The first law of thermodynamics states that energy is conserved. It is one of the most fundamental laws of physics and not one that you would expect many physicists to challenge, so it comes as a surprise to find that a growing number of cosmologists and relativists are doing just that. Of course any law of physics is subject to experimental verification and as new realms of observation are opened up we should require that previous assumptions including conservation of energy are checked. But the subject under question is not new physics in this sense. It is the classical theory of general relativity. Whether general relativity is correct is not the issue, although it has withstood all experimental tests so far. The question concerns whether energy is conserved in the classical theory of general relativity with or without cosmological constant as given be Einstein nearly 100 years ago. This is a purely mathematical question.
It has indeed been said that too much ink has been spilt on this subject already, but the fact is that the wrong conclusions are still drawn. It does not matter how well-respected the cosmologists are or how many people have read their textbooks, the fact is that they are wrong. Energy is conserved in general relativity. There are no ifs or buts. The mathematics is clear and the errors in the thinking of those who think it is not conserved can also be traced. It is time to put the record straight.
Not all the cosmologists are so bold as to state directly that energy is not conserved, but some are. Here are some examples of the kind of things they do say:
“there is not a general global energy conservation law in general relativity theory” – Phillip Peebles in Principles of Physical Cosmology
“In special cases, yes. In general — it depends on what you mean by ‘energy’, and what you mean by ‘conserved’.” – John Baez and Michael Weiss in the Physics FAQ
“The energy conservation law is an identity in general relativity” – Felix Klein
“the local conservation laws, integrated over a closed space [..] produce nothing of interest, only the trivial identity 0 = 0“, John Wheeler in Geometrodynamics.
“a local energy density is well-defined in GR only for spacetimes that
admit a timelike Killing vector” – Steve Carlip on sci.physics.research
“Energy is Not Conserved” – Sean Carroll on Cosmic Variance
“Energy is not conserved in cosmology. As is always the case
with confusing stuff in cosmology, this is covered well in Edward
Harrison’s COSMOLOGY textbook.” – Phillip Helbig on usenet
Many other statements have been made to the effect that conservation of energy in general relativity is only approximate, quasi-local, trivial, non-covariant, ambiguous or only valid in special cases. They are all wrong. Energy is conserved in general relativity.
Discussions about conservation of energy in cosmology often arise when people write about redshift of the cosmic background radiation. Individual photons are not created or destroyed as they travel across space, so if they are redshifted they are losing energy. Where does it go? The answer is that it goes into the background gravitational field. The presence of the CMB affects slightly the rate at which the universe is expanding so there should be an energy term for the expansion rate. This is what happens for particles moving in other types of background such as electric and magnetic fields so it should work for gravity too.
Since the discovery of “dark energy” the level of confusion has become worse. People visualise dark energy as a constant density of energy that pervades space. If space expands then there should be more of it, so where does the energy come from? The anser is the same as for radiation. The dark energy, or cosmological constant as it used to be known, affects the expansion rate of the universe. The gravitational component of energy has a contribution from this exapnsion and the rate changes to conteract the amount of dark energy being added so that total energy is constant.
To make the case for energy conservation in general relativity sound, we need a valid mathematical formula for it in terms of the gravitational field (the metric tensor) and the matter fields. This problem was initially tackled as soon as general relativity was proposed by Einstein. The mathematician Emmy Noether was asked to look at the problem and she solved it eloquently by stating her theorem relating symmetry to conservation laws. Although the theorem is well-known to physicists it is not often appreciated that it was formulated to tackle this specific problem.
Noether’s theorem tells us that if a physical law derived from an action principle is invariant under time translations, then it has an energy conservation law. In fact the theorem provides a formula to derive an energy current whose divergence is zero. Such a current can always be integrated over a region space to provide a total energy whose rate of change is equal to the flux of the current from the surface bounding the region. This is exactly what we mean by conservation of energy.
For example if we take Maxwell’s equations in special relativity such invariance applies and we can derive a formula for the energy current. Of course in special relativity time is not absolute and there are different concepts of time dependent on an observers velocity. This means that we actually get an infinite number of energy conservation laws, one for each possible velocity. Conveniently this boils down to a single energy-momentum tensor that gives the energy current for any choice of the time coordinate in a reference frame. The same tensor can be used to provide momentum and angular momentum conservation laws. It is all very intuitive and nice!
What about the case of general relativity? Invariance under time translation still holds and general relativity is derived from the Hilbert action principle so Noether’s theorem can be applied to the gravitational field along with any matter fields to give a total conserved energy current, but there is a technical hitch. The Hilbert action includes second derivatives of the metric tensor as well as the first, and Noether’s theorem only deals with the case where there are first derivatives. The usual solution applied in the early days of relativity was to modify the Hilbert action in a way that removed the terms containing the second derivatives without affecting the dynamics of the Einstein equations derived from it. Noether’s theorem could then be applied. The only snag was that the procedure could not be made gauge invariant so the energy-momentum quantities derived did not form a covariant tensor as they did for special relativity. Sometimes they are called the energy-momentum pseudotensor. The solution works but some people just don’t like it. They complain that the pseudo-tensor can be made zero at any point in spacetime for example. It is not really a problem but people did not expect it so they complain about it.
The source of the problem (which is not really a problem) can be traced to the fact that the spacetime symmetry group in general relativity is bigger than it is in special relativity. Instead of just a choice of time coordinate for each velocity of an inertial reference frame, you have one for any choice of motion whether inertial or not. This gives a much larger set of conservation laws and with the extra choice you can always make the energy and momentum of the field zero at any given event in space and time.
The choice of time coordinate can be associated with a contravariant vector field that generates the time translation. We should expect the formula for our energy from Noether’s theorem to have a dependency on this field. Trying to express it as a tensor is not really appropriate and that is what causes the confusion.
Modern Covariant Solution
It turns out that there is a more general version of Noether’s theorem that can be used even when the action includes terms with second derivatives. This provides a more modern approach to the derivation of an energy current that has a dependency on the time translation vector field. Since it does not require any manipulations of the action the result is a covariant local expression. I am avoiding formulae here but you can look up, the answer in arXiv:gr-qc/9701028. This paper does not take into account the cosmological constant but that is not a problem. The conditions for Noether’s theorem still apply with the cosmological constant term in place and the derivation of this more general case is a straightforward exercise left for the reader.
So the outcome is that there is a local covariant expression for the energy current in general relativity after all. This is exactly the thing that many cosmologists claim does not exist, but it does, and energy conservation holds perfectly with no caveats.
To finish off let’s take a look at some of the specific things that cosmologists and relativists have been saying and debunk them one by one in the light of the solution we now understand.
Energy Conservation in general relativity is approximate NOT
It is sometimes claimed that energy conservation in general relativity is only approximate. On further examination of what is meant we find that the person who thinks this only knows of (or only accepts as valid) the extension of the covariant energy-momentum tensor from special relativity to the general theory. This tensor includes only contributions from the matter fields and not the gravitational field. Its covariant divergence is zero just as required for a conserved current vector, but unfortunately it is a symmetric tensor and you can not integrate a divergenceless symmetric tensor to get a conserved quantity in curved spacetime. That only works for vectors and anti-symmetric tensors. Because of this people say that the conservation is only approximate.
It should be clear now where the error in this argument lies. The energy-momentum tensor does not include contributions from the gravitational field and energy conservation cannot be formulated without it. Of course your energy conservation law is only going to be approximate if you neglect one of the fields that has energy.
The correction is to include the gravitational field either by using the pseudotensor method or by using the more modern derivation of the current as a function of the time translation vector field.
Energy conservation only works in special cases in general relativity NOT
The cause of this false claim is once again the use of the energy-momentum tensor. For some special cases the gravitational field has a killing vector that indicates that it is static in some specific reference frame. If you contract this killing vector with the energy-momentum tensor you get an expression for an energy current that is conserved. That’s very nice but nothing unusual. It is normal that you can get a conserved energy in a fixed background field which is static. The same happens for other fields such as the electromagnetic field. If the energy in the background field in not changing then the energy in the rest of the system can be conserved too without adding the energy from the background field.
Just because energy conservation is a bit simpler in special cases does not mean that it does not work in more general cases, which it does of course.
Another special case often cited is an asymptotically flat spacetime. You can work out the total energy and momentum and it takes the form of a familiar energy-momentum four vector in the asymptotic limit. Very nice, but again just a special case while the general case also works perfectly well.
Energy conservation in general relativity is trivial NOT
This particular version of the energy conservation “problem” in general relativity goes back to the early days when Noether, Einstein, Klein, Hilbert and others were investigating it. Klein claimed that the conservation law that Noether’s theorem gave was an identity, so there was no real physical content to the law. This claim has been echoed many times since, for example when Wheeler claimed that the law reduces to the trivial result 0 = 0 for closed spacetimes.
In addition to her well-known theorem, Noether had a second theorem that elaborated on what happens when there is a local gauge symmetry rather than just a global symmetry. In this case you can derive Bianchi type identities that provide formulae for currents that are conserved kinematically, even if the equations of motion are not. You can say that such a current is trivially conserved. The formula for the energy current derived from Noether’s theorem is not such a quantity, but it is the sum of two parts one of which is trivially conserved and the other of which is always zero when the field equations apply. For some people this is enough to make the claim that energy conservation is trivial in general relativity.
That this makes no sense is easily seen by considering any other gauge field and its conserved charges. For example, electromagnetism is a gauge theory that conserves electric charge. Because of Noethers second theorem the expression for the electric current can be written as the sum of a term depending only on the electromagnetic potential whose divergence is explicitly zero, plus a term which is obviously zero when Maxwell’s equations hold. This is exactly analogous to the case of energy conservation in general relativity. Nobody claims that this makes charge conservation trivial in the classical theory of electromagnetism so they should not make such a claim for energy conservation in general relativity.
I have debunked some of the major claims about energy conservation in general relativity that people use to justify the idea that there is something wrong with it. There are others but they are all just as shallow and easy to deal with. If you come across anyone making such claims, please just refer them to here and hopefully we can put an end to this nonsense.