Energy Is Conserved (the maths)

Judging by the comments on the previous article I have not yet succeeded in convincing anyone that energy is conserved. Luboš Motl has posted a response contradicting my viewpoint and agreeing with an older blog post by Sean Carroll. Fortunately I have an advantage, I’ve done the maths and the outcome is clear and unambiguous. Since my detractors are people who understand equations I should have no trouble convincing them if I take a more technical approach. So, no more analogies, let’s start with Einstein’s equation.

 G_{ab} + \Lambda g_{ab} = \kappa T_{ab}

G_{ab} is the Einstein tensor given by

G_{ab}  = R_{ab} – \frac{1}{2} R g_{ab}

R_{ab} is the Ricci curvature tensor, g_{ab} is the metric tensor, \Lambda is the cosmological constant, \kappa is a gravitational coupling constant and T_{ab} is the energy-stress tensor for matter.

A time translation is generated by a time-like vector field \xi^a with a small parameter \epsilon 

\Delta x^a = \epsilon \xi^a

All the tensor quantities have corresponding transformation rules and the field equations are covariant under the transformation. Using Noether’s theorem a formula for the corresponding conserved energy current can be derived. The details of this calculation are quite lengthy and can be found in  arXiv:gr-qc/9701028 (without the cosmological constant which is easy to add in.) The result obtained is

J^a = J_M^a + J_{DE}^a + J_G^a 

Where J_M^a is the energy current from the matter contribution given by

J_M^a = \xi^b T_{cb} g^{ca}

J_{DE}^a is the dark energy component given by

J_{DE}^a = \frac{-1}{\kappa}\xi^a \Lambda

and J_{G}^a is the gravitational contribution given by

J_G^a = \frac{-1}{\kappa} {\xi}^b   G_{cb} g^{ca} + K^a

K^a is the Komar superpotential given by

K^a = \frac{1}{2\kappa} (\xi^b ;^a - \xi^a ;^b) ;_b

Given the Einstein field equations we can eliminate the matter term, the dark energy term and the first part of the gravitational term to leave just the Komar superpotential

J^a = K^a

Then it is easy to see that

J^a ;_a = K^a ;_a = 0

Since the current is divergenceless it defines a conserved energy. Some people claim that this result is trivial. Clearly it is not because it requires the gravitational field equations to prove conservation of energy. You should not make the mistake of defining energy as just the Komar superpotential, even though it is equal to that when the dynamics are taken into account. Energy must be defined as a sum over contributions from each field including the gravitational field, and dark energy.

The energy contribution from matter J_M^a = \xi^b T_{cb} g^{ca}  is a sum of contributions from each type of matter field. It includes all the non-gravitational forms of energy including heat, electrical energy, rest mass equivalent of energy, radiation etc. This form of the law of energy conservation in general relativity tells us that the energy from all these contributions plus the gravitational contribution and the dark energy contribution is conserved. In other words energy can be transformed from one form to another but is never created or destroyed. There is nothing approximate, trivial or ambiguous about this result. It is energy conservation in the same old form that we have always known it but with the contribution from gravity included.

19 Responses to Energy Is Conserved (the maths)

  1. Luboš Motl says:

    Dear Phil, you’re surely joking. The task to find a conserved energy is not to find “any” divergenceless vector that is mainly a function of some arbitrary new vector field “xi”.

    Of course, there exist many divergenceless vector fields in spacetime and one can engineer many of them. They have nothing to do with “energy”. They’re not functions of (just) the configuration and its measurable quantities.

    They’re functions of a newly added variable field “xi” that has nothing to do with physics. Note that the definition of the “Komar superpotential” only depends on “xi” and the metric. It has nothing to do with “energy” as we know it.

    Moreover, when you consider the Minkowski spacetime and xi is simply (1,0,0,0) – the normal time translations – your energy density vanishes automatically. That’s surely no good for an “energy density”.

    Also, you’re wrong that the vanishing of the divergence of the Komar potential requires Einstein’s equations. Just try to calculate it yourself. You don’t need any of them. Clearly, Einstein’s equations link the curvature to the matter’s stress-energy tensor, but because the latter is not present in the Komar superpotential, the former can’t be, either.

    At any rate, one can prove the vanishing of div(Komar) without knowing any dynamics whatsoever.

    If you want to argue that you can also add (or you have added) some vanishing terms to the Komar superpotential that are “nontrivially vanishing”, it’s like claiming that the height of a skyscraper is very deep physics because I can define it as

    H = height(skyscraper) + anomalies in SO(32) type I string theory

    where the second term nontrivially cancels by the insights of Green and Schwarz that sparked the first superstring revolution in 1984. Would you agree that such a statement would be completely idiotic? What’s the difference between this idiotic statement and your statements? You can’t make a thing being nontrivial by randomly adding another, unrelated term that is zero because of some “deep” reasons.

    Even if you could, it’s not really as deep as you think.

    You just seem completely out of touch here. What you’re writing has nothing to do with physics. It’s a meaningless formal masturbation with no results and no relationships to the concept of “energy” of any type that a physicist could recognize. Why don’t you just agree with the obvious fact that one can’t define any non-vanishing conserved notion of energy in generic geometries described by general relativity?

  2. Philip Gibbs says:

    Lubos, surely it is you who is joking here? :)

    I did not just pull out a random expression and show it was a conserved current. It is exactly what you get when you apply Noether’s theorem directly to the Hilbert action. The details of that calculation are in the paper cited. I would quote the whole thing here but it is quite long and it was hard enough to get wordpress’s buggy blogtex to work with just these equations.

    I did not say that you need the field equations to show that the Komar superpotential has zero divergence. I said you need it to show that the original expression for the current has zero divergence. Once you have used the field equations to reduce it to the Komar superpotential it follows that it has zero divergence because this is obviously the case for the superpotential. I have tried to explain why you cannot just ignore everything except the Komar superpotential because you need to regard the energy as a sum over contributions from all sources.

    In the case of Minkowski spacetime there can be no matter since matter makes curvature, so of course the energy is zero in this case. I would not expect a good example using Minkowski space in the context of general relativity, it could never be anything but trivial.

    The field \xi does not have to be physical. Energy is a relative concept and the choice of \xi is what it is relative to.

    From the language in your last paragraph I’m thinking we may be reaching the point where it would be better for us to agree to disagree ;)

  3. Ulla says:

    A remarcable paper – congratulations.

    “Modern attempts to improve on it can only really be said to be successful in specific cases such as asymptotically flat space-times.” … if I understood Lubos right this is a 2-D world without gravity. This is special relativity and quantum world, maybe with antigravity (Pauli exclusion).

    Your calculation was without cosmological constant. Also that constant is changing along with the other changings (Lubos).

    “we can eliminate the matter term, the dark energy term and the first part of the gravitational term to leave just the Komar superpotential”
    it requires the gravitational field equations to prove conservation of energy. It is energy conservation in the same old form that we have always known it but with the contribution from gravity included.”

    “the radiative (photonic) energy is transferred to the gravitational field which has an increasing (negative) total energy (topology is that of a three-sphere.). ” – This energy goes to gravitation in interaction, dissipation, that is entropy. The changing from 2-D to 3-D. Verlinde is right.

    “energymomentum transferred by gravitational waves and the contribution of gravitational self energy to mass” – so gravity is split?

    “the energy is only zero dynamically, not kinematically. I.e. it is only zero when the gravitational field equations are satisfied.” But in Big Bang there was no energy? So energy must have been created? But it is an oscillation? This oscillation happens no more, because of the gravity (dissipation, 3-D)?
    This sounds not convincing in my ears.

    An emerging gravity is not possible in your theory?

  4. Ulla says:

    Verlindes talk
    much thunder in the air, I could not see it myself now.

  5. Philip Gibbs says:

    Ulla, interesting questions and I need to clarify a few things.

    Lubos and I talked about asymptoticlly flat spacetimes as if its obvious what that means, but it may not be to everyone. We are talking about hypothetical models in the usual 4D spacetime where spacetime becomes flat and empty in the limit as you go out to spacial infinity. An example would be the usual static or rotating black hole solutions. Also any system of stars in a cluster but isolated in space. In such a system you can define the ADM mass by integrating the Komar su[perpotential over all spacetime. There is an exact integration which means you can write this as an integral over a boundary and then take the boundary to infinity. This is dealt with at length in more advanced relativity texts such as Walds “General Relativity”

    The cosmological constant is usually regarded as constant, but there are theories in which it is allowed to vary as Lubos mentioned as an aside. If it is constant the conservation of energy works in the way I described. If it varies you could still have energy conservation provided it varies according to some generally covariant dynamical law derived from an action principle.

    In a closed cosmology the total energy must always be zero. This means you escape the “problem” that energy needs ot be created from nothing. In an infinite universe the idea of total energy may not make sense so it is a moot point. Another valid response to this question is to say that there is no time before the big bang so it does not really make sense to say enegy is created at that time anyway.

    Your other points about Verlinde’s theory are interesting but beyond what I want to cover here.

    • Ulla says:

      Look at this video.
      The gravity is essential to this question, I think.

      The model is an analogy but not hypothetical at all. When we look at condensed matter physics it is most real. The dimensions scale down.
      Thanks for the expl.

  6. Philip,

    the Komar potential is divergenceless because it is completely analogous to a U(1) current associated with the U(1) gauge field defined by a vector potential now identified as a time like vector field ksi. Therefore its conservation is automatic and poses no conditions on metric. You can take any ksi so that you get anything you want. The infinite-dimensional Lie-algebra of volume preserving flows is in question, which as such might be physically interesting although it is not symmetry algebra.

  7. Philip Gibbs says:

    Matti, I am glad you find the analogies helpful. Some people think that GR is in some way fundamentally different so that these analogies dont apply, but actaully the analogies are valid and help show when reasoning is going wrong.

    It is slightly better to compare with a non-abelian Yang-Mills gauge theory than U(1) because the gauge field is then charged. This is analogous to the gravitational field having its own energy.

    In a U(N) gauge theory you would define the total charge as an integral over terms that include the charged particles and also the gauge fields. In GR the energy is an integral over terms for the matter fields and also the gravitational field, which is the gauge field.

    In U(N) gauge theory you can take the expression for the charge current and apply the field equations to simplify it to an expression involving just the gauge potentials. This expression is then explicity divergence free because it is itself the divergence of an anti-symmetric form. In GR you can also use the field equations to reduce the energy current to an expression involving just the metric. This is the Komar superpotential and it is explicitly divergence free because it is also the divergence of an antisymmetric form.

    In the U(N) gauge theory you can determine the charge inside a boundary by integrating the flux of just the gauge field over the bounday. This is analogous to the way ADM mass-energy is determined by an integral over a boundary based on just the \xi field and the metric.

    So you can see that everything we are doing is exactly analogous to what happens in a non-abelian gauge theory. Most physicists have a good sense of how this works for ordinary Yang-Mills gauge theory so by caomparison they should understand what i am trying to say about energy conservation in GR.

    You also say that you can choose any \xi and get any energy you want. That is like saying that is specioal relativity you can choose you reference frame and get any energy you want. It is true but it just shows that energy is a relative quantity. If you are worried about the dimension of the space of choices being infinite, just compare again with the U(1) or U(N) gauge group where the choice of gauge is also infinite dimensional. It does not constitute a theoretical problem, just a practical inconvenience because you have to fix the gauge in a clever way to do a calulation.

    I hope you find these analogies useful. Some people think they are specious and just want to discuss the maths of GR directly. I am OK with it either way.

    By the way there is no need to look for volume preserving diffeomorphisms. In the standard cosmological models you would find that impossible because total volume is increasing. It is more useful to look for a \xi field that generates synchronous reference frame, but that is not essential. (Look up “synchronous frame” if you are not familiar with them)

  8. Phil, a comment about how one might get non-conserved but well-defined Poincare charges is asymptotically Minkowskian space-time. One must however give up the dream about Noether charges. What is left is purely Lie-algebraic approach. Entire Poincare algebra is needed asymptotically since otherwise one has no hopes about quantum counterparts of Poincare charges. One can also consider the extension of Poincare algebra to an infinite-dimensional Lie algebra with generators approaching Poincare algebra generators asymptotically. The following argument gives some hopes about this. The formulas of the following text might be difficult to read without latex in spine. At my blog you find an easier-to-read version.

    a) You would start with the identification of vector fields $j^{Ia}$ defining infinitesimal translations, rotations, and boosts in asymptotic regions. In this region they define asymptotic Killing vector fields satisfying


    and the currents

    $(G-lambda g)^{ab}j_b$

    are asymptotically divergenceless because Killing vector field property is true and G and g are divergenceless in covariant sense. If you can continue j^I to entire space-time uniquely, you get well-defined Poincare charges, which are however not conserved.

    b) You must replace Killing vector field property with something weaker and the condition that $j^I$ define flows conserving only four-volume instead of distances is a natural generalization. This implies the condition

    $\nabla\cdot j^I=0 $

    and the infinite-dimensional Lie-algebra of volume preserving vector fields is obtained.

    c) A further condition is needed and this is very natural. You must be able to define global coordinates along the flow lines of the vector fields in questions. This requires

    $j^I = \Psi \nabla \Phi.$

    $\Phi$ defines the coordinate. This kind of vector fields are known as Beltrami fields.

    d) In asymptotic region $\Phi$ would represent either a counterpart of linear $M^4$ coordinate, rotation angle around some space-like axis, or hyperbolic angle around time-like axis. In the asymptotic region $\Psi$ would be constant for translations in the asymptotic region. For the rotations around a given axis the orthogonal it would reduce to the orthogonal distance $\rho$ from that axis. For the Lorentz boosts around given time-like axis to the orthogonal radial distance $r$ from origin in the rest frame defined by that axis.

    Let us look what volume preservation and Beltrami property give.

    a) By simple calculation you obtain

    $\nabla^2 \Phi + \nabla \Psi\cdot \nabla \Phi=0.$

    This is massless field equation with an additional term which might relate to massivation. If one has two solutions with same $\Phi$, one obtains the condition

    (\nabla \Psi_1-\nabla \Psi_2)\cdot \nabla\Phi=0 ,

    which suggests that you must have

    $\nabla \Psi\cdot \nabla \Phi=0 $

    quite generally.

    b) The physical interpretation would be obvious. The solutions describe as special case the modes of massless gauge field. $\Phi$ defines the counterpart of a pulse propagating to local light-like direction and $\Psi$ defines a local polarization vector orthogonal to it. There are also solutions which do not allow this interpretation and corresponds to the functions $\Phi$ and $\Psi$, which are relevant in the recent case.

    c) The solution set is quite large for a given $\Phi$. One can replace $\Phi$ with an arbitrary function of $\Phi$. Same applies to $\Psi$. Linear superposition holds true. One can also form the Lie-brackets for given $\Phi$ and one finds that they vanish. Therefore one has infinite-dimensional Abelian algebra. The natural interpretation is as commuting observables corresponding to polarization direction and propagation direction.

    d) Can one obtain unique continuation of j^A from the asymptotic region to the interior so that unique conserved Poincare charges would exists for asymptotically Minkowskian space-time? The radiative solutions are the problem. If the condition that the radiative part vanishes in the asymptotic region implies that it vanishes, there are no problems.

    Minkowski space serves as a good test bench. In this case functions $\Phi(p\cdot m)$ are simplest propagating pulses: here $p$ is light-like momentum. The condition that they vanish in all directions including the propagation direction in which p\cdot m is constant indeed implies that the multiplying function $\Phi$ vanishes. By choosing $\Psi$ so that it vanishes far away does not allow to achieve the condition. Hence there are hopes that one can define non-conserved Poincare charges in asymptotically flat space-times. One can however imagine presence of
    light-pulses which are emitted and absorbed and thus exists in a finite volume of space-time and they might course problems.

    e) In the case of non-vanishing cosmological constant one would obtain infinite energy and the contribution to the charge would be the charge assignable to the vector field defining time translation. This does not favor cosmological constant.

    As a matter fact, one ends up with the Beltrami fields from a general solution ansatz for a solution of field equations in TGD. The interpretation is that one has the analog of Bohr quantization for solutions of the extremely nonlinear counterpart of Maxwell’s equations coupled to classical gravitation via induced metric. Only the superposition of solutions corresponding to same function $\Phi$ is allowed. They represent pulses of various shapes and different polarizations propagating in a particular local light-like direction.This conforms with what one knows about outcomes of state function reduction. These solutions have 3- or 4-D $CP_2$ projection. So called massless extremals with 2-D $CP_2$ projection have same physical characteristics. Cosmic strings and $CP_2$ vacuum extremals with Euclidian signature of metric describing massless particles are also basic solutions and the topological condensation of $CP_2$ type vacuum extremals to a space-time sheet with Minkowskian signature of the induced metric creates around itself a solution described by $\Psi$ and $\Phi$ meaning that particle picture implies field picture.

    What comes in mind is that these Abelian algebras might be relevant also for the construction of solutions of field equations in General Relativity.

  9. Phil, still two comments.

    The volume preserving flows – in particular Beltrami flows allowing to assign global coordinate to the flow lines, preserve 4-volume. Not 3-volume. Therefore they are well-defined in any metric.

    The analogy between GRT and gauge theories of course exists but the fact that energy momentum tensor field is tensor rather than a collection of vector fields breaks the analogy.

    I agree that conserved Poincare charge exist classically. The question is whether or not GRT allows it. My claim is that this is not the case and that
    sub-manifold gravity is unavoidable if one believes in four-momentum conservation.

  10. Still a little comment. The Abelian algebras defined by Beltrami flows, and perhaps also large algebras generated by them via commutators might be relevant also for the construction of the solutions of field equations in General Relativity. The construction of deformations of an existing metric by adding gravitons is what comes in mind first. The scalars Ψ would define polarizations in a given background metric used to build polarization tensor and the functions Φ could be used to build the analogs of plane waves. One would obtain gravitons and also gauge bosons localized in transversal directions. The algebra formed by the Beltrami flows could thus play a role analogous to Kac-Moody algebras.

    What is interesting that one could always interpret a many-graviton state as a background to which one can add new kind of gravitons! This all is of course speculation but because these algebras allow a concrete interpretation as classical representations of elementary bosons, I would not find it completely surprising if an algebra related directly to the metric would play a fundamental role in the quantization of General Relativity.

  11. Ervin Goldfain says:


    In your previous posting I made the point about the importance of defining the context when talking about conservation laws in GR. It is clear that your derivation holds true, but it does so with certain caveats. Specifically,

    1) If the reference frame is chosen such that no distinction can be made between coordinates (such as an isotropic/axially symmetric frame), then \xi a;b = \xi b;a and the Komar superpotential, along with the resulting overall current vanish. One can say that, in such cases, the conservation law is “trivial”.

    2) The conservation law in its differential form holds only for isolated systems. It cannot hold as such if the system is open. It follows that, if it is unrealistic to model the Universe as an isolated sytem, the conservation law you wrote down cannot hold at the cosmological scale of the Universe.

    3) Likewise, if the Universe is not in thermodynamic equilibrium, the conservation law may or may not apply. This is because the evolution of certain dynamical systems that are out of equilibrium no longer respects the action principle and the rules of Lagrangian theory.



  12. Philip Gibbs says:

    Ervin, glad you are still thinking about this. I plan to post more on the subject later, but for now let me take each of your points.

    1) It is true that the Komar Potential is zero in the standard cosmological models. However, even in this very idealised case energy conservation is not “trivial”. The individual energy in the matter fields and gravitational fields can be measured and calculated. They are not zero and they are not constant, but they add to zero because of the dynamical equations.

    I hope to do a separate post for the cosmological case some time to illustarte this point with the equations.

    Note also that the energy current is only zero on average over distance scales in which the universe can be treated as homogeneous. Locally the energy of the matter fields is concentrated in stars, planets, dust and dark matter, while the gravitational energy is mostly a feature of the universe’s expansion so it is spread more evenly over spacetime, with isolated concentrations around black holes and very massive objects.

    2) You can treat open systems provided you are able to measure the flow of energy across the boundary of the system. You can only consider the total energy of the whole universe if it is closed or if it is asymptotically flat. In other cases you can still consider energy conservation over regions of space provided you have the flow over the boundaries under control. This is just a practical consideration and does not invalidate the law of energy conservation.

    3) It does not matter if the system is not in thermal equilibrium. Energy conservation is exact even at the microscopic level where heat is just a form of kinetic energy and the dynamics is Langrangian. You do not need to consider thermodynamics at all to describe energy conservation exactly. It is only when you start to talk about the second law of thermodynamics and measure entropy that you need to think in terms of equilibirum systems.

    Feel free to come back on these points.

  13. Ervin Goldfain says:


    I realise that these topics are quite involved and we could easily spend countless hours debating them. I don’t have the time to elaborate and I assume that you don’t have either.

    Just one observation:

    You say “Energy conservation is exact even at the microscopic level where heat is just a form of kinetic energy and the dynamics is Langrangian.You do not need to consider thermodynamics at all to describe energy conservation exactly. It is only when you start to talk about the second law of thermodynamics and measure entropy that you need to think in terms of equilibirum systems.”

    This, in itself, deserves a comprehensive discussion. It relates to the point I raised before about intrinsic irreversibility and time-asymmetric MICROSCOPIC processes. These topics cover a lot of ground from Ilya Prigogine’s approach to persistent scattering and the decays of K-mesons to models of non-Markovian quantum systems, to non-extensive statistical physics and so on. For example in Prigogine’s work, statistical physics and irreversibility are manifested precisely at the microscopic level and require a revision of traditional views on conserved quantities which may become otherwise ill-defined. The introduction of fractal topological spaces is able to regularize divergent integrals in QFT, remove anomalies and lead to a consistent interpretation of concepts. What is important to remember is that many of these theoretical frameworks have been already supported by observational evidence.

    Also, there are non-extensive statistical models of inflationary cosmology that interested readers might wish to look into.



  14. Philip Gibbs says:

    Ervin, the ideas about intrinsic irreversibility would go beyond the general scope of the discussion which is about classical general relativity and field theory. I agree they are interesting ideas, especially if the neutrino results hold up but they would not be covered by the classical derivation of energy conservation I have given which assumes a Langrangian formulation.

    It is true that my time is limited as anybodies, but I want to make sure that everyone can see that there are no counter arguments to what I have said that I have not considered and discounted.

  15. Bill K says:


    Put me down as agreeing completely with the man from the Czech Republic. Total energy is meaningful in General Relativity only in special cases, such as spacetimes which are asymptotically flat.

    I’m waiting to see your promised post on this issue for cosmological solutions, but I bet you’ll just show us one of the Einstein equations and claim it represents energy conservation!

  16. Philip Gibbs says:

    I promise I’ll do that case Bill but I can’t promise you’ll like the solution.

  17. Lawrence B. Crowell says:

    Here is a potential problem I see. The current you have identified as J^a~=~\xi^b{T^a}_b is defined on a spatial manifold $latex\Sigma^3$ and for two such spatial boundaries we may define the current flux through those two regions

    $latex\int d\Sigma^aJ_a~=~\int d\Sigma^aJ\xi^b T_{ab}~=~\int d^4x\nabla_a(\xi^b{T^a}_b)$

    The covariant derivative will give the continuity condition on the momentum-energy tensor, but we have for the vector field


    which is going to be coordinate dependent.

  18. [...] a previous post I gave the equation for the Noether current in terms of the fields and an auxiliary vector field [...]


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