Gravity is mediated by particles of even spin, namely spin=2. It's possible to show that particles of even spin mediate attractive interactions, while particles of odd spin mediate repulsive interactions. Additionally, the number of distinct charges and how these can be composed is, also, known. Cf. https://arxiv.org/abs/gr-qc/0411026

Have a look

Preprint Modified General Relativity and dark energy

Stam Nicolis

so, I understand from your sentence that MGR by Gary Nash (which accounts for galaxy trajectories correctly) according to your opinion violates the energy conditions that ensure stability of matter

Galaxy motion is sensitive to the Newtonian limit of general relativity, to the effects of dark matter and insensitive to the effects of dark energy, since the scale of any galaxy is, simply, too small to be affected by dark energy. So it is only sensitive to the fact that gravity is attractive, but can't explain why it can’t be repulsive, also, which general relativity and all its consistent modifications can. For example, supergravity does describe the effects of a spin-1 field, that can affect the geometry of spacetime. This field would mediate a repulsive interaction, however it can’t exist in isolation of the effects of the attractive force mediated by the spin-2 graviton. Its effects can be probed by looking for violations of the equivalence principle, since it does couple to the fields that describe matter, beyond the energy-momentum tensor. Dark energy may violate the equivalence principle, needn’t do so, however. Scalar-tensor theories violate the equivalence principle, not, however, the property that gravity is only attractive, because scalar fields have spin 0, which mediates attractive interactions only. These scalars, similarly, can't exist in isolation of the spin-2 graviton, either. (There can exist interactions, mediated by spin-2 particles that are NOT gravitons, even though they do affect the spacetime geometry and can't be identified with matter fields; however they similarly can't exist in isolation.)

Stam Nicolis: Modified General Relativity (MGR) is General Relativity (GR) with the inclusion of Einstein's missing symmetric connection-independent tensor that describes the energy-momentum of the gravitational field. It took over a century to find that tensor in 2019. It followed from the Orthogonal Decomposition Theorem which was built from the classical theorem of Berger and Ebin 1969 on the decomposition of symmetric tensors on a compact Riemannian manifold that was adapted by Gil-Medrano and Amilibia 1994 to a paracompact Riemannian manifold and then to a paracompact Lorentzian manifold by G. Nash 2019 to produce the Orthogonal Decomposition Theorem (ODT):

An arbitrary non-divergenceless (0,2) symmetric tensor wαβ in the symmetric cotangent bundle S 2T∗M with smooth sections of compact support on an n-dimensional paracompact boundaryless time-oriented Lorentzian manifold (M, gαβ) with a Levi-Civita connection can be orthogonally decomposed as wαβ = vαβ + Φαβ where vαβ represents a linear sum of symmetric divergenceless (0,2) tensors and

Φαβ = 1/2£Xgαβ + £Xuαuβ

where the timelike unit vector u is collinear with one of the pair of regular vectors in the line element field (X, −X) and X is not a Killing vector.

You said "Galaxy motion ... is insensitive to the effects of dark energy, since the scale of any galaxy is, simply, too small to be affected by dark energy." That is incorrect because dark energy as developed in MGR scales with the gravitating mass of any object. Then the Newtonian force can balance the dark energy force and the power 4 Tully-Fisher relation appears from the modified Newtonian radial force equation for weak fields (see MGR and dark matter Int. J Mod. Phys D 2023).

Because MGR contains Φαβ that is constructed from Lie derivatives, it does not depend on the Levi-Civita connection and is therefore invariant under free fall when the connection vanishes; it has the same value when expressed in terms of partial derivatives as it does with covariant derivatives. The equivalence principle is obeyed and local gravitational energy-momentum exits, contrary to GR.

If you apply the ODT to a general symmetric tensor that satisfies the spin-2 Klein-Gordon equation and is divergenceless and traceless, you will see that the Einstein equation strips the metric out of the tensor equation leaving only non-Lovelock tensors as spin-2 tensors. Thus, the metric does not satisfy the spin-2 equation as a field variable and gravitons described by the metric cannot exist. The metric does not act as a force mediator of gravity; it is the field that the curvature of spacetime is calculated from, which replaces the notion of gravitational forces in GR/MGR. One must remember a tensor that is a small perturbation relative to the background Minkowski metric is not the full metric and that the Einstein equation fundamentally contains the full metric.

MGR is not a scalar-vector-tensor theory of gravity. It does not introduce an arbitrary vector field into GR. Rather it explicitly uses the line element vector field that is already a fundamental part of all Lorentzian metrics. The Lie derivative of the Lorentzian metric and the collinear unit vectors u is taken along the flow of the line element vector as given by Φαβ.

, Dear Gary, Gary Nash

I think Stam Nicolis erased one of his replies because I did not invent the sentence "violates the energy conditions that ensure stability of matter"

Dear Stefano. Thanks for clarifying that as we don't want any "misconceptions" to develop on any thread.

Gary

Chian Fan: GR/MGR does encompass both positive and negative gravity. Negative gravitational energy is paramount to cosmic inflation in both GR and MGR. In GR, the false vacuum borrows an enormous amount of negative gravitational energy from the vacuum to create the false vacuum that causes the brief inflation of the Universe. Whereas in MGR rather than assuming the Big Bang evolved from a point in the maximally symmetric spacetime, that singularity can be avoided by assuming an enormous gravitational energy existed before the Big Bang that was contained in an infinitesimal spherical ball of radius ϵ = a(t = 0). The expansion of the sphere requires a closed Universe, so κ = 1. The value of ϵ = 2.63 × 10−34 is determined from the typical quantum calculation of the vacuum energy density (see preprint MGR and dark energy on ReseachGate). After inflation ends, there is a transition phase from negative to positive gravitational energy.

You say: "Schwarzschild spacetime is a spherically symmetric solution of GR, can spherical symmetry be extended across the "event horizon" to r=0?" In MGR, there is no singularity at r=0 because it explicitly depends on the nonvanishing line element field vector.

Gary Nash

Dear Stam Nicolis , Stefano Quattrini and Gary Nash

Thanks to Stam Nicolis for providing references of S. Deser's "How Special Relativity Determines the Signs of the Nonrelativistic, Coulomb and Newtonian, Forces "[ https://arxiv.org/abs/gr-qc/0411026 ], and "supergravity" concepts.

Thanks to Stefano Quattrini for providing the reference of Gary Nash's "Modified General Relativity and dark energy" https://www.researchgate.net/ publication/380065979_Modified_General_Relativity_and_dark_energy

Thanks to Gary Nash for bringing the MGR theory to us. A theory that can include positive and negative gravity and avoid entering a singularity is always more like reality. But Riemannian manifold is a bit difficult for me and I may need time to learn it.

Best Regards, Chian Fan

It is likely that the observable universe is in a state of negative mass or negative energy state. Therefore, it is highly likely that the accelerated expansion of the universe is an anti-gravity (negative gravity) phenomenon.

1.The first result of Friedmann equation for accelerated expansion was negative mass density

Nobel lecture by Adam Riess : The official website of the Nobel Prize

Refer to time 11m : 35s ~

https://www.nobelprize.org/mediaplayer/?id=1729

=====

Negative Mass?

Actually the first indication of the discovery!

=====

HSS(The High-z Supernova Search) team : if Λ=0, Ω_m = - 0.38(±0.22) : negative mass density

SCP(Supernova Cosmology Project) team : if Λ=0, Ω_m = - 0.4(±0.1) : negative mass density

*This value is included in a paper awarded the Nobel Prize for the discovery of the accelerated expansion of the universe.

In the acceleration equation, (c=1)

(1/R)(d^2R/dt^2) = -(4πG/3)(ρ+3P)

In order for the universe to expand at an accelerated rate, the right side must be positive, and therefore (ρ+3P)must be negative. In other words, a negative mass density is needed for the universe to expand at an accelerated rate.

They had negative thoughts about negative mass (negative energy). So, they discarded the negative mass (density). They corrected the equation and argued that the accelerated expansion of the universe was evidence of the existence of a cosmological constant. However, the vacuum energy model has not succeeded in explaining the value of dark energy density, and the source of dark energy has not yet been determined.

They introduce negative pressure, which hides the negative mass density in the negative pressure, but this does not mean that the negative mass density has disappeared.

ρ_Λ + 3P_Λ = ρ_Λ + 3(-ρ_Λ) = - 2ρ_Λ

If we expand the dark energy term, the final result is a negative mass density of -2ρ_Λ.

2. Logical structure of the standard cosmology

We need to look at the logic behind the success of standard cosmology.

Let's look at the equation expressing (ρ+3P) as the critical density of the universe.

In the second Friedmann equation (c=1),

(1/R)(d^2R/dt^2) = -(4πG/3)(ρ+3P)

Matter + Dark Matter (approximately 31.7%) = ρ_m ~ (1/3)ρ_c

Dark energy density (approximately 68.3%) = ρ_Λ ~ (2/3)ρ_c

(Matter + Dark Matter)'s pressure = 3P_m ~ 0

Dark energy’s pressure = 3P_Λ = 3(-(2/3)ρ_c ) = -2ρ_c

ρ+3P≃ ρ_m +ρ_Λ +3(P_m +P_Λ)= (1/3)ρ_c +(2/3)ρ_c +3(−2/3)ρ_c= (+1)ρ_c + (-2)ρ_c = (−1)ρ_c

ρ+3P ≃ (+1)ρ_c + (-2)ρ_c = (−1)ρ_c

The logic behind the success of the ΛCDM model is a universe with a positive mass density of (+1)ρ_c and a negative mass density of (-2)ρ_c. So, finally, the universe has a negative mass density of “(-1)ρ_c”, so accelerated expansion is taking place.

The current universe is similar to a state where the negative mass (energy) density is twice the positive mass(energy) density. And the total mass of the observable universe is the negative mass state.

3. So, what can correspond to this negative mass density?

When mass or energy is present, the negative gravitational potential energy produced by that mass or energy distribution can play a role. The gravitational potential energy model is a model in which +ρ and -ρ_gp exist, and shows that |-ρ_gp| can be larger than +ρ.

Looking at the mass deficiency effect (negative mass effect) caused by binding energy,

● ----- r ----- ●

m ------------ m

When two masses m are separated by r, the total energy of the system is

E_T=2mc^2 − Gmm/r

In the dimensional analysis of energy, E has kg(m/s)^2, so all energy can be expressed in the form of (mass) X (velocity)^2. So, E=Mc^2 holds true for all kinds of energy. Here, M is the total mass or equivalent mass. If we introduce the negative equivalent mass "-m_gp" for the negative gravitational potential energy,

−Gmm/r=(−m_gp)c^2

E_T=2mc^2−Gmm/r=(2m−m_gp)c^2=(m^∗)c^2

The gravitational force acting on a relatively distant third mass m_3 is

F=−G(m^∗)(m_3)/R^2=−G(2m−m_gp)(m_3)/R^2=−G(2m)(m_3)/R^2 + G(m_gp)(m_3)/R^2

That is, when considering the gravitational action of a bound system, not only the mass in its free state but also the binding energy term (-m_gp) should be considered.

In the general case, the value of gravitational potential energy(gravitational self-energy) is small enough to be negligible, compared to mass energy mc^2. However, as more mass is collected, the ratio of (negative) gravitational potential energy to (positive) mass energy increases.

In the case of Moon, U_{gp - Moon} = ( - 1.89 x 10^ -11)M_{Moon}(c^2)

In the case of Earth, U_{gp - Earth} = ( - 4.17 x 10^ -10)M_{Earth}(c^2)

In the case of the Sun, U_{gp - Sun} = ( - 1.27 x 10^ -4)M_{Sun}(c^2)

In case of a Black hole, U_{gp - Black - hole} = ( - 3.0 x 10^-1)M_{Black - hole}(c^2)

At R=2GM/c^2,

U_gp-Black-hole = -(3/5)(GM^2/R) = -(3/5)(GM^2/(2GM/c^2)) = -(3/10)Mc^2

In the case of a black hole, the negative gravitational potential energy is 30% of the positive mass energy (mass energy when the masses forming a black hole are free state). It can be seen that the magnitude of the gravitational potential energy is not negligible compared to the mass energy.

What would happen in a universe with more mass?

4. For the observable universe, calculating mass energy and gravitational potential energy

Use a critical density ρ_c=8.50x10^-27[kgm^-3]

[Result summary]

At R=16.7Gly, (-M_gp)c^2 = (-0.39)Mc^2

|(-M_gp)c^2| < (Mc^2) : Decelerating expansion period

At R=26.2Gly, (-M_gp)c^2 = (-1.00)Mc^2

|(-M_gp)c^2| = (Mc^2) : Inflection point (About 5-7 billion years ago,

consistent with standard cosmology.)

At R=46.5Gly, (-M_gp)c^2 = (-3.04)Mc^2

|(-M_gp)c^2| > (Mc^2) : Accelerating expansion period

Even in the universe, gravitational potential energy (or gravitational action of the gravitational field) must be considered. And, in fact, if we calculate the value, since gravitational potential energy is larger than mass energy, so the universe has accelerated expansion. Gravitational potential energy model accounts for decelerated expansion, inflection point, and accelerated expansion.

As the universe ages, the range of gravitational interactions increases, and thus accelerated expansion appears to have occurred because the negative gravitational potential energy increased faster than the positive mass energy.

As the universe grows older, the range R of gravitational interaction increases. As a result, mass energy increases in proportion to M, but gravitational potential energy increases in proportion to -M^2/R. Therefore, gravitational potential energy increases faster.

Therefore, as the universe ages and the range of gravitational interaction expands, the phenomenon of changing from decelerated expansion to accelerated expansion occurs.

Preprint Dark Energy is Gravitational Potential Energy or Energy of t...

Stam Nicolis "Galaxy motion is sensitive to the Newtonian limit of general relativity..."

The calculation of galaxy motion is based on a retarded relation between the movement of the gravitational field sources and the presence of according fields. The retardation is assumed to be according to the speed of light.

But if the fields are co-moving with its non-accelerated source, the field values are instantaneously matching to the source position without any retardation.

Only a field source acceleration is subject to a retardation. But this is an effect some orders of magnitude smaller.

The thread question

“【NO.40】Doubts about General Relativity (5) - Should there be "negative gravity" in General Relativity?”

- really looks as is rather vague one. The “negative/[“repulsive”] gravity” indeed is considered/published in conventional journals in mainstream cosmology, when the authors attempt to explain the cosmological effect “accelerated Matter’s space expansion”, which rather probably correctly is experimentally observed.

At that this effect is “explained” already in mainstream cosmology, by introducing in the GR equations of "lambda term"; what is possible in the GR, but has rather indirect relation just to gravity, which acts in gravitationally coupled systems of bodies as action on the bodies of “spacetime curvature”, which is created by the bodies’ masses; and, at that, this curved spacetime obtains energy to force bodies to move along “geodesics”;

- while the lambda-term relates to some “dark energy” that has no relation to the stress-energy tensor hat is created by masses, i.e. no relation to gravity.

Really “dark energy” is nothing else than some really completely ad hoc mystic “energy”, which has no any relation to everything that is observed, besides cosmology, in Matter and considered by physics,

- though the postulated in the GR interactions in systems “mass-spacetime-mass” are equally ad hoc mystic ones, which have no any relation to everything that is observed besides gravitational interactions in Matter and considered by physics.

Nonetheless even on this standard mystic landscape some “negative gravity” looks as tooo strange something.

All the transcendent/mystic strangenesses above in the mainstream principally logically inevitably follows from the fact, that in the mainstream philosophy and sciences, including physics, all really fundamental phenomena/notions, first of all in this case “Matter”– and so everything in Matter, i.e. “particles”, “fundamental Nature forces” – and so “fields”, etc., “Consciousness”, “Space”, “Time”, “Energy”, “Information”, are fundamentally completely transcendent/uncertain/irrational,

- from any transcendence by using really rigorous logical consideration principally only some other transcendence can appear.

The phenomena/notions above can be, and are, really rigorously scientifically defined only in framework of the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

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

- and mire concretely relating to Gravity

[which is, of course, fundamentally nothing else than some fundamental Nature force, as that Weak, Electric, Strong/Nuclear Forces are, and fundamentally isn’t some “spacetime force”. Matter’s spacetime is 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), which fundamentally cannot be “contracted”, “dilated”, “curved”, “expanded”, etc.]

in the Shevchenko-Tokarevsky’s 2007 Planck scale informational initial physical Gravity model, more see https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces, where also the initial Electric Force model is presented,

At that it turns out to be that at least 3 fundamental Gravity, Electric and Nuclear [more see the initial model in https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force/comments, recently edited a bit version appeared] act by the same scheme, Matter is rather simple logical system.

More see the links,

Cheers

Negative energy is binding energy. According to General Relativity, more compactly arranged matter has a smaller gravitational mass than the same matter distributed over a larger volume. A photon has twice the gravitational mass compared to the rest mass of a particle with equivalent energy. This follows from the solution of Tolman, Ehrenfest, and Podolsky for the directed flow of electromagnetic energy and from quantum gravity, L B Okun 2008 Phys.-Usp. 51 622. Now, if we consider the annihilation of an electron and a positron whose gravitational masses, according to the CERN experiment, coincide, it turns out that their total gravitational mass is half the gravitational mass of the resulting gamma quanta. Let's take a sphere filled with electron-positron pairs that annihilate simultaneously. According to Birkhoff's theorem, the gravitational mass of the sphere right after this does not change. This means that during annihilation, additional particles with negative gravitational mass must be emitted.

https://link.springer.com/epdf/10.1007/s12648-021-02104-2?sharing_token=Rsa-0JjSn3MjZWLzxz6JlPe4RwlQNchNByi7wbcMAY5Yb5WOp5mY7YYxfnBXwPCRDmZHgE1DZeIA6fnbvOqBP1sIaK8dXgxq0-x5UMFsakk0gwHrb-ALEs164wM0KKRToVn7NW0VzHyrmm4xEEs0EKRrI15oEmJpE9aZj2kwT0k%3D

Recent SS comment in https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics/comments

- is relevant to this thread discussion.

Cheers

Check this out https://www.researchgate.net/post/Exploring_Photon_Emission_from_Quantum_Vacuum_Fluctuations_and_Its_Implications_for_Negative_Mass_Generation#view=669a0d4f494fc5e17b07ab77

Preston Guynn "Since general relativity theory has been thoroughly falsified..."

In what world are you living? General relativity has been uncountably often verified and never falsified. All alleged falsifications finally turned out to be caused by misunderstanding general relativity.

As long as we recognise that the world consists of time, space, matter and energy, the fundamental thought of general relativity, the "mass/energy- spacetime structure" relation, cannot be wrong. All that is lacking in GR is an explanation of the mechanism by which this relation arises. Physicists say that "Tµν-Gµν" is unique, and this should be true in mathematical form. But physically, the spacetime structure corresponding to Tµν expressed by Some/µν may not be unique and simplest. There are other simpler ways of describing the spacetime structure that still perfectly reflect Tµν and are consistent with cosmological observations.