But they are right. ;-) Let me explain.

**What is energy?**

In different physical situations, we use different formulae for "energy" but we always want the "same convertible currency" that may be summarized as follows:

Energy is the scalar quantity that is conserved as a result of the time-translational invariance of the laws of physics.This deep relationship between symmetries of the laws of Nature - in this case the invariance under the translations in time - and the conservation laws was discovered by Fraulein Emmy Amalie Noether - who is worshiped by young mathematicians such as the Gentleman on the picture.

Note that the quantity that is conserved because of spatially translational invariance is known as the momentum; it combines with the energy into a D-dimensional vector in relativity. The quantity conserved because of the rotational invariance is nothing else than the angular momentum. In some cases, only the rotations around some axes may be respected; correspondingly, only some components of the angular momentum are conserved.

The electric charge is linked to U(1) transformations in field theory (which change the complex phases of charged fields) and we may also mention many other laws, including discrete symmetries. Parity - a quantum number whose eigenvalue must be either +1 or -1 - is multiplicatively conserved in theories that are left-right symmetric, and so on.

**Quantum mechanics and Noether's duality**

Quantum mechanics seems much more complex than classical physics but it actually offers us a much more crisp and transparent explanation of Noether's relationship. The evolution in time is generated by the Hamiltonian which contains all the "dynamical" information about the physical system.

The evolution respects a symmetry if the Hamiltonian commutes with the generator of this symmetry; in that case, it doesn't matter whether you apply the symmetry transformation before or after the time evolution. But if the generator of the symmetry commutes with the Hamiltonian, it is conserved - because Heisenberg's equations link its time derivative to its commutator with the Hamiltonian.

The generators of symmetries are nothing else than the observables that are conserved if these symmetries are respected. In particular, the Hamiltonian generates the translations in time and it is conserved if the physical laws don't change in time i.e. if they're time-translationally symmetric. ;-) Also, the translations are generated by the momentum and the rotations are generated by the angular momentum.

**Energy in mechanics and field theory**

It's useful to recall some basic formulae for energy. In mechanics, the energy is typically the sum of the kinetic energy - "mv^2/2 = p^2/2m" - and potential energy "U(x,y,z...)". Their sum is conserved - it is the total energy - but the terms are not preserved separately. Energy may be changed from one form to another. The energy can also be used as the defining formula that tells us everything about the time evolution of any measurable quantity; in that case, we usually call it the "Hamiltonian" instead of "energy" because Hamilton found the Al Gore Rhythm to deduce the equations of motion for any quantity from the formula for the energy, the so-called Hamilton's equations.

Alternatively, one may use the action - the integral of the Lagrangian; you may guess that Lagrange has discovered Lagrange's equations. In simple cases, the Lagrangian only differs from the Hamiltonian by a minus sign in front of the "potential energy" terms.

In field theory, "v^2 = (dx/dt)^2" in the formula for the kinetic energy must be replaced by the sum of squared derivatives of some fields - now "phi(x,y,z,t)" rather than "x(t)" - with respect to all spacetime coordinates "x,y,z,t..." rather than just "t". And there are degrees of freedom at each point of space - so the Lagrangian is not just one expression but rather an integral of a similar expression over the space.

But the basic logic remains unchanged.

Energy may be transformed from one form to another. All phenomena in space follow the dynamical laws that can actually be derived from the formula for the total energy - the Hamiltonian.

*Oliver Toussaint, Paul de Senneville: Balad for Adeline, played by Richard Clayderman*

**Energy in special relativity**

I have already implicitly mentioned that in special relativity, space and time are unified into spacetime. Consequently, translations in space and translations in time are unified into translations in spacetime. Their generators, the momentum and the energy, are also merged into a single vector in spacetime - the energy-momentum vector - that transforms nicely under the Lorentz transformations that mix the space and time.

There's nothing really complicated here. Noether's logic still applies. The formulae are similar to the non-relativistic ones: you must just realize that the total kinetic energy of an object of mass "M" at velocity "v" is no longer "Mv^2/2". Instead, it is

ENote that there's a new huge term, "E=Mc^2", the most famous expression due to Einstein. It's the latent energy hiding in the rest mass "M". However, because this term above is constant, it doesn't affect the (Hamilton or Heisenberg) equations of motion. In Schrödinger's approach to quantum mechanics, the constant term only adds an extra superfast rotation of the phase to all parts of the wave functions - but the overall phase can't be measured (the same is true for a phase that you may produce in Feynman's path-integral approach to quantum mechanics)._{kinetic}= Mc^{2}/ sqrt(1-v^{2}/c^{2}) =

= Mc^{2}+ Mv^{2}/2 +

+ 3/8 v^{4}/c^{4}+ ...

The first subleading term is the well-known non-relativistic kinetic energy but there are also corrections that become non-negligible when the velocity "v" approaches "c"; I have only showed you the first one which is enough if "v" is much smaller than "c" but you still want to be by "one class" more accurate than your non-relativistic friends.

In special relativistic field theories, the energy is still exactly conserved - because the physical laws are still exactly time-translationally invariant. Moreover, it is conserved locally: the conservation law can be reduced to a continuity equation.

**Energy in general relativity**

But this article should have been about the energy in general relativity. We're exactly in 1/2 of the article and I still haven't answered the main question. Does energy exist in general relativity? Is it nonzero? Is it exactly conserved? Is it approximately conserved? Can it be written as an integral of the energy density over space?

Well, most of these answers are No, at least morally. But let's look at them more carefully. The precise answers will depend on what you mean by energy and what situation you consider.

General relativity allows the space and time to get curved. So it is no longer the case that the objects are moving in a translationally invariant background. Most backgrounds are not translationally invariant. That's a reason why Noether's argument fails in its simplest form.

For example, you may study the evolution of particles and fields - including electromagnetic fields - in the background of an expanding cosmology. I mean the Big Bang cosmology. Because the history of the Big Bang is not invariant under translations in time, Noether's theorem tells you that the energy of the objects will not be conserved in general.

And indeed, you can check that it's not conserved. For example, photons with wavelength "L" and energy "E" become photons with wavelength "K.L" and energy "E/K" if the Universe expands by a factor of "K". It can't be otherwise: note that the number of "peaks" of the wave arranged along the visible Universe can't change - e.g. because of a "Z_K" symmetry - so the wavelength has to grow proportionally. And the photon's energy is inversely proportional to the wavelength.

So the energy of radiation will go down as "1/K".

On the contrary, the energy stored e.g. in the cosmological constant will expand as "K^3". Why? Well, the energy density carried by the cosmological constant is constant during the cosmological evolution - that's why the "cosmological constant" is called a "cosmological constant". ;-) But the volume of space is literally expanding so the total energy is increasing proportionally to the volume.

I deliberately wrote "K^3" in order to appease the people who don't realize that there are extra dimensions of space or those who do realize that there are extra dimensions but who also know that only 3 of them have been expanding in the recent 13.7 billion years. ;-)

So the conventional formulae for the energy of objects propagating upon the background geometry explicitly lead to non-conserved quantities and we can see that they're not conserved. Only the dust (with no pressure) would conserve the energy ("E=mc^2") in an expanding Universe but no physical evolution will guarantee that everything stays in the form of "exact dust". Can we be less hostile? Yes but we will have to be limited in various ways.

**How can we resuscitate some traces of conserved energy in GR**

The first way to undo the damages is to consider a spacetime that resembles that of the special relativity in some key ways. Well, obviously, if you ban or neglect any effects of curvature, which also includes any gravitational waves that could carry the energy away, you're back to special relativity and you can define a conserved energy. However, can you allow the spacetime to get curved and find some conserved energy?

The answer is either "Approximately Yes" or "Exactly Yes given some additional assumptions". What do I mean?

Well, the answer "Approximately Yes" is pretty clear. If the spacetime curvature is small in some proper counting, the violations of the energy conservation law will be small in an analogous way.

The other answer, "Exactly Yes given some additional assumptions", is more interesting. You may consider spacetimes that asymptotically (at infinity, very far from all the matter) converge to a flat Minkowski space (or anti de Sitter space, or another standardized space); note that our expanding Universe doesn't satisfy this condition. However, you may allow this spacetime to get curved arbitrarily in the bulk as long as the curvature agrees with the laws of physics.

Will the energy be conserved in such a physical system?

The answer is Yes because the physical laws for everything that exists in such a Universe are translationally invariant (in time and probably in space, too). So Noether's theorem implies that there must exist a conserved energy (and probably momentum). It's called the ADM energy. You can measure it by the coefficient of the "1/r" gravitational potential (extracted from the "g_{00}" component of the metric tensor) at infinity.

However, it's important to realize that the translational invariance was only guaranteed at infinity where the spacetime was "rigid", too big (infinite) to be affected by the finite matter somewhere in the middle of the space (the asymptotic region at infinity is surely too big if there are at least 3 spatial dimensions). Consequently, we find the conserved quantity and nothing else. In particular, the ADM energy can't be canonically written as an integral of an energy density.

There exist many candidate formulae how this total ADM energy may be divided among individual places in space and time but none of them is better than the other formulae - and none of them transforms naturally under the general coordinate transformations that general relativity accepts as its group of symmetries.

**Vanishing energy in general relativity**

In special relativistic field theories, there exists a simple general way to derive the energy density or, more accurately, the whole stress-energy tensor: it can be calculated as the derivative of the Lagrangian density with respect to the metric tensor.

It's pretty much the same quantity that you can reconstruct via Noether's procedure. For ordinary field theories, it reduces to the "kinetic plus potential" terms and so on. It is conserved.

Can't you use the same trick to derive the energy density in general relativity, too? Well, you can. But you get zero. The variation of the action with respect to the metric tensor gives you something that must vanish because the metric tensor is a dynamical degree of freedom in general relativity and the action must be stationary with respect to all the dynamical degrees of freedom - which now includes the metric tensor, too!

The stress-energy tensor that you deduce as this derivative of the Lagrangian with respect to the metric tensor is the Einstein tensor minus matter's stress-energy tensor. The latter is the conventional part associated with non-gravitational fields (and particles) that you know from special relativity. However, the former term is related to curvature and it is a kind of the energy of the gravitational field.

The punch line is that the sum/difference that you deduce as the full derivative of the Lagrangian vanishes! It vanishes because its vanishing is equivalent to Einstein's equations of general relativity! When Phil Gibbs talks about a conserved energy, claiming that the photon's energy was converted to some kind of gravitational energy, it's very clear what his formula for the gravitational energy has to be so that the total energy is conserved. Its density must be given by the Einstein's tensor so that the total energy is always exactly zero!

So if you define the energy in this way, it's just zero. Now, is it trivial or non-trivial?

It depends what you mean by "trivial". ;-) The fact that such a combination has to vanish is kind of deep. After all, the Wheeler-DeWitt equation of quantum gravity may be written as "H=0": the total energy vanishes, too. The vanishing of the total energy or the energy density is a profound property of general relativity that I will discuss later.

However, once you know that general relativity is sophisticated, the insight is *trivial* and unhelpful for any task in which you want to solve or analyze actual situations. The vanishing of the energy will tell you nothing about the situations you care about. If a conserved quantity is always zero, it can never tell you that something "cannot happen". The constraint that "zero has to remain zero" is not terribly constraining. That's why it really tells you nothing nontrivial about what "can happen", either. :-)

Clearly, if you don't have any asymptotic region of your spacetime, everything can happen. Any Universe can arise from anything or nothing. Inflation produced our whole Universe from a tiny seed; it's the ultimate free lunch cooked by Alan Guth. So if your energy is conserved, the energy of anything must be equal to the energy of anything else, for example to the energy of nothing, so it must be zero. :-)

**These statements don't mean that the Wheeler-DeWitt equation is vacuous.**

Why? Well, I have just tried to convince you that because the "total" energy density is zero, you cannot use the energy conservation as a useful tool to find answers to your physical questions about particular situations. You have to solve the equations of motion in different ways.

However, the Wheeler-DeWitt equation is potentially deep because in some cases, you don't have any "different ways". The equation becomes the key dynamical principle and there's nothing else to solve. Well, in some sense, the Wheeler-DeWitt equation is nothing else than Einstein's equations that you encountered elsewhere - potentially simplified and reduced in a minisuperspace formulation.

But if something like the Wheeler-DeWitt equation is the only equation that defines what can happen in a quantum gravity system, your task actually becomes harder, not easier. You must artificially reparameterize your physical states according to a new observable (e.g. the total volume of the Universe at some "moment") that will play the role of your time (recall that there's no preferred choice of coordinates in general relativity) - and with this new "gauge-fixed" choice of your time, you will find out that the Wheeler-DeWitt "H=0" equation becomes non-trivial.

But that would take us too far. The main lesson here is that general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime. That's why the corresponding energy conservation law justified by Noether's argument either fails, or becomes approximate, or becomes vacuous, or survives exclusively in spacetimes that preserve their "special relativistic" structure at infinity. At any rate, the status of energy conservation changes when you switch from special relativity to general relativity.

And that's the memo.

My knowledge of physics is sub-elementary but...

ReplyDeleteYou said (with M = rest mass),

E_kinetic = Mc² / sqrt(1-v²/c²)

Shouldn't that be

E_total = Mc² / sqrt(1-v²/c²)

and

E_kinetic = (Mc² / sqrt(1-v²/c²) - Mc²

I once defined Joementum as proportional to (E_total/c)-p, but I was told that it's not part of the Lorenz-covariant 4-momentum and that it's not vector-additive. My lame response was that linear energy isn't scalar-additive. Was I wrong about that too? (They even got on my case about the phrase "linear energy" but I swear I saw in an old Taylor & Wheeler text book).

Dear ForNow,

ReplyDeleteyes, you're right. In normal texts, the kinetic energy is a term for the difference "M(total)c^2 - M0 c^2" only.

I just hate this convention because according to relativity, this separation of the energy into pieces is completely artificial.

Energy is additive which is the technical principle behind locality - the independent existence of weakly interacting regions or subsystems.

Best wishes

Lubos

To be fair the WdW equation is just the statement that time reparametrization (i.e., the gauge transformation generated by H) is trivial in a diffeomorphically invariant theory.

ReplyDeleteI've read Gibbs' claim and from what I gather it seems he's confusing local conservation with global conservation in GR. The 1st law of thermodynamics (and the 1st law of Black Holes thermodynamics, incidentally) only works with

stationarysystems (be they a box of gas or a space-time). In either case the observer has a (aproximate or not) time-like Killing vector field to relate her energy to. Quasi-static processes in GR have been studied thoroughly during the 60's and the analogue of Poynting's theorem needs an asyntotically static spacetime to work.Dear Lubos:

ReplyDeleteWhat do you think about this paper?:

http://arxiv.org/abs/1007.1750

Best wishes

Dams

Hi Dams, I consider all papers trying to build on variable speed of light (click for more!) to be symptoms of complete misunderstanding of modern physics. Sorry I won't read beyond 1/2 of the abstract because that would be a waste of time which would be bad especially because I am still kind of busy.

ReplyDeleteThe main lesson here

ReplyDeleteis that a lesson?

well....in your personal cosmology?

general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime

according to relativity....

this separation of the energy into pieces is completely artificial.

Energy is additive which is the technical principle behind locality - the independent existence of weakly interacting regions or subsystems.

Love your site Lubos! I would like to see you write a book ala Randall, Suskind, Kaku, for purchase. Any thoughts?

ReplyDeleteLubos I'm just wondering something. I understand that energy conservation can be violated if time translational invariance doesnt hold. My only question is where does the energy either go or come in from? I mean the math might work out fine but im just wondering what physical mechanism (I assume all energy has some sort of physical existence or essence) leads to the disappearance or introduction of energy to make conservation untrue? Do theorists have any ideas, maybe within string theory or loop quantum gravity? And if so, could there still be a global conservation law in the entire multiverse or?

ReplyDeleteDear Pete, well, when something isn't conserved, then it makes no sense to assume that it is conserved. A thing's not being conserved is the default, generic situation. When it's not conserved, it can be created and/or destroyed by physical processes.

ReplyDeleteYour question is exactly like the question that Feynman's father asked to his son: were the photons emitted by the atoms in the atom ahead of time? Click the link for a funny answer involving an analogy with a "word bag'. There's no word bag!

When something like energy isn't conserved, it's normal for any physical process to create it or destroy it, much like one may create words by a mouth, entropy, or anything else. There's nothing extra to be explained here.