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This question already has an answer here:

Some people say that dark-energy density remains constant in an expanding universe. However this is something that's still rather hypothetical. It's rather atypical too, in that we have no examples of non-conservation of energy. If you move towards the $E=hf$ photon it appears blue-shifted, but it hasn't actually gained any energy. It hasn't changed at all, instead, you changed. If you drop a brick, gravity converts potential energy into kinetic energy. But it doesn't create any energy. It doesn't do work on the brick, instead you do work on the brick when you lift it up. In similar vein the ascending brick doesn't lose any energy. Nor does the ascending photon. See Einstein talking about red-shift in the digital papers hosted at Princeton. Note that he doesn't say light changes frequency as it climbs out of the gravitational field. He says it's emitted at a lower frequency. There are no perpetual motion machines.

However some people say things like "any reasonable definition of energy that has been found leads to non-conservation in many GR scenarios". Only I don't know of any example of non-conservation of energy. Can you provide any? Either in a GR scenario, or in any other scenario?

I will award a 200-point bounty on this question. I will award it to the answer that I think is the best. If I don't much like any of the answers, I will still award the bounty to one of them.

Edit: now the question has been closed I have to withdraw the offer of a bounty I'm afraid. Apologies to those who have answered.

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marked as duplicate by Danu, HDE 226868, user36790, Wolpertinger, knzhou Oct 22 '16 at 20:48

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ This question is a response to a comment of mine. $\endgroup$ – Danu Oct 22 '16 at 13:57
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    $\begingroup$ Possible duplicate of Is the law of conservation of energy still valid? $\endgroup$ – Danu Oct 22 '16 at 13:58
  • $\begingroup$ @Danu : it isn't a duplicate. The other question isn't asking for examples of non-conservation of energy, and the answers don't give any. You said "any reasonable definition [of energy] that has been found leads to non-conservation in many GR scenarios". So give some. Answer this question. $\endgroup$ – John Duffield Oct 22 '16 at 14:08
  • $\begingroup$ I suggest you read the answers to the linked question. In particular, Lubos' answers gives the FLRW metric as an explicit example. Therefore, the answers to the linked question do indeed provide examples of non-conservation in GR and hence answer your question. $\endgroup$ – Danu Oct 22 '16 at 14:17
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    $\begingroup$ It's unclear what this question is asking. If you're just asking "is energy conserved or not in mainstream GR", then it's an exact duplicate of the other question. It appears that you think it's not a duplicate because you reject mainstream GR and the examples given. In that case, you should explicitly say what you find wrong with those examples and GR in general, otherwise the question is impossible to answer. $\endgroup$ – knzhou Oct 22 '16 at 20:51
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Energy is the conserved current associated with the time-symmetry in time-independent Lagrangians. For problems whose Lagrangians lack that symmetry, there will be no such conserved current. Attempting to define one will produce a time-varying quantity instead.

In Galilean physics time-asymmetric Lagrangians are associated with problems that include e.g. friction terms. In these cases, though, it is always possible to consider "larger" problems that account for the friction dynamically rather than statistically, so energy conservation can always be recovered in principle. This is because the background manifold upon which the problem lives is itself time-symmetric.

In general relativity, however, there is no guarantee that the background will be time-symmetric. In other words, there are perfectly good solutions that lack timelike Killing vectors, because the gravitational field varies in time. In such cases even defining energy is ambiguous: there is no symmetry to compare to. One could in some cases define the energy in terms of a timelike Killing vector associated with a related time-symmetric spacetime, but in this case the energy won't be conserved.

The basic implication of this is that energy is not conserved in any spacetime with a time-varying gravitational field. Strictly speaking, that means energy conservation fails anytime anything moves such that its quadrupole moment varies (of course, in almost all practically interesting cases the magnitude of the failure is small).

We normally deal with asymptotically flat problems, however. In that case, it is possible to recover some notion of energy by using the timelike Killing vector at spatial infinity to define a total energy for the whole spacetime. This allows one to say things like "gravitational radiation has carried $X$ solar masses away from a black hole merger", by approximately treating the binary black hole system as an entire spacetime. However, one cannot pinpoint the location of the GW energy flux to specific points of the binary in such a way that the energy accounting works out, because the timelike Killing vector does not exist at those points. If one tried to do that they would find the specific points had "energy" which was not accounted for by the GW flux.

In other words, the energy of a gravitationally-radiating object is in some sense conserved globally but not locally. This is basically because the energy density in the gravitational field itself cannot be defined in a covariant way. But local conservation of energy is what is really important: if I were to show you a perpetual motion machine here on Earth, it would be small comfort to learn that energy somewhere in the Andromeda Galaxy were decreasing to power it.

Therefore, here is my first example of a failure of energy conservation:

-Anything radiates gravitational waves for any reason.

The most important example of a problem which is not asymptotically stationary is the FLRW expanding Universe. There is no timelike Killing vector anywhere and thus no conserved energy, even globally. If we consider a volume of space at two times $t_0$ and $t_f$, $t_f > t_0$, filled with, say, light, the total energy within the volume at $t_f$ will be less than that at $t_0$, since the photons have been redshifted. If there's a cosmological constant, the energy will actually increase: there is now "more space" to fill with cosmological constant.

Note this is importantly different from a photon ascending through a static gravitational well. While the photon is redshifted in this case, one can reconcile the lost energy in terms of the static gravitational potential of the Schwarzschild solution. But no such constant potential exists in the FLRW universe.

Edit: OP views the FLRW Universe as an invalid solution to the EFEs, since it is homogeneous and isotropic. The arguments given above apply, however, equally well to the Kasner metric

$ds^2 = -dt^2 + \sum_{j=1}^{D-1} t^{2p_j} [dx^j]^2$

which describes an expanding/contracting family of universes which are neither homogeneous nor isotropic (the expansion rate is different in different directions).

Edit2: OPs suspicion of the FLRW solution appears to be motivated by the assumption it describes the "entire Universe". It is important to point out that while this is certainly the most important application of the FLRW solution, it is not the only one. In particular, the FLRW solution can be smoothly joined to the Schwarzschild solution in order to describe a finite ball of homogenous and isotropic pressureless matter, which will expand or contract. Of course real matter has pressure, but up to this approximation this is in principle something you could build in your living room. Within the ball, energy will not be conserved.

Edit 3: Symbolic demonstration of the dependence of energy conservation on symmetry (following http://www.blau.itp.unibe.ch/newlecturesGR.pdf).

The stress-energy conservation law

$\nabla_\mu T^{\mu \nu} = 0$

expands to

$g^{-1/2} \partial_\mu (g^{1/2} T^{\mu \nu} ) + \Gamma^\nu_{\mu \lambda} T^{\mu \lambda} = 0.$

The energy-momentum density $T^{\mu 0}$ is neither conserved nor covariant, due to the Christoffel term above (the Christoffel symbols are not a tensor). We can attempt to form a conserved current by contracting with some vector $V^\lambda$; thus we seek $V^\lambda$ such that

$\nabla_\mu(T^{\mu}_{\lambda} V^\lambda) = 0$.

This will correspond to energy-momentum conservation if $V^\lambda$ is timelike. The above expands to

$\frac{1}{2} T^{\mu \nu} (\nabla_\mu V_\nu + \nabla_\nu V_\mu) = 0$.

So we will have energy conservation iff the bracketed term vanishes. The bracketed term is Killing's equation: it vanishes iff $V_\nu$ generates an isometry, i.e. if the Lie derivative of the metric along $V$ vanishes: $L_V g_{\mu\nu} = 0$. But no such $V_\nu$ exists in general.

To show energy is not conserved in a particular case, pick any non-stationary spacetime you like, and form the above current using any timelike vector field you like. You will not obtain a conserved quantity.

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  • $\begingroup$ Re this : "Note this is importantly different from a photon ascending through a static gravitational well. While the photon is redshifted in this case, one can reconcile the lost energy..." There is no lost energy. The photon frequency doesn't change. You and your clocks go faster when I lift you up and do work on you. I add energy to you, so relative to you the photon appears to have lost energy. It hasn't. Instead you've gained it. In similar vein when you send a 511keV photon into a black hole, the black hole mass increases by 511keV/c². $\endgroup$ – John Duffield Oct 23 '16 at 11:37
  • $\begingroup$ Can I add that I reject your assertion that a gravitational wave is an example of the non-conservation of energy. I consider them to be an example of the conservation of energy. $\endgroup$ – John Duffield Oct 23 '16 at 11:54
  • $\begingroup$ It can be understood as a realization of global conservation of energy, in spacetimes that happen to be asymptotically flat. The gravitational wave flux at infinity balances the energy flux from spacetime as a whole. But there is no Gauss' law for gravitational waves: the gravitational wave flux from a particular region of spacetime does not in general balance the change in energy within that region. That is what I mean. $\endgroup$ – AGML Oct 23 '16 at 18:50
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    $\begingroup$ Perhaps a simpler way of seeing this is to note that "energy" is only one component (the time-time component) of the stress energy tensor. That whole tensor is covariantly conserved by hypothesis, but its components will not be individually unless there are special symmetries. I think you also may be conflating two different questions: 1. is energy conserved in nature? 2. is energy conserved in GR?. The answer to question 2 is unambiguously no due to the above argument. There is more wiggle room for question 1, if you believe GR is incorrect or incomplete. $\endgroup$ – AGML Oct 23 '16 at 19:07
  • $\begingroup$ The reason for the distinction is that question 2 is purely theoretical. Question 1 seeks an actual experiment whose results can only be explained by appeal to failure of energy conservation. The latter is a lot harder to find, but the former is quite easily demonstrated. $\endgroup$ – AGML Oct 23 '16 at 19:10
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The CMB must be the poster child for this. At recombination the temperature was about 3000K, now it's about 2.7K: the photons of the CMB have lost a lot of energy. So either energy is conserved in which case there would need to be a lot more of them now (what process created all these extra photons?) or energy is not conserved (because, of course, spacetime is not symmetric under time translation).

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  • $\begingroup$ Converted from a comment. $\endgroup$ – rob Oct 22 '16 at 19:43
  • $\begingroup$ ...or you're missing a contribution to total energy in your accounting... $\endgroup$ – Christoph Oct 22 '16 at 21:22
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    $\begingroup$ As the original author of this, I'd suggest looking at @AGML's answer below which incorporates the same result with hugely more detail. $\endgroup$ – tfb Oct 22 '16 at 22:31
  • $\begingroup$ Alternatively, energy is conserved and those photons have the same energy now that they used to have. Or energy is conserved and those photons have lost energy whilst something else has gained it. $\endgroup$ – John Duffield Oct 23 '16 at 11:38

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