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The introduction of the principle of conservation of mechanical energy has been tremendously useful from the practical point of view. But...

Consider the case in which we shoot an electron up in the stratosphere, it travels up to a certain height and then it stops when its $KE = 0$. We say, according to that principle, that lost energy is stored as $PE$. Wiki goes probably too far, misleading students, stating that energy is stored in the body. More correctly, we say it is stored in the system and is measured by the distance from the ground.

Now suppose that when the electron stops it meets a positron; they annihilate into a photon of equal energy, no more nor less. It would appear evident that its $KE$ has been drained out by $g$ and definitely destroyed. Is this correct?, if you think it is not, where has primitive $KE$ gone?

If this is correct, does the same conclusion apply to a photon leaving a supermassive body and being redshifted to $0$?

Is the answer different in Newtonian and in relativity physics?

Please say also if, in principle, it is acceptable/accepted that: a) energy is not conserved (destroyed), and b) energy can be created. If the answer is different please explain the reasons.

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  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/19216/2451 and links therein. $\endgroup$ – Qmechanic Nov 6 '14 at 11:03
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    $\begingroup$ This isn't really a GR question, but since the GR tag was included: physics.stackexchange.com/q/2838 $\endgroup$ – Ben Crowell Nov 6 '14 at 16:53
  • $\begingroup$ Emphasizing anna v's answer - the positron-electron annihilation produces two photons, which conserve the momentum of the original electron-positron pair. The center of mass of the two approaching particles is an easy/natural frame of reference from which to analyze this collision and conversion from mass-energy to photonic-energy. The central point of the question was whether this set of gravity-bound particles, which stored up GPE, which then converted into photonic energy, violated the conservation of energy when the mass-energy and GPE converted into 2 photonic quanta. $\endgroup$ – Thomas Lee Abshier ND Jun 26 '18 at 6:12
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bobie, notice one thing. Gravitation as such is energy from nothing. Where does the energy to make things move in the gravitational field come from?

The conservation of energy here refers only to a "system" body-source, in which an external energy is working against the field. So if it produces movement of the body away from the source of the field, the field will (be able to) produce equivalent movement back. But this external force is not contributing to the growth of gravitational energy within the source. At least we know nothing about it. Because gravitation is capable of attracting a body that has never before moved away from it. It can bring a body from infinity to the source of gravitation. Where does this energy come from?

As we can see the Sun shining, physicists feel compelled to deliver a theory explaining the source of the energy needed to produce this electromagnetic radiation. And this energy is said to being used up. Now, what theory explains the source of energy the Sun needs to attract the Earth? (Or to curve spacetime, if one prefers the modern gravity theory.) And is going to get used up one day? I haven't seen an attempt to even pose such questions. Perhaps because we have this law of energy conservation that seems to have solved the problem. Well, perhaps it has solved a problem, but not this problem.

So back to your problem. Movement away from the source of gravitation is not producing any additional energy in this source. It only allows the source of gravitation to use the energy it already has to produce movement. The source of gravitation has this energy anyway, whether we are considering a body you just moved away from it, or a body that has always been far away. Unless we learn it is expiring, gravitation needs to be considered an infinite source of energy. The movement against gravitational field to gain potential energy always requires an external input of energy. And if the body is allowed to move freely under the influence of the field, this energy invested in the movement away from the source of gravitation can then be "retrieved".

But consider a case when you move a body away from one source of gravitation, far enough for it to get close to another source of gravitation that will make the body move toward it. What happened then? Yes, you got the energy back, but from a different "system". Was it conserved then? The original system did not "get it back", did it? It seems that the potential energy created within the first system has been "lost" to it. Well, potential energy is not an energy really. It is just a displacement from the source of the field, and the greater the displacement, the greater kinetic energy the body can potentially achieve once you "let it go". It is only a potential to gain energy, and not energy per se.

So, gravitation does not obey the law of conservation of energy. It is creating energy et nihilio as far as we know. And the problem of annihilation of a moving electron-positron pair (which must move anyway in order to meet) is a separate issue.

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  • $\begingroup$ this is a very misleading anwer , and the last paragraph is wrong within the known physics presently. $\endgroup$ – anna v Jun 26 '18 at 5:35
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The idea of partitioning energy into different forms like "mechanical energy" or "chemical energy" and such is actually arbitrary.

More or less by definition, energy is that which is conserved unter time translations by Noether's theorem. If what you call "mechanical energy" has changed, then there is another term in the Noetherian energy that has changed correspondingly, such that the total energy is conserved.

Questions like "Has the kinetic energy been destroyed?" are, from the Noetherian viewpoint, essentially meaningless, since there is no well-defined notion of kinetic energy to begin with - you have a Lagrangian describing the system (note that there is no algorithm to find the Lagrangian, you essentially "guess" it so that the equations of motion predict the correct physics as measured), and from that it follows that there is a quantity called energy that is conserved. You can say that the terms proportional to the derivatives of the dynamical variables are "kinetic energy", and the rest are "potential energy", but there's nothing saying anything about the conservation of these individual quantities in a general setting. In some settings, you might be able to say more, but in general, you should not expect individual summands in the total energy to behave in any particular way.

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Consider the case in which we shoot an electron up in the stratosphere, it travels up to a certain height and then it stops when its $KE = 0$. We say, according to that principle, that the lost energy is stored as $PE$.

This has been experimentally verified - the electron falling back gains the kinetic energy it lost going up. The concept of potential energy, energy that may be potentially retrieved, has been very successful, and the theoretical framework developed as described by ACuriousMind has been validated again and again and again, i.e., it has not been falsified in the Newtonian framework.

Now suppose that when the electron stops it meets a positron, they annihilate into a photon of equal energy, no more nor less.

NO, two photons are produced. Momentum is a conserved quantity (in the well-validated theoretical framework), and in the center of mass system of an electron-positron, momentum would not be conserved with just one photon. Two particles, at least, are needed for conservation of momentum.

Now we are far away from Newtonian physics. We are in the realm of quantum mechanics and special relativity. The special relativity total four-vector describing the electron plus positron before scattering and annihilating, and after, is invariant - it does not change with the interaction. Thus, the potential energies of the electron and the positron will be taken into account by the energies and directions of the two photons. This is because their center of mass is continuous with the center of mass of the original $e+e-$ pair. The two photons have an invariant mass-energy equivalent that will be affected by the gravitational field - changing the direction of their decay to compensate for the energy. The energy balance will be reflected in the frequency ($e=h\nu$) of the photons.

It would appear evident that its $KE$ has been drained out by $g$ and definitely destroyed. Is this correct?, if you think it is not, where has primitive $KE$ gone?

$KE$ is continuously transmuted to potential, to chemical, to rest mass, to... Kinetic energy is not conserved. It is the total four-vector components of the system, $(E,p_x,p_y,p_z)$ that must be conserved, and they are. Lorentz invariance has been validated experimentally - innumerable times.

If this is correct, does the same conclusion apply to a photon leaving a supermassive body?

In black holes, photons are attracted by the gravitational field too, and conservation of four-vectors always holds.

and being redshifted to $0$?

This redshift business is the realm of General Relativity. In GR even though conservation of four-vectors holds locally, talking of redshifted photons takes us out of locality and into the Big Bang model, General Relativity frame, and the expansion of space itself. This is a research area both experimentally and theoretically and one where to balance energies we have "invented" dark energy and dark matter, so as to keep the conservation of local four-momentum vectors, which is another story.

Please note that I am asking this question in Newtonian physics. Is the answer different in Newtonian and in relativity physics?

Of course it is different, as I explained above.

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It would appear evident that its KE has been drained out by g and definitely destroyed. Is this correct?

This is from a different perspective than the other answers and is not so much an answer as an extended comment on the above quoted question.

It occurs to me that creation and destruction are, in some sense, absolute. In your thought experiment, all observers agree that the electron and positron are destroyed and a photon is created.

Likewise, it seems to me, the (genuine) creation or destruction of energy (if that is possible) must be absolute, i.e., observer independent

But kinetic energy is observer dependent. Observers in relative motion don't agree on the kinetic energy associated with the electron and, indeed, in the rest frame of the electron, there is no associated kinetic energy.

In this frame, which is freely falling, what KE has been definitely destroyed?

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Posting another view on the already nice answers.

Conservation of (mass-)energy is a principle in physics. Feynman used to say, (Feynman lectures on physics) that when various processes are studied, one finds that energy is not conserved, but then looks under the carpet or in waste bin and finds another form of energy which when taken into account makes the whole energy indeed conserved.

In this sense energy is transformed into various forms, and if all those forms are taken into account the compatibility condition of mass-energy conservation holds.

Special relativity (among other things) just added another form of energy, the mass. In this sense, SR takes into account creation-annihilation of particles as forms of energy (or transfer of energy) based on masses.

That is why one says the mass-energy is conserved.

i would say that it is not a religious belief that energy should be conserved under all plausible cases in general at any cost (see below). But neverthelesss the principle (which is always evolving) has indeed been found to hold and make physical investigation easier.

A physical process has various compatibility conditions (also refered sometimes as conservation laws, for example in the context of the theorem of Noether in Lagrangian formalism) which effectively express correlations and/or conditions of symmetry and/or connections with the rest of the environment and/or boundary conditions and so on..

The conservation of mass-energy is one of them, and one that is constantly evolving (by taking into account new and different forms of energy).

Now when one goes into general relativity (standard GR) and general covariance and riemaniann spaces, some energy transformations are not covariant. For example the energy-momentum tensor of the gravitational field itself (either it is not conserved or the conservation law is not covariant). So in this sense the previous approach to the energy conservation principle is not applicable.

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Is Energy Always Conserved?

We have a large body of experimental evidence (First Law of Thermodynamics), and rationally confirmatory theoretical evidence (Noether's Theorem) to validate the absolute, always, universal principle of energy conservation.

Bobie tried to construct a plausible scenario whereby energy was not conserved. I do not believe his example was successful, but it was clever - a good effort. He attempted to find a contradictory phenomenon, a single example that disproved the theory. But, a careful analysis, using known theoretical principles (GR & QM) will unmask this problem's obscuring complexity and reveal it merely hiding the apparently missing energy - just as Feynman warned.

In this example, the energies of mass and Gravitational Potential Energy are stored in the system of mass and gravitational field, which was subsequently converted into a system of photonic energy and gravitational field. A thought experiment validation (possibly using gravitational redshift considerations) will confirm the totality of conversion of the energetic reactants (GPE plus the mass-energy of electron and positron), into equivalent energetic products (GPE and two photons).

At the present time, our concept of energy is largely defined by mathematical abstractions and examples which we have come to recognize as members of the family of energetic phenomena. We rely on the strong causal inference implied by the deductive proofs of mathematical logic to justify the necessity of energy conservation.

Concerning the more ethereal/abstract question, as to whether energy must be conserved, this question will someday be answered on a more intuitive level when we have a more ultrastructural concept of what energy "is".

When we have a more concrete, abstract-tangible conception of what energy is, we will clearly see the element of commonality which gives the family of energy-phenomena their distinguishing identity-characteristic.

We currently have an intuition of energetic conservation, but we don't have a firm grasp of the deep and particular essence of energy. Someday, the theories and models of space, time, and energy will mature to a level where all energetic phenomena will be as identifiable and definite to our imaginative-rational consciousness as ordinary mass is to our senses and experience.

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The KE is converted to PE. The photon-planet system still has potential energy since there is gravitational red shift.

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  • $\begingroup$ The electron's KE has converted into GPE at the top of the arc. The electron then reacts with a positron, converting its mass-energy & GPE into two photons with the energetic summation reflected in the $E=h\nu$ of the two photons. The problem starts with an electron-positron system and becomes a photon-planet system. The question is whether the energy before and after the conversion is conserved. If not, then this is the single counter-example needed to disprove the validity of energy conservation. But, it does not disprove conservation, it only leaves the mechanism of conservation ambiguous. $\endgroup$ – Thomas Lee Abshier ND Jun 26 '18 at 6:41
  • $\begingroup$ The question to @Asaf is whether the Gravitational Red Shift is the process that gives explanatory logic and mechanism to the migration of energy from the two-mass system to the photonic system. Possibly, conducted as a thought experiment, measuring/calculating the expected frequency of the photon emitted from the annihilated electron (which was going up, and hence the trajectory of the photon would be radial) would correspond to the expected conserved value of the energy, as modified by the processes of the "gravitational redshift". $\endgroup$ – Thomas Lee Abshier ND Jun 26 '18 at 6:53

protected by Qmechanic Nov 6 '14 at 12:56

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