newtonian mechanics - Can energy be created and destroyed?

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.

Answer

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.