"I think that this problem reflects the incompleteness of General relativity. "

Yes for sure GRT really misses a lot. The problem of the free falling clocks which have to maintain the same rate is something related to the problem of non accountable gravitational energy. Feynman tried to put a patch on the issue of the gravitational energy  but wasn't very successful either.

GRT works well wherever problems are stationary or where the system has a definite Hamiltonian, cannot cope with transients.

The problem is of paramount importance since it is reasonably something which impairs the unification of QPhisycs and Gravitation. Newtonian gravitation is conservative globally and locally, GRT is not, quantum Physics is based strictly on energy and momenta conservations. Not so few are still attempting to use Newtonian gravitation to try to make the unification with QPh, while they are using strings to try to avoid GRT and unify Gravitation and QM.

"One way to prove that gravity is a true force is if a gravitational field can be shown to possess energy density."

It is easy. The resulting E is true scalar. The energy is linked with it's use. Before Einstein's discovery E = m c^2 were was little useful energy. So, please specify the receiver of energy, then insert them all over the spacetime. So, You get the field E = E(t,x,y,z), which is physical. Thus, is independent of the coordinate system. What do you think? Yes, the field of receivers will modify the gravity wave and spacetime, making the result E useless. Then, one can use following philosophy. The E will be not the actual energy, but the "conditional" energy: "the value, which would get single receiver, if you put it there at t and in the point x,y,z during 1 second". All rights reserved: please include my name in your authors' list and tell me publication story.

General Relativity not allows a definition of a local energy-tensor for the gravitational field itself, only global definition of gravitational energies are possible. I think that this problem reflects the incompleteness of General relativity. 

Dear Andrea Addazi, you have formulated the state of global research. The local research can have the solution to any problem. This called local, because there always someone against something. The scientific philosophy can very well solve any global problem, but for the particular philosopher and his "school". To my previous comment now. Take the temperature in the town, you do measure it with thermometer. Our thermometer is not perfect. So the resulting field T(t,x,y,z) depends on the choice of the device. But nevertheless we say in peer-review journal: "the temperature of the town is T". Apply this philosophy to the gravity field. Thank You and thank to all contributers, thanks for the paper. Bye.

Dear John, your question focuses a very important aspect of the actual understanding of our universe. In order to answer to this question it is necessary to start from quantum systems encoded by quantum super Yang-Mills PDEs, say (YM), as I formulated in DOI: 10.1016/j.nonrwa.2012.02.014. There all the quantum energy is represented by the quantum Hamiltonian. It depends also from the quantum graviton as well from quantum e.m. fields. This is what happens at the microscopic level. Taking the classic limit of such a situation, namely by restriction on the classic limit, say (YM)c, of (YM), taking into account that (YM)c, is contained in (YM), we get that also at the macroscopic level gravitational energy is contained in the macroscopic Hamiltonian as well as e.m. field.

By conclusion we can answer in the affirmative to your question..  

Since many of you answer the question in the positive, I will give an opposite answer.

It is based on the fact that gravitational fields and electromagnetic fields are basically different, viz. the elecromagnetic field exhibits no self-interaction like the gravitational field. This is also one of the reasons why the wavefunction in QM is not a physical field.

I believe Andrea's answer is the consensus opinion within current theory, that the energy is not localizable.

Erkki's answer is true, but over simplified.  The EM field has energy density, but only interacts with charge.  Since a linear moving or static EM field is uncharged (or has balanced charge if you prefer) it doesn't interact with itself.

Since energy is mass, and gravity interacts with mass, then if there is energy in the gravitational field, there has to be self-interaction.  This is one source of difficulties in quantum gravity, but not the insurmountable one as I understand.

A force through a distance is energy.  You can turn that around.  If the gravitational field is energy, then it is capable of exerting a force through a distance.  Therefore, I would say Erkki inadvertently also gives a positive answer.  If the gravitational field self-interacts, it must have a local energy density.  So why can't we find it in current theory?

 Instead of asking where the energy is (i.e. what is the density function), ask what mass moves (or resists motion) when you push on something in a gravitationally coupled system.  You have two or more masses and some non-gravity fields among them which have energy and therefore mass.  You push on one of the masses.  Is it the only one that moves?

No.  For example, place an object near an event horizon.  It is stuck there due to extreme time dilation.  But you can push on it and the whole system including the black hole will move with inertia M+m (where M is the mass of the black hole and m the mass of the stuck object), with no consideration of time dilation.

The closer m is to M, the more coupled M and m are, and the more you will feel resistance from M and see motion of M when you push on m.

So... WHERE is the combined mass m+M?  It is anywhere in space that you can push and get resistance.  The potential says where it potentially is.  But there is no actual resistance, or actual energy, until you put some mass in the potential, i.e. put m near M.

If my last speculation about potential vs. actual energy is taken literally, it suggests there might not be a gravitational self-interaction.  This would yield a barely distinguishable grr=1/gtt of something like (1+GM/rc2) instead of (1-GM/rc2)-2.

I have a question related to John's ... Is the scale of a gravitational system, i.e. the universe, an observable?  Or is it a gauge symmetry?

I realize in talking about the expanding universe, we are talking about an observable.  But is there another scale factor.  If the universe varies in size S(U) according to grr (the spatial coefficient of a metric), speaking informally, then the ratio S(U)/grr would be undetectable, and therefore a gauge symmetry.

The simplest example of a gauge symmetry is in fact gravitational potential.  We can have potential V*=V+k where k is any scale factor, and since energy is only a matter of differences, V* and V are observationally indistinguishable.  So therefore we cannot speak meaningfully of an energy density in the potential field.  It is unobservable.  We cannot discover k.

Similarly, if S(U)/grr is undiscoverable, then grr is undiscoverable, the gauge symmetry factor.  Any energy density related to grr is then also undiscoverable.

"I think that this problem reflects the incompleteness of General relativity. "

Yes for sure GRT really misses a lot. The problem of the free falling clocks which have to maintain the same rate is something related to the problem of non accountable gravitational energy. Feynman tried to put a patch on the issue of the gravitational energy  but wasn't very successful either.

GRT works well wherever problems are stationary or where the system has a definite Hamiltonian, cannot cope with transients.

The problem is of paramount importance since it is reasonably something which impairs the unification of QPhisycs and Gravitation. Newtonian gravitation is conservative globally and locally, GRT is not, quantum Physics is based strictly on energy and momenta conservations. Not so few are still attempting to use Newtonian gravitation to try to make the unification with QPh, while they are using strings to try to avoid GRT and unify Gravitation and QM.

Erkki's remark on the diversity between gravity and e.m. fields is absolutely correct. In fact at the quantum level quantum graviton is structurally different from quantum e.m. field ... However this does not justifies his conclusion to answer in the negative to the John's question since both contribute to the quantum Hamiltonian. Furthermore, about the generalized opinion that energy is not localizable, let me underline that the observed quantum Hamiltonian is a local field defined on solutions of observed (YM). Therefore also the quantum mass has a local meaning, identified by the mass-gap of such solutions. Similarly it  holds for the other quantum  energy components.

I'm surprised no one mentioned the York-Brown quasilocal energy.  Maybe the question should be reformulated as one about whether there is a unique gravitational stress-energy tensor.  Since you can always choose coordinates where the connection terms locally vanish this is typically considered to be no.  I think that restricting oneself to tensors defined by the gravitational metric is excessively restrictive.  There are coordinate frameworks that have well defined translational symmetries that give well defined and conserved energies.  The debate is then about how to extract these results in terms of "physical" observables and if the space and timelike nature of these coordinates is preserved for various situations.  I have expanded on this sort of formulation and it is being reviewed.  It is kind of like choosing a coordinate system with the Earth at the center of the universe.  It seems unnecessarily complicated but it does give another valid description.  

"The problem of the free falling clocks which have to maintain the same rate"

otherwise an equivalence principle is violated. If clocks don't, it does not destroy the General Relativity, see: Einstein has found the LAW by wrong way. It is same as his mistake "cosmological constant", which nevertheless has described the Universe expansion. Moreover, if you put the large cabin to be free, but between two dense planets, then, the situation is just like free fall but with zero velocity. Then the time dilation is applied: parts of cabin, which are close to planet have slower clock rate.

Dear Dimitri,

"if you put the large cabin to be free, but between two dense planets, then, the situation is just like free fall but with zero velocity. "

Yes it should be like that. No matter the configuration  the time goes slower getting close to ponderable masses.

Consider infact that the time is delayed (Greater stress tensor) at the Dead Sea level (closer to the center of Earth)  but gravitation is weaker.

If that didn't happen there would be a simple way to show that the travelling photon is massive or rather it is attracted by Earth and the time dilation does not have influence on the gravitational Redshift.

But it seems that in GRT Energy-momentum stress tensor is strictly local so the absurd thing comes out:" I can restrict myself porperly at the center of Earth where the time-rate I measure is the one of the deep space"..totally unacceptable.

Energy conservations and global properties are unfortunately neglected in GRT, and that makes it just OPTICS. THe stress tensor of GRT has to be changed in something else. The idea of the HYPERMEDIUM which is stressed is a genial idea and I share it, but the Einstein Stress tensor is not suitable to describe it properly in the general case.

In the book, GRAVITATION, by Misner, Wheeler, and Thorne (in the 1970s), the idea of localized gravitational energy is criticized for several reasons. To paraphrase them: a viable and unique gravitational stress-energy tensor cannot be defined. This argument rests on the Equivalence Principle: by transforming coordinates over to a free-falling frame of reference there must be at least one point where the effects of gravitation vanish and cannot be detected. However, the components of the Riemann curvature tensor for space-time cannot be made to vanish by a coordinate transformation. Transforming coordinates to a free-falling reference frame cannot transform away the curvature of space. In other words, the gravitational field cannot be transformed away by coordinate transformations. We are then faced with the question: Why would the gravitational effects appear to vanish in a free-falling reference frame such as the elevator of the imaginary experiment posed by Einstein and Infeld? One interpretation is that this effect is due to a physical cancellation of gravitational and inertial accelerations.    

We can determine electric field energy density because there are two kinds of charge, and a zero reference can be found.  But gravitational potential is a gauge quantity, that is the zero point is not observable.  Since energy density would have to be related to potential, I assume, how could this be overcome?

I did a simple web search, and this is an often asked question and a much researched topic.  Of the references I found, at least several look interesting.

There are many others of course, and several forum discussions.  Does anyone think we have added anything new?

Article Can the energy density of gravitational field be interpreted...

In three papers posted on the arXiv I have advocated the hypothesis that the energy density of the gravitational field is is simply the sum of the principal pressures of the energy momentum stress tensor for all matter and fields other than gravity.  This easily follows for instance from the Einstein equation for gravity as soon as one realizes that the whole purpose of any gravitational theory is simply to find the Jacobi curvature operator which characterizes all tidal accelerations.  The Einstein equation gives the trace of the Jacobi curvature operator and the usual boundary conditions then give the whole spacetime metric which then gives the Jacobi curvature operator (which contains the exact same information as the Riemann curvature operator).  The idea that the energy of the gravitational field is not localizable because of the Einstein equivalence principle is completely incorrect, as true gravity is due to spacetime curvature and cannot ever be transformed away.  Thus, the whole quasi-local program which has lead to numerous different definitions of quasi-local energy of the gravitational field should be viewed as providing interesting invariants for analyzing spacetime models, but not the true energy except in very restricted circumstances.  In a paper in the Annals of Physics, Fred Cooperstock and I have advocated replacing quasi-local energy with spacetime energy momentum which is truly invariant as an integral over regions of spacetime.

Maurice, I suppose it would be too much to expect if maybe one of your papers provides a complete and clear summary and is, say, less than 10 pages?  Most of the papers on this subject run around 30 pages, as daunting as the original paper presenting GR.

To: E.J. Zampino.

Thank you, but there is no point in a free falling body, where the tidal stress is absent. Don't ignore the tidal forces!

Dear Maurice,

"the energy density of the gravitational field is is simply the sum of the principal pressures of the energy momentum stress tensor for all matter and fields other than gravity"

I thought it was like that, that Einstein meant... otherwise it's gravity recalling gravity...but I think you are right...

The stress tensor is given by the energy content of matter and fields, Eddington says the curvature is due to the ACTION, energy density over four dimensional space-time.

The problem is that radiation is missing...

"The Einstein equation gives the trace of the Jacobi curvature operator and the usual boundary conditions then give the whole spacetime metric which then gives the Jacobi curvature operator (which contains the exact same information as the Riemann curvature operator)."

I'm ignorant, can you explain simply the difference?

"The idea that the energy of the gravitational field is not localizable because of the Einstein equivalence principle is completely incorrect, as true gravity is due to spacetime curvature and cannot ever be transformed away."

I think so too.  Whatever gravitation is classified, as a field or not, it defines an interaction, it alters the state of systems (free falling bodies or pressure under my feet). If the theory is able to cancel it away, such thing is so wrong that should have been sufficient to invalidate the theory itself.

The whole mess about photons, mass and energy is due to the fact that the equivalence between mass and energy runs via photons. If an elementary particle annihilates, then its mass is completely converted into photon energy. That does not mean that the photon takes over the mass of the elementary particle! Photons encode their energy in their frequency. The passage of the photon takes time. During that time a certain number of wave tops pass a point along the path of the photon. The speed of a photon in free space is constant. So, if the duration of passage is also constant, then each wave top represents a bit of energy. That bit of energy corresponds with a bit of mass in the annihilated elementary particle. Thus, where the photon consists of separate wave fronts, will the particle consist of separate "elements" that each represent a bit of mass.

What can these elements be? My best guess is that these elements are embedding locations. These locations are positions where the particle can or could be detected during one generation or annihilation cycle. Together these locations form both a coherent swarm and a hopping path. The swarm can be characterized by a normalized continuous location density distribution. Since the locations are points the location density distribution is also a probability distribution. It conforms to the squared modulus of the wave function of the particle.

With other words: Each embedding location in the swarm representation of an elementary particle corresponds to a wave front in the photon that is generated at the annihilation of the particle.

(One dimensional wave fronts keep their amplitude when they travel through space. In this way they can travel billions of light years without weakening.)

Commonly photons consist out of strings of wave fronts. A photon with the lowest possible energy contains only a single wave front. In this simplified model the passage of a photon takes a fixed amount of progression steps. This also indicates that emission and absorption of photons takes a fixed amount of progression steps. It is the same amount as the passage of the photon takes.

According to this simplified model, elementary particles are recurrently regenerated, but usually this does not result into emission or absorption of strings of wave fronts. Instead each swarm element that is discarded is immediately regenerated at a slightly different location. In that case the exchanged wave fronts are singles. These single wave fronts are not photons. They take much less progression steps for being generated or absorbed.

The question addresses several, important and difficult, issues.

1. Inertial forces are usually seen as pseudo forces : they are created when a material body changes its motion, and proportional to the change in the momenta. But that means that gravitational forces do exist, as well as a gravitational field. And one of the tenets of GR is the equality between the inertial mass and the gravitational charge.

2. In the usual formalism of GR it is not easy to deal with the gravitational field, which is fully derived from the metric. In the fiber bundle formalism it is simpler, more general, and the gravitational field can be represented by the strength of a connection, similar to what isdone for the other field (such as the EM field, the strength is the 2 form F). For the gravitatiional field the 2 form F is actually the Riemann tensor. Using this tensor it is then possible to define a density of energy of the gravitational field in a way similar to the other field (the scaar product of F).

3. The third issue is the definition of energy itself. In Relativity the energy of a particle can be defined from the mass and the square of the velocity, and the measure of the energy by an observer then leads to a split emphasizing the kinetic energy. For force fields this is leass obvious, however we have a similar splitting, which uses the velocity of the observer (oriented as his time axis), which shows that the density of energy for the gravitational field has two comonents (one transversal corresponding to the time axis and another rotational) with opposite signs. Which could explain  that the gravitationl field as we measure it is so weak. Notice that the usual scalar curvature, when computed from the Riemann tensor, shows also this kind of decomposition.

Yes the gravitational field has an energy density covariant with the electric field energy density - in terms of a 4D spin-stress energy momentum density tensor formulation I have posted here on ResearchGate at the bottom link, titled: Dirac-Einstein-Maxwell Density Functional Theory Equation of State. As you can see reference [27] cites a NASA engineer J.C. Kolecki: A Possible Scalar Term Describing Energy Density in the Gravitational Field. NASA Math and Science Resources (2010) 

Abstract: Photon and electron-positron are first two trace matrix element invariants of R4 spin-stress pressure gravitoelectromagnetic (GEM) delta functional compact wavetrain integration energy density tensor equation of state (EOS). Conserved angular momentum Hermitian self-adjoint operator coupling of Einstein-Maxwell equations is inertially covariant inversely compressive/dispersive of cosmological constant vacuum energy density, wherein every Hilbert state is of greater than zero energy density. Establishing simple hydrogen-like s-orbitals spin-weighted gauge group, rendering in pascals as shown in online constructive proof, exceeding Yang-Mills mass gap solution requirements. In present density functional theory (DFT) there is only one universal wavefunctional domain and range, extending the paradox of quantum mechanics via stationary frequency domain Dirac delta functional hypercomplex GEM energy density distribution, and range (i.e., image) of inverse Fourier transform time domain observer-participant experienced phase evolution of EOS delta functional wavetrain integrations, along holomorphic linear functionals spacetime event spin-weighted quaternion group world line trajectories thru the domain. Establishing the wavetrain integration basis for the Poincar´e group representations length-contractions, time-dilations, red-shift, blue-shifts, lacking in standard model. Establishing Navier-Stokes Lagrangian drifter smoothness via GEM EOS differentiable spacetime energy density, contrary to nondifferentiable spacetime of standard model as proven by Feynman, wherein currently the standard model universal wavefunction variations are diverging into supersymmetry particles. Finite element analysis basis is established for general interdisciplinary physics, in particular photon-electron gauge group basis for rapidly emerging field of functional magnetic resonance imaging (fMRI) studies, establishing a continuous evolution of smooth operators conserved angular momentum Noether probability current origin to the Libet-Von Neumann observer-participant experience.

There is an online constructive proof in the Maplesoft Application Center, Quantum Mechanics section, and I'm currently working on the 2015 version of the manuscript.

https://www.researchgate.net/profile/David_Harness3

Dear Mr. Macken

Aleksei Bykov said: >>"Real" things in physics are observable things. Gravity is definitely observable

The gravitational field does have an energy density, like any other field. In contrast to the electromagnetic field, that does not, itself, carry electric (or magnetic) charge, the gravitational field, indeed, carries energy and momentum, hence spacetime can have curvature, in the absence of matter, if the cosmological constant doesn't vanish. 

This is, also, one way to understand how Einstein's equations are non-linear equations for the metric, since the metric couples to the energy-momentum tensor, thereby expressing the equivalence principle. 

However, since a gravitational theory is invariant under diffeomorphisms, the definition of such a density and, indeed, the energy-momentum tensor, is quite subtle and here the work of Brown and York, mentioned  in previous mssages, is relevant. Since the AdS/CFT correspondence, further insight has been gained-cf. for instance,http://arxiv.org/pdf/hep-th/9902121.pdf

(This is, in fact, similar to the situation for Yang-Mills theories, where the gauge field is, also, charged under the symmetry group.)

Hans-G. Hildebrandt makes several statements that are philosophical, interesting, and irresistible for further discussion.  With some I agree and others I don't.

Re: "Gravity is definitely observable." - So you might first think, but Einstein's theory is built on a foundational assumption that gravity is NOT observable.  This is called the Equivalence Principle.  It has made for 100 years of havoc in approaching gravity any way other than the field equation because if you try to derive gravity in a differential volume - necessary for almost any approach - you find at the end whatever you've got is not observable.  For an explanation with a diagram, see the first page of https://www.researchgate.net/publication/273764795_Coordinate_potential_approach_to_space-time_curvature

Re: "The reason of the electromagnetic force is known, the reason of gravity not." - Well you can describe the electromagnetic force in terms of virtual photon exchanges, is that what you mean?  But no one knows how EM energy is condensed into particles with different charges and other kinds of forces (strong, weak), so basically we know nothing at the end of the day.  We just have more elaborate nesting of formula.  Something Hans hits on in his next remark.

Re: "It's wrong simply to take a theory and some mathematical formulae and say: The problem is solved. In reality, however, the solution is the problem." - This is a statement of opinion, or perhaps you'd consider it moral judgement, and is not answerable by scientific inquiry.  I happen to agree with it.  And I think most of the argumentation over gravity, which you see a lot of on RG threads, boils down to this.  Those who accept the field equation as "reason enough" are content with GR.  But the field equation actually creates more mysteries than it solves for those of us who try to understand things.  It posits the question of how matter-energy can create space-time, and if space-time is not to be taken for granted, then what are its mechanics?  This is what is difficult about quantum gravity, not the renormalization problem and other trivia.  I am very close to being able to describe a constructive gravity in which a universal action creates a pseudo space-time by shortening radial rods and dilating clocks.  But it is only relative.  Or relational if you prefer that term.  It is not substantive, and cannot create wormholes, nor the interior of a black hole.  However, if we gave up that one philosophical notion of Einstein, about matter creating space, then the field equation or something like it would still be valid and would merely have boundary conditions that restrict solutions to those describable with pseudo space-time (mappable to Euclidean coordinates).

Re: "From my point of view gravity acts only between atomical structured matter." - Hmm... a light ray would have no mass?  Then if two large masses, one anti-matter, were brought together and annihilated 100% (not actually very easy to accomplish due to radiation pressure), the gravitational field would vanish at all distances?  I don't really think so but it is a bit difficult to run the experiments.

Re: "Has anyone observed the gravitatitional load between particles, e.g. neutron-neutron, pion-pion, pion-neutron et cetera?" - Of course not, it's 40 orders of magnitude too small ... however, the gravitational trajectories of neutrons have been measured.  If we merely assume gravity is universal and therefore reciprocal, it follows that neutrons generate a gravitational effect as well as respond to it.  We can observe the gravitational effects of neutron stars.  I didn't mention this above because I supposed Hans would declare a neutron (or quark) star to be atomically structured matter.  But I don't see what difference it makes.  I believe you could have a box of photons with a certain mass (due to the photon energy since they have no rest mass) and it would have the appropriate gravitational effect.

Dear Mr. Shuler!

You know, that your response is exactly the repetition of the old common positions. But in my answer is clearly visible: We have to find real ways to solve all the problems in sience, espacially in physics. The time has gone to make theories about the reality like Einstein and others and all the experimets has to prove the theories and bring us not closer to the objective reality, means cognition.

What is the value of all theories about particles and nuclei, if someone show us, that all of them created respectively constituted by electrons and positrons as basic building blocks of material matter? It's possible!

Or someone find out the real cause of the gravitatitional load. I think, it's possible. What's about the old theories of gravity?

The finding is: The gravity is 42 orders of magnitude smaller than electrostatic force. Both of them are very similar. The question is: Why is that so? and not: What states the theorie?

(I would like to make it clear: I am not a fan or an enemy of Einstein. His greatest accomplishment was to formulate the equivalence between energy and matter,  in my opinion the equivalence between energetic matter and material matter. But this is another story...)

Robert,

"Then if two large masses, one anti-matter, were brought together and annihilated 100% (not actually very easy to accomplish due to radiation pressure), the gravitational field would vanish at all distances? "

Once I asked such a question, it regarded fermion and positron and somebody said it was a stupid one... for sure It was not. 

I think that gravitational waves would be produced in order to balance the mass defect and in any case they will be present during the approaching phase of matter antimatter which will have a formidable acceleration in the last instants.

"I believe you could have a box of photons with a certain mass (due to the photon energy since they have no rest mass) and it would have the appropriate gravitational effect."

I firmly belive that "radiant energy" by itself does not warp space, or if it does it, it does it with higher order law than  "tied energy".... the transient of matter-non matter, is managed with an electromagnetic and gravitational energy-momentum flux which are the leftovers of the fermions.

Matter and antimatter have the *same* mass (consequence and expression  of the CPT theorem) so the gravitational field of the combined system would not vanish. There isn't any ``mass defect', nor would they accelerate significantly, just due to the fact that it's matter and antimatter. Cf. positronium, that has a finite lifetime. After ``annihilation'', i.e., for electrically charged particles, transformation to photons, the energy and momentum, which are conserved, would be that of the electromagnetic field, that does curve spacetime , also (cf. below).

Photons don't have mass-they have energy and momentum. A box of photons requires a box, i.e. boundary conditions. The mass of the field configuration will, still, vanish-in flat or curved spacetime-the energy and momentum, however, will not. This can be checked  by computing the energy-momentum tensor of the field configuration: the mass is the integral of the time-time component of the energy-momentum tensor evaluated on the boundary.  Of course, if the boundary conditions explicitly break Lorentz invariance, then the photons will have a mass, as is the case of a waveguide. 

Electromagnetic energy definitely does curve spacetime, since the energy-momentum tensor of the electromagnetic field can and does lead to a metric that's not flat-as can be checked by solving the Einstein equations with such a right hand side-and,as was found, first by Reissner and Nordstrom, can lead to the formation of a black hole, that carries charge, as well as mass. The spacetime of the Reissner-Nordstrom black hole is asymptotically flat. 

While it is possible, at any given point, to define an inertial reference frame, i.e. coordinates of a flat spacetime,  if spacetime is, indeed, curved, it is not possible to do so *globally*-that's the insight from Einstein's general theory of relativity. It's in this way that the equation of motion of matter has a geometric character and forces are expressed through geodesic deviations, i.e. that matter, subject to forces, follows  trajectories that are *not* geodesics of the spacetime, since the geodesic equation then acquires a right hand side (however these equations *can* be cast in the form of geodesic equations of *another* spacetime.)

Incidentally, these are all standard topics, treated in any course on general relativity and electromagnetism. 

In summary, gravity is, indeed, a  force, mediated by a gauge field, just like the other three forces; the difference is that, while for the other interactions the force is transmitted by excitations that carry spin 1, gravitational interactions are transmitted by excitations that carry spin 2. This has, indeed, been confirmed experimentally by the measurements and calculations of Hulse and Taylor, since the agreement between observation and calculation of the variation of the period of binary pulsars (now there have been more discovered) isn't possible otherwise. The fact that the spin is even can be shown to express the property that gravitation is always attractive.

Incidentally, the discovery of the Higgs boson, whose spin is 0, heralds the discovery of a new, attractive, force, mediated by the Higgs, that's, of course, very short-ranged, since the Higgs is very massive.

Stam,

"There isn't any ``mass defect', nor would they accelerate significantly, just due to the fact that it's matter and antimatter. "

sorry for the "mass defect" which is usually meant as something else. I meant just presence of mass before annihilation and total absence after.

That they won't accelerate significantly you have to show me why it is not, because two charged bodies exert a considerable force when they are close to eachother.

Not if they're in a bound state, e.g. positronium (the simplest case; same holds for any other such bound state).  Once more, the vaguer the statement, the more difficult it is to understand (e.g. ``formidable''-with respect to what scale?). The fact that a bound state is, indeed, possible, shows that outside probes do not perturb the constituents more than the state itself-else there wouldn't be such a bound state.  Any more than one would talk about ``formidable'' relative acceleration between the electron and the proton in the hydrogen atom.   Indeed the only difference is, apart from the mass scale, that, for positronium, the 2-or 3-photon state, is accessible, consistently with known internal conservation laws, for the hydrogen atom it isn't. And the calculation by Pauli and Fock, for the ground state of the hydrogen atom, goes over for positronium-only the reduced mass has a different value. 

So if one had a hydrogen atom and an antihydrogen atom, they wouldn't exert any force on one another, that would be different from that of two hydrogen atoms, if they were ``far enough'' apart. And it's possible to calculate the cross section towards the final state of photons-the lifetime of this system isn't zero.

Yes ok between neutral mass atom and anti atom there would not be any acceleration.

Re Hans: "your response is exactly the repetition of the old common positions"

Alas, none of the "old common physicists" will acknowledge the positions I take, and since you don't either, we have a nice syllogism going here that you are an old common physicist with old common positions.  :D

Re: "Stefano" I think that gravitational waves would be produced in order to balance the mass defect

Conservation of energy ⇒ Eemitted radiation = Mnucleus/c2

⇒ Eemitted radiation = ( Mnucleons - MassDefect ) / c2

⇒ no change in total mass before/after and no grav wave due to mass defect

Re: "Stefano" and in any case they will be present during the approaching phase of matter antimatter

It seems so, yes.  I have only started looking into gravitational waves recently, but they seem to result from scattering events, of which this is one.

Re: "Stefano" I firmly belive that "radiant energy" by itself does not warp space

An article of faith?  OK, we'll let the bishops decide.

Re: "Stefano" or if it does it, it does it with higher order law than "tied energy"

Sounds sort of like double or nothing, a gambler's technique.

Re: "Stam" Photons don't have mass-they have energy and momentum.

Last I heard, E=mc2 ⇔ m=E/c2 so if they have energy they have mass.  There is a similar argument for momentum.

I am leaving this thread, guys.  If you have something to say, you can message me, but it is getting too far out for a [cranky] "old common physicist" who doesn't get the time of day from other old common physicists.

E=mc^2 only for a massive particle in its rest frame; for a massless particle, E=|p|c E' = |p'|c, for any two frames, related by a Lorentz transformation. And, while E/c^2 does have the dimensions of mass, from the invariant relation, E^2-|p|^2c^2=(mc^2)^2=E'¨2-|p'|^2c^2, E and p (viz. E' and p') aren't Lorentz invariants, m is;  therefore E/c^2 depends on the reference frame and can't be assigned, in general,  any meaning of mass, any more or less than |p|/c can.   There's a difference between dimensional analysis and physics. It's the other way around: from the fact that m is a Lorentz invariant, it follows that there exists a Lorentz transformation that takes the  4-vector, (E,pc), to the form (mc^2,0,0,0), for a massive particle-and that no such transformation exists for a massless particle. 

Of course an atom and an antiatom would have (a very weak) gravitational attraction-just like two atoms-consistent with the equivalence principle.

Experimental studies of antihydrogen are, naturally, very challenging: http://alpha.web.cern.ch/ and http://asacusa.web.cern.ch/ASACUSA/asacusaweb/main/main.shtml

Gravitational field cannot have energy density because gravity cannot be mediated by a particle. Energy of the quadrupole gravitational waves may have nothing to do with the gravitational attraction.

If anybody is still interested in the original question, the answer is simply no. It is well-known that it is impossible to define a local energy density for the gravitational field in general relativity. The best we can hope for is to define the total energy contained in a given region, however we still don't have a good measure of that either. This kind of measure of energy/mass is known as quasilocal mass, and in fact there are several inequivalent definitions in the literature; reconciling these is still an open problem. There is a detailed review article by Szabados on this, if you are interested.

See: http://relativity.livingreviews.org/Articles/lrr-2009-4/

Stam Nicolis said: Experimental studies of antihydrogen are, naturally, very challenging.

The experimenters have gathered a few dozen of antihydrogen. (My informatinon.) I hope, this amount is sufficient for such experiments. There are also many other problems and I think, it's almost impossible to measure the weak gravitationally interactions.

In my opinion between atoms and ainti-atoms acts anti-gravity because they have inverse structures.

Dear Stephen,

Yes for GRT you are right. But GRT tries to represent gravitation and it has issues.

Stefano,

GR is considered one of the most well-tested physics theories we have, and it's passed every one of those tests with flying colours. Whatever else happens in physics, we know it must reduce to GR in appropriate limits. As far as the experts are concerned, there is no debate about this.

 Robert:

" I firmly belive that "radiant energy" by itself does not warp space

An article of faith?  OK, we'll let the bishops decide.

From Roger Penrose and quantum mechanics "Road to reality":

I can split the wave of a single photon in two parts, (the entire wave function is the sum of these two parts) and recive the photon again along two different paths. This allows me to acquire the photons   in two different places (NON LOCAL), though if I choose to acquire  the photon in one place, I won't find it in the other one (energy conservation).

How can a local stress tensor be able to account  that light have followed a path or the other, if It is actually impossilbe to affirm which of the two paths the light was following?? This for me is sufficient to deny the possibility for light to partecipate to the local stress tensor. When the light will be absorbed, then and  only then it will partecipate to the stress tensor as "tied energy".

Re: "Stefano" or if it does it, it does it with higher order law than "tied energy"

Sounds sort of like double or nothing, a gambler's technique"

It is unlikely that it posesses such a possibilty, but an hypothesis can be dared. 

A relation between a spherical zone of space time and the wave function which is likely to be in that spherical volume, expanding at the speed of light, could influence the stress tensor in that  volume, that would mean a very tiny and variable contribution compared to the tied energy of the photon.

gravity force is one of the conservative forces, that can be drived from a potential field,j ust like the electric and magnetic forces, however, its magnitute  is much lower than the others, so i expect small amount of  enegy density for it.  i think any basic physics book gives you the real equation for it. e.g.

a candidate gravitational energy density term may now be constructed and written as uG = g2/(8 pi G)

Near the surface of the earth

g = 9.807 m/sec2.

Also G = 6.672 X 10-11 (nt m2)/kg2

so that uG = 5.736 X 1010 j/m3.

The subtleties  of defining what a conserved quantity might be, when the gauge fields  are charged, i.e. transform non-trivially under the symmetries, are well-known. Unfortunately, many times words have taken over calculations. It doesn't matter whether a quantity is *called* ``quasi-local'', or by some other term, what does matter is the consistent mathematical description, which does exist-cf. the work by Brown and York for the energy-momentum tensor of gravity, (cf., also, the work of D. Bak,D. Cangemi and R. Jackiw, http://arxiv.org/abs/hep-th/9310025 )- and has been generalized, in fact, many years ago, by, for instance, B. Julia and S. Silva, http://arxiv.org/abs/gr-qc/9804029, where the issues that have been raised here, have been clarified.  So it's useful to learn how these issues have been resolved and go on from there.  History of physics is fascinating-but it's distinct from physics.

Gravitational energy localization has been a subject of controversy off and on since  1917.  Formally there is no problem applying Noether's first theorem and inferred a conserved quantity from each rigid translation symmetry of the Lagrangian (adding a constant to a coordinate, say).  One gets what has been called the Noether operator---Bergmann did this in 1958, Trautman in the 1960s, Schutz & Sorkin in the 1970s, etc.  It's a high-tech analog of the energy-momentum pseudo-tensor.  The difficulty for most people has been how to make sense of the multiplicity of conserved quantities, their lack of interconnections, the vanishing at a point or along a worldline in a certain coordinate system, etc.  Such problems are sufficiently old that people have long since given up and tried a lot of other things more recently.

However, in my view the problem is that people simply haven't been able to take the Noether mathematics literally, because they tacitly wrongly assumed that there should be just one energy rather than infinitely many.  Noether's first theorem associates a conserved current with each rigid symmetry.  GR has infinitely many rigid symmetries of the Lagrangian, because any time-like vector field gives a notation of time translation.  So why not just accept the idea that there are infinitely many energies?  Once one accepts a multiplicity of energies, most of the usual objections to pseudo-tensors disappear, just as one's puzzlement at finding a contradiction between "John is short" and "Juan es grande" disappears if Juan and John are different people---rather that the same person described in different languages.  If coordinate systems are like languages, then transformation laws are like translation.  Hence I see no great difficulty in believing in gravitational energies in GR, as long as one notices the plural.  That's what the math wants to say.  See my  arXiv:0902.1288 published in GRG. 

"GR has infinitely many rigid symmetries of the Lagrangian, because any time-like vector field gives a notation of time translation"

This is the problem. Such metric is not real and brings to paradoxes.

Stam,

"The subtleties  of defining what a conserved quantity might be, when the gauge fields  are charged"

These are artifaxes. Newtonian dynamics works better that Geometrodynamics, though in the first the speed of light is considered infinite in the second the energy does not circulate at all because QM is not integrated.

 Stephen,

“...GR is considered one of the most well-tested physics theories we have,...”

- that isn’t so. An example – see the paper

Gravitation, photons, clocks. Available from: https://www.researchgate.net/publication/231149107_Gravitation_photons_clocks

This paper just shows, that the GR is wrong as that the experiment Pound and Rebka had shown – because of the GR claims that the  “approach… that the photon reddens because it loses the energy when overcoming the attraction of the gravitational field… is misleading and only serves to create confusion in a simple subject…”;  and so, according to the GR,  all measured “gravitational time dilation” value appears because of different clock rates if the clocks are on different heights over Earth only.

But this claim immediately means that in the GR gravitational energy of the system “Earth+a clock” is twice larger then Newton had said and anybody can measure.

In the reality only half of the measured value is indeed because of “gravitational time dilation” – in fact  because of the gravitational mass defect; when  other half is just because of photons energy losses when they travel between points with different gravitational potentials.

More – see “The informational model – gravity”  http://vixra.org/abs/1409.0031

Cheers

Article Gravitation, photons, clocks

Stephen,

go and ask to Stephen Crothers, he will be a bit harder than me on GRT, saying that it is very wrong.

This link regards something quite interesting about GRT which a Nobel Prize recently said.

Research General Relativity: In Acknowledgement Of Professor Gerardus...

There is something that has come to me as I read over some of the work here.  All of this is based on the general theory of relativity.  Any time we talk about gravity we talk about it.  General Relativity has the ability to work in all kinds of universes not just the one that looks like ours.  In my opinion this implies that it could not be complete as a complete theory would apply to reality and not to our imagination. 

I have said it many times humans can perceive beyond reality.  This is why we have not solved the dilemma with quantum mechanics and general relativity.  They will never resolve the issues as they can not match reality in the forms they are proposed in today.

Relativity changed time and light, and quantum mechanics ignores the quanta.

To resolve these issues we need new thinking not the thinking that got us into this place to start.

In 1995, I. Ciufolini and J.A. Wheeler, published "Gravitation and Inertia" (Princeton University Press). Take a look at page 15.  Modern gravity gradiometers can be used to measure the gradient of a gravity field between two nearby points that may be only a few tens of  centimeters apart. The detection of  the gradient of a gravitational field also implies the existence of the gravitational field.  It then becomes possible to distinguish between the free-falling cabin of a spacecraft in the gravitational field of the earth and the cabin of a spacecraft which is in the far-field (approximately flat space-time ) region from the earth.

    There is also a temporal restriction on the applicability of the Einstein Equivalence Principle.  I. Ciufolini and J.A. Wheeler point out that for a test particle in orbit around a central mass, the domain of observation in time must be limited to values that are small compared to one period of revolution in order to assure that the equivalence principle may be applied.

It's useful to learn the technical issues about general relativity, from textbooks or lectures and from solving technical, not philosophical,  problems. Personal beliefs and issues of language or style aren't relevant. A good starting point may be found here: http://www.staff.science.uu.nl/~hooft101/lectures/gr.html

And it's the *technical content* that's relevant, not the fact that 't Hooft has won the Nobel Prize in Physics. Anyone can understand the technical issues, by learning and study and there's no need to appeal to any authority.

History of physics is fascinating, but is distinct from physics so it would be useful, when discussing issues that pertain to the latter, not to pay attention to the former-but to what has been understood to be the correct, technical,  way of studying the topics, as has been shown by calculation and experiment. 

Regarding the supposed paradoxes from the fact that GR has infinitely many rigid symmetries of the Lagrangian:  what paradoxes do you have in mind that do not have analogs along the lines of "John is short" and "Juan es grande"?  (There might be some, but not as many as most people think.)  If one assumed that John and Juan are the same person (like there being just one energy) and that these statements should be equivalent under translation (like having a tensor transformation law), then there is a problem, because the same person is short and large, which is a contradiction.  But if John and Juan are different people, then there is no problem.  Many of usual objections to pseudo-tensors disappear once one accepts that distinct time-translation symmetries give different conserved quantities.  We all know that the world contains many people named John or Juan.  But we often tacitly assume, without argument and contrary to the natural reading of Noether's first theorem, that GR has just one energy.   

Stam,

yes so we have to talk about some unphysical situations that are predicted by the GRT, though considering that there are a good amount of experiments where GRT has been successful in,  but so far no experiment has been set up on purpose to find its disconfirmation.

Once more-there aren't any *unphysical* situations, that are predicted by general relativity. There are many mathematical issues, of course, if one wishes to *prove* certain statements. However these are way beyond what's been discussed here.  As with any theory, it has a well-defined domain of validity. Since, in addition,  general relativity is a *gauge* theory, care must be taken to define gauge-invariant quantities. How to do this is now *known* and it has nothing to do with experiments. (By definition, the way any experiment is set up is to eliminate everything that's expected by the theory-that's what's called ``background''- and, once that's done, to check, whether there's anything more, at statistical significance, what's called ``signal''.) Cf. the linked lecture notes and the paper by B. Julia and S. Silva.

Dear Professor Stam Nicolis,

"Once more-there aren't any *unphysical* situations, that are predicted by general relativity. "

I would like to talk about "speed of osclillators in a graviational field", and to discuss with you some unconvincing (wrong) predictions of GRT.

According to GRT and according to the SEP (Strong Equivalence Principle) as well, free falling osclillators or atomic clocks (in a gravitational field)  should posess the same clock-rate. During their free fall, provided that they don't free fall (or free rise) from the same apogee but different heights, atomic clocks are supposed to posess the same pace as the pace of the atomic clock in deep space (the reference pace far from masses). This results also applying the Peinlevè Gullstrand metric of GRT. 

If so far you agree we can continue, otherwise you can clarify better which is the position of GRT, or correct the innapropriate or inadequate language I'm using.

Free-falling objects are objects that move on geodesics of the spacetime metric and everything can be calculated from that. Objects that don't move on geodesics satisfy corresponding equations (the same equations, with non-zero right hand side, or an, equivalent, equation for a different metric, or by changing the boundary conditions, for instance) that can, also, be solved. 

But these calculations must be done-not talked about. There's no way to guess at the answer. Cf., once more, the lecture notes on general relativity. These are standard exercises.

The assertion: “…gravity force is one of the conservative forces, that can be derived from a potential field, just like the electric and magnetic forces….” is not true in GR.

The problem arises because in relativity theory the force law, the momentum law and the energy law are not compatible. Hence the classical field related to the force of gravity is not commensurate with the classical fields associated with electromagnetism. In a sense this is the reason to entail tensorial descriptions. Moreover, generalizations to QM field theories are still open.

Since we do not know the details regarding pair production and pair annihilation it is also not possible, within present physical theories, to draw some analogies between matter fields and the electromagnetic field. I would think that since the above problems are still open, one cannot answer the question posed by John Macken in the positive, see also my earlier posting.

No problem for calculations, the Schwartzshild solution can be used without problems in the next posts, but now I need just an additional answer.

So can it be said that a free falling atomic clock follows an approximately geodesic motion, and its non exact behaviour (approximated) is due only to the tidal effects of the field on the body? 

A free falling atomic clock, that would be a single point, follows *exactly* (not approximately) a geodesic-that's what the words ``free falling'' *mean*.  If this clock isn't a point, however, then it's a number of points, so what its motion means  must be made more precise. One might think to argue as follows: The total  motion can be described by the motion of these points in the spacetime in question. The total kinetic energy is a sum of terms (m_i/2)g_μν(x) (dx_i^μ/dτ)(dx_j^ν/dτ)δ^ij, where i and j label the points of the clock; and the total potential  energy is V(x_i^μ). This potential energy *seems* to express the fact that the clock as a whole isn't a set of independent particles. The parameter τ labels the worldlines of the particles that make up the clock. From the Lagrangian one obtains the (coupled) equations of motion and solves them, eventually, numerically. Since the points that make up the clock don't coincide, there will be differences between the frames attached to each point, that are described by the equations of motion. 

However one must think a bit more carefully. The expression thus obtained does not seem to have a consistent flat spacetime limit. Another reason to be wary that is one would like to make sure that this approach is consistent with the equivalence principle, whereby the correct coupling is given by g_μν T^μν, with T^μν the energy-momentum tensor of the particle system that makes up the atomic clock. Another way to see the difficulties is to realize that, with a finite number of particles, the interactions are non-local. Also, that there's no reason that the parameter of each worldline is the same. 

One then, finally,  after some thought, finds that it's not possible to write a potential term for an interaction of a finite number of particles, that's relativistically invariant:  V=0 for a finite number of particles is the only consistent choice. (Indeed, this is known as the ``no interaction theorem'' in this context, cf. for instance, http://link.springer.com/article/10.1007%2FBF02749856 for an elementary proof.)  One can introduce a non-zero potential  for a field, however-a configuration of an infinite and not fixed number of particles.  So it's possible to describe a ``point-like'' atomic clock, or a clock made up of an infinite number of particles unambiguously. In the latter case the clock is a field, with appropriate boundary conditions for the field configuration. 

Dear Erkki,

"The assertion: “…gravity force is one of the conservative forces, that can be derived from a potential field, just like the electric and magnetic forces….” is not true in GR"

this is the biggest and unsolvable problem..

It's neither a big problem, nor an unresolved problem, as any course shows-and as is the case for Yang-Mills theories, too, where the gauge fields are charged under the group and the current has similar properties to the energy-momentum tensor in gravity.  It was solved by E. Noether-cf. the paper by B. Julia and S. Silva.

All this is *known* and taught and has been learned and can be learned-not, however, by chatting on discussion fora, but by systematic study. These issues are all understood, so it's useful to get them out of the way, if any progress is to be made on issues that are *not* understood, or much poorly understood-e.g. how to describe interactions consistently, for gravitational theories (how to do this for Yang-Mills is, of course, known).

A nice result, established by Ehlers and Geroch, is how well a ``small'' body follows a geodesic: arxiv.org/abs/gr-qc/0309074

"where the gauge fields are charged under the group and the current has similar properties to the energy-momentum tensor in gravity"

We are talking about  something described by the Poincarè Group (YM theory) which has a finite set of local infinitesimal generators (7) against something (GRT) which has an infinite series of local infinitesimal generators, due basically to the principle of equivalence /gravitational energy not localized. 

I think it is quite difficult to reconcile them I think.

Feynman in his "Lectures on Gravitation" showed that it was necessary to introduce the term of the Gravitational energy otherwise the GRT was missing something locally.

"A free falling atomic clock, that would be a single point, follows *exactly* (not approximately) a geodesic-that's what the words ``free falling'' *mean*. " 

Yes. In such case we can assume that the free falling atomic clock is a good approximation of a clock along a geodesic. Details about geodesic motion of more complex body is just a complication to confuse the gist of the problem.

The atomic clock can be considered like a single point under a very good approximation. I cannot distinguish between the results given by my atomic clock and a much smaller atomic clock (given by a single atom or better a single decaying particle, like a muon). The 10-18 s, which is the present maximum accuracy of atomic clocks,  won't be enough to make a distinction between the "normal" device and the point like one,  if their centers of mass are undergoing the same rule of motion.

Two of these atomic  oscillators are fixed at different heights radially in a gravitational field, such that H is the reciprocal distance of their center of masses. They are sincronised and immediately free fall.

I will continue if there are not objections to the previous.

One solves the geodesic equation in the given spacetime, that's all. Then one notices that two spacetime trajectories, with the same endpoints, enclose an area.If spacetime is curved this has consequences. 

More precisely: between any two spacetime points, there's one geodesic; obviously some care must be taken, but that's well known, to avoid ``going the other way around''. So any other curve between the same points isn't a geodesic. The action is the proper time. The geodesic equation expresses the condition that the trajectory extremizes the proper time and a calculation of the second variation shows that the extremum is a maximum. Therefore any other trajectory between the same two points leads to a different, lower value of the proper time and, thus, a mismatch, that's exactly the time dilation factor.

The geodesic motion is just the HAMILTON's least action principle applied , so coherent  results will be found according to energy conservation:

 the clocks do not maintain their sync, being static in a graviational field (as it results from the Pound and Rebka experiment) freq1=freq2(1+gh/c2).  They they will delay in  the same way in free fall while maintaining the same distance. They will be in sync only at the arrival of the sync signal before the free falling begins.

There's no point using words that are ambiguous instead of doing calculations, that are described in all courses and textbooks and are unambiguous. This is *spacetime* not space and time separately and energy conservation doesn't have anything to do with this-precisely, since, in curved spacetime, energy is *locally* not *globally* conserved and, as explained in 't Hooft's course and, more generally, in Julia and Silva, one must do some calculations. Variational principles aren't limited to that of classical mechanics, so the ``action'' for *this* problem is the proper time. 

For the case at hand the solution is simple: compute the length of the geodesic, from spacetime point A to spacetime point B-that's the proper time, measured by a freely falling observer. Compute the length of the spacetime curve, from A to B, that's *not* a geodesic-it's some other parametric curve. This is the spacetime trajectory of an observer that's not in free fall in that spacetime-but, in fact, can be shown to be in free fall in *another* spacetime, with a different metric (expressing, thereby, the equivalence principle). The lengths of the curves are the proper times measured by each observer. They are different and the one that's not a geodesic is shorter. Therefore two clocks, that were synchronized at A are no longer synchronized at B. This was, indeed, what was measured by Pound and Rebka. 

Indeed the uniqueness of the geodesic in spacetime expresses the absence of closed time-like curves in the spacetime in question. 

Dear all,

Several established scientists have made assertions about gravitation as a conservative force, similar to E. Nihal Ercan, but nobody did react. Then Stefano says this is the biggest unsolvable problem followed by Stam’s “It's neither a big problem, nor an unresolved problem..”.

Obviously there seems to be no consensus regarding even trivial knowledge about gravitation. To me such a lack of agreement must be corrected before one starts discussing cutting-edge improvements of energy-momentum tensor of gravity, and also referring to Yang-Mills- not to mention string-brane theories.

A general criticism that I have regarding modern p-branes, while being spatially extended strings, originate in mathematical concepts that appear to conflate matter with its mathematical description, i.e. do not exhibit the proper fabric for a self-referential examination – the latter, as we have agreed upon, being one of the distinguishing properties in the comparison between gravitation and electromagnetism.

"Obviously there seems to be no consensus regarding even trivial knowledge about gravitation. To me such a lack of agreement must be corrected before one starts discussing cutting-edge improvements of energy-momentum tensor of gravity, and also referring to Yang-Mills- not to mention string-brane theories."

I agree..

I would like to talk about the experiments performed and possible thought experiments testing GRT. If math predicts something it has to be tested. Math alone is not physics at all.

The test of GRT so far performed regard massless entities like photons, or relations between oscillators. Or in the last cases isolated systems in which can be given for granted that the energy is conserved, both locally and globally.

1) gravitational Redshift, Harvard Tower experiments

2) Local position invariance GP-A experiment

3) Radar Signals delayed by gravitation, Shapiro Delay

4) Mercury Perihelion (sun-mercury isolated system)

5) Hulse and Taylor binary stars (gravitational waves)

"The lengths of the curves are the proper times measured by each observer. They are different and the one that's not a geodesic is shorter. Therefore two clocks, that were synchronized at A are no longer synchronized at B. This was, indeed, what was measured by Pound and Rebka. "

In the attached paper emerges a different view of the gravitational time dilation much closer to Quantum Mechanics which is the theory according to which the atomic clocks behave.

"in curved spacetime, energy is *locally* not *globally* conserved and, as explained in 't Hooft's "

Yes this is how it should be. But according to GRT energy is only globally conserved, like in the case of the Pseudotensor, an "escamotage" found to account for energy conservation in GRT.

Article Gravitation, photons, clocks

The way to resolve any technical issue is to calculate-once it's understood what to calculate. For general relativity this has been the subject of textbooks and courses available everywhere. Once this background is agreed upon can any discussion make sense-all the disagreements in this and similar threads is that there isn't any prior agreement on what's considered known and understood and what's not. There's no point in discussing detailed aspects of general relativity, if a course hasn't been assimilated first. While the technical content isn't subject to any consensus-any discussion must have some agreed upon conventions, otherwise it's metaphysical. 

According to general relativity energy and momentum cannot be globally conserved, because the energy-momentum tensor can only be covariantly conserved-this is known as the Bianchi identity in general relativity. Incidentally, it is this fact that shows that gravity, as described by the dynamics of the metric tensor, couples to energy and momentum-i.e. that the metric tensor, as a gauge field, is charged and transforms non-trivially under the gauge group.

There's no point in imagining how things should conform to what one might think is the case-the mathematical formulation is unambiguous and can be understood from the fact that coordinate transformations are a local symmetry. Therefore, how to define conserved quantities requires the care that is appropriate in such situations. While, historically, Bianchi's work and Noether's work predated Yang-Mills and much had to be understood in other ways, now there's no point in repeating history-now it's understood what's the  correct way to formulate *these* problems and solve them and gravitational physics is studying other issues. 

 If one reads the nice paper by Okun et al. one finds what's expected. Time dilation doesn't have *anything* to do with quantum mechanics, it's a classical effect of spacetime geometry. The quantum description of matter and its non-gravitational interactions, that is relevant for the description of how an atomic clock works, turns out, indeed, to be consistent with what's expected, when measuring time under circumstances where classical gravitational effects are appropriate. 

Dear John, I have seen that people answering to your question

'Does a gravitational field have energy density like an electric field?'

arrived to discuss whether the gravitational field is conserved ... locally and/or globally ...

You know my answer to your question, given many posts ago, and I do not consider useful to repeat it ...

However, with respect to the local/global conservation law of energy of the gravitational field, I think that it is worthy to precise some important facts that I believe are completely forgotten in this discussion.

In fact, when one talks about conservation laws of gravitational fields at the macroscopic level, today any serious scientists refers to the Einstein PDEs, say (Ein). Now, this equation being invariant for time-translations admits a 3-differential form, say w, on (Ein), such that its differential dw=0 on all the solutions V of (Ein). This is what one calls energy conservation law. How it is possible to obtain the global energy content of the initial and final Cauchy data and the global evaluation of w can be understood by looking to my paper http://arxiv.org/abs/1206.4856. Then one understands that under some particular conditions, interesting the boundary of V, (we consider compact solutions) one can have the global conservation of the energy content of the initial Cauchy data with the final one. In this sense the global energy-conservation law is related to the geometric structure of the solution.

Other considerations are philosophical talks only !

Stam,

I think that you paint a too positive picture insinuating that everything is more or less understood and what is left is more or less a technical question that sooner or later will be completely solved. This was indeed also the situation at the end of the ninteenth century just before Planck and Einstein.

Time dilation should be commensurate with both classical and quantum. We need to know what is the fundamental characteristics of a black hole if QM is correctly taken into consideration. Moreover, how do you describe the details of particle-antiparticle annihilation and pairproduction from fundmental physics?  We do not even know in detail how an 8 m long photon gets into an Angstrom size atom.

There are many "simple" things we do not know – things that we ought to know, but do not since the microscopic-, the macroscopic- and the cosmologic domains are formulsted by theories that are not consistently developed by present day physicists – not to mention the problems occurring within the biological sphere.

Everything is definitely not understood and many conceptual challenges exist-only not in the issues discussed here. Special relativity and general relativity don't pose conceptual issues-for physics; general relativity does for mathematics, though these issues are way beyond what's been discussed here.  Classical probes of spacetime-i.e. the quantitative description of how classical objects move in spacetime and how spacetime reacts to their dynamics is understood. This includes black holes, as solutions to Einstein's equations. Time dilation is nothing more or less than the consequence of solving the classical equation of motion of a test particle in a given spacetime.  

While the description of the properties of quantum probes, the subject of condensed matter and high energy physics, is, of course, work in progress, how such quantum objects probe classical  spacetime is understood. This includes black hole entropy, for a special class of black holes. How spacetime reacts to quantum probes  is not understood: this is the issue of  quantum probes of a time-dependent spacetime, where Hawking radiation plays a role and how the resolution of spacetime singularities may be described.  But what we've been discussing here is the subject of courses in undergraduate and beginning graduate programs in physics, that are the foundation for attempting to frame new questions. 

The point, of course, being, that since the classical properties of spacetime are understood, it's possible to learn, from how quantum probes behave, about the properties of the probes-in the case discussed here about atomic clocks.

"Special relativity and general relativity don't pose conceptual issues-for physics; general relativity does for mathematics, though these issues are way beyond what's been discussed here."

As far As GRT is concerned there are no "math issues" in GRT, as also affirmed in other threads by Charles Francis who is an expert.

The GRT doesn't give so good prediction in astronomy for example. Predictions given with the FGT, field gravitation theory (THIRRING, KALMAN and FEYNMAN), studied by Yurij Barishev who is an astronomer and according to Others too, behave better than the ones given by GRT. So the issues are related to Physics. 

The situation in GR is closely analogous to that in non-Abelian gauge theories in the following sense: The Yang-Mills (YM) current is, as a consequence of the inhomogeneous YM-equation, only covariantly conserved (vanishing covariant divergence). (For this one has to use an identity that follows from the gauge invariance of the YM action, similar to the contracted Bianchi identity that follows from the diffeomorphism invariance of the Einstein-Hilbert action.) Therefore, one can in general not define a conserved YM charge density. Similar to the energy-momentum pseudo-tensor in GR (Einstein, Landau-Lifschitz), one can introduce a pseudo-current (that transforms inhomogeneously under gauge transformations), and use this for special situations to define a global YM charge. For this the system has to be isolated, such that the YM gauge fields rapidly become asymptotically pure gauges (asymptotically flat in GR). 

So, in contrast to the Abelian case of electrodynamics, on has in YM-theories no conservation law for the current. What is divergence-free in the usual sense is the sum of the YM-current plus a pseudo-current that can be transformed away by a gauge transformation in any given space-time point. The indicated analogy reflects the fact that GR can be considered as a special gauge theory, in a sense that has been described precisely in the past for instance by A.Trautman.

James, if there are too many "energies", one would tend to conclude that none of them really exists, but that they all are only some mathematical artefact.  

Then, we would like to have an energy in quantum theory - what else should be the operator which defines evolution?  Of course, this is closely related to the problem of time of GR quantization.  

A straightforward solution would be to introduce a preferred time and preferred spatial coordinates into GR, which is surprisingly easy - the natural candidate, harmonic coordinates, is well-known and widely used.  To incorporate them into the Lagrange formalism is easy too - harmonic coordinates are simply special scalar fields.  And then one obtains a unique Noether conservation law for energy and momentum, like in http://arxiv.org/abs/gr-qc/0205035

Getting back to the original question: Does the Gravitational field have an energy density? Yes. It does. In his 1992 paper, H. Yilmaz shows that there is a gravitational stress-energy tensor that must be added to the matter stress-energy tensor on the right-hand side of the Einstein field equations. This stress-energy tensor has a specific form expressed in terms of a generalized tensor potential. The solution of the field equations yields a metric that contains exponential functions of the Newtonian gravitational potential. This solution will remove the space-time singularity at the center of a Black hole and also removes the event horizon. The gravitational energy density can be integrated over all 3-space for a point mass particle: the integral is convergent.      

Could you please give a more accurate reference to the paper of Yilmaz?

Regards

If one actually *does* the calculations in general relativity-as in any other field-one finds that proclamations before doing the calculations aren't as useful as doing the calculations first and providing the interpretation afterwards-e.g. the statement that general relativity can't describe time-dependent phenomena is immediately seen to be false; and there isn't any problem of freely falling objects-or with accelerating objects, for that matter.  The *proofs* of these statements can be found in the textbooks on the subject. There's a difference between ``statement'' and ``proof''. That many people, over time, may have had difficulties does not mean that these difficulties have remained. Once more, technical content trumps history. Similarly, while the definition of an energy density for the metric is a subtle issue, that requires quite a bit of work, that's been quoted here before, that the metric tensor does have self interactions is much easier to understand and apprehend. 

A solution that would ``remove the singularity'' would, already, have to evade the assumptions of the Hawking-Penrose singularity theorem, so it, surely, means something else. The Hawking-Penrose paper is available here: http://rspa.royalsocietypublishing.org/content/314/1519/529 and can be read with standard knowledge of mathematics. However, already, intuitively, it's clear that the only way to stop the formation of a singularity, due to gravitational collapse, is by including forces that can cancel the attraction. And a known example is provided by extremal black holes-where electrical forces (for Reissner-Nordstrom) or angular momentum (for Kerr), can lead to interesting configurations, where the singularity, while present, can be avoided. However, quite recent work in the mathematical aspects of these problems has led to the discovery of new kinds of instabilities, cf. http://arxiv.org/abs/1206.6598, for instance.

(Incidentally, while, in general relativity, a freely falling observer in a spacetime of a non-extremal black hole, will, apparently, encounter the singularity (though what this means isn't clear, since a singularity implies additional data that would resolve it), in finite proper time, an observer, accelerating outside the horizon, will never perceive the singularity.) 

Boundary terms for the energy-momentum tensor, (prior to the generalization in the work of Julia and Silva, already quoted), were studied by Gibbons and Hawking, cf. here: http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/GibbonsHawking3.pdf, which may clarify what ``avoiding the singularities'' actually means. 

Answer for Stefano Quattrini : there are three real good references-

1.       H. Yilmaz, Nuovo Cimento, 107B, No.8, 941,(1992)  

3.       C.O. Alley, “Investigations with LASERS, Atomic Clocks and Computer Calculations of Curved Spacetime and the differences between the Gravitation Theories of Yilmaz and of Einstein,” Frontiers of Fundamental Physics, Edited by M. Barone and F. Selleri, Plenum Press, New York, 1994, Pg.132.

Also; right here on Research Gate, under Zampino, E.J., is a downloadable paper "Gravitation without Singularities or Event Horizons." I wrote this paper a long time ago to capture everything I went through to understand better what is in reference 1. Reference 1 is not always so transparent.

 It's incorrect to state that a theory of gravitation must describe event horizons. What the Hawking-Penrose theorems imply is that a theory of gravitation, that satisfies certain regularity assumptions, will, inevitably, produce singularities-how these may be resolved is beyond the domain of classical gravity.  It's a conjecture by Penrose (the ``cosmic censorship conjecture'', that has been proved in many, though not all, cases-cf. http://www.aei.mpg.de/~rinne/presentations/140224oppurg.pdf) that such singularities are always ``hidden'' behind event horizons (where behind, for extremal black holes, means on the horizon), so that observers outside cannot access the singularity.As mentioned in the presentation, it is possible, however, to prove that, under appropriate conditions, naked singularities are inevitable, i.e. not hidden behind an event horizon.

It is also incorrect that the Schwarzschild metric is an approximate solution to the Einstein equations, involving a Taylor expansion. If one imposes spherical symmetry, and no other fields, one can prove that the unique solution of Einstein's equations is the Schwarzschild solution and the metric functions are exactly given by the usual expressions,  Cf. for instance,  http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html

If one introduces electric charge, and the corresponding electric and magnetic fields, one finds the Reissner-Nordstrom, spherically symmetric, black hole. 

Much is, also,  known about the stability of black holes to perturbations-cf. for instance,http://www.ctc.cam.ac.uk/activities/adsgrav2014/Slides/Slides_Holzegel.pdf

for a summary (beyond the new instability for extremal black holes, mentioned earlier). 

In fact it's useful to recall that a major difficulty in trying to define what the energy-momentum tensor of the gravitational field could be is that, if one applies the usual definition, that T_μν=(δS/δg^μν)/(2(det g_μν)^(1/2), one finds that, in the absence of matter, this is nothing but the Einstein equations of motion in vacuum, therefore, in the absence of matter, one finds that it vanishes. How one should resolve this issue may be found, for instance (this by no means an exhaustive list), in the work of Gibbons and Hawking cited above, in the work of Brown and York and in Julia and Silva, cf. also Balasubramanian and Kraus, for instance and, as mentioned many times, boils down to treating properly the fact that the metric transforms non-trivially under general coordinate transformations and there is a corresponding difficulty in defining the current  in any gauge theory-that has, however, been understood how to do properly (cf. Straumann's reply). It is misunderstanding this fact that has led to much confusion. 

The answer to the question ``does gravity exert gravity?'' is, without any qualification, affirmative, since the work of Einstein: the metric does couple to all sources of energy and momentum and, thus, to itself, since any field carries energy and momentum. Cf. the review article, http://arxiv.org/abs/1409.7871, that does discuss gravimagnetism, also. While massive objects have an energy in their rest frame, E=mc^2, massless objects, also carry energy. In general relativity, as has been stressed, how to define the energy density is subtle, so the energy density quoted at  the Earth's surface refers to something completely different than what's discussed here. Massive objects *do* affect spacetime and do lead to measurable effects, like the period variation of binary pulsars, that's explained by gravitational wave radiation.

Thierry De Mees

" I also suppose that everyone can agree that the action of attraction upon a mass is exerted by that (detected) gravity. So, work has been exerted.

If one can agree upon that, then there must be a gravitational energy density present at that time."

The gravitational energy density exists and can be seen also as a local consequence of the warpage of the hypermedium (space-time). Eddington said in his space time and gravitation:

"Density multiplied by volume in space gives us mass or, what appears to be the same thing, energy. But from our space-time point of view, a far more important thing is density multiplied by a four-dimensional volume of space and time; this is action."

There is an energy density in any case locally, othewise there would not be any action, no motion, gravitation would not be able to move anything.

But according strictly to GRT such energy cannot be localized for the Einstein principle of equivalence, (which I gently refute). The EEP  is not the EP or Newton EP which was experimentally demostrated by Lorand Eotvos. The EP can be for sure be incorporated into Gravitaton, regardless of the theory.

The EEP is the derivation Einstein made, assuming the WEP (direct consequence of the Newton's second law for falling bodies) and the Local Lorentz Invariance which gives rise to other problems.

It is interesting how necessary is to get rid of the EEP for the correct coupling. See the publication below.

Article Breaking of the equivalence principle in the electromagnetic...

The energy density of graviton field was calculated in the paper http://vixra.org/abs/1503.0127 . And the energy density of charged particle field was calculated in the paper http://vixra.org/abs/1503.0126. Both fields are parts of vacuum field. The graviton field is the reason of gravitational field, and the charged particle field is the reason of electromagnetic field. The energy densities of the fields are in the same proportion as mass proton and electron. But energy density of gravitational field is not the energy density of graviton field, since the energy density of gravitational field is a function of body mass and distance. It is a measure of that part of graviton field energy density which involved in gravitational interaction between the body and test particle and contained in the volume of the particle.

Of course the gravitational field has energy density and stress-energy tensor, see https://en.wikiversity.org/wiki/Gravitational_stress-energy_tensor .

Stefano, you say "It is interesting how necessary is to get rid of the EEP for the correct coupling".  

Here, the answer can be given:  The EEP (but not the Strong EP) is compatible with local energy and momentum conservation laws.  This can be seen at the example of my theory http://arxiv.org/abs/gr-qc/0205035 which has local energy and momentum conservation laws but, on the other hand, has also the EEP.  

Since the 19th century, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature - a unified field theory.

Classical unified field theories are attempts to create a unified field theory based on classical physics. In particular, unification of gravitation and electromagnetism was actively pursued by several physicists such as Faraday and Einstein. Einstein believed there was a link between the need to resolve apparent paradoxes of quantum mechanics and the need to unify electromagnetism and gravity. Classical unified field theories were unsuccessful, but we can unify quantum field theory with gravity by adjusting some concepts of quantum mechanics.

Today’s physics is outspread between macro and micro worlds. General Theory of Relativity very well describes Macro world, while Quantum Mechanics very well describes probability in micro world. But problem occurs when we want to unify these two theories into the one that would be able to describe each phenomenon in the Universe.

In standard model graviton is a massless with spin two. But same as photon, there are several experimental searches for massive gravitons that result an upper limits on the graviton mass.

To define graviton, we should consider to a photon that is falling in the gravitational field, and revert back to the behavior of a photon in the 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. When some gravitons are around a photon (or other particles) they convert to color charges and enter the structure of photon. Color charges around particles/objects interact with each other.

There are many layers of gravitons around a photon. The first layer is close to the photon, so that its gravitons interact with charge and magnetic fields in the structure of photon. The second layer interacts with the first layer and third layer and so on. Therefore; when a photon is falling in the gravitational field of the Earth, two layers of gravitons are applied to it, first layer up (at high h) and second down (at high h-dh). In down layer, the density of graviton is greater than up, so the photon falls and its energy increases. So, we can define graviton relative to electromagnetic energy.

To Norbert Straumann:  "What is divergence-free in the usual sense is the sum of the YM-current plus a pseudo-current that can be transformed away by a gauge transformation in any given space-time point."

Indeed.  But why consider one coordinate system at a time and transform from one to another?  If one considers them all at once, then one has an infinite-component entity (an "object" in the classical sense of Nijenhuis), a few components of which vanish at any given point, but nothing very mysterious happens. 

To Ilja Schmelzer:  "James, if there are too many "energies", one would tend to conclude that none of them really exists, but that they all are only some mathematical artefact.

Then, we would like to have an energy in quantum theory - what else should be the operator which defines evolution? Of course, this is closely related to the problem of time of GR quantization."

I don't see why having many energies implies having too many.  The Hamiltonian already has much of this ambiguity:  one has to slice the space-time and label the slices (time coordinate), stipulate sameness of place over time (shift vector) and spatial labels on each slice (spatial coordinates). 

I agree that people "tend to conclude" as you indicate, but only as a habit. 

Dear Hossein, from your post I see that your intuitions are well directed ... So I suggest you to look to the following Wikipedia link:

http://en.wikipedia.org/wiki/Talk%3AQuantum_gravity

to understand that the future of the quantum gravity is already here.

My best regards.

Agostino

From Dirac 1975, General Theory of Relativity page 62.

"It is not possible to obtain an expression for the energy of the gravitational field satisfying both the conditions:

i) when added to other forms of energy the total energy is conserved

ii) the energy within a definite (three dimensional) region at a certain time is dependent of the coordinate system.

Thus in general gravitational energy cannot be localized.

The best we can do is to use a pseudo-tensor which satisfies the condition (i) but not (ii)

thus Only the total energy and momentum are conserved and these are for isolated systems"

This means that it is not possible in any case localize according to GRT the gravitational energy.

 GRT is not Gravitation is only a theory, and since the Gravitational field is conservative, GRT may have problems. The fact that it can be reduced to Newtonian is something wihch raises some doubts.

Dear Agostino

Thank you for you kindly suggestion. Maybe foloowing link be interesting for you.

http://gsjournal.net/Science-Journals/%7B$cat_name%7D/View/5408

The fact that general relativity does have a Newtonian limit doesn't raise doubts-it shows how Newtonian gravity can be consistently recovered. Far from this raising doubts, in fact it's a necessary consistency check, that *removes* doubts that the somewhat counterintuitive properties of general relativity-since they are so far removed from direct experience-are artifacts.  And it's necessary for consistency of the theory, since Newtonian gravity should be the approximation of any gravitational theory, when masses move slowly with respect to the speed of light and  spacetime can be taken as flat.  And is presented in all textbooks on the subject. This isn't a metaphysical debating issue-it's an essential point, since approximations must be consistent.

The gravitational field in the Newtonian approximation is conservative, since it's described by a scalar potential; when relativistic effects are taken into account it's not. And the reasons are subtle, but can be deduced from the equation of motion of a test particle-the geodesic equation. In summary, while, in the Newtonian approximation, time translation invariance is a global symmetry, therefore energy is defined as a globally conserved quantity, when relativistic effects are taken into account and  Poincaré invariance is a local symmetry, time translation invariance is a local  symmetry and energy is covariantly conserved. The fact that it has these new properties, doesn't mean that it isn't a useful quantity-just that there are  new consequences, that  must be taken into account. 

All this is *known*-at least since 1918 and the work of Noether-and is subject to technical analysis, not philosophical speculation-technical terms have *meaning*, and aren't just subject to grammatical and syntactical rules.

So it is possible to assign to the  gravitational field in general relativity a quantity that does have the properties of an energy density, in the Newtonian approximation, where the concept of an energy density is well-defined-and which has new properties, when this approximation is no longer valid, in a way that can be proved, mathematically, to be consistent and can be tested in experiments. The tests aren't, necessarily, ``direct'', so many, subtle, issues must be worked out-but these have been worked out. 

Incidentally, the second condition that Dirac, apparently, mentions, is that the the energy, within a three-dimensional region should be *independent* of the coordinate system. This, as noted, isn't possible, in general relativity, in particular and, more generally,  in theories where general coordinate transformations are symmetries.

"And the gravitational field in the Newtonian approximation is conservative, since it's described by a scalar potential; when relativistic effects are taken into account it's not, because it's not described by a scalar, but by a tensor-the metric tensor in general relativity, or by additional fields in scalar-tensor extensions or in supergravity."

I agree. Einstein in his book at the beginning, introducing the stress tensor in relation to the acceleration, accounted for such reduction to Newtonian gravitation as the necessary condition for his theory to be valid and accepted.

 I would like to stress the fact that the gravitational time dilation is something which occurs also in the weak field limit and is to a very good approximation modeled in function of the Newtonian potential at the first (1+gH/c2) or second order.

I go back to one of the open questions:

According to GRT free falling bodies should present the same clock-rate wherever they fall from, the same clock-rate as the one  at infinite distance from masses (1sec/1sec), if we can neglect the tidal effects due to the extension of the atomic clocks.

This cannot be in Agreement with the experience.

The statement `` free falling bodies should present the same clock-rate wherever they fall from'' is meaningless, as stated-hence confusion and misunderstanding. 

What is meaningful is the following, a standard textbook exercise: 

A freely falling body follows a geodesic, i.e. a solution of the geodesic equation, that gives x^μ(λ), the spacetime trajectory between two spacetime points and λ is the ``affine parameter''-the trajectory has reparametrization invariance. The only observable is the proper time, the *integral* from spacetime point A to spacetime point B, of 

-(g_μν(x)(dx^μ/dλ)(dx^ν/dλ)^(1/2)

over the value of the affine parameter, that attains its extremum, a maximum, for x^μ(λ) the solution of the geodesic equation, with boundary conditions that the trajectory pass from the points A and B. This proper time, the integral, is readily seen to be invariant under reparametrization of the trajectory-and this is the physically crucial property.

(Metric convention: +---; the -+++ convention requires an additional minus sign in the square root, in order that it be positive for time-like curves and the proper time always be real in this case.  Units: c=1,m=1 for a massive particle. The global minus sign has physical meaning: it's needed to ensure that physical particles, not ghosts, propagate. It also, ensures that the extremum is a maximum. )

The expression ``proper time'' means, precisely that this is the time that has passed, for the observer, traveling along x^μ(λ), from A to B. 

Now take any other, time-like, parametric curve, y^μ(σ), that, also, passes from the spacetime points A and B and compute the proper time, i.e. the integral of the quantity above, along this curve. This curve describes the spacetime trajectory of an observer that is *not* freely falling-it doesn't satisfy the geodesic equation. The value of the integral is the time that has passed for the observer, traveling along the curve y^μ(σ). 

The value obtained will be different, namely,  *less*, since this curve *isn't* an extremum, in particular a maximum, of the functional in question. That's all there is to it. That's what ``proper time'' *means* and what time dilation means. Much of the misunderstanding comes from trying to reconcile these statements with Euclidian geometry, by drawing curves on paper. The missing ingredient is the metric: that's what distinguishes space from spacetime. 

The geodesic equation is a second order differential equation, therefore its solution, given the boundary conditions, that it pass by two distinct spacetime points, is unique-as long as the spacetime does not allow closed time-like curves. 

If the test particle is massless, one must introduce the vielbein and obtain the corresponding action from there. 

These calculations are valid for any given metric-they refer to test particles. So for the Minkowski metric they express the resolution to the ``twin exercise'', for instance. 

The calculation, also, is valid for a point-like object; for objects that are not point-like, special care is required-cf. papers linked to in previous messages.

Dear John Macken

We are working within a paradigm essentially defined by Newton; before him, the paradigm was another: it was considered that once matter attracts matter, all the matter of the universe had to be collapsed in one place, which had to be the Earth – therefore the center of the Universe. Which, as observed, rotates around Earth in 24h, the stars being dragged by celestial spheres. Local motions were explained by epicycles on deferents.

Now, cosmic observations has led to the conception of a universe where stars move away dragged by dark energy; local motions are explained by dark matter – the role of celestial spheres is now played by dark energy and the one of epicycles by dark matter. The distant universe is made of unknown entities while the nearby one by matter, in both cases.

In the present paradigm, energy conservation holds in mechanics but fails outside it – for instance, photons evanesce (the density of radiation decreases with the 4th power of expansion but this only accounts for the 3rd power) and neutrinos are just a mathematical parameter required to hold an energy conservation that is not measured.

Your question is quite pertinent but you will not find the answer within current paradigm. All this is quite clear but can only be assumed once a new paradigm is established.

Alfredo and Everyone,

I asked the question about whether a gravitational field possesses energy density to see if there was any support for this idea. As you might know, I have developed a model of the universe based entirely on the properties of 4 dimensional spacetime.  The attached recently published paper shows how this fundamental concept yields particles, fields and forces including gravity.  For example, charged particles and an electric field have never before been characterized as a quantifiable distortion of spacetime.  One discovery is that photons experience the same impedance (Zs=c3/G) as gravitational waves.  These equations lead to the concept that photons are quantized waves propagating in the medium of the "spacetime field" (explained in the paper).

The reason for this introduction is that I have recently expanded the model and it indicates that gravity has both a non-oscillating component and an oscillating component.  Related equations have successfully yielded equations which show previously unknown relationships between the electromagnetic force and the gravitational force.  Some of these equations are in the paper below, but I have more recently developed many new equations showing different aspects of this connection.  These same equations say that the gravity generated by a fundamental particle has both a Compton frequency oscillating component and a non-oscillating component that strains spacetime (produces curved spacetime). 

Knowing the frequency, amplitude and the impedance of spacetime it is possible to calculate the energy density of a gravitational field and it turns out to be Ug=Gm2/8πr4.  The total energy external to radial distance r is Eext = Gm/2r.  These equations are similar to what other people have obtained, but they were generated using an entirely new approach that starts with the properties of spacetime.  I can also show how this energy density interacts with the spacetime field to produce the weak gravity curvature of spacetime. 

 Once I got this, I wanted to gauge how much resistance I would encounter if I suggested that a gravitational field possesses energy density.  This energy density in the gravitational field comes from the energy reduction of the particles in a gravitational field.  For Example, an electron in a gravitational field possesses slightly less energy than the same electron in zero gravity.  The energy density in a gravitational field has been compared to what I call the “interactive energy density of the spacetime field" and shown to produce the weak gravity curvature of spacetime.

Chapter Spacetime Based Foundation of Quantum Mechanics and General Relativity