Thank you.  The article by Baez and Weiss is helpful, but doesn't answer my question, which really refers to calculation of energy density in the case of a weak field -- in which case the nonlinearities that arise in GR can (I think) be ignored.  The main message from the articls seems to be that there is no consensus on the "right" definition of energy density in the context of general relagivity.

Thanks for that link too, but it does not seem to address energy density of the field.

Dear Steve,

There is no such concept. The situation is quite analogous to the one in non-Abelian gauge theories, where there is no reasonable definition of "color charge density" (in contrast to the Abelian case of ED you mentioned). For suitable asymptotic conditions one can only define global Yang-Mills charges. This issue comes up again and again, but that's the way it is.

I just had reviewed paper on this topic for Math.Sci Net by S-T.Yau and associates. I suppose, you know who is S-T Yau  https://en.wikipedia.org/wiki/Shing-Tung_Yau  

The story begins with the ADM paper  https://en.wikipedia.org/wiki/ADM_formalism   But lately there was some additional progress, e.g. look up at latest papers by Yau and associates and references therein

It's not a question of the existence of a ``reasonable'' definition for the energy density of the gravitational field-there doesn't exist any such definition at all. And the reason is that energy is the conserved quantity related to time translation invariance. Since time translation invariance is a  local, not a global symmetry in gravity, there isn't any way to define a gauge invariant energy density, which is, by definition, a local quantity. Any gauge invariant quantities are global quantities, with respect to the gauge group, which, for gravity, means, they're non-local. This holds, therefore, not only for energy and its density.

In addition, gravity is non-linear and in that sense is similar to a non-Abelian gauge theory (indeed, it is  a non-Abelian gauge theory, with non-compact gauge group, the group of diffeomorphisms.). It is the non-compactness of the gauge group that makes it different qualitatively from the non-Abelian gauge theories that describe the other interactions.  

The Euclidian action of the gravitational field-which would be the energy density of a field configuration in contact with a bath for a usual field theory and would, thus, be bounded from below, if the system were stable-is, indeed, unbounded from below, in general. One way to understand this is from the fact that the gauge group is non-compact; another way is to understand that gravity is an attractive force, and leads, generically, to the formation of black holes. So it's not true that the energy (not energy density, which isn't a gauge invariant quantity) of a gravitational field is positive. For this to be the case, a positive cosmological constant is needed.

Thank you, that is helpful.

Stam, I think you misunderstood my point. In non-Abelian gauge theories the Yang-Mills current  is only covariantly conserved, very much like the energy-momentum tensor in GR. Similarly to Einstein's  pseudo-tensor (or the one by Landau-Lifschitz), one can  introduce a pseudo-current ( t'Hooft) which is conserved in the usual sense, and define with it for certain situations (asymptotically pure gauge fields) total Yang-Mills charges. I just wanted to emphasise this close formal analogy. (Of course, the physics is entirely different.) I used this analogy in some talks to particle physicists, in order to explain why in GR there is no general energy conservation law. There are, of course, also other reasons, like the lack of translation symmetry or the equivalence principle. I hope we agree on this.

Professor Straumann,  of course we agree. What I wanted to stress is, precisely, the difference between covariant conservation, which occurs when a gauge symmetry is present and global conservation, when a global symmetry is present. There can't be any gauge invariant energy density in gravity, precisely for this reason-so my reaction was to the qualification ``reasonable'' in your previous comment. No qualification is required. 

Does the electromagnetic field energy density retain its meaning in GR?

It does, as a source term for the Einstein equations. That's how electromagnetic waves affect spacetime. 

The energy momentum tensor of matter, defined as the fields that cannot be identified as  part of the gravitational action, by field redefinitions, does have a physical meaning-though the identification of energy and momentum is, still, tricky, for the reasons discussed above.  For the gravitational field itself, i.e. in the absence of any such matter, one way to see the problem for it is that, by the definition of the Einstein equations, deduced from the Einstein-Hilbert action, it vanishes identically.  

The standard rule of transferring local physical laws from SR toGR (unambiguous for SR-laws of first differential order) also supplies the energy-momentum tensor of classical matter models, and that it is covariantly conserved,  thus not a conservation law in the ordinary sense. This is no surprise since the gravitational field acts on matter. (Such considerations may be taken as an exact expression of Einstein's heuristic "principle of equivalence".)

Dear Steve,

I add something partly said by Prof. Straumann et al. 

GR does not allow an energy density of the Gfield, since does not allow a localization of the gravitational energy. The Einstein EP forbids such localization, but thanks to the Pseudotensor the energy is conserved globally for isolated systems. GR states that gravitational energy exists but In regions of a globally minkowskian metric  where gravitation is globally negligible, in the far surroundings of distribution of masses.

In other words in GR there is no room for gravitational energy where the space-time is not flat, close to the masses. The degrees of freedom of GR cannot allow the curved space-time and the gravitational energy coexist locally.

There are  theories of gravitation such as FGT (Poincarè, Feynman, Thirring, Kalman) in which the spacetime is not Riemaniann which  allow  the localization of the gravitational energy. These theories though have other kind of issues.

For 100 years after the failed attempt of Kaluza and Klein to unify EM and Gravitation, there has been no successfully attempt to build a satisfactory theory  on higher dimensions.

It is interesting the attached paper published recently in the European Journal of Phisics. The attempt to insert the gravitational energy together with the EEP (non flat metric) in the flat space time of SR   gives birth to an unlikely exponential law.

Article Non-linear energy conservation theorem in the framework of S...

Thank you, those comments are very helpful.  Field energy density seems to be an important concept, and it appears that there have been many attempts to construct a version of General Relativity in which field energy density has a legitimate meaning.  If such a theory were constructed that met the requirements of general relativity and produced versions of Maxwell's equations and Einstein's equations whose quantitative predictions did not conflict with experimental and cosmological observations, what importance would it have?

A comment to what Stefano wrote. Thirring and several other authors (including Feynman) started with attempts to develop a gravitational theory in the framework of SR, where the gravitational field was originally a massless spin-2 field (discussed in 1939 by Pauli and Fierz). The consequent development of the theory showed that it is possible to eliminate the flat Minkowski metric, leading to a description in terms of a curved metric which has a direct physical meaning. The originally postulated Lorentz invariance turns out to be physically meaningless and plays no useful role. The flat Minkowski spacetime becomes a kind of unobservable ether. The conclusion is inevitable that spacetime is a pseudo-Riemannian (Lorentzian) manifold, whereby the metric is a dynamical field, subjected to field equations.

Let me also include a quote of Einstein from 1954 (one year before his death):

``I see the most essential thing in the overcoming of the inertial system, a thing that acts upon all processes, but undergoes no reaction. This concept is, in principle, no better than that of the center of the universe in Aristotelian physics.''

Dear Prof. Straumann,

"The flat Minkowski spacetime becomes a kind of unobservable ether. The conclusion is inevitable that spacetime is a pseudo-Riemannian (Lorentzian) manifold, whereby the metric is a dynamical field, subjected to field equations."

I agree that a non detectable Aether should emerge from the FGT which was initially conceived by Poincarè who supported the presence of a medium for the EM waves.

Yes the metric has to be Riemaniann if the EEP holds. There are at least a couple of  points still given for granted regarding the Einsten EP.

a) the EP (inertial and gravitational mass equality) has been tested in a static/stationary experiment (EOTVOS) to very high accuracy. Though It has  been given for granted that it holds locally also in a non stationary situations as EEP in order to allow the description of non stationary motion in GR.

b) The experiment of atomic clocks in accelerated frames has never been performed so far. The proof regards the foundation of the Einstein EP tested in dynamical, non stationary condition. It is demonstrated in  SR that twin clocks experiencing same proper acceleration do not stay in sync, they delay according to VX/c2 where V is their final speed after the accelerated motion and X is their reciprocal distance.

“…the EP (inertial and gravitational mass equality) has been tested in a static/stationary experiment (EOTVOS) to very high accuracy. Though It has to be given for granted that it holds locally also in a non stationary situations …The experiment of atomic clocks in accelerated frames ha s never been performed so far…”

 - the acceleration with rather large probability doesn’t relate to some “deviations from EP” [which holds in the statics]. That is another thing that when a body is accelerated to a speed V its inertia – and so “inertial mass” becomes be essentially uncertain since depends on the angle between impacted force and the speed directions; besides – at body’s motion with a large probability some gravitomagnetic force appears , what can be interpreted as some “changing of the gravitational mass”.

 “…in the theory of SR that twin clocks experiencing same proper acceleration do not stay in sync and delay according to VX/c2 where V is their final speed …”

 - that isn’t so in the reality and even in the SR; the integral slowing down of moving clocks on the way from start to a distance X  is determined by the reverse Lorentz factor and the time elapsed interval when the speed isn’t zero – i.e. the clocks’ slowing is determined by the instant [absolute] speed’s value  only.

 Cheers

Dear Steve Mcgrew,

to calculate the energy density of the gravitational field you can use Covariant theory of gravitation. The time component of the stress-energy tensor in the theory is the energy density of the gravitational field. See the link .  

https://en.wikiversity.org/wiki/Gravitational_stress-energy_tensor

The energy of the gravitational field is positive - because the binding energy of the surrounding vacuum medium of 'virtual' fermion/anti-fermion pairs is negative. (Chemical convention for binding energy). Gravitation results from local  reduction of that negative binding energy of the vacuum state due to imposition of matter in atomic state causing bodies to be pushed together by greater pressure beyond than between.( J/m3 = Nm//m2 = Pa, units of pressure). When fully bound the vacuum is undetectable but when partially bound the inertial mass and charge of the fermions controls 'our' world of the atomic state. E=mc2 gives c = sqrt(E/m) - choose your volume to measure the energy and mass densities.  Newton used same formula for speed of sound; use it with binding energy of electron and positron at 1.02MeV to get value of c.  Simhony's electron positron lattice (epola) model of 1973 explains it all. [See archive of his original website at epola website and download his booklet and paperback]

. 

www.epola.co.uk

You can calculate the energy of the gravitational field of treating flat "capacitor" of two heavy plates.

You can borrow a result of the Landau-Lifshitz "field theory" of the book.

and

https://www.researchgate.net/publication/305765985_Popravka_k_nutonovskomu_zakonu_tagotenia

or

https://www.researchgate.net/publication/305308805_Einstein%27s_exact_equation_and_energy-momentum_tensor

"So, if we accept the equivalence E=mc² in all cases, there should be an attraction between the gravity field and the surrounding masses. "

Yes, the correction is small, but it is of the same order as the Schwarzschild solution. And if we consider it in Einstein's equation, we obtain new results.

https://www.researchgate.net/publication/303844514_Gravitational_field_equations_and_relativistic_homogeneous_gravitational_field

Preprint Einstein's exact equation and energy‐momentum tensor

Preprint Поправка к ньютоновскому закону тяготения

Preprint Gravitational field equations and relativistic homogeneous g...

Thierry,

You cannot make appear a mass without making a gravitational field for it too. The quantum fluctuation is based on the fact that gravitational field, mass field, fluctuates at the same rate. But maybe you can gather mass quickly enough by 1-dimensional transitions so that 3-dimensional space have not as much time to form gravitational field spherically correct. You cannot observe it at short distances and at spherical gravity field, but possibly with fly-bys.

That could be the phenomenon we know as dark mass (or dark matter). When galaxy clusters collides, there is formed a new deeper gravitational well and maybe serial signalling is needed to transfer information for the big form of the gravitational field to take its new form which takes many million years. We observe "dark matter halos" going through each other...

Dear Valery,

you pointed out that there is an issue in the way energy is treated by GR. Your exponential metric is in the line of the exponential metric suggested in the link I've shown. The metric in GR and SR in order to satisfy locally the conservation laws, have to take a different form which is not the one derivable in the case of GR from the Einstein Field Equations.

DM effects can be interpreted as the result of grain boundary defects in a polycrystalline bound electron-positron lattice acting as the medium of the 'vacuum'.

I agree naturally on the fact that we have no observations on any matter behind dark mass phenomenon. The question is very open.

"How can we calculate the energy density of the gravitational field? "

The energy density of the field is known.

By Newtonian theory -     Landau L D, Lifshitz E M The Classical Theory of Fields  Vol. 2 (4th ed.). Butterworth–Heinemann (1975).

But this does not give the desired result. See

https://www.researchgate.net/publication/305765985_Popravka_k_nutonovskomu_zakonu_tagotenia Sorry, only in Russian.

Preprint Поправка к ньютоновскому закону тяготения

The discussion has, apparently,  left the declared subject of the thread-which is good. Regarding dark matter: It doesn't have to do with the bending of light,   but with the rotation curves of galaxies, (as well as other effects; cf. Planck data, e.g. here: https://arxiv.org/abs/1502.01589 .  While, historically, evidence for dark matter was obtained  from measurements of rotation curves, this is, no longer, the only way to probe its presence.).  While the particle content is not known, it is known that it isn't matter that has strong, or electromagnetic, interactions with ordinary matter. Current searches focus on whether it has weak interactions  with ordinary matter-beyond the gravitational ones.

It may not-in which case, unless it has hitherto unknown, non-gravitational interactions, with new forms of matter, to be discovered in particle accelerators, it may be impossible to discover its particle content, but only determine  its equation of state. 

Any other attempts to describe the effects of dark matter,  without invoking new forms of matter, involve either modifying gravitational interactions explicitly (e.g. MOND)-but this requires fine-tuning, in order not to disagree with known measurements-or by assuming additional effects, such as mass for the graviton. This, also, requires fine-tuning, of course, and has issues with instabilities that are under investigation. 

Good question, to answer this question, first of all we need to know what made the gravitational field.

With regard to the exchange particles concept in the quantum electrodynamics theory and the existence of graviton, we should present a new definition of graviton. To define graviton, let’s consider a photon that is falling in the gravitational field, and revert back to the behavior of a photon in the gravitational field. But when we define the graviton relative to the photon, it is necessary to explain the properties and behavior of photon in the gravitational field. The fields around a "ray of light" are electromagnetic waves, not static fields. The electromagnetic field generated by a photon is much stronger than the associated gravitational field. When a photon is falling in the gravitational field, it goes from a low layer to a higher layer density of gravitons.

We should assume that the graviton is not a solid sphere without any considerable effect. Graviton carries gravity force, so it is absorbable by other gravitons; in general; gravitons absorb each other and combine. During the photon is falling in the gravitational field, its energy (mass) increases. According to dW=dmc^2, the force of gravity performs work on the photon, so the mass (energy) of the photon and its frequency increase.

In interaction between gravity and photon (blueshift), when gravity acts on photon and gravitons enter the photon, gravitons do change the intensity of electric and magnetic fields which belong to photon. So, gravitons behave so that they are carrying the charge and magnetic effects in the structure of photon. When gravitons enter the photon, the intensity of electric and magnetic fields increases, but photon has no electric effect. So, there should be two groups of gravitons one that behaves like electric field and the other one that neutralizes the electric effect of other group. So, a group of gravitons behaves like positive electric field and the other one behaves like negative electric field and they neutralize each other’s electric effect. But they are moving, so a group of gravitons behave like magnetic field, and the intensity of two vertical electric and magnetic fields increases. So, gravitons are either color charge or color magnet. When a photon shifts to blue in the gravitational field, gravitons convert to electromagnetic energy. In fact gravitons convert to color charge and magnetic color and enter electric and magnetic fields of photon. It is acceptable because when photon is falling in the gravitational field, the intensity of its electric and magnetic fields increase. So, a photon is made up of color charges and magnetic color that have linear speed equal c with photon motion and nonlinear speed in the structure of photon, so they move faster than light speed.

https://www.researchgate.net/publication/309153372_What_is_CPH_Theory?ev=prf_pub

https://www.researchgate.net/publication/308960493_Generalization_of_the_Dirac%27s_Equation_and_Sea?ev=prf_pub

Article What is CPH Theory?

Data Generalization of the Dirac’s Equation and Sea

In my own work I have found that Maxwell's equations and Einstein's gravitational field equations can be derived together by variation of a single action constructed from a complex energy-momentum tensor in which the electromagnetic field provides the real part and the gravitational field provides the imaginary part. The source term is complex. The imaginary source component has the effect of a mass density, and is the sum of the energy-momentum density of the electromagnetic field and the energy-momentum density of the gravitational field, where the gravitational field energy is negative (because it results from the square of an imaginary quantity).

The real component of the source term - the source of the electromagnetic field - is particularly interesting. It contains only cross-terms. For example, it contains the term E dot G which has the effect of a charge density.

The resulting field is nonlinear because of the source terms, but the source terms are significant only in the case of extremely strong fields. In the case of ordinary field strengths, the electromagnetic field and gravitational field are effectively uncoupled because the source terms become extremely small. However, near a point charge or a point mass, and particularly near a charged point mass, the effects should become significant.

I would like to work with others to explore the possible usefulness of this idea. I hesitate to call it a "theory" because its consequences have not yet been worked out far enough to know if it leads to conclusions that conflict with observations and experimental results.

Steve, make sure that Rainich was the first one  who solved  joint Gravity and Maxwell's fields. This happened in mid of 1920's. His work  was sitting dormant till it was noticed by Wheeler and Misner in 1957 https://en.wikipedia.org/wiki/Geometrodynamics      Further developments you can find in beautiful PhD thesis by Edward Anderson   https://arxiv.org/pdf/gr-qc/0409123.pdf   For a quick introduction read this    https://jayryablon.files.wordpress.com/2008/03/wheeler-geometrodynamics.pdf    Good luck!

In fact, the solutions of the Einstein equation do not allow us to find the elements of the energy-momentum tensor. See, for example, 1. Einstein A. Tiber Gravitationwellen. Sitzungsber. preuss. Akad. Wiss., 1918, 1, 154-167.

This breaks the beauty of GR.

I will try to explain this and find a way out of this situation.

Einstein's Equation

Abstract

According to the principle of the general relativity theory, the gravity field equation should contain the field energy as a source of the field itself. Including the field energy-momentum tensor into the Einstein’s equation brings extra unknown quantities to the equation. Such equation is not suitable for a metric finding; however it allows – based on the known metric – calculating the whole energy-momentum tensor of both matter and gravitational field. As the gravity field metric, the metric of continuous field can be used, parameters of which are found from the generally covariant one-parametric equation. Here, the solutions are given of the equation for the spherically symmetric stationary problem. One of the solutions coincides practically with that by Schwarzschild for weak fields, while the other one describes an expulsive field.

https://www.researchgate.net/publication/319328979_Einstein%27s_Equation

Article Einstein's Equation

Tierry (De Mees) , please read my paper  Newtonian limit of Einsteinian gravity: from dynamics of Solar system to dynamics of stars in spiral galaxies  listed in my papers on Research gate. You will notice after reading that I am on the same page with you on this topic

Dear Steve, Your question is a very good one. As previous posts have pointed out, it is not possible, in general, to localise the energy of the gravitational field. But there is the pseudo stress energy tensor, (usually called t) whose time-time component, if added to that of the matter SE-tensor T, integrated over all space is conserved. It's most clearly described in Dirac's text book, in my opinion.

Now what is strange -- and had me confused for some time -- is that, as you point out, in Newtonian gravity the energy density is negative and is -|g|^2 / 8 pi G, while (in the weak-field limit) t^0_0 isthe same but positive. It is all consistent, however, as the matter part of the conserved thing involves sqrt(g_00) times the mass density and sqrt(g_00) = 1 + Phi for weak fields, and gives a negative contribution.

Nicholas Kaiser,

Good point. I expected this, but was too lazy, to work out the details.

In the Newtonian limit, the energy density of the gravitational field is calculated

Hi Valery, That's a nice find. This expression for the energy density of gravity also follows directly from Poisson's equation (if you integrate L+L's W by parts you get half the density times the potential; the usual result). But Steve's original question was, I believe, how one reconciles this (being negative) with the positivity of the component t^00 of the pseudo-tensor for the field. L+L comment on this also in the footnote to this problem (attached).

The half is significant, in this light. If you look at L+L's action [...] here as kinetic (1st term) minus potential energies, then it's clear they're taking latter to be the integral of ρ φ (no half!) plus a positive term from the field. The same thing happens in the conserved total pseudo-tensor √g(T+t) where there is a (1 + φ) in the Jacoboian factor that multiplies the density ρ in T^00 (in the weak-field limit).

Dear all,

I reproduce here what Nicholas Kaiser wrote a few hours ago (so far he is the only one who did the work I recommended, with the result that had to be expected):

Dear Steve, Your question is a very good one. As previous posts have pointed out, it is not possible, in general, to localise the energy of the gravitational field. But there is the pseudo stress energy tensor, (usually called t) whose time-time component, if added to that of the matter SE-tensor T, integrated over all space is conserved. It's most clearly described in Dirac's text book, in my opinion. Now what is strange -- and had me confused for some time -- is that, as you point out, in Newtonian gravity the energy density is negative and is -|g|^2 / 8 pi G, while (in the weak-field limit) t^0_0 is the same but positive. It is all consistent, however, as the matter part of the conserved thing involves sqrt(g_00) times the mass density and sqrt(g_00) = 1 + Phi for weak fields, and gives a negative contribution.

I hope that some of you convince themselves that what Nicholas wrote is correct.

Sorry, confused the text.

Dear Nicholas Kaiser

> 

according to what emerges from GR, the ED should be positive, but unless you want to infringe the conservation laws, the energy hence the energy density must remain negative...

GR has issues in localizing the gravitational energy...or rather cannot do that. To be sure of that you can have a look.

https://www.researchgate.net/post/Is_the_non-locality_of_the_gravitational_energy_a_serious_problem_for_General_Relativity

To put it simple, for "weak fields": When two equal elemental electric charges reduce their distance, the electric field configuration changes and the increase of the potential energy can be associated with an increase of the positive (local) field energy (proportional to the local electric field strenght squared). The charges remain unchanged in this process. When two equal masses approach, the field configuration changes in a completely analogous way, and one may assume that the energy of the gravitational field is positive and therefore increases. However, when the two masses approach, their *mass* (energy "mc^2") embedded in the mutual gravitational field *decreases* by an amount which is twice as big as the increase of the field energy. This way, the total energy of the system decreases when the masses approach, whereas approaching equal charges requires energy.

Steve Mcgrew "Doing the same for the gravitational field yields an energy density proportional to the negative of the square of the gravitational field.  Yet, it seems to be accepted that the energy density of the gravitational field is positive.  Can anyone explain this to me?"

Yes, the energy density of the gravitational field is negative. The plate example impressively shows this as follows:

The volume between the plates of area A is field free. But the sum of field filled volume and empty volume A*d is constant if we modify the distance d between the plates.

If we increase d, by ∆d, we increase the field free volume by A*∆d. We then reduce the field filled volume by -A*∆d.

By increasing the plate distance against the gravitational force F we add the energy F*∆d to the gravitational field. This energy density is contained in the modified volume which leads to F*∆d=volume*energy density.

This leads to energy density = F*∆d/(-A*d). Because F*∆d is positive, the energy density is negative.

As a summary, the fact that the volume between the plates is field free and that the sum of field free and field filled volume does not change, if we move the plates, proofs the negative energy density of the gravitational field.

Einstein solved this problem in 1913. The energy density of the gravitational field is positive, but approaching two masses reduces the energy of the system because the masses get smaller, in contrast to electric charges which stay the same.

Andreas Aste "The energy density of the gravitational field is positive, but approaching two masses reduces the energy of the system because the masses get smaller, in contrast to electric charges which stay the same."

Sorry, but this is blatant nonsense. Only the relativistic mass m is variable. The difference to the rest mass m0 exclusively corresponds to the kinetic energy. (m=γm0). γ is the Lorentz factor γ=1/sqrt(1-v²/c²)

The total mass of the system keeps unchanged because the loss of gravitational field energy is compensated by the gain of kinetic energy.

Obviously, you don't understand what I mean. The masses approach, but you consider a static configuration then. A negative energy density of the gravitation field would lead to instability, by the way. In full relativity there is no localizable energy density of the gravitational field.

Just read

https://arxiv.org/pdf/1408.3594.pdf ,

and try to understand why the field energy gives a positive contribution in eq. (17).

Andreas Aste

It is so easy to understand that the gravitational field energy is negative. You only should consider that the gravitational field increases its strength if it provides energy. Two bodies, which attract each other, increase their common field and transfer field energy to kinetic energy.

If this is not plausible, you can look at a bank account, which increases its absolute amount, when you withdraw some money. This account must be in the red.

I see now that you lack any deeper understanding of field theories. When a positive electric charge approaches an other negative charge, both charges stay the same, and the positive field energy of the (static) system decreases. If you let the masses move freely during this process, a part of the positive field energy is transformed into kinetic energy. The field energy is always positive, but it becomes smaller. But two masses approaching each other in a scalar theory of gravity increase the field energy which is also positive in general (the field lines look like in electrodynamics between equal repelling charges), but at the same time the masses get smaller such that the total energy of the (static) system also gets smaller. That is Einsteins solution. Also in general relativity, a mass or energy quantum like, e.g. a photon, disappears when it approaches the event horizont of a black hole; an electric charge stays the same. Anyway, gravity is not (purely) scalar and the notions you use are of limited use. And field theories with negative energy are unstable and inconsistent. Please read and understand the paper I mentioned above.

Andreas Aste "try to understand why the field energy gives a positive contribution in eq. (17)."

Equation (17) has been derived with a Lagrangian energy density:

"Lagrange density associated with gravity, we would call the energy density L = H = 1 /(8 π G) ∇Φ · ∇Φ from (10), and this agrees with the field energy density we got in (6)."

However, the energy density expression in (6) is negative. Could you explain why the sign suddenly becomes irrelevant?

Andreas Aste "And field theories with negative energy are unstable and inconsistent. Please read and understand the paper I mentioned above."

Look at:

The energy density W of the gravitational field is W=W0-g²/(8πG). g is the gravitational accleration, G is the gravitational constant, W0 is an omnipresent cosmic field which ensures that the gravitational energy density remains positive.

There is nothing unstable or inconsistent in the formula for the energy density of the gravitational field.

As reported before, the paper from Joel Franklin has an issue. It is inconsistent in respect to the sign of the gravitational energy density. But there are also more inconsistencies.

So if you change the sign of the potential of the harmonic oscillator from x^2 to -x^2, you get a meaningful theory? This is an analogue to what happens when you introduce negative definite field energies. It is interesting that you are correcting Einstein. The discussion stops here, sorry.

Andreas Aste

Sorry, but it only is an allegation that I would propose a meaningful theory based upon an absolute negative energy density. An offset leads to a positive energy density.

However do you really deny that gravitational fields get stronger if they deliver energy?