We have been discussion the law of conservation of energy in the context of classical general relativity. So far I have not been able to convince anyone here that the maths shows that energy is conserved. Lubos Motl and Matti Pitkanen have posted some contrary arguments on their blogs to add to the old one by Sean Carroll. We have also been trading points and counterpoint in the comments with Ervin Goldfain joining in, also in disagreement with me. To avoid going over the same arguments repeatedly we have agreed to disagree, for now.

If you think such a discussion about Energy in physics seems off the wall, think again. This subject and related issues concerning gravitational waves have occupied physicists for years. Some well-known names in the world of science have exchanged some heated words and still not everyone agrees on the outcome.

But it is too soon to end our debate. There are still a few more points I want to make. It was said that my claim in favour of energy conservation means that I am “convinced that all relativists are wrong”. This is not the case. Historically many relativists have been on my side. This is actually a debate that began as soon as general relativity was formulated by Einstein. Einstein in fact developed the first complete formulation of energy conservation in GR, but Hilbert objected. The argument has raged ever since with as many different views on the subject as there have been relativists and cosmologists. Amongst those who have accepted the law of energy conservation and produced their own formulations are Dirac, Landau, Wald, Weinberg and of course Einstein himself, so to say I am contradicting all relativists is far from true.

It has also been said that all the textbooks show that energy is not conserved in general relativity, except in special cases. This is also not true. Most GR textbooks do not tackle the general formulation of energy conservation in GR. They just deal with special cases such as a static background gravitational field with a killing vector. This does not mean that energy conservation only works in special cases as some people claim. The textbooks just don’t cover the general case. Some textbooks do cover it but by using pseudotensor methods (e.g. Dirac, Weinberg, Landau & Lifshitz) A few textbooks do suggest that energy is not conserved, e.g. Peebles, but these are the minority.

I am going to recount some of the history of the debate. To keep it orderly I’ll give it as a timeline of events with my own contribution immodestly tacked on the end. We start in 1915 with conservation of energy a well established concept recently unified with the conservation of mass by Einstein. The World is at war and Einstein is about to publish his general theory.

**July 1915: **Einstein lectures on his incomplete theory of general relativity to Hilbert, Klein and possible Noether at Göttingen, convincing them that his ideas are important.

**October 1915:** Albert Einstein publishes a tentative equation for general relativity with being the Ricci curvature tensor and being the covariant generalization of the energy-momentum tensor.

**November 1915:** Einstein realises that his previous equation cannot be right because the divergence of the energy momentum tensor is zero as required by local energy conservation. To correct it he writes the new equation . These are the Einstein Field equations which work because the left hand side has zero divergence due to the Bianchi identities.

**November 1915:** David Hilbert publishes a calculation showing how the Einstein Field Equations can be derived from a least action principle. In fact his work is dated prior to Einstein’s but they had been in communication and it is reasonable to give the priority for the equations to Einstein and for the action formulation to Hilbert.

**1916:** Hilbert publishes a note with an equation for a conserved energy vector in general relativity.

**1916:** Einstein publishes a full formulation of energy conservation in general relativity in which a pseudotensor quantity is added to the energy-momentum tensor and another superpotential term to give a conserved energy current.

**1916:** Einstein predicts the existence of gravitational waves which will carry away energy and momentum from orbiting stars. He derives the quadrupole radiation formula to quantify the rate at which energy is dispersed.

**July 1917:** Oscar Klein points out (with help from Noether) that conservation of Hilbert’s energy vector is an identity that does not require the field equation.

**1917:** In response to Klein, Hilbert publishes an article questioning the validity of energy conservation in general relativity. He says that the energy equations do not exist at all and this is a general characteristic of the theory.

**1917:** Writing to Klein, Hilbert says that general relativity has only improper energy theorems. By this he means that the pseudotensor methods are not covariant.

**1917:** Klein writes to Einstein making the claim that energy conservation in general relativity is an identity. This is based on Hilbert’s energy vector.

**1917:** To construct a static cosmological model Einstein introduced the cosmological constant as an extra term in his field equations.

**March 1918:** Einstein writes back to Klein explaining that in his formulation of energy conservation the divergence of the current is not an identity because it requires the field equations.

**July 1918:** Emmy Noether publishes two theorems on symmetry in physics. The first showed that symmetry in any theory derived from an action principle implies a conservation law. In particular, energy conservation is implied by time invariance. The second shows that in the case of gauge theories with local symmetry such as general relativity, there are divergence identities such as the Bianchi Identities.

**1918:** Felix Klein uses Noether’s theorems to derive a third boundary theorem to show why the conservation law of energy in general relativity must take a particular form that he considers to make it an identity.

**1918:** Einstein comments on the power and generality of Noether’s theorems but does not accept the conclusion that energy conservation is an identity.

**1919:** Arthur Eddington measures the deflection of starlight by the Sun during a solar eclipse. The observation confirms the prediction of general relativity and provides massive press publicity for the theory.

**1922:** Arthur Eddington expresses skepticism about the existence of gravitational waves saying that they “travel at the speed of thought”.

**1922:** Friedman finds cosmological solutions of general relativity that describe an expanding universe.

**1927:** Lemaitre predicts an expanding universe

**1929:** Edwin Hubble observes the expanding universe in galactic redshifts. This led to Einstein dropping his cosmological constant.

**1936:** After working on exact solutions for gravitational waves with Rosen, Einstein concludes that gravitational waves can not exist, reversing his 1916 prediction. This sparked a vigorous twenty year debate over the reality of gravitational waves.

**1936:** After working with Robertson, Einstein eventually concedes that gravitational waves do exist. Rosen who had departed for the Soviet Union did not accept this concession. He never changed his mind even as late as 1970.

**1951:** Landau and Lifshitz publish “The Classical Theory of Fields” as part of a series of textbooks on theoretical physics. It deals with energy and momentum in general relativity using a symmetric pseudotensor. The symmetry means that they can also show conservation of angular momentum using the same structure.

**1955:** Rosen computes the energy in exact gravitational wave solutions using pseudotensors and finds the result ot be zero. He presents this as evidence that gravitational waves are not real.

**1957:** Herman Bondi introduced a formalism now known as Bondi Energy to study energy in general relativity and gravitational waves in particular. This work was very influential and formed a turning point in the understanding of gravitational waves and energy in general realtivity.

**1957:** Weber and Wheeler find a gravitational wave solution that does transmit energy.

**1957:** Richard Feynman describes the sticky-bead thought experiment to show that gravitational waves are real. The idea was popularised by Herman Bondi and finally led to the general acceptance of the reality of gravitational waves.

**1959:** Andrzej Trautman gave the formulation of energy conservation for the special case where a static background is given by the existence of a killing vector field.

**1959:** Komar defined a superpotential for general cases whose divergence vanishes as an identity. The superpotential uses an auxiliary vector field similar to the killing vector field in Trautmans theory, but the Komar field does not need to satisfy any special conditions so the solution is more general. The Komar potential has the advantage over pseudotensor methods that it is expressed in a covariant form. However, the zero divergence of the superpotential is an identity.

**1961:** Arnowitt, Deser and Misner formulate the ADM mass/energy for systems in asymptotically flat spacetimes

**1961:** In his book “Geometrodynamics” Archibald Wheeler says that energy conservation in a closed universe reduces to a trivial 0 = 0 equation.

**1964:** Weber begins experiments to try to detect gravitational waves.

**1972:** Steven Weinberg uses a pseudotensor method to show energy conservation in his textbook “Gravitation and Cosmology”

**1974:** The discovery of the Hulse-Taylor binary pulsar shows that gravitational energy is radiated as originally predicted by Einstein.

**1975:** In his concise introductory text to general relativity Dirac derives a pseudotensor using Noether’s theorem to prove energy and momentum conservation.

**1979:** Schoen and Yau prove the positive energy theorem for ADM energy. (A simpler proof was given by Witten in 1981)

**1993:** Phillip Peebles in his book “Principles of Physical Cosmology” claims that energy conservation is violated for the cosmic microwave background

**1997:** Philip Gibbs shows that if Noether’s theorem is generalised to include second derivatives of the fields and is applied to the symmetries generated by a vector field, a conserved current with a covariant current can be derived. The current which has an explicit dependence on the vector field is equal to a term that is zero when the Einstein Field Equations are satisfied, plus the Komar superpotential.

**1998:** Observational evidence (Riess, Pulmutter) leads to the reintroduction of the cosmological constant, now called dark energy

For further historical details and references on the Klein-Hilbert-Einstein-Noether debate see “A note on General Relativity, Energy Conservation and Noether’s Theorems” by Katherine Brading in “The Universe of General Relativity” ed A.J. Fox, J. Eisenstaedt.

A good read on the history of gravitational waves is “Traveling at the speed of thought” by Daniel Kennefick

Hi Phil,

I don’t mean to sound like a parrot but this is a great post from which I am learning quite a bit about this issue over the history. This is also very pertinent to the PSTJ Focus Issue. After all, we are seeking truth!

In an earlier post on gravitational wave, I commented that there is evidence that gravity is an aspect of quantum entanglement thus it is instantaneous (implying no gravitational wave). Then you stated that “[if it is this is so], on the theroetical side you would also need an alternative explanation for the obeserved effect by Hulse-Taylor on the speed of binary pulsars].

This bring to my question which I ponders when I go to the beach and may turn out be due to my ignorance:

As moon circles Earth, it causes high and low tides which means that moon does work and loses its kinetic energy through gravitons (gravitational radiation). Assuming no nonlocal mechanisms which compensate this loss, why the moon has not falling on Earth?

Huping

Huping Hi, the high and low tides cause ordinary mechanics effects that do result in a loss of energy and gradual effect on Earth Moon rotation. The gravitational wave effect is much smaller for planets.

Hi Phil,

I’m going compound on my ignorance here: How is the ordinary mechanics effects mediated here? or what mediates/transmits the mechanical effects?

Huping

It is mediated by gravity but not by gravitational waves. The energy is lost mostly to heat generated by the tidal movement rather than radiated gravitational energy.

Hi Phil,

Thanks. What you say makes sense with respect to the energy is lost through heat generated by the tidal movement.

I mis-spoke when I mentioned gravitational radiation in my 1st reply and then imporperly phrased my follow-up question.

I meant to ask: what is mediating the energy transfer between the moon and the Earth (my present view is that gravity is an aspect of quantum entanglement and not mediated by graviton)? I know this is not related to the main topic.

I may not be seeing the full depth of your question, but different ansers might be given in different theories.

The tidal forces can be understood in the purely Newtonian approximation where we would just say that the Moon’s tidal forces act on the Earth to distort it and then small extra gravitational forces act on the Moon due to the distortions in the Earth’s shape. These forces change its orbit slowly.

If you want to describe it in terms of general relativity you would have to turn this into a statement about the variations in the curvature of apcetime instead of forces.

In the context of some possibly hyporthetical theory of quantum gravity you might just say that the effects are mediated through virtual gravitons, with all the usual caveats about the reality of virtual particles.

I hope at least one of these is getting nearer to what you are looking for :)

And if you believe in something deeper like entropic gravity or entanglement based effects then I am interested but someone else may have to provide the answer :)

What would your answer be?

Huping, a few things wrong here. Well, almost everything. :-)

General Relativity is a classical theory and has nothing at all to do with quantum entanglement. In turn quantum entanglement does not imply instantaneous action at a distance. This false issue was cleared up by Everett.

In the Earth-Moon interaction, it is the Earth which loses energy (slower rotation) and the Moon which gains energy, moving farther away not nearer. This energy transfer takes place in the near zone and has nothing at all to do with gravitons!

Hi Bill K,

On your statement “General Relativity is a classical theory and has nothing at all to do with quantum entanglement. In turn quantum entanglement does not imply instantaneous action at a distance. This false issue was cleared up by Everett.”

I’d like you to consider the following:

We have not only theorized but also experimentally proven that gravity is the manifestation (an aspect of) quantum entanglement. Please see:

http://www.neuroquantology.com/journal/index.php/nq/article/view/140

http://ptep-online.com/index_files/2007/PP-09-03.PDF .

Also see:

http://vixra.org/abs/1001.0011

http://prespacetime.com/index.php/pst/article/view/6

So, it is my opinion that there is no gravitational wave since as the manifestation of quantum entanglement, gravity is instantaneous just as Isaac Newton hypothesized. Of course, other researchers also pointed out that there is no gravitational wave.

On your statement “In the Earth-Moon interaction, it is the Earth which loses energy (slower rotation) and the Moon which gains energy, moving farther away not nearer. This energy transfer takes place in the near zone and has nothing at all to do with gravitons!” O.K., thanks for the free education (I mean it).

So, instead of moon hits Earth, it will run away from Earth! Please also allow me to compound on my ignorance, if the energy transfer has nothing to do with the graviton, what is mediating the energy transfer (my present view is that gravity is an aspect of quantum entanglement and not mediated by graviton)?

Huping

Hi Bill,

The first link is not working, so try this one:

http://www.neuroquantology.com/journal/index.php/nq/article/view/108

Huping

Sean Carroll doesn’t seem to understand the mathematical equations of physics in a physical let alone historical sense, when he claims that energy isn’t conserved in the field equation of GR. The entire basis for the difference in predictions between Newtonian mechanics and general relativity is due to the inclusion of the principle of conservation of mass-energy in general relativity. Although most textbooks prefer to ignore the elementary physical basis of general relativity, there are statements of the facts at places like http://www.mathpages.com/home/kmath103/kmath103.htm which states:

“It’s interesting [to look at] the process by which Einstein arrived at the final field equations of general relativity. The simplest hypothesis involving only the metric coefficients and their first and second derivatives, is that the Ricci tensor R_(mn) equals the stress energy tensor T_(mn), but then we notice that the divergence of T_(mn) does not vanish as it should in order to satisfy local conservation of mass-energy. However, the tensor T_(mn) – (1/2)g_(mn) T does have vanishing divergence (due to Bianchi’s identity), so we include the “trace” term -(1/2)g_(mn) T to give the complete and mathematically consistent field equations of general relativity. … the inclusion of the “trace stress-energy” in the expression for the Ricci tensor was the key to Einstein’s self-consistent field theory of gravitation. … the extra term was added in order to give a divergenceless field.”

Penrose makes the same point in his discussion of the development of general relativity in his popular book “Road to Reality” (2004) and in other essays. So it’s absurd of Carroll to ignore these facts and go in for promoting ignorance. Without the adjustment to satisfy conservation of energy, Einstein would have had the false prediction of Newtonian gravity for the deflection of light, etc., because the Ricci curvature giving the gravitational acceleration would be equal to the stress-energy tensor in all circumstances. It’s the extra term which causes the essentially important departures from the predictions of Newtonian gravity for relativistic velocities. Simply building a curved spacetime model for the Newtonian law merely reproduces the Newtonian predictions.

Unlike Carroll, I respect general relativity precisely because of its inclusion of the conservation of energy which led to checkable predictions, but general relativity still certainly has a lot of problems. E.g. the stress energy tensor isn’t used to represent particles as the source of gravity: instead you need to put in an artificially smoothed source of gravitational fields like a perfect continuum fluid, so that you get a smooth curvature in the Ricci tensor instead of of discontinuities! So the whole thing is a bootstrap con: the smooth curvature from the Ricci tensor is an artifact of the false smooth model for the distribution of particulate matter and energy which is causing the “curvature”.

I also don’t like the way that the gravitational field is formulated in terms of second-order (rank-2) differential equations whereas Maxwell’s equations are first order rank-1 equations, stemming from the curls and divergences of “field lines”. If general relativity was formulated in terms of gravitational “field lines” diverging outward from particles, instead of curvatures in spacetime, then there would be no difference in rank; alternatively if Maxwell’s equations were formulated in terms of spacetime curvatures rather than field lines, they would be second-order not first-order equations. There is a physical confusion present in efforts to unify electromagnetism and general relativity, with people thinking that the rank of the equation is physically caused by the spin of the field quanta, rather than by historical chance in the way the field is described.

Nigel it’s good to hear from someone who agrees with me on the energy conservation issue. I don’t think I agree fully with you on the last two paragraphs but let’s count our blessings instead.

You know I actaully agree with Sean on the majority of things he writes. I just don’t agree with his understanding of energy conservation. I like what he says about entropy except I dont like his view that time is a fundamental variable. Likewise I agree with a huge amount of what Lubos says ,but of course we spend more time arguing about the things we disagree on rather than discussing things where we agree.

The big surprise for me is that people who have spent so much time studying and working with relativity still don’t agree on such a basic issue as energy conservation in GR. I can understand that we disagree on what lies beyond the limits of current experiments, and I can understand that there are some differences of opinion over interpretations of quantum theory and other stuff that is basically metaphysical. But how come we can’t come to agreement on what is essentially a mathematical aspect of a well established classical theory? It should be possible to form a concensus on this but it isn’t happening.

I feel I’m not qualified to discuss this. But it is interesting. Maybe I can add some reflektions too? At least I hope so.

If only 3% of the energy is in form of mass, and gravity is about only that fraction only, then what force keep the other fraction of energy together? If susy is true then there are no other force, when gluons become ‘neutralized’ by the mirror image?

The curvature is a result of gravity and anti-gravity? The idea about a split gravity would need some explanations (to me)?

Also the informational content (negentropy) that can be seen as entanglement. Although the QM should not be included, but how can these questions be kept apart? What kind of relation has information and gravity? Information is the source, gravity the result?

Is information only a QM/entanglemental question? No, it is relaxation too. So it is classic (gravity). Is information also conserved? The black hole research says no.

In QM/entanglement the information is maximized.

[…] “We have been discussion the law of conservation of energy in the context of classical general relativity. So far I have not been able to convince anyone here that the maths shows that energy is conserved. Lubos Motl and Matti Pitkanen have posted some contrary arguments on their blogs to add to the old one by Sean Carroll. We have also been trading points and counterpoint in the comments with Ervin Goldfain joining in, also in disagreement with m … Read More“ […]

Hi Phil,

I think a new thread works better. Earlier, in responding to my questions, you stated the below:

[quote]

I may not be seeing the full depth of your question, but different ansers might be given in different theories.

[I] The tidal forces can be understood in the purely Newtonian approximation where we would just say that the Moon’s tidal forces act on the Earth to distort it and then small extra gravitational forces act on the Moon due to the distortions in the Earth’s shape. These forces change its orbit slowly.

[II] If you want to describe it in terms of general relativity you would have to turn this into a statement about the variations in the curvature of apcetime instead of forces.

[III] In the context of some possibly hyporthetical theory of quantum gravity you might just say that the effects are mediated through virtual gravitons, with all the usual caveats about the reality of virtual particles.

I hope at least one of these is getting nearer to what you are looking for

[IV] And if you believe in something deeper like entropic gravity or entanglement based effects then I am interested but someone else may have to provide the answer

What would your answer be?

[unquote]

The above are well said and you’ve “taken the wind out of my sail.” My choices are:

1. Entanglement based effect in [IV] as this is our own pet theory and experimental result (this is self-serving and self-promotion :-), I know).

2. The lastest entropic gravity in [IV] should work from a phenomenological perspective since entropy is connected with quantum entanglement as we speculated in 2006, e.g., see here http://cogprints.org/5259/ . Jonanthan recently speculated on “Do Thermodynamic Entropy and Quantum Non-Locality Have a Common Basis” (see Vixra.org, I put no link to avoid the spam filter).

3. Newtownian view in [I] works classically since Newtonian Gravity is instantaneous and consistant with our entanglement based view.

4. GRT view of spacetime curvature in [II] may also work as a phenomenology (to say the least) if one can tie the spacetime curvature to the wave function and/or quantum entanglement as we suspect that this can indeed be done! Our colleagues with GRT expertises should take heed of this!

So you see, Phil, my answer is not that simple as the manifestations of truth are multifacet as it should be :-)

I may write this whole thing up with more details, if time & energy permit.

Huping

As the classic gravity is only about 3 % of total the really important question is- in what form is gravity in quantum, non-local world, 97 % of all energy? What is the geometry of that world? If there is superpositions and non-local geometry there is also infiniteness?

The important question is – what happens to that world when we get the ‘one state only’ condition? Is the energy changing?

In my mind this question cannot be solved by classic physic alone. Distortions and following relaxations are also gravity and a form of entanglement, seen as network building. But before that is the quantum world, full of uncertainty.

Hi Ulla,

Thanks for your thoughts and comments on these intriguing issue.

Huping

The mechanism of the gravitational energy exchange and possibility of entanglement relate rather closely in TGD framework were zero energy ontology and hierarchy of Planck constants bring in essentially new physics also in astrophysical scales. Dark matter, dark energy and the anomaly known as Allais effect related very interestingly to this. Also the identification of quantum theory as a “complex square root” of thermodynamics emerges naturally in zero energy ontology.

The energy change between say Earth and Moon could be seen as taking via the exchange of virtual gravitons. In TGD framework and in zero energy ontology gravitons and in fact particles-virtual or not- have a classical description. Feynman’s original idea that virtual particles are real rather than only a fictive notion assignable to perturbation theory conforms with TGD views: now the off mass shell particles are pairs of on mass shell particles associated with worm hole throats and since they momenta can be non-parallel quantum mechanically and also the signs of their energies can be opposite one obtains continuous spectrum of off mass shell momenta. Besides it one obtains manifest finiteness (due to very strong kinematical constraints due to on mass shell property and momentum conservation at vertices) and unitarity in terms of Cutkosky rules.

One of the not so well-known anomalies associated with gravitation is Allais effect (Allais was Nobelist in economy). During solar eclipse when Earth, Moon, and Sun are at same line, so called paraconical penduli begin to behave strangely. No standard explanation have been found and the effect seems to be extremely sensitive to the position at Earth and to also to the time of occurrence for the eclipse. This can be understood qualitatively if Allais effect is quantum mechanical interference effect for virtual gravitons and the pendulum serves during ecliple as a highly sensitive interferometer sensitive to the precise distances between the three bodies (see this.

This kind of interference effects require however very large value of Planck constant. In TGD framework the hierarchy of Planck constants makes them possible and also implies quantum entanglement in all scales (see for instance this). For instance, for space-time sheets mediating gravitational interaction between two bodies with masses M and m the value of Planck constant would be hbar_gr/hbar= GMm/v_0, v0= 2^{-11} or harmonic or sub-harmonic of it according to the findings of Nottale. Nottale ended with the idea from the observation that planetary orbits in solar system and also for exoplanets seem to obey Bohr rules. He did not believe in genuine quantum effect but in TGD framework hierarchy of Planck constants, which now follows from the basic structure of the theory rather convincingly, implies it.

Dark energy would be associated with these space-time sheets and because Compton lengths are proportional hbar_gr and hence enormous , the density of this astroscopic quantum phase would be essentially constant in cosmological scales and would give rise to effective cosmological constant. Cosmological constant could be also understood in terms of magnetic energy momentum tensor if with these space-time sheets correspond to magnetic flux tubes. Nottale ended with his idea from the observation that planetary orbits in solar system and also for exoplanets seem to obey Bohr rules. Nottale did not believe in genuine quantum effect but in TGD framework hierarchy of Planck constants, which can be now argued from the basic structure of the theory rather convincingly, implies it.

One of the interesting implication of zero energy ontology implies

that time-like entanglement between positive and negative energy parts of zero energy states defines what might be interpreted as a “complex” square roots of density matrix equal to positive diagonal square root of density matrix and unitary S-matrix. Quantum theory becomes square root of thermodynamics. Thermodynamics or at least its formal structure would be basic aspect of quantum states, not only fictive description of reality by replacing it with ensemble.

Hi Matti,

Interesting explanations/interpretations from your TGD perspective on these fascinating issues.

Thank you.

Huping

Sorry for the repetition of text about Nottale’s role in the post. It was not intended but Nottale certainly deserve it.

A comprehensive history, including the Philip Gibbs entry haha.

Concerning the pseudotensors, well, they’re not tensors. So if you use the tools based on the main principle of GR, the equivalence principle, and go to the local inertial frame, your Landau-Lifshitz tensor will reduce to the multiple of the Einstein tensor – and it is not conserved covariantly.

So the non-tensorial part of the LL pseudotensor really measures the violation of the conservation laws. Its key defect is that it depends on your choice of coordinates, even on their nonlinearities.

For any theory that violates the energy conservation, you may calculate “how much” and integrate this quantity to construct a version of energy that is conserved, thus de facto claiming that the energy conservation is saved in any theory. This is of course a vacuous procedure because the actual energy or energy density is not “locally there” in time (or space and time); instead, you must look how you got there and how the coordinates connect the initial and final state.

The Landau-Lifshitz pseudotensor is doing the same thing in the case of GR. So it’s just a quantification by how much the energy conservation law is violated – but because this quantification depends on all details of the coordinate system, the “conserved quantity” is not an objective one. It also fails to be a quantity that can be obtained by adding local measurements.

Dear Philip,

the charge conservation in electromagnetism is not trivial – it is not vacuous – because if you actually modify the Lagrangian (so that it is not U(1) invariant), the current expressed by the same formula (in terms of charged fields, for example) will no longer satisfy the continuity equation.

If the current is not expressed in terms of other fields, then your question makes no sense because it is an external parameter, not a dynamical field. But you won’t find solutions of Maxwell’s equations for the currents that violate the continuity law.

However, on the other hand, the Landau-Lifshitz pseudotensor in general relativity identically satisfies the vanishing of its divergence, using the partial derivatives, regardless of the choice of the Lagrangian and regardless of the equations of motion, so it is trivial and vacuous. I am baffled that you still don’t understand the difference.

This is probably the “critical difference” you don’t see and you’re asking me to reveal for you.

Cheers

LM

Lubos, You are moving the goalposts :)

You previously said that the Laundau-Lifshtiz pseudotensor was trivial because the total current derived from it is the sum of two parts, one which is zero under the field equations, and another that has divergence zero as an identity. I do not agree that this is a condition for it to be trivial because this is also true for the Noether current for electric charge, even in flat spacetime with no gravity.

You dont like the pseudotensor because it is not covariant, but I have shown that the Noether current for enargy is covariant when derived directly using the vector field that generates the diffemorphisms.

All these formulations of energy conservation in GR work correctly to provide predictions about energy that are verified by observations. They are not trivial because they require the field equations to show that the current is conserved when it is written in a form that expresses energy as a sum of contributions from different physical fields.

I am sure you can continue to find things you dont like about the equations, but they work.

We probably cant go much further with this particular thread. It has been an interesting exchange even if we did reach much agreement. Feel free to add a last word. In a couple of days I will post more details about how the energy conservation works out nicely for real cosmology and you may want to comment further on that when I do.

Dear Philip,

indeed, there is an analogy with the electric charge conservation *in GR*. After all, the electric charge may be realized as the momentum along an extra circular dimension in Kaluza-Klein theory; you know, the extra-dimensional momentum is p=Q/R.

And indeed, what I say about the energy-momentum conservation in general backgrounds *does* apply to the electric charge as well. Its conservation law in GR in general backgrounds is completely vacuous.

Only if you have backgrounds where you have an asymptotic region at infinity where the field strength normally vanishes, one can have a nontrivial (nonzero) conserved value of electric charge that can be also seen from the Q/r behavior of the electric potential at infinity (taking the flat space as the major example).

Such a definition of the electric charge is completely analogous to the ADM mass. After all, e.g. the Kaluza-Klein theory unifies them into a 4+1-dimensional energy-momentum vector.

If you have compact spaces, the total electric charge has to be zero, as long as the potential is single-valued, in analogy with the total energy in GR.

However, even when the total charge is zero, one can define the charge density in an observer-invariant fashion. That’s because the charge density is a properly transforming tensor (well, a vector), unlike the Landau-Lifshitz pseudotensor.

In the comment above, I was comparing the conservation law for energy in curved backgrounds of GR (also) with the electric charge conservation in a flat spacetime. The former is trivial, the latter is not.

Best wishes

Lubos

No, Philip, it’s just not true. The electromagnetic current is just j^mu and not “the sum of one part that is zero according to the field equations, and a seocnd part whose divergence is zero as an identity.”

The two cases are analogous but you completely misunderstand the analogy. Even in the electromagnetic KK case, it is true that the electric charge conservation law disappears unless the background preserves the U(1) isometry.

One more example why the Kaluza-Klein picture actually answers all these conceptual questions.

The charge conservation follows from the U(1) symmetry, via Noether’s duality, and the U(1) symmetry may be interpreted as the isometry of the extra circular coordinate, one shifting x5 by a constant.

This U(1) symmetry holds whenever I can assume that the 4+1-dimensional spacetime is the circle bundle over a 3+1 dimensional spacetime. If this is not the case, the U(1) charge conservation simply fails.

This is completely directly translated to the case of the energy. The energy is conserved as long as you may consider all your configurations to be perturbations of a time-translation invariant background. In a general background where the time-translational invariant starting point is too far and useless, e.g. in the FRW cosmology, there won’t be any nontrivial conservation law, much like there is no p^5 (charge) conservation law in 4+1-dimensional spacetime that are not simple bundles with the U(1) isometry.

This is not just an analogy: in the KK theory, the energy and the charge are just two components of the same vector. It’s very clear what you have to assume for a nontrivial conservation law to exist: the whole physics must be a perturbation of a background that is U(1) or time-translationally invariant, respectively.

All this stuff is so clear, Philip!

Best wishes

Lubos

No, Philip, I am not moving any goalposts. You are confused about “what is included” in the object/current that is conserved. In electromagnetism, you have the equation

partial_mu F^mu nu = j^nu

But the current that is conserved is simply j^nu, the real current, not just the difference (j^nu – partial_mu F^mu nu) which is tautologically zero (and which would be of course “conserved” as well).

The analogy of j^nu in GR is clearly T^mu nu, and the analogy of partial_mu F^mu nu is clearly the Einstein tensor. But unlike the electric case where j^mu is conserved with partial derivatives, T^mu nu is simply not conserved with partial derivatives in GR. The only way to make it conserved is to subtract the Einstein tensor term, and perhaps add the non-covariant terms if you are not in a locally inertial frame. But then the construction is trivial.

In a properly chosen, local inertial frame, the whole Landau-Lifshitz tensor plus the matter part – the total stress-energy pseudotensor that should be conserved – is zero. So whenever the conserved pseudotensor is nonzero, its non-vanishing is just a coordinate artifact. That’s the reason why the LL tensor plus the multiple of T mu nu is trivial and doesn’t say anything about a conserved nonzero quantity.

You can’t make this thing with the electric current. If j^nu is nonzero, you can’t go to a gauge where it is zero.

The vacuous character of the LL tensor plus the multiple of T mu nu is not just that it’s ugly because it’s not a tensor: the problem is that its physical portion is simply zero, given conveniently chosen coordinates (e.g. convenient and natural “gauge” for the diffeomorphisms which is a gauge symmetry of GR). It’s not the same thing. You keep on saying that it’s the same thing because you’re naive, you don’t understand the difference, and I am not sure whether you want to understand the difference.

Best wishes

Lubos

But you

cando exactly the same thing with electromagnetic charge, so if this condition implies triviality for enrgy in GR, the same must apply to electric charge in EM. This was the point I made in my first post, second last paragraph.Lubos: “the Landau-Lifshitz pseudotensor in general relativity identically satisfies the vanishing of its divergence, using the partial derivatives, […] regardless of the equations of motion”

This is not the case according to the wikipedia article you linked to earlier http://en.wikipedia.org/wiki/Stress-energy-momentum_pseudotensor . Under the heading “verification”, point 3, they say that the field equations are required to show that the divergence vanishes.

Dear Philip, fine, I said it slightly incorrectly. I didn’t mean the partial divergence of the Landau-Lifshitz pseudotensor; I meantt the partial divergence of the Landau-Lifshitz pseudotensor plus the appropriate multiple of T_{mu nu}. That’s the combination that’s conserved.

One can simplify this combination and reduce the expression to a simpler term that only depends on the metric – and its partial divergence vanishes due to antisymmetry as explained in the second part of the point 3. This is why it’s trivial.

You can’t make such an operation – that would make the conservation law trivial – in the case of the current in electromagnetism.

Sorry I can’t let that be the last word :)

Yes, field equations are

partial_mu F^mu nu = j^nu

and current is just

j^nu

(because the gauge field has no charge in abelian guage theory.) But then

j^nu = (j^nu – partial_mu F^mu nu) + (partial_mu F^mu nu)

first part is zero by the field equations, second part has zero divergence because F^mu nu = partial_mu A_nu -partial_nu A_mu which has zero divergence.

So like I said, the current is the sum of one part that is zero according to the field equations, and a seocnd part whose divergence is zero as an identity. This is exactly the same as the condition you used to claim the current from the pseudotensor is trivially conserved. [ For the electric current you have to make the expression more complicated before you can see that it is trivial :) ]

This also works for non-abelian guage theories where the gauge field has charge which makes it look even more similar to the gravitational case.

Your points about Kaluza-Klein just underline the fact that the two cases are analogous. Another reason is the boundary theorem of Klein which says this must happen in general for any gauge theory. You dont have to have the EM case of a curved background to justify the similarity.

The fact that the two cases are analogous just shows that you cannot say that the conservation law is trivial in GR and not in EM. They are analogous in all relevant points. Either they are both trivial or both not trivial depending on your definition of trivial.

I agree that this is very clear.

The source of the electromagnetic field is not the field itself. That is in some ways what I find strange about Phillip’s statement above. This is the case with general relativity. The field is curvature that may be determined by a source that in natural units is equivalent. The “charge” in general relativity is mass-energy, which is equivalent to the field. With electromagnetism the charge is given by the roots of U(1), ±1, and are not equivalent to the energy of the electromagnetic field.

The source of the electromagnetic field is the charged particle fields (e.g electrons). In GR the source is the fields of the matter fields and the gravitational field. If you generalise electromagnetism to non-abelian Yangs-Mills then the source of the charge is the charged fields (e.g. quarks) and the gauge field itself (e.g. gluons) so the analogy is perfect. The energy of the elctromagnetic field was not mentioned.

Although everyone here is arguing on the side of Lubos, this does not reflect the most common opinion among people who have researched this issue in detail. Unfortuneatly most of what has been written on the subject is not openly available online. However you can read an article by Katherine Brading in one of the Einstein study series volumes. It is available as a sample in Google Book Search http://tiny.cc/4likl

Katherine describes the debate between Klein, Hilbert and Einstein and sides with Einstein as I do. She uses the same elctromagnetic analogy to support her case.

I prefer arguments based directly on the GR case itself. There is always the danger that analogies will be misunderstood. In the case of Lubos, he understood the analogy but simply accepted that even charge conservation is trivial in GR. I have never seen anyone else go that far before. Most people cave in at that point because they accept that electric charge conservation is not trivial. Of course this is all relative to your definition of “triival”.

I dont like pseudotensors either. They are not covariant and come in diverse forms. That makes them hard to get a grip on intuitively.

Despite this they were used to make the original predictions about energy radiated from binary systems and the answer has been conformed experimentally, so they are not a total loss.

The turning point in the understanding of energy in GR was the work of Bondi. I think it was he who introduced the time translation vector into the formalism to provide a covariant approach. This led to the Komar superpotential and ADM energy. However, this does not help clarify the question of whether energy conservation is in some sense trivial. My small contribution clears that up. I know you dont accept it yet but I am not finished. :)

Lubos, firstly I appologise for misleading you about which Klein was involved in the debate. It was actualy Felix Klein not Oskar Klein. I had written both names in different places and have now corrected the text.

Felix Klein was another great mathematician who also did some physics, but not as much as Oskar. Felix died in 1925. Oskar was also working at that time but was much younger. Felix knew a lot about non-euclidean geometry but gauge theories came later.

Coming back to zero energy, it is more than just a total number. The physical content of the law of energy conservation is that it exists in different physical forms and can change from one to another so long as the total is constant. In general relativity the total might be zero, but the seperate energy contents of gravity and matter are not zero and can change. Furthermore the energy is not zero locally except in special artificial cases such as a standard cosmology with no local perturbations. In the real universe energy flows from one place to another. Even when only gravity is present without matter this happens through gravitational waves. This is a testable physical phenomena, not just a mathematical fantasy.

I am well aware that in quantum gravity the Hamiltonian and any other diffeomorpohism operator should annihilate states and this is imposed as a constraint. This reflects the fact that the total energy is zero under certain assumptions. It is not inconsistent with what I have just said about the individual components of the energy for different forms of matter and gravity being non-zero and changing, nor is it inconsistent with the local flow of energy.

Lubos, the meaning of trivial in this context is certainly not that it is simple. The tensor calculus here is very easy. It is stuff I learnt while at secondary school. If you told me you learnt it at primary school I would not have reason to doubt it.

The interpretation Einstein used when replying to Klein was that it is not an identity (i.e. trivial) because it requires the field equation to show it. My definition is that it is not trivial because it tells us physical facts about energy transformation and propogation.

If you want to tell me that energy conservation in general relativity is trivial first answer me this yes/no question. Do you also think charge conservation is trivial in classical electromagnetism? If you answer “yes” then we have been at cross purposes all along because you mean something very different from what I thought. I you answer “no” then my followup question is: what is the critical mathemacial difference between energy conservation in general realtitivty that does not apply to charge conservation in classical electromagnetism? Perhaps from that point we can determine what it is that we see differently.

Dear Phil,

I think that just like many others among physics fans, you are just confusing physics with formalism. Physics of GR is given by the dynamical theory, Einstein’s equations, that contain all the information about the evolution of spacetime and its verification.

So there’s nothing such as an “experimental verification of pseudotensors”. Such a notion makes no physical sense. “Pseudotensors” are not a theory in the physical sense; they are not a modified version of GR that can be tested. They are just mathematical objects that you can use in your calculation of X or Y. But as long as you are doing general relativity, what you’re ultimately computing is GR and its problems, not pseudotensors or Komar superpotentials that can be at most more or less helpful intermediate auxiliary tools.

Again, general backgrounds such as the FRW cosmology simply don’t preserve energy. They explicitly violate it. Just try to define any nonzero object – and compute its actual value in Joules for the current visible Universe and for it used to be 10 billion years ago. You know, I want you to say that it is 1.6 x 10^{80} Joules today, and so on, instead of the vacuous babbling that you are doing 99.9% of your time.

When you do so, you will see that both numbers are either zero or they are not equal. This is the kind of fact that physics should be answering and it does answer it completely unequivocally: the energy is simply not conserved in GR as long as it is defined to be nonzero. No amount of masturbation with pseudotensors or the names of 3rd rate physicists such as Bondi or Komar can change anything about it.

Cheers

LM

Despite your quotation marks I did not use the phrase “experimental verification of pseudotensors”. I agree that such an expression would be meaningless. What I said was that pseudotensors have been used to do calculations. Then the answers from those calculations have been verified in experiments. I would not defend pseudotensors any further than that because I am not a fan of pseudotensors. I prefer my own covariant expression for the current.

I know I have to show how the cosmological case works. I will do it in a later post. It is not complicated and I would do it now, except I have a family event to prepare for that is taking up my time. I don’t want to write something too rushed.

I think it is ironic that you say I am confusing physics with formalism. That is exactly the accusation I would have directed at Hilbert and Klein when they said that energy conservation is an identity and it is improper because it uses pseudotensors. I would side with Einstein who had a better understanding of the physics over the formalism when he said that it is not an identity because it requires the field equations, and the calculations give physical answers.

Dear Philip,

whether some mathematical objects are used to make a prediction X or Y has no implication for the question whether the energy is conserved. Mathematics can be right or wrong and it is pure logic; it says nothing about the validity of physical theories.

You haven’t answered my question what is the numerical value of your conserved energy for our current visible Universe (I can tell you exactly what slice and volume I am talking about), so I assume that you agree that no such objective nonzero conserved energy can be associated with slices and you’re just trying to obfuscate this obvious point.

It is indeed ironic if you would direct criticism against two mathematical great physicists – greater than anyone else in your list except for Einstein – criticism that accurately captures the mistakes of your own approach.

Best wishes

Lubos

Dear Lubos, I agree that pure mathematics does not prove anything certain about real physics. However, I stated from the outset that this was a discussion about the validity of energy conservation within classical general relativity. From that point of view it is essentially a mathematical question although not quite stated rigourously.

Perhaps one way to make some progress would be for us to try to agree on what would constitute a valid law of energy conservation in classical GR so that we would then just need a mathematical proof to settle it. ;) (the smiley because I know it could never be that simple)

The case for a standard cosmology I’ll do in detail later. The short answer is that the total energy is zero when the field equations are satisifed for this ideal case. I have already stated that. I disagree that this makes the energy law invlalid in some sense. Furthermore, for the actual observable universe it will only be zero on average over large enough scales where the homogeniety and isotropy is a good enough approximation. This is easy to see by considering the ADM energy of an isolated system which is not zero. This would be a very good approximation to any star system that was sufficiently isolated.

As for criticising great mathematical physicists, at the time only Einstein would have had the intellect to do it, but with nearly a century of hindsight starting with Einstein’s own comments it is no big deal. Nobody can be assumed to have done everything right no matter how great. I am not the only one to reach this conclusion.

Dear Phil,

the energy conservation in a physical system always has some mathematical component but it is a physical question. The mathematical component is pretty much trivial in most cases. It is never hard to decide whether energy is conserved or not.

If an energy can be shown to be zero in the whole class of realistic or semirealistic configurations, then its conservation *is* trivial whether or not everyone “easily” understands why it is so. The law doesn’t say or imply anything.

And there are good reasons why GR has no meaningful energy conservation law e.g. in compact geometries or FRW cosmology. In normal systems, the Hamiltonian generates translations in time. But in GR, translations in time are just a subset of all diffeomorphisms – which become the local algebra of gauge transformations, and have to annihilate all physical states.

So unless you have some asymptotic region that separates “large diffeomorphisms” from others (large diffeomorphisms don’t have to keep physical states invariant) – and you still need the time translations to remain a symmetry for the energy to be conserved – it’s just true that the energy has to be zero because it’s a generator of a gauge symmetry which must vanish because they’re constraints.

Hilbert was great but Oskar Klein may have been the most reliable visionary in theoretical physics of the 1920s and 1930s.

His understanding of gauge symmetries in particular – and they are essential in this debate about energy conservation in GR – was decades ahead of everyone else. That’s because he was able to think about physics in physical terms. Yang-Mills symmetries were thought by him as parts of (higher-dimensional) diffeomorphisms which is what they are, either exactly or morally. That’s why he was almost able to design an SU(2) x U(1) gauge theory of elementary interactions in the 1930s – while people had lots of conceptual trouble with this even in the 1960s.

You’re focusing on some irrelevant technicalities of 3rd class importance and moreover you seem to misunderstand them. You’re missing the whole big picture and you incorrectly determine who was right in this energy debate.

Best wishes

Lubos

Dear Philip,

just to be sure that you don’t “revoke” my statements. I do insist that Oskar Klein was a great thinker when it comes to gauge symmetries – of general relativity and what we call Yang-Mills theories today – who was decades ahead of the colleagues.

I don’t know whether you view him as a participant of this energy debate and what the criterion was or is for someone to be relevant but he surely did understand these matters more than others. Well, Felix Klein was a pure mathematician – who also studied non-Euclidean geometry, among other things. But that’s a different question from physics.

The constraints have to vanish not only in a quantum theory; they have to vanish in a classical theory, too. The quantum version of all these statements only differs by “hats”. You’re just confused if you think that these issues don’t exist in the classical theory.

Well, energy is surely often written as a sum of several terms – not just in GR – which are however not conserved separately. So the individual pieces are not terribly interesting. Only the total (conserved) energy can be useful to say something about physics.

In GR, however, the energy density (and the whole stress-energy non-tensor) defined by the Landau-Lifshitz pseudotensor may be written – and usually is written – in the way that doesn’t depend on the matter at all. It can be written purely in terms of the metric tensor and its derivatives. So it has nothing to say about the matter fields and its conservation is a trivial identity that doesn’t depend on the dynamics of matter at all. So there’s not even an “interesting contribution to the conserved energy in GR coming from matter fields”. Do you dispute any of these propositions?

Best wishes

Lubos

I agree that Oskar Klein was a pioneer of many things in theoretical physics and made wonderul contribution to gauge theory. I also dont know if he entered into the energy or gravitational wave debate at any point.

I agree that the value of the Hamiltonian is zero in classical GR too. But again this just indicates that the total energy is zero, it does not reduce energy conservation to a trivial result. I disagree when you say that the individual parts are not interesting. The change of energy from one kind to another is the whole essence of the law of conservation of energy. If you think only the total value is intersting then even in ordinary mechanics the law of energy conservation would say nothing, but it does say something because it is about energy changing forms while staying constant in total.

Often people simplify the pseudo tensor using the field equations so that it no longer shows the matter terms. This is the same as just using the Komar superpotential in my expression for the current. It can still be useful if you just want to know the total ADM mass of a system, or the flow of energy in gravitational waves, but to understand energy conservation in the standard cosmological model you must look at the contribution from each type of matter, dark energy and gravity separately, so you should not apply that simplification upfront.

We seem to be homing in on the point at which we disagree at least.

Dear Philip, let me isolate the most problematic sentence of yours:

“But again this just indicates that the total energy is zero, it does not reduce energy conservation to a trivial result.”

A statement that you can call “energy conservation” may be viewed as a “result” of some derivations based on the equations of motion etc. And that’s fine with me. You may say that it’s nontrivial because you may find it hard to verify that the divergence of the Landau-Lifshitz pseudotensor is identically zero. I personally don’t find it difficult but it’s OK. It’s a subjective opinion.

But what really matters in the “triviality discussion” as long as it is a discussion about objective science and not about sentiments connected with “difficulty of formulae” is something else. In genuine physics, the derived energy conservation law is not the end of the story. Quite on the contrary: it is the beginning of the story. It is a tool that can be used to say something about the physical phenomena in actual situations. If you release an apple from the height “h”, you can use the conservation law to calculate something, namely the velocity near the ground, v=sqrt(2gh) or whatever it is.

It’s this part of the energy conservation law – the actual content of the law – that is trivial in GR or any other theory where the conserved quantity vanishes. It’s because if there were valid answers, the answers would always be the same because the total energy is always the same. You seem to be misinterpreting the word “trivial”. It’s not some statement about the lacking difficulty of proving something – which is a purely subjective matter. (And yes, I think it’s not difficult even in this sense.)

What the word “trivial” means in physics is that it doesn’t tell you anything about the physical systems. And that’s the case of any conservation law as long as the conserved quantity can be shown to be universally zero (whether you find the proof extremely easy as I do, or hard as you apparently do). Such a conserved quantity doesn’t divide the possible configurations into classes or superselection sectors. It keeps the same “whole class of all configurations” which is a trivial procedure. It doesn’t contain any physical insights or tools and it can’t be used to learn anything about the physical systems.

Again, you’re also wrong that the energy in GR as defined by the LL pseudotensor has a “matter part”. According to the standard definition, it is a purely gravitational object:

http://en.wikipedia.org/wiki/Landau-Lifshitz_pseudotensor#Definition

It’s expressed using the metric and its derivatives only. So it can’t tell you anything about the behavior of the matter. Its conservation can’t tell you anything about gravity, either, because the conservation law is a mathematical tautology. You still seem to be missing all the key points here.

But even if you wrote the tensor in a way that depends on the matter fields, it would change exactly nothing as long as the definition would be equivalent to the standard one. That’s what “equivalence” means. It’s “equal value”. If something is useless to make physical predictions, all equivalent things are equally useless.

Cheers

Lumo

The ADM equation NH = 0, for H ~ Tr(K^2) – (TrK)^2 + R(3), is really a constraint. Given a lapse function N = sqrt{g_{00}} this lapse function defines a contact manifold for the spacetime with this metric. There is also the momentum constraint as well N^iH_i = 0 for N^i the shift function. The lapse and shift functions are Lagrange multipliers which extremize the H and H_i on the contact manifold — or set them to zero. Of course the constraint H = 0 gets converted into the Wheeler DeWitt equation for the spacetime wave functional HΨ[g] = 0.

This H = 0 is then somewhat different from a standard conservation law. It is tempting to think of N as a time, for it is a scale factor on a coordinate time. Yet the lapse function is a variable with determines a constant action, or in the Hamiltonian perspective a contact manifold. This is not something which defines a translational action which defines a constant of the motion, here being energy. The ADM or WD equations define H with no reference to a time, so this H = 0 or HΨ[g] = 0 are not dynamical equations or wave equations in the standard sense.

This touches on what I indicated yesterday. Quantum field theories are defined by local variables which are Noetherian. The translation in q defines a momentum p that is conserved, and a time translation defines a Hamiltonian with some constant spectra over eigenvalues. Yet we are not able to derive the same situation for gravity according to local variables, and so standard quantization techniques are not sufficient. Our understanding of quantum gravity is incomplete, in spite of the massive volumes of papers written on a canonical quantization of gravitation using the ADM formalism.

One can assign to any action Noether currents. One can also take the curvature scalar for Einstein’s equations and assign it to Noether currents. The conservation of the Noether charges tells how much action changes in the corresponding transformation. As I explain at my blog, a highly interesting family of 4-volume preserving deformations is obtained as Betrami flows for which the flow parameter extends to a global coordinate. These flows could be used in asymptotically Minkowskian space-time to define non-conserved Poincare charges.

What is interesting that this algebra extends to infinite-dimensional algebra and that the flows have interpretation in terms of polarization and four-momentum generalized to local quantities so that the basic observables of gauge theories would relate directly to space-time geometry iniversally. It might be interesting to find whether they might allow to generalize the quantization in Minkowskian background to that in general background in general coordinate invariant manner without any ad hoc identification of counterparts of Minkowski coordinates. An analog of Kac-Moody type algebra would be in question.

Philip & Lumo,

In thinking about this I get the sense there is something lurking behind this debate. I always thought of pseudotensors as junk-math, but a thought occurred to me which might be of some interest. This amounts to the question of whether one can specify a field in a general spacetime in the same manner one does within flat spacetime. This seems to be an aspect of a general question on how one can define local fields on spacetime, and whether this sort of quantization can work in gravity. The inability to localize energy in any region of spacetime makes it difficult to define energy to begin with. If you have two spatial surfaces connected by timelike and null geodesics it seems reasonable to think the energy on the early surface is pushed to the subsequent surface by the diffeomorphisms of spacetime and all the data is preserved. This is contained in the continuity equation D_aT^{ab} = 0 for D_a a covariant derivative. So in order to project out a vector from the momentum energy tensor you contract with e_aT^{ab} and you get a summation of components of the momentum energy tensor. So I should be able to integrate between these two spatial surfaces to define in a Gauss law (like) manner the mass-energy scalar quantity defined in this region should be similarly defined. However, once I do a Stokes’s law calculation things fall apart, for the basis vector I choose to contract with the momentum-energy tensor with gives inhomogeneous connection terms. The domain of integration is itself a field subject to the diffeomrophisms of spacetime. This means the attempt to localize this mass-energy in this region is dependent upon the frame you choose. This appears terribly unacceptable. To get around this problem one must consider a large region which encompasses the region of gravity dynamics that is in an asymptotically flat spacetime.

Pseudotensors and the construction Philip advances are on affine spaces or constructed from a particular frame bundle on a tetrad. These are defined on a particular section of the frame bundle or tetrad bundle. In doing this the diffeomorphisms of general relativity are in a sense broken. This is a bit like what happens in electromagnetism where one chooses a particular gauge condition on A and then compute the magnetic field B = -curlA. The pseudotensor is of the form

t_{ab} = -T_{ab} + k’(g_{ab}g_{cd} – g_{ac}g_{bd})_{,cd}.

The derivatives of the metric terms contain all sort of connection coefficients, but what does permit these gadgets to have some utility is that on an inertial frame they are zero locally. The second order derivatives of the metric are a form of the delocalization of mass-energy, which locally remove the delocalization in T_{ab}. So we are locally demolishing something unpleasant by introducing nontensor elements which cancel out this problem This gadget then permits one to define the continuity current as ∂_at^{ab} = 0 and avoid delocalization problems.

Is this a hopeless procedure? It clearly can’t hold universally. It also illustrates an interesting aspect of local field theory in spacetime and why a theory with gauge invariant local fields can’t describe quantum gravity. A part of the problem is that given a field φ it is not in general locally gauge invariant, because in a general spacetime φ = φ(x) x is not gauge invariant. Conversely, for x a point in spacetime M, where M transforms under Diff(M) of GR, it is not possible in general to have a field that depends on x be invariant under the gauge-like symmetries of the spacetime. So attempts to unify the two notions are not amenable in a straight forward way. So one could in a local patch introduce this little gadget and avoid the problem. Yet the local region, where we have a locally defined field φ, say for this field the scalar field with its T_{ab} defined by its lagrangian in the standard way, will result in obstructions. There must then exist some more general form of geometry, something akin to sheaf theory. The fields defined in these local patches are extended across the manifold M with different t_[ab}’s in a way which contains more data than in φ. So while φ exists on the manifold, its local properties are insufficient to describe all field theory, in particular quantum gravity. So locality of fields is lost, and the gadget g_{ab}g_{cd} – g_{ac}g_{bd})_{,cd} which forms the pseudotensor is then something involving a conformal symmetry, such as on a string world sheet. This gadget, whether or not is gives a real conservation of energy or not, might then be some sort of structure which from the target map onto spacetime which defines quantum amplitudes of spacetime.

Thanks for these interesting points which I can’t respond to adequately now due to lack of time.

Phil,

I understand your lack of time and I apologize for continuing to dwell on the same topic. My point won’t take too long to develop.

You say:

“The physical content of the law of energy conservation is that it exists in different physical forms and can change from one to another so long as the total is constant. In general relativity the total might be zero, but the seperate energy contents of gravity and matter are not zero and can change.”

These arguments apply exclusively to closed mechanical or statistical systems where energy conservations stands. Homogeneity and isotropy may be justified for cosmological models at sufficiently large scales but time-symmetry is certainly not. To reconcile the cosmological arrow of time with energy conservation, one has two options:

1) consider that the Universe is open and in contact with a larger dynamical structure acting as a reservoir.

2) draw the conclusion that the concept of energy fails to stay meaningful at the Universe scale.

Option 1) is ruled out but the very assumption of considering the whole Universe which, by default, contains the reservoir. It follows that one is only left with option 2).

Cheers,

Ervin

Always happy to continue the discusion with you

“…time-symmetry is certainly not”

The gravitational field

when treated as a backgroundis not always time symmetric, but the gravitational fieldis not a background. It is a dynamic entity in its own right just as the matter fields are. The field equations overall are diffeomorphism invariant and energy conservation comes from timelike diffeomorphisms. So energy conservation holds in all cases including the expanding coosmological solutions. I’ll provide the relevant equation in a few days when I have more time.If for some reason you see this differently I would like to understand why. I feel we are talking past each other and would like to get to the bottom of it if possible.

Phil,

Cosmological time arrow is an objective reality embodied in the expansion of Universe, which has no reverse replica. So I feel that this inherent time-asymmetry does not care if field equations are or are not diffeomorphism invariant.

Ervin

The arrow of time may be thought of as subjective. Entropy is defined by a coarsed graining, which is subjective. The arrow of time, so called unidirectionality, is a bit of a thicket, for evolution of fields by gauge symmetries does not exhibit this property.

“a physical system always has some mathematical component but it is a physical question. The mathematical component is pretty much trivial in most cases.”

Said by Lubos.

The question is if mathemathics is something real or not. Everything can be expressed bt maths. But math as a tool to reveal reality is forceful. You must consider what pattern in the energy makes it worth to be selected for. If you have two pattern, both as dense, the same mass, gravity, gauge forces etc. but one is selected more willingly (by what) than the other, then what is the real force then? Energy itself? No, it is the information in the energy, the consciousness if you want, the negentropy, entanglement, networks. And that is quantal. A discussion without that aspect is sort of meaningless. As I understood the discussion is held in this format to reduce it, and that’s ok, as long as it is realized.

I guess spinors are exactly that networking in energy, and spinors have an informational content, but how can this information be measured?

Entropy is subjective, do you know what that means? It is a subjective diamond :) a zero energy cone. A subjective time of arrow is a subjective time, meaning we have two times, one objective, universal and one as a subjective cone :) Make the braidings then. I guess Ervin asked if this subjectivity has any meaning. It has, because it drives the evolution. The evolution is made in steps out of consciousness.

And I want to tell Lubos I can see he is able to discuss without violence, at least in a limited way. Continue with that.

I also want to ask him to stop his talk of crackpots. It is of no use. nobody knows more than someone else, because nobody knows particularly much about this field. This crackpot-behaviour is just childish. When I began to comment on his blog he could barely stand the name TGD, now he discuss with Matti. Some progress is made :) Philip Gibbs is a good site because of his kind attitude. I hope Lubos would learn something.

That was my memo :)

I don’t believe it is correct to simply characterize coarse-graining, and hence entropy and the arrow of time, as subjective. Granted, coarse-graining, as a feature of our descriptions, has a subjective aspect. But at the same time, and here I quote Gell-Mann and Hartle, “in the modern quantum mechanics of closed systems, some measure of coarse-graining is inescapable because there are no non-trivial, probabilistic, fine-grained descriptions” (this is from their abstract for “Quasiclassical coarse-graining and thermodynamic entropy,” available on the arXiv and in Phys Rev A, 2007). This indicates that coarse-graining and entropy have an objective aspect as well; and so, I think that Ervin Goldfain’s point stands. We can even go so far as to say, using T. Padmanabhan’s recent result that an objective cosmological arrow of time can emerge from equations of motion that are time-reversal invariant (see his “Why does the universe expand?”, on arXiv), that this arrow of time “does not care” if a system’s equations of motion are invariant under time-reversal.

It is important to recall that symmetries present in the equations of motion are not necessarily respected by their particular solutions. Boundary conditions often lead to solutions that break the original symmetry of the action. Likewise, spontaneous symmetry breaking chooses a vacuum that fails to obey the original symmetry (classical example of a ferromagnet breaking of SO(3) rotational invariance below its critical temperature).

We face a similar situation in relativistic cosmology: diffeomorphism invariance of field equations is not reflected in the time-asymmetry of cosmological evolution. Whereas it makes sense to talk about energy conservation in the context of Einstein field equations, the concept of energy may become ill-defined if cosmological arrow of time is accounted for at the global scale of the Universe.

Ervin

Yes, most solutions do not have the symmetry of the equations. However, it is the symmetries in the equations that lead to the conservation laws. The conservation laws still hold when the solutions do not have the symmetry, even if the symmetry is spontaneously broken.

To use your example, magnets cannot be used to violate the law of conservation of angular momentum, even though a feromagent breaks rotation symmetry spontaneously.

In cosmology the conservation of energy and momentum holds, even though the temporal symmetry is not present in the solution (i.e in the real universe) You just have to take into account the energy in all fields including the gravitational field to see that they are conserved.

I still have some hair left :) but the scratching is familiar.

TGD is very complex, and it has these traits discussed here, also the zero energy ontology, earlier discussed, suits here well? But Matti can surely defend this better than me. I will not even try, because he will be angry, afterall I am a novis.

My problem is that I am a biologist and not a mathematician or physicist. All this math is difficult for me, and so I try to look only in relations.

GR as an emergent field out of quantum physic sounds good. So, instead of adopting quantum to classic physic, maybe it would be an idea to look at classic physic with quantum eyes? The problem is then that we don’t know how gravity looks like in QM. Is it there at all? Maybe it is a mistake to search it there? Superpositions and entanglement is though, and some force must make it. I was chasing the ‘informational’ force, or holography. Teller has some interesting reports on ‘informational’ force, some negentropy. In that way gravity could be emergent, when superimposed states in matter relax and energy is released (as in chemical reactions). This would make gravity to an ‘energy-deficit’.

When we dive in water the water makes a pressure against us. Water is an entangled quantum liquid. In the same way the pressure from Universe can make gravity emergent (suits the Sun – Moon dilemma?) and give mass to ‘entangled’ (by different chemical, magnetic, electric forces, also gauge forces) particles. But then we meet the problem of ‘ether’ again. As I see it Einstein only prooved ether wasn’t material, it could be virtual or dark.

Wilchek talks of ‘Grid’ instead of ether. I guess it would correspond to a zero point structure or Dirac Sea etc. A zero point ‘pulls’ evenly in all directions. A zero point could be a soliton or some monopole. Then we have the duality there. Matrix would follow this ‘Grid’ as spinors, and the math could be p-adic?

http://web.mit.edu/physics/people/faculty/docs/wilczek_space06.pdf

The problem in this discussion is that it ignores the geometry of that Dirac Sea, and so the gravity force is problematic. In this Sea can be very much energy, as the vacuum problem tells us. But Philip wanted it so.

Sorry for my simple words.

To Ulla,

I encountered Matti P.’s TGD some years ago. I guess I found myself scratching my head more than anything else. TGD has elements to it that seem right. The use of p-adic numbers in roots of polynomials does play some role in physics. The leaves of associators and associahedra in what I work with are determined by prime factors and p-adic systems. These roots are involved with establishing braid groups (quantum groups or ordered elements in S-matrices) from underlying non-associative structures. P-adic numbers play a role in other aspects of physics and zeta functions. However, I will confess that I am not an expert on this subject and these considerations lie in the future TBD domain of things if I manage to get that far. It was over 10 years ago that I communicated with Matti on these issues. I will say that he seemed somewhat resistant to suggestions outside the domain of his work.

With respect to what I wrote above, I think that gravity is not fully quantized at all. I think it is only quantizable in a tractable manner to the one or two loop level. The problem is of course the standard problem that as one computes higher loop corrections the Ricci curvatures of the problem increase which results in a nest of infinities. This seems virtually impenetrable, and to my mind it suggests that there is some different physics so that general relativity is some sort of emergent field effect. Further, quantum mechanics appears to be completely linear, and a recent report on quantum optical measurements suggests that different paths in quantum physics interfere pairwise and not higher as might be expected in a nonlinear quantum theory:

Ruling Out Multi-Order Interference in Quantum Mechanics

Urbasi Sinha, Christophe Couteau, Thomas Jennewein, Raymond Laflamme, and Gregor Weihs

Science 23 July 2010: 418-421.

The entropy associated with gravity, whether that be with black holes or the Verlinde entropy force (really a sort of Birkhoff theorem result with entropy) suggests that gravity is emerges as an effective field through some entanglement entropy. This entanglement may involve states in our universe or cosmology on a Dp-brane are entangled with inaccessible states in the “bulk.” For energy in the universe largely in an equipartion E = nkT/2 this entropy E = TS is then a measure of inaccessible energy, which is a property of gravity — we are not able to localize gravity and to define conservation of energy.

To Lawrence

Interesting.

“suggest that mass-energy is some sort of topological index in cosmology or quantum gravity on a global scale.”

This index (in gravity) depends on the quantum entanglement? So the mass is ‘a dance following energy field from infinity’. What rules determines the massivation? A duality, a magnetic monopole in the center?

I hope you see the similarities with TGD :D Only because of the differences in dimensions and layout you physicians refuse even to have a look on what Matti tells you?

Excuse me again.

Louise Riofrio talks for an conservation of energy, and a diminishing light speed instead of an accelerated enlargement of the Universe. What is your opinion of her hypotheses?

Another thing. Magnetism is then only borrowing energy from the environment. Magnetism is behind phase transitions and work in the same way as entropy (temp). So a magnetic wave is eg transfer of energy only? Also pressure works in the same way?

If energy is conserved gravity should work at the same principles? A gravity wave is an energytransfer?

Thank you, this was great. Ulla.

This is then all times together, past and future, the energy is conserved? All zero energy light cones together. Into infinity?

Ervin, Now that I have more details on your view of the situation I will try and cover these points in my next energy post when I show how energy conservation works in our expanding universe.

Phil,

This is precisely the core of our disagreement.

Even if you take into account the energy of all fields including the gravitational field, the loss of time-translation invariance leads to an ambiguous interpretation of the energy concept at the cosmological scale of the Universe. Specifically, if you choose to define energy as the scalar conserved as a result of time-translational invariance of the laws of physics, then the unavoidable conclusion is that this scalar either becomes ill-defined or fails to stay constant at the Universe scale.

It is ultimately a matter of choice and context underlying the definition of concepts such as time-symmetry and energy. This is the reason why we hold divergent views on this matter.

Cheers,

Ervin

Phillip,

This debate has entered some interesting ground. The connection to Kaluza-Klein theory indicated by LM and the formal analogue between charge and ADM mass raises some interesting points. The compactification of a space, say it is the real line, amounts to a periodic sequence of points on that line which are identified with each other. This gives the usual association x ~ x + 2πR, where R is the radius of that compactification. The momentum of a particle spinning around a circle defined this way is p = ħ/πR. Further, a particle running around this circle at a velocity v = c will have an energy E = ħc/R, or a mass equivalency m = ħ/Rc. This mass-energy is associated with the field or its potential and the charge. In the case of electromagnetism the charge is a source of the gauge field, and that field is determined by the a gauge symmetry and …, well we know the script. This can be extended to other gauge fields with more complex compactification schemes, or orbifolds that strings wrap around.

Does this have an analogue with gravity? I think it does, and I will give an outline of how I think this works. We have this coup in physics called the AdS-CFT correspondence. The CFT may be compactified, and as with the periodic case of the line onto the circle, there is a discrete structure. The discrete structure defines a Klein group as G/Γ, where the group is the AdS group or the CFT sphere S^n. So suppose we have (AdS_nxS^n)/Γ the quotient hits on AdS_n or S^n. On a Dp-brane level the two quotients are D1 plus D5 plus a duality in 4-dim of a pp-wave and a Taub-NUT spacetime. The Taub-NUT spacetime has instead of a mass a NUT parameter, which is analogous to a magnetic monopole. The source of the plane parallel (pp) wave is ordinary mass — ADM mass. The Taub-NUT spacetime embeds a discrete structure on hyperboloids, which under this duality on the AdS defines a conformal compact condition for the AdS as an Einstein space, or a de Sitter spacetime of one dimension lower. This is a combination of the T and S dualities, where the S duality defines a magnetic monopole charge g according to an electric charge q by a Bohr-Sommerfeld condition qg = nħ.

This then have parallels to gauge theoretic construction. This may not necessarily indicate how to generally define energy conservation in spacetime. However, it does suggest that mass-energy is some sort of topological index in cosmology or quantum gravity on a global scale. This does not necessarily mean that one can define mass-energy for local systems. The reason may be that this index for mass-energy is defined as some entanglement entropy of our cosmology with the so called multiverse. These Taub-NUT spacetime can be thought of as Misner-Dirac strings bolted to Dp-branes, and mass-energy is globally defined according to a Gauss law on branes. What an observer finds locally is that mass-energy is from a partial trace on the density matrix of states and this has a higher entropy, or uncertainty in defining mass-energy.

Within general relativity energy conservation can’t be established. On balance Lubos has the upper hand in this argument. What Philip argues for is really some conservation law on a set of “patches,” where the lack of covariant tensorial properties of these pseudo-tensors is small on any patch.

I guess I never thought of water pressure as a case of entanglement. Entanglements on the large of course can happen. A state of 0 and 1 for an atom can be entangled with a spin or some other element. By computing the interference term between 0 and 1 are taken up by the spin, here a switch that turns on a device. This further entanglement then puts the superposition into the device state, which leads to the Schrodinger cat being in two different states of alive and dead by this entanglement. So one can of course entangle a quantum system with a large system, and this does happen with measurement.

I spent some time looking at TGD material, and I still am left scratching my head. The space Matti works within is CP^2xM^4, which has eight dimensions. I am not sure why he chooses this space. Now there are some dimensions of interest, 1, 2, 4, 8 for the Cayley numbers. These correspond to the reals, complex numbers, quaternions, and octonions. Clifford algebras on a 3-fold system exist in the corresponding 3, 4, 6 and 10 and similarly on the 4-fold system 4, 5, 7, 11, which correspond to this Cayley number sequence. So I am not sure where Matti’s scheme fits in here. The superspace theory in 11 dimensions is M^4xM^7, where M^7 has a G_2 holonomy.for a system of Killing spinors. If Matti had CP^3xM^4, this would conform to a possible 10 dimensional superspace. Then somehow p-adic number enter the picture, but I don’t remember how.

What Philip argues for is really some conservation law on a set of “patches,” where the lack of covariant tensorial properties of these pseudo-tensors is small on any patch.

I have said that I don’t like the pseudotensor methods either, but it is not because they are wrong. There is nothing incorrect about working with co-ordinate patches. A single co-ordinate patch could cover the entire observable universe, possibly even the entire infinite universe if it is infinite. It is only if the topology of the universe is different that multiple patches would be needed. There is no sense in which the lack of covariant tensorial properties is small. There is no approximation introduced by using the pseudotensor. Everything is exact.

Even though the pseudotensor methods work I prefer the covariant approach that uses an auxilliary vector field to specify the time translation diffeomorphism. I described this at http://blog.vixra.org/2010/08/08/energy-is-conserved-the-maths/ I think the idea of the auxiliary field was introduced by Bondi. It makes it much clearer how the energy is relative to the choice of time translation. With this approach you can even deal with the closed universe case without having to worry about the problem of patches. That is one reason I prefer it.

Somehow I get the impression you are starting from the wrong end with the dimensions. Your problem should seek the math, not math should choose the problem. What yoy say only confirm you haven’t studied very well.

What I say about liquid entanglement is in agreement with what Huping said. But Lubos did not at all like the idea of ‘entanglement’ in Nature :) There are interesting science done.

To see gravity as some form of entanglement or negentropy is reasonable, I think.

wman,

Coarse graining is subjective in that one chooses how to lump microstates into macrostates. In information theory this has connections with Bayesian prior estimates. If you define a macrostate for a system as Ω, by some ensemble method, the entropy is S = -k*log(Ω). Now if this macrostate has some error so that a better estimate is Ω’ = Ω(1 + x) then log(Ω’) = log(Ω) + log(1+x). So if the phase space estimate involves a mole of atoms n = 10^{23}, then the phase volume is 10^n, and so log(Ω) ~ 10^{23}. Now if our error part (1 + x) is even several orders of magnitude off the logarithm “covers our tail,” or this contribution is negligible. It is in this sense that coarse graining is subjective, but does work well enough to give accurate answers.

Lawrence,

Coarse-graining is a well-established procedure of Renormalization Group theory whereby one systematically integrates out all microscopic fields above a cutoff scale. There is really nothing “subjective” about this procedure which necessarily leads to a loss of information and a corresponding increase of entropy.

Ervin

That is consciousness :D

And information is lost says Ervin. Information cannot be lost says other. That is a problem.

Again, it is subjective, for the cutoff scale is chosen by the analyst. It is not mandated by nature.

Lawrence

That the cutoff scale is chosen by the analyst is not universally true. In many case the cutoff scale is determined by the physical problem at hand.

Take for example the Fermi theory of weak interactions. It is the low-energy limit of electroweak model having massive bosons as high-scale degrees of freedom. Fermi theory can be regarded as an effective field theory that disregards the exchange of massive bosons in an energy range upper limited by the Fermi constant (GF ^(-1/2)= 293 GeV). In this case, Fermi constant sets the natural cutoff scale.

Ervin

Ervin,

Similarly some critical point or exponent can provide a cutoff. By saying this is subjective it is not meant to mean there is no “reason” for a coarse graining. It is called subjective because there is nothing which computes the cut off in some explicit way. The analyst chooses a coarse graining or a cut off based scaling or on some auxilliary criteria.

Lawrence,

Thanks for clarifications.

Cheers,

Ervin

Ulla,

Let me briefly explain what information loss is in the context of coarse-graining.

I am going to consider a real-life case and this is a digital camera. A digital camera contains a light sensor (imager) built from a rectangular array of square detectors called pixels. If you have a camera with, say, 2N x 2N pixels, the digital image you capture has higher resolution than if your camera has only N x N pixels. What this means is that camera with 2N x 2N pixels is able to resolve finer detail of the object you are taking a picture of. Why? Because fine detail in the image is averaged out (or coarse-grained) within each pixel in the array. The larger the pixel size, the larger the amount of coarse-graining.

The bottom line is that the loss of detail (or optical information) in a camera with N x N pixels is larger than the corresponding loss of detail in a 2N x 2N camera. To this extent, we say that the entropy associated with the N x N camera is larger than the entropy associated with the 2N x 2N camera.

Hope this helps.

Ervin

I hope you see the subjectivity in that process :) You talk of what you measure.

This discussion has been very helpful for me. Wait impatiently for next article of Phil.

That depends on how well the first choice was done. What kind of energy is in that choice?

Symmetry is always assumed for the vacuum field, but it may be a false assumption. The symmetry can also be a first choice. Looking at the new data from neutrino-antineutrino measurements (sigma 3-level) hint at asymmetry. Then the time-aspect is important. Can we get energy from the past, or maybe from the future :)? Energy conservation can have a long time-line.

Multiverses can have an impact, but about that we know almost nothing. Then we will have to discuss an open Universe.

The initial choice of scale may be subjective (arbitrary) but the outcome of the Renormalization Group process is scale independent. Coarse graining is only the first part of the process, the second part being rescaling. Rescaling redefines dynamical variables and parameters of the system in a way that preserves the dimension or symmetry of the internal space.

A comment to Lawrence who was scratching his head with the motivations for the choice of M^4xCP_2.

The first manner to end up with this choice is the condition that standard model quantum number spectrum and classical gauge fields are geometrized in terms of sub-manifold geometry.

Second motivation comes from the physics as geometry of world of classical worlds vision and the observation that the Kaehler geometry of loop spaces is unique and possesses maximal group of symmetries. The conditions for the geometry of the space of higher-dimensional objects are even more restrictive and one has excellent reasons to expect more or less unique imbedding space and space-time dimension. Physics would be unique just from the condition that the “world of classical worlds” exists. One can understand the reasons 4-D Minkowski factor and 4-dimensionality of space-time surfaces from extended super-conformal invariance (3-D lightcone boundary is metrically 2-D and possesses extended conformal symmetries). But why just CP_2?

This highly unique 8-D space-or actually its CP_2 factor, must be distinguished in some manner and a good guess is that its symmetries have a number theoretical interpretation. Classical number fields fit nicely with the dimensions of the basic objects of TGD. The subspaces of their complexifications with Minkowskian signature turn out to play a key role in number theoretical formulation of the theory. I assign these subspaces the attribute “hyper”. Space-time surfaces are postulated to be hyperquaternionic sub-manifolds of hyper-octonionic imbedding space:: the challenge is to prove that this condition is equivalent with preferred extremal property for Kaehler action.

Hyperoctonionicity and quaternicity are defined for tangent space gamma matrices in octonionic representation so that imbedding space coordinates are not hyperoctonionic or space-time coordinates hyper-quaternionic. CP_2 geometry can be understood purely number theoretically and its isometries relate closely to the subgroup SU(3) of octonionic automorphisms. One has M^8-M^4xCP_2 duality analogous to spontaneous compactification in string theories. No physical compactification of course occurs.

One can say that the world of classical worlds is the space of hyper-quaternionic sub-spaces of the local complexified octonionic algebra defined by the local Clifford algebra of the imbedding space.

Matti,

A few points come to mind. The Hopf fribration for 8 dimensional manifold is a sphere S^8, and the S^7 –>S^{15} –> S^8 defines the number of ways one can link S^7 sphere in S^{15}. This tends not to involve and 8-dimensional manfold that is a projective CP^2 with a 4-manifold at every point.

I would be happier if the CP^3 were contained in CP^3, or was the result of a projection on C^3 and the manifold prior to this action is C^3xM^4. This would have potential for a lot of structure I might be interested in.

Phil,

I may be wrong here but I have the feeling that the debate has gotten out of hand. It clearly lost its focus as some commentators are talking about issues that have no direct bearing on the subject matter.

Cheers,

Ervin

:) Possibly true, but it’s no big deal. I will keep the moderation light here. The main thread will probably pick up again with the next post.

Phillip,

What you say about patches is close to the mark. The conformal completion of the AdS is an Einstein spacetime or a Minkowski spacetime. The conformal completion is a patch on the AdS, where T^{00} = const is defined. So applications of these methods are not completely out of line.

The use of a time translation diffeomorphism works, but it leaves some questions in my mind. For Delta x^a = eps*X^a the various current terms are J^a = X_bT^{ab}. The covariant derivative contains X^b_{;c} which is not necessarily zero unless it is a Killing vector. Now one thing to consider is J^aJ_{b;a}, or the covariant derivative of this current along the current. This would then be

J^aJ_{b;a} = X^cX^d_{:a}T^{ac}T_{bd}.

Now for the stress-energy proportional to the metric, say T^{ac} = Λg^{ac} this covariant derivative of the current along the current gives a geodesic equation X^cX^d_{:c} = 0. In this case this is a conservation of a J^2, which for dust T^{00} = ρ is conservation of energy. For pressure terms the expression is a bit more complicated, though similar to the invariant mass interval in special relativity.

OK, but dont forget that X_bT^{ab} is just the contribution to the current density from the non-gravitational matter and radiation fields. Gravity is a non-abelian gauge theory so the gauge field has energy too. In other words terms from the gravitational field must be added to the energy current. You can compare this to charge in non-abelian Yangs-Mills if it helps.

Of course the Einstein field equation comes from the Bianchi identity:

R_{ab[cd;e]} = 0

Contracting on the first and third index gives the expression according to the Ricci curvature, and then another contraction gives the G^{ab}_{;b} = 0. Of course we could define a current J^a = X_bG^{ab} and repeat the analysis with J^aJ_{b;a} = 0. However, that only gets us to the problem Lumo pointed out. However, not all is lost. We do have the reverse form of the Einstein field equation

R_{ab} = T_{ab} – 1/2Tg_{ab},

And so if this trick works with the T_{ab}’s it works with the R_{ab} and its Ricci scalar trace. So we don’t need to do this trivial matter of looking at G_{ab} – T_{ab} = N_{ab}, which is trivially zero to start with.

Lawrence, you are edging closer to what I have in mind but there are two more points I need to make.

Firstly, your N_{ab} is not “trivially zero”, it is only zero when the field equations hold. These are not trivial because they contain all the gravitational physics. It is only fair to call a result trivial if it has no physical content. This would require it to hold independently of the field equations.

Secondly, there is another term that needs to be added to the current density. It is called the Komar superpotential and it is not always zero. These are then all terms you get in the current density when you use Noether’s Theorem.

[…] viXra log News Blog of Science and Mathematics « Energy Is Conserved (the history) […]

The currents written according to J^a = X_bT^{ab} are extended into J^a – -> J^a + K^a for this Komar potential

K^a = c(X^{b;a} = X^{b;a})_{;b}

The covariant derivatives are X^{a;b} = ∂^bXa + Γ^a_{cd}g^{cb}X^d. Without the covariant terms this would be in fact div-curlX which is identically zero. So this potential (potential vector?) is not determined by “curlX” or should we say dX. Hence K^a is similar to a gauge term, we might say it has some cohomology, where under the boundary operator, or covariant boundary operator K^a_{;a} = 0 — which is trivially true as there is an anti-commutation in ab with X and the two derivatives ;a and ;b commute. So that appears to be some sort of identity.

Then the issue is what is the meaning the K^a in a local patch. As I see it the K^a represents the difference between the vector flows in two patches, say on their overlap. The difference between the X^a’s say X^a(f_i) – X^a(f_j) is determined by the difference in the presheaf structure or F_{ij} = f_i – f_j. There is some sort of cohomological meaning to this and I think this is the best way to look at it.

Sorry about that equation above it should read

K^a = c(X^{b;a} – X^{b;a})_{;b},

and c = constant.