Hossein> According to F=-dU/dx, does force (in fact bosons) and energy convert to each other?

I don't understand what your question is supposed to mean; so I will ask a different one:

Is it possible to have a force, with related potential energy, in classical relativistic physics which converts potential energy to mass?

The answer to this is "not quite". It is possible if you allow motion in the imaginary time direction. In that case I think you can construct a world line (in complexified Minkowski space) which starts by moving backwards in time, takes a little detour in the imaginary direction, and comes back with a changed sign of velocity, such that it can continue to move forward in time. With the world line always obeying the "classical" equations of motion.

Physically the whole process can be interpreted as the creation of a particle-antiparticle pair, converting the energy in a force field (i.e., potential energy) into mass and kinetic energy, through a tunneling process.

Dear Kåre

Your question is an interesting question. As you mentioned, energy (potential and kinetic, attachment) has mass, and mass itself is just another form of energy.

http://www.physicsoftheuniverse.com/topics_relativity_emc2.html

When you push to a body (you are applied force on it), you are transferring energy and momentum to it. In fact force converts to energy, my question is how we can analyze mechanism of this process?

In fact force converts to energy, my question is how we can analyze mechanism of this process?

It is unclear to which level you want to understand how this happens. But to me the analysis is completely straightforward in terms of four-vectors p and four-forces F, with the proper form of Newton's second law being

dpmu/dt = Fmu, with mu = 0,1,2,3, and p0 = E/c.

There is the requirement that pmu Fmu = 0 for proper forces (with the scalar product determined by the Minkowski metric), to ensure that

 pmu pmu = (mc)2.

With this definition electromagnetic forces are proper, gravitational forces are not.

PS. I would not denote all kinds of energy as mass; that is in my ears an abuse of the word. E, E/c, E/c2 are essentially the same "thing" (i.e. energy), expressed in different units.

Dear Kåre

"It is unclear to which level you want to understand how this happens. " In general form.

I do not disagree with you, we can take a new look at physical phenomena.

I take a shot with mass m and shooting it with velocity v upward. Shot takes kinetic energy. The action of my hand on shot is applying force on it, only. But shot gets energy. In the other word, in during dt, n photons (as energy) leave my hand, convert to force and applies on shot, its momentum changes (Newton's second law). The photons enter into structure of shot and convert to energy again.

"With this definition electromagnetic forces are proper, gravitational forces are not." What is different between electromagnetic forces and electromagnetic energy?

In the other, which particles carry electromagnetic forces and which particles carry electromagnetic energy?

Dear Hossein and Kåre,

I have difficulty in understanding both of you. At which level do you treat this issue, classical, or quantum?

The formula F = -dU/dx is for a classical field (I mean, non-quantum). For a classical field, e.g. electromagnetic field that we treat according to Maxwell's equation, if the field transfers to a body a few photons or absorbs from a body few photons, the macroscopic characteristics of the field, F and U in this particular case, don't change.

To the contrary, a quantum field is described by formulas in which we take in consideration the number of particles - the particles that are the carriers of the respective field. The interaction of such a field with an object is done by that the field transmits to the object, or absorbs from the object, such particles. In this case we can't use anymore the macroscopic formula F = dU/dx.

So, would you be kind, both of you, to tell us whether your discussion is at quantum level, or at classical level? It seems to me that you jump from one level to another, however, if I missed something from what you said or I misunderstood, I apologize.

I don't claim that I answered the question, I am only trying to say that some concepts appropriate to one level may be inappropriate to the other level.

With best regards,

Sofia

In the picture of particle collisions (like photons being absorbed or emitted) we never talk about force or power, but of transfer of momentum and energy (which can be considered as short time integrals of applied force and power). I.e., transfer of four-momentum, where energy and momentum play equally important roles as part of a single geometric object. 

Classically, electromagnetic forces (Lorentz forces) are linear in the electromagnetic fields, while electromagnetic energy is quadratic in the fields (but electromagnetic stresses are also quadratic). Photons don't carry forces, they carry four-momenta.

This shows that the wave and quantum particle paradigms of electromagnetism are very different. They must be attached to specific physical situations, described mathematically, for an unambiguous treatment.

In summary, your focus on energy and force as comparable qualities is fundamentally wrong; the correct qualities to compare are energy and momentum, which transform into each other under Lorentz transformations.

Dear Sofia

Thank you for interesting detail, I also agree with you "that some concepts appropriate to one level may be inappropriate to the other level."

According to quantum level, there are two opposite electric fields around the electron and positron. They apply electric force on each other (please consider that force is not same as energy). Electron and positron move toward each other, in fact they get kinetic energy. Please describe this phenomena by using quantum mechanics concepts such as virtual particles and real particles.

Best Regards

Hossein

Dear Kåre

Is classical concept of force (field) replaced by boson (that carry forces) or no?

For detail please see my last comment.

Dear Hossein,

Unfortunately, I am no specialist in quantum field theory (QFT) and quantum electro-dynamics (QED). So, I'll tell you the little that I know, and it may be things well known to you.

When two oppositely charged particles are accelerated or decelerated due to the electromagnetic (e.m.) interaction, they exchange photons with the electromagnetic field. The field either looses photons which are absorbed by the charged particles, in which case the energies of the latter increases, or, the field absorbs photons, in which case the particles' energy decreases.

But there are two things on which people well trained in QED may have some explanations: 1) how works the linear momentum conservation in these exchanges, 2) whether the photons exchanged with the e.m. field are virtual.

Many practical calculations are formulated and intuited in terms of Feynman diagrams, where four-momentum is transferred from one part of the system to another by exchange of particles (bosons or fermions). This is a nice and consistent picture, but it is completely delocalized in space-time. And the particles never carry forces, they carry momenta.

It is another matter how a space-time description of interactions between two macroscopic objects can be derived from this picture, with their associated concepts of force and power. I cannot recall having seen details of such a derivation, beyond calculations of an effective action. What seems clear is that the exchange of (effectively) bosons is required to obtain a macroscopic force. For some types bosons with non-zero spin, like photons and gravitons, the number of degrees of freedom (spin polarizations) involved in the force calculation must be larger than the number of propagating degrees of freedom. The Coulomb force (resp the Newton gravitational force) cannot be describe by the exchange of freely propagating photons (resp gravitons).

Dear Hossein and Kåre,

Force is defined as the change in the linear momentum per unit time. By applying the formula of the derivative of the operators in the quantum formalism, we obtain an operator for force. Thus, the concept of force is not alien to QM.

Now, the e.m. field surrounding charged particles consist in photons, and as these photons carry energy, so they carry linear momentum. By gaining/loosing energy from/to the e.m. field, in form of photons, the charged bodies gain/loose also linear momentum.

However, the acceleration/deceleration of an electron in the e.m. field is a phenomenon that we observe at macroscopic scale. It's not one single photon that the electron absorbs/looses, but many photons, eventually, an undefined number of them. So, in principle we can calculate an average of the force operator, though, it would be at the classical limit at which ħ is practically zero.

There is indeed a problem what to do with the spin of the charged particle when absorbing a photon, because the photon has angular momentum ħ. The addition of spin ħ/2 and angular momentum ħ should produce either ħ/2 or 3ħ/2. Since the electron spin is only ħ/2, a final angular momentum 3ħ/2 for the electron would be possible only if it would perform some additional rotational movement.

Dears Sofia and Kåre

QED rests on the idea that charged particles (e.g., electrons and positrons) interact by emitting and absorbing photons, the particles of light that transmit electromagnetic forces. These photons are virtual; that is, they cannot be seen or detected in any way because their existence violates the conservation of energy and momentum.

Also, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume.

Feynman diagrams give a simple visualization of what would otherwise be a rather arcane and abstract formula.

Scientists are trying to identify the secrets of space, but the big questions remain unanswered so far.

For example: What is the nature of the vacuum? What Fields made up of really?

Theories are our understanding of phenomena in order to understand the laws of nature. So when the old theory cannot respond to new questions, we need to review them and think beyond the old theories. We should do base new theory on the old theories that are empirically validated.

Is there a way to explain virtual photon (in fact interaction between charged particles) without using the uncertainly principle? My answer is yes, we need a new definition of quantum fields. This definition should be able to describe the relation between field and its source. For example relationship between charged particles and their fields, relationship between matter and graviton (more than Newton's gravitational law shows).

Hossein> These photons are virtual; that is, they cannot be seen or detected in any way because their existence violates the conservation of energy and momentum.

This is not the standard way to look at it, when describing processes by momentum space Feynman diagrams: Four-momentum is exactly conserved, but the particle is not necessarily on-shell. I.e, it does not necessarily satisfy the energy-momentum condition  p2 = m2c2. It is a poetic way of speaking which appeals to intuition.

Hossein> What is the nature of the vacuum? What Fields made up of really?

I think these are mostly metaphysical questions, about which it is rather useless to speculate. At least before some explicit results have been calculated, which one wants to interpret.

Hossein> This definition should be able to describe...

But that is what current quantum field theories do! The question is whether it is the right description. Or rather, what are the limits of this description, and which paradigm should replace it? Many of the sharpest brains in theoretical physics has converged on superstring theory as the answer to the last question.

Dear Kåre

"it does not necessarily satisfy the energy-momentum condition  p2 = m2c2. It is a poetic way of speaking which appeals to intuition." It is not my view, this is the unrealistic attitude result of the Standard Model to elementary particles.

In the standard model, the uncertainty principle is a golden key to solve the undescribeable problems.

For example, in QED, the electric repulsion between two electrons could then be understood as follows: One electron emits a photon and recoils; the second electron absorbs the photon and acquires its momentum. Clearly the recoil of the first electron and the impact of the second electron with the photon drive the electrons away from each other. So much for repulsive forces. How can attraction be represented in this way? The uncertainty principle makes this possible. The attraction between an electron and a positron may be described as follows: the electron emits a photon with momentum directed away from the positron and thus recoils towards the positron. This entails a degree of definiteness in the momentum of the photon. There must be a corresponding uncertainty in its position - it could be on the other side of the positron so that it can hit it and knock it towards the electron.

"This is not the standard way to look at it..." I agree with you, physical progress in the last two centuries, due to the non-standard look is created. For examples, please consider to aether concept and absolute refrence frame before the relativity, and Dirac equation for anti-particle concept.

In my view, for approaching to a new understanding of nature, we need to passing of limitation of light speed and uncertainty principle.

Hossein> this is the unrealistic attitude result of the Standard Model

Physicists not only love poetry; they also know how do the mathematics of the Feynman diagrams. With experience the realistic physics behind the poetry can also be intuited.

Hossein> Clearly the recoil of the first electron and the impact of the second electron with the photon drive the electrons away from each other. So much for repulsive forces. How can attraction be represented in this way?

You have a much-too-classical view of quantum processes! (That is an unrealistic attitude.) In fact, by considering the exchange of only one single particle it is impossible to decide whether it corresponds to a repulsive or attractive force. The computable information behind the poetry is a complex number, the probability amplitude for the process to happen. It is only by analyzing interference terms between amplitudes corresponding to several exchanges that the sign of the force can be decided (directly from the Feynman amplitudes). An alternative way is to consider the classical limit, which is possible with bosons. Simple rules have emerged from such analyses.

The Standard Model is of course consistent with the Heisenberg uncertainty principle. It seems to me that you have limited, and sometimes misunderstood, knowledge of the scientific fields you want to criticize and improve.

 Dear Hossein Javadi,

I think that we can use the relation F=-dU/dx only if particle has internal structure, that is if it consist of aether "particulate",

"According to F=-dU/dx ..."?, I thought F = ma(!). Sometimes we have a "potential", mostly there is none. Force is a concept of classical mechanics, this "potential" in QED is obscure (gauge dependent) and it's derivation is the definition of the e.-mg. fields.

Dear Kåre

I admit that at the quantum level, focusing and describing the behavior of a particle is not similar the large objects (in classical mechanics). For example, look at the shape of snowflakes. Depending on the temperature and humidity of the air where the snowflakes form, the resulting ice crystals will grow into a myriad of different shapes.

In addition, there is more conditions when snowflakes are forming. These conditions can be categorized in this way. The number of water molecules in crystals, different force of gravity and the forces exerted on a crystal by seeds around it. But we know the Macroscopic rules apply to them.

Kenneth Libbrecht, professor of physics at the California Institute of Technology, has made extensive observations of how water molecules get incorporated into snow crystals. In his research, he has observed that the most intricate snowflake patterns are formed when there is moisture in the air. Snowflakes produced in drier conditions tend to have simpler shapes.

http://earthsky.org/earth/how-do-snowflakes-get-their-shape

Feynman diagrams are (only and only) pictorial representations of the mathematical expressions describing the behavior of subatomic particles. They do not describe the mechanism of interaction between charged particles.

But we can describe it by generalize laws from the classical mechanics to quantum mechanics and vise versa.

There are various theories in physics, but nature is unique. This is not nature's problem that we have various theories; nature obeys simple and unique law. So, we should improve our theories.

Dear Alexander

In my theory, sub atomic particle such as photon, electron, quarks and even W, Z bosons and gluon have internal structure.

Dear Anton

In my view we can ignore the force and use the concepts of quantum (boson and energy) to describe physical phenomena.

Also, we need a new definition of a (acceleration) in F=ma.

dear Hossein, you can never ignore Newton's first law and you can never describe the earth's orbit quantummechanically.

Hossein> we need a new definition of a (acceleration) in F=ma

What about rewriting Newton's second law in a more sensible form instead?

dpmu/dt = Fmu, with mu = 0,1,2,3, and p0 = E/c, and t eigentime,

(as I have mentioned earlier in this thread).

Hossein>  is not nature's problem that we have various theories

No. And we do have a connected and consistent theory of almost everything, the union of the Standard Model of particle physics with the General Theory of Relativity. But one reason we use different paradigms to describe nature is that this makes us able to use a sufficiently simple and sufficiently accurate mathematical model for the problem at hand.  I guess one could describe the elliptic motion of the earth around the sun by use of the Schrödinger equation for hydrogen-like atoms, using properties of the high order Rydberg states and Laguerre polynomials. But that would put too much focus on irrelevant properties, hence be a wrong paradigm.

Quantum mechanics may have some influence on the formation of snow-flakes, but it is not practically possible to model this by ab initio computations using the time dependent Schrödinger equation involving thousands of water molecules. One have to find another paradigm for a useful model.

If you need a new satellite antenna on the roof of your house, you don't blame your house for not having one already, and start tearing it down to build a new house (for which you have no materials). Or would you? Why should that be a good strategy for extending a theory of physics?

Dear Kåre

I agree with you that "... it is not practically possible to model this..." but we can improve our understanding of nature law.

Motion is an intrinsic property of physical existence. But there is a problem about concept of acceleration in theoretical physics.

First of all, what we know about acceleration? And what is the definition of acceleration? In physics, acceleration is the rate at which the velocity of an object changes over time that in classical mechanics is given by Newton's second law with a=F/m. In special relativity an accelerating particle has a world-line which is not straight. This is not difficult to handle. The 4-vector acceleration can be defined as the derivative with respect to proper time of the 4-velocity. It is possible to solve the equations of motion for a particle in electric and magnetic fields, for example. Accelerating reference frames is a different matter. In general relativity the physical equations take the same form in any co-ordinate system. In special relativity they do not but it is still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems. This is done by introducing a metric tensor.

http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

However, Newton and Einstein define the acceleration regardless of the structure of particles (in classical mechanics and relativity). This definition belongs to Newton era or macroscopic level era. It should be noted that the interaction between large objects (e.g. collision of two bodies) under the action of the quantum layer (in fact sub quantum level) has been done. Thus, according to quantum mechanics and mass–energy equivalence E=mc^2 , we must redefine acceleration. It means we should review the relativistic Newton's second law.

I have reconsidered relativistic Newton's second law.

https://www.researchgate.net/publication/280491440_Reconsidering_relativistic_Newton%27s_second_law_and_its_results?ev=prf_pub

Article Reconsidering relativistic Newton's second law and its results

Dear Anton

In geenerally I agree with you. In addition, we can from a new approach, turns out to merge the fundamental principles of quantum physics, relativity and classical mechanics through a new definition of quiescent state of particles like photon, and attempts to present the reasons and the possibilities of the existence of the superluminal speeds. At the beginning of the 20th century, Newton’s second law was corrected considering the limit speed c and the relativistic mass. At that time there has not been a clear understanding of the subatomic particles and basically there was little research in high energy physics, if we ignore the zero rest mass of photon, much better and more real physical phenomena may be investigated. The speed of the created particles is a function of the internal interaction and the mechanism of creation of subatomic particles, and the external forces that are exerted on them.

https://www.researchgate.net/publication/280150715_Modern_physics_problems_and_solutions?ev=prf_pub

https://www.researchgate.net/publication/279185909_Graviton_and_Newton%27s_second_law?ev=prf_pub

Article Modern physics; problems and solutions

Article Graviton and Newton's second law

dear Hossein, "The speed of the created particles is a function of the internal interaction and the mechanism of creation of subatomic particles, and the external forces that are exerted on them."

This is certainly not the case, the "speed" of a MASSIVE particle is a matter of frames: In its restframe this particle has speed 0, (=ZERO!)

Dear Anton

"In its restframe this particle has speed 0",

Does "its restframe" mean that frame is based on particle? If yes, so, every particle even photon relative to its restframe has speed o (=ZERO).

If I misunderstood "its restframe", please correct me.

a photon is not a massive particle and it has no "frame", it exists on the light cone. this lightcone is a singularity in Minkowsky space and separates the two domains time like (=massive particles) and space like, the tachyons, spin, and more ...

Dear Anton

I know, in modern physics photon is a massless particle (that is an assumption only).

I have analyzed in the Minkowsky formula in real and virtual space-times.

https://www.researchgate.net/publication/270339919_Interactions_Between_Real_and_Virtual_Spacetimes?ev=prf_pub

Article Interactions Between Real and Virtual Spacetimes

Anton,

I know that the excepted line in science today is that a photon has no mass or any internal frame but this is a slap in the face of Albert Einstein and his work on the mass energy equivalence.   So in science today is it okay to say that both or neither of the associated theories are incorrect?   This is one of the fundamental flaws in science today.  We can not have it both ways and call ourselves logical rational people. 

If something and I do not care what you call that something, has energy it must carry mass with it.  This means that a photon must have mass and there by be a physical thing.  There can be no such thing as "pure" energy.  Energy is how we tell something is real.

I am puzzled by the scientific community allowing this lie to be used for more than one hundred years.  We can not have it both ways and still say we understand science.

Dear Hossein, consider S = Q/T, where Q is the heat transfer, a massless form of energy ...  No, energy must not carry mass. On the other hand, for decades it was standard thinking that neutrinos were massless!!! Now, we must believe, they carry some "tiny" mass, so .001eV, only to organize the solar neutrino problem by oszillating.

So, why not, your idea is not better or worse, just say you need photon oszillation on the "Planck scale", far away from any experiment, and "YOUR" photon has 10-20eV mass, a kind of "inverse" Planck energy. Maybe you'll become a famous physicist.

Dear Anton

Thermal energy has mass.

Einstein said: "A hot iron is heavier than a cold one." Page 59 of:

https://books.google.com/books?id=Q_uEqifzpNEC&pg=PA59&lpg=PA59&dq=Hot+iron+heavier+than+iron+is+cold+Einstein&source=bl&ots=P59QRDME3C&sig=CrF6kL1IGVgmcmULQo_t0a-HOhk&hl=en&sa=X&ved=0ahUKEwjd-c_8oM3KAhVr_HIKHbkHDtUQ6AEIHDAA#v=onepage&q=Hot%20iron%20heavier%20than%20iron%20is%20cold%20Einstein&f=false

Hot water is actually a little bit heavier than cold water because as Einstein told us E=mc2.

http://www.thenakedscientists.com/HTML/questions/question/2946/

For more detail please see:

https://www.researchgate.net/publication/279531060_Graviton_physical_time_and_thermodynamics?ev=prf_pub

Article Graviton: physical time and thermodynamics

o.k. lifting a mass in constant gravity g: additional mass: mgh/c². total mass: m(1+gh/c²).

repeating this n times: m(1+gh/c²)n --> m(1 + ngh/c²), increases to infinity.

1) Result: in a constant gravity field it is impossible to compare masses.

2) lifting a rotating mass, the rotation slowes down because mass increases and angular momentum conservation. the same: angular momentum cannot be compared.

impossible, potential lifting energy cannot increase mass.

Anton @

What kind of medication are you on? The energy budget for particles in a gravity field is high-school curriculum. At least it used to be, last time I checked (after you must have passed all exams).

I agree whit Kåre Olaussen 

@Kare: it is Hossein's claim that ALL kind of energy transfer results in mass increase, f.i. heat transfer: S = Q/T. I doubted this and brought as counterexample: mass lifting in a constant gravity field:

it is plain arrogant and ignorant that my Etot = mc² + Epot = mc² + mgh --> mc² + Mliftc², Mlift = mgh/c², and finally: Etot = m(1+ gh/c²)c², or: lifting mass increase: mlift(1+gh/c²), or, after n liftings: m(n) -->  m(1 + ngh/c²), the original mass m increases to an arbirary high amount is on a basic high school level 

Dr.  Olaussen: YOU call this "high-school curriculum"? is this the Norwegian level?  Next: it is possible that YOU BOTH did not understand my thought experiment. 

"after you must have passed all exams.": Dr. Olaussen, I'll tell you, "passing exams" means only the person passed, that's it. On the other hand, arguing with "exam passing" demonstrates that this person ability of judging quality is only developed on a very lowel.

##########

@Hossein Javadi: your answer "I agree whit Kåre Olaussen" shows that you didn't understand a word of that what I wrote above!

Dr. S.

Anton @

It is possible that I misunderstood your question. Which is about what kind of energy contributes to gravitational attraction, right? And yes, there are aspects of this question going beyond high-school physics. Some of them have been discussed on another running Q&A thread.

One problem is that energy is not an invariant quantity, hence answers can depend on how you choose coordinate systems. But a rather natural choice is to choose local Lorentz frames at rest in Schwarzschild coordinates. My conclusion for this case is that all kinds of energy, including gravitational potential energy, will gravitate. That heat, other kinds of kinetic energy, and local potential energy between atoms in a material must be included has always been fairly obvious. What I have been uncertain about is whether the gravitational potential energy also contributes.

But I again stress that the above statements are related to a specific, although natural, choice of coordinates. Coordinate dependent questions have a tendency to lead to endless empty discussions.

Dear Anton

"...shows that you didn't understand a word of that what I wrote above!" I understood.

You have wrriten: "potential lifting energy cannot increase mass. "

Shooting shot

I take a shot with mass m and shooting it with velocity v upward. Shot takes kinetic energy. The action of my hand on shot is applying force on it, only. In the other word, in during dt, n photons (as energy) leave my hand, convert to force and applies on shot, its momentum changes (Newton's second law). The photons enter into structure of shot and convert to energy again. So, shot takes kinetic energy equal 1/2 mv^2 to upward.

As the shot goes upward, loses some of its kinetic energy (mass decreases). Instead, its potential energy increases.

Note that the potential energy depends to the position of the two objects (eg, shot and earth) ratio together. But when the shot falls toward the earth, its energy (or mass) increases.

https://www.researchgate.net/publication/280626794_Unified_Force_Energy_and_Mass?ev=prf_pub

Article Unified Force, Energy and Mass

"What I have been uncertain about is whether the gravitational potential energy also contributes"

Exactly this is the question. For simplicity I tread this in a thought experiment on the most simple possible level:  a mass m is lifted in constant gravity g at h meters: will mgh be transfered into mass increase as pointed out above? Further, this process may be repeated n times, in addition with rotational mass. In case this gravity can be switches on and off, then there cannot be a mass increase, because it had to vanish after switch off. Result: a gravitational field cannot been switched on and off.

Anton @

You are not very precise (I feel) in your descriptions, but your questions (the way I interpret them) are interesting and valid. I can say that, because I have very recently investigated the same issues -- a fact which may have forced your statements into the cage where my own thoughts are enslaved (a poetic way of saying that I may have misunderstood you completely).

You include additional (interesting) concepts, like suddenly turning gravity on and off. This may be mathematically implemented to some extent, within the theoretical concept I have in mind.

But, when you say that a mass is lifted to a higher level, you must take proper account of the energy used to perform the lifting. This can be analyzed to a quite satisfactory degree in static geometries. Such analyses, within their coordinate dependent interpretations, led to my conclusion that gravitational energy must also gravitate. Otherwise, Perpetuum Mobile of the first kind would be possible. Such analyses go beyond GR, in their assumptions that all possible local forms of energy may be transformed into each other, with perfect efficiency. The locality condition made me unsure about the gravitational energy for a very long time --and I still think this is a matter of consistent interpretation within the (natural) coordinate system being used. 

For many purposes, one may consistently work within the no-backreaction scenario. I.e., the assumption that movement of (localized) energy-momentum has no effect on an existing gravity field. But the assumption of no-backreaction should always be kept in mind, before paradoxes are declared.