Is energy really conserved?

In high school I was taught energy was conserved. Then I learned that nuclear reactions allow energy to be converted into mass. Then I also heard that apparently energy can spontaneously appear in quantum mechanics. So, are there any other caveats with the conservation of energy?

• It's unfortunate that the accepted and highly upvoted answer is completely wrong re general relativity. – Ben Crowell Apr 14 '18 at 17:51
• Quantum mechanics is not an exception unless you misinterpret the uncertainty principle. Special relativity is not an exception if you generalize conservation of energy to conservation of mass-energy. The only actual exception is general relativity, although the explanation of this in Daniel's answer is completely wrong. See physics.stackexchange.com/a/2856/4552 for a correct explanation. – Ben Crowell Apr 14 '18 at 17:54

The topic of "Energy Conservation" really depends on the particular "theory", paradigm, that you're considering — and it can vary quite a lot.

A good hammer to use to hit this nail is Noether's Theorem: see, e.g., how it's applied in Classical Mechanics.

The same principle can be applied to all other theories in Physics, from Thermodynamics and Statistical Mechanics all the way up to General Relativity and Quantum Field Theory (and Gauge Theories).

Thus, the lesson to learn is that Energy is only conserved if there's translational time symmetry in the problem.

Which brings us to General Relativity: in several interesting cases in GR, it's simply impossible to properly define a "time" direction! Technically speaking, this would imply a certain global property (called "global hyperbolicity") which not all 4-dimensional spacetimes have. So, in general, Energy is not conserved in GR.

As for quantum effects, Energy is conserved in Quantum Field Theory (which is a superset of Quantum Mechanics, so to speak): although it's true that there can be fluctuations, these are bounded by the "uncertainty principle", and do not affect the application of Noether's Theorem in QFT.

So, the bottom line is that, even though energy is not conserved always, we can always understand what this non-conservation mean via Noether's Theorem. ;-)

• More on (lack of) energy conservation in GR: blogs.discovermagazine.com/cosmicvariance/2010/02/22/… – coneslayer Nov 12 '10 at 16:43
• Off topic: do you have any source on the spelling of "Nöther"? It's not even used in her own works. – The Vee Mar 28 '18 at 10:59
• This answer is full of incorrect material. Noether's theorem does not produce anything useful when you try to apply it to GR, and the lack of a conserved measure of mass-energy in GR has nothing to do with global hyperbolicity or lack thereof. For example, mass-energy is not conserved in the standard cosmological spacetimes, even though they are globally hyperbolic. – Ben Crowell Apr 14 '18 at 17:48
• It's also not true that energy fluctuates in quantum mechanics. Energy is strictly conserved. – Ben Crowell Apr 14 '18 at 17:48
• Noether's theorem relates energy conservation to four translation invariance. If energy is not conserved, you are simply interacting with an external system. – my2cts Jul 23 at 17:08

Then I learned that nuclear reactions allow energy to be converted into mass.

That would be the opposite and in any case, mass is energy (and energy is mass), so converting one into the other does conserve energy.

Then I also heard that apparently energy can spontaneously appear in quantum mechanics.

For a very short time, given by Heisenberg uncertainty principle. And that's not a violation of the conservation of energy.

So, are there any other caveats with the conservation of energy?

Why "other" ? There's not any problem with the conservation of energy.

• @Cadric: A caveat doesn't necessarily mean an exception. It can include qualifications as well – Casebash Nov 6 '10 at 10:44
• @Casebash: OK I should really check in a dictionary before answering. – Cedric H. Nov 6 '10 at 11:03
• If you are using Firefox, try Wikilook – Casebash Nov 6 '10 at 11:07

Energy is always conserved without any caveat.

With the advent of special relativity, mass and energy are considered equivalent. In other words, they are represented by a vectorial quantity called energy-momentum vector. Before relativity there were separate laws which have been unified. It is a very fundamental law that is connected to some basic empirical properties of the universe, like the fact that the laws of physics do not change over time.

Energy cannot spontaneously appear in quantum mechanics -- however it cannot be precisely measured and this allows for energy fluctuations. The important difference is that although the total amount of energy can change, this is for very brief amount of time, after which the original quantity is restored. So the energy fluctuation can be considered virtual. You don't get energy out of nothing and energy is still conserved.

• neither. I mean that energy can increase or decrease locally for an amount of time which is inversely proportional to the fluctuation size, and that any fluctuation must disappear within that time frame. The overall balance of energy is constant. – Sklivvz Nov 6 '10 at 10:56
• en.wikipedia.org/wiki/… – Sklivvz Nov 6 '10 at 10:57
• they can have very real consequences as long as they don't violate the conservation of energy, for example, Hawking radiation (en.wikipedia.org/wiki/Hawking_radiation) – Sklivvz Nov 6 '10 at 11:18
• Or for example, that's the creation of virtual particle as mediators that explain the interactions between particles. – Cedric H. Nov 6 '10 at 11:22
• General relativity is definitely physics. – j.c. Nov 12 '10 at 21:50

Usually, but not always, Energy is a conserved quantity as the other answers have explained.

But the following clarification is important:

In the BB framework where space expands, or in the dual 'shrinking matter' (almost dual) (comoving framework), the ratio of matter/space is not invariant and energy is not conserved, i.e. Nöether's Theorem does not apply. It is well known that photons lose energy as they propagate. I cannot find an argument to explain why particles should not also lose energy (as they are matter waves).

The other relevant point is that energy can be destroyed, canceled, annihilated, as proved by experiment described here: real-live-antilaser, paper, and here a tentative discussion.

Another example: what is the energy radiated by two dipoles centered in the same frequency and in phase opposition? It is zero. The same happens with two photons in similar conditions.

It is well known that the cosmological redshift of light is usually interpreted(*) as a decrease in energy because the photons wavelength is increasing with time.

The equations for the interference of parallel polarized light are: see Kostya answer here and substituting Delta with Pi:

$\vec{E}=iE_{0}(cos(wt)+cos(wt+\pi))=0$ ; $I\vec{=|\vec{E|}^{2}}=0$.

Light cancels when the instantaneous sum of the E and B vector field components equals 0, which is a fact long known (since Maxwell ?). (See superposition principle or interference.)

Another two situations that should make us think about our energy conservation assumptions:
accelerated charges radiate (discussion at matpages)
moving bodies in a gravitational field radiate gravitational waves (see MotionMountain free e-book, ch 18-Motion in General Relativity)
(*) I do not share the usual interpretation, but that one is the official.

I cannot remember of any other situation where energy is not conserved. I'm not including possible Dark Energy issues (see CosmicVariance)

Those four exceptional cases should make us think about our energy conservation concepts.

• -1 for simply being completely incorrect – Colin K Mar 4 '11 at 18:53
• Thanks Colin. I understand your valuable arguments. You made clear that I've to substantiate more my answer. I will do that. – Helder Velez Mar 5 '11 at 12:02
• When "moving bodies" radiate energy, their kinetic energy decreases. – DWin Oct 11 '14 at 1:46

protected by Qmechanic♦Apr 14 '18 at 18:09

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