relativity.

It would be nice, but ready to defend?

The c calculation would be ackward, almost 0/0

Dear 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."

No it is not irrelevant, it becomes slightly inaccurate because the mass equivalent of the field energy is not included in the poisson equation. But this inaccuracy does not change the principle that the potential energy of a mass assembly in its own field is contained in the field energy. The only point is, that the partial integration becomes tricky if the field mass contribution is taken into account.

Joel Franklin exactly points to that circumstance in his article. As I understood, your work also considers the contribution of the mass equivalent of field energy in the context of general relativity.

My point here only is that the partial integration shows that the potential energy and the field energy is the same.

Dear Stefan Bernhard Rüster

You also misunderstood me. I only want to show that the partial integration proofs the equivalence of field energy and potential energy.

The potential energy is just another expression for the field energy. In terms of physics this shows that there is just one appearance of energy and this is the field energy.

The field energy can be expressed as potential energy. But this does not mean that mass contains that energy in a somehow compressed form.

No, the energy is and remains distributed in space as an energy density. But this energy density can be expressed as potential energy, which then can be associated to a mass.

But this association does not mean that mass has swallowed potential energy pills.

Stefan Bernhard Rüster "Mc² = mc² + Epot"

We can extend the funny statement to:

"But this association does not mean that mass has swallowed potential energy pills with the wondrous capability to automatically adapt their energy content to the individual situation of the mass." It is even more wondrous that those pills would not have inertia and gravitational mass.

Just the equality relation derived by the partial integration shows that there is no term Epot, because the total potential energy is contained in the gravitational field.

The appendix to the energy pill statement points to causality problems, which would arise if we stay with the potential energy concept.

We finally have to accept that there is no potential energy contained in the mass.

This is indeed in contradiction to essential physics lectures, which have reached an extremely high confidence level, so that fallacies virtually must be excluded. But ...

Valery Borisovich Morozov

Hopefully the intense discussion between Stefan and myself for you does not lead to the impression that the thread, you created, has been kidnapped or misused.

At least I think the "... BUT" in your introduction is met with this special discussion. Further consequences depend on an eventually reachable consens in that discussion.

Stefan Bernhard Rüster "Mc² = mc² - Ebind (*)"

With this statement you enter the discussion about Epot=-Ebind.

If we consider M as the mass of a system. Then of course the binding energies, which adhere the components of the system together, belong to the system. However, just this binding energy is located in the fields, which imply the binding.

Epot=-Ebind just maps the relation between potential energy and field energy revealed by the partial integration.

Consideration of "Epot" excludes fields and considering "Ebind" includes fields. But "Ebind" is not the only contribution which must be included if M denotes the mass of a system. We also must consider the energy content of radiation, temperature, and pressure. In a somehow extended view we can collect all these contributions in the term "binding energy".

But I think you would admit that collecting all this contributions in the term "Epot" would not be acceptable.

Ok, we continue the discussion about field energy and potential energy in the thread "The ultimate reason for the gravitational force"

I beg to differ:

• 100 years of variations of GR developed‡

• Gravity Probe B successful

• a Lagrangian approach toward unifying GR and the SM†

† much more work is needed in this direction

‡ a correct model of the graviton needs development

sgm, 2023/OCT/05

Salvatore Micheal

.....

"a Lagrangian approach toward unifying GR and the SM†"

This is not the result. An attempt to disfigure general relativity.

No connection with general relativity and observed gravity.

This is not physics at all. The fact that a huge number of people get pleasure from it and money does not mean that they are doing something useful.

"Gravity Probe B successful" is the success of the 1915 theory!