In fact, Friedman-Lemaitre expansion equations can be established from Newtonian arguments. So I am inclined to say that it is incorrect to state that energy is not conserved!

Dear Alain,

first, FLRW expansion model is an EXACT Einstein’s field equation for the case of homogeneous all-space matter distribution. It is a mathematical fact that Friedmann equation COINSIDE with Newtonian equation of the finite homogeneous ball is a real riddle – why the exact relativistic equation governing the dynamics of the whole Universe is exactly Newtonian equation of the expanding material ball for any radius (not small)? (see mathematical derivation e.g. arXiv: 0810.0153)

Second, the classical Newtonian form of the relativistic Friedmann’s equation is the true cause of the violation of the relativistic part of the energy-momentum conservation in FLRW cosmology. This is clear from the solutions of the Friedmann equations (motion plus continuity) for time dependence of the energy content of any co-moving ball of matter having equation of state p = w e (were p – pressure, e – energy density, w – constant):

E(r) = const x S^{-3w}, where E is the energy content of a co-moving ball of radius r, and S is the scale factor (see Baryshev & Teerikorpi, Fundamental Questions of Practical Csmology, Springer, 2012, p.290). This formulae demonstrate that only in the case of “dust” having w = 0

the conservation of energy is fulfilled, for the case of photons w = 1/3 and the energy of photon gas in any finite co-moving ball is not conserved (disappeared nowhere as Harrison noted). In this sense the observed CMBR is a manifestation of disappearing of the initial energy of hot photon gas of the Big Bang.

Dear Tamara,

first, of cause it could be that you know general relativity much better than Hilbert, Einstein, Landau, Harrison all together, but they discussed such problem as absence of energy-momentum tensor (EMT) of the gravity field in GR. Please read the paragraph 101 “The energy-momentum pseudotensor” of the book Landau&Lifshitz “The classical theory of fields”, Pergamon Press, 1971, where you will find the mathematical difference between ordinary divergence of EMT in Minkowski space and covariant divergence of EMT in curved space (the history and the physical significance of the problem see e.g. in arXiv:0809.2323). The reason of the violation of the energy-momentum conservation in Friedmnn cosmology is the absence of the energy-momentum of the gravitational field in GR.

Second, for all fundamental physical interactions there is well defined energy-momentum conservation by using Gauss theorem for any finite local volume, and this is a fundamental property of the Minkowski space. It does not exist in Friedmann model just because it lost the concept of the energy of the gravitational field itself.

Third, there is a strict mathematical derivation of the formula (see e.g. my preceding answers to Alan) which demonstrate the energy non-conservations in any local volume of the Friedmann model, so it is very clear why the energy is not conserved in the expanding Universe as a whole.

Dear Yurij, the problem is the lack of an energy momentum tensor for gravity. This was the first thing that Einstein set out to derive and he failed, so he went to Riemann theory. The question has gotten some answers by now.

Yurij Baryshev

Energy and momentum are not conserved if one takes into account the effect of the expansion of the universe on the local invariance of all physical laws, in particular Schrodinger equation (quantum mechanics). Then the only way to conserve local invariance under the expansion of the universe is to allow for the speed of light to decrease with the expansion. See Nguyen in arXiv.

Another more physical possibility to get conservation laws in cosmology is just to reject the expansion of space as a real physical process.

In my view the energy should go to gravitation. But then one should first have a proper definition of the gravitational energy momentum tensor. That the Landau-Lifshitz pseudo tensor can be properly generalized was shown by Babak and Grishchuk 1999.

I also made some comments about it, Europhysics Letters 2008.

Thanks Theo for your comments. Yes, I agree with you that the energy-momentum is the corner stone of Theoretical Physics. In its turn energy contain all other basic physical quantities such as space, time, mass, action, force, work, quanta of energy, and all them based on the symmetry of the Minkowski space.

This is why curved space-time of GR is a radically new approach to physics, which contains such paradoxal things as absence of the field energy, black holes, superluminous space expansion, continuous criation of energy in expanding space

(see discussion in Topic Cosmology and Gravity Physics).

It is explained in our book BT2012 (Springer) and in paper arXiv:0809.2323, that Grishchuk geometry-field approach has havy problems with the gravity field energy

(and the energy-momentum tensor). For example their energy density of the static gravitational field is negative: T_oo < 0.

The only way to have energy-momentum tensor of the gravity field and its conservation is the Field Gravity approach - the Richard Feynman tensor field in Minkowski space (see his book “Feynman Lectures on Gravitation”,1995) and calculations of new predictions for experiments/observations in aXiv:0809.2328.

General Relativity is a good theory actually tested in a weak field approximation

(no observational proof of strong gravity horizon), so GR could be an approximate

physical gravity theory as geometrical optics in electrodynamics.

Dear Yurij

You say "For example their energy density of the static gravitational field is negative: T_oo < 0." I always thought that the energy of the gravitational field is in fact negative. By a thought experiment two masses m (very small size than M) and M, m

Dear Antonio, unfortunately you confuse classical Newtonian potential energy (which is negative) with the relativistic energy desity of the gravitational (or electromgnetic) field,

which is the T_00 component of the energy-momentum tensor of the field!

Please read differenses e.g. in the textbook Landau & Lifshits "Field Theory" 1971.

All fundamental physical fields with integer spin (photon -spin 1; graviton - spin 2)

has positive T_00 component (e.g. static electric field has T_oo =+ E^2/8pi ).

It is because it does not take into account inflation.

Dear Yurij: I always thought that physical relations expressed within relativity theory, speed of light c finite and velocities close to c, coincide with classical Newtonian physics as one goes to velocities v

Dear Yurij,

The short answer to your question is that in General Relativity (GR) the energy-momentum is not locally conserved but it is conserved globally.

Elaborating a little bit on this point, for classical and quantum mechanics in flat space there is a result called the Noether theorem stating that for each symmetry there exist a conserved quantity e.g. for the time and space translation symmetries the associated conserved quantity are respectively energy and momentum. The Noether theorem in this simple form doesn't hold for GR, this is due to the fact that GR is a theory invariant under diffeomorphism (general change of coordinates).

On the other hand it is still possible for a general spacetime to define global conserved quantities, this is achieved using the Arnowitt Deser Misner (ADM) formalism, for the original work see [1]. In the ADM framework the globally conserved charges like the energy are defined in terms of the isometry group of the spacetime at spatial infinity.

This is for example the way the mass, angular momentum and charge are defined for a charged and rotating Kerr black hole.

1) Arnowitt, R.; Deser, S.; Misner, C. (1959). "Dynamical Structure and Definition of Energy in General Relativity". Physical Review 116 (5): 1322–1330 http://prola.aps.org/abstract/PR/v116/i5/p1322_1

Dear Lorenzo,

yes you are right the Noether theorem is the fundamental basis of the conservation laws in modern field theory of all fundamental interactions.

Hilbert (1917) was the first who noted that "I contended ... in general

relativity ... no equations of energy ... corresponding to those in orthogonally invariant theories". Here "orthogonal invariance" refers to theories in the flat Minkowski space.

The history of the energy-momentum tensor (EMT) definitions in gravity theory and two ways for a solution of this conceptual problem is presented in my paper (see arXiv: 0809.2323).

Arnowitt Deser Misner (ADM) formalism does not solve the energy problem in GR, they wrote: . “Thus a statement of the dynamics is meaningful only after a set of coordinate conditions have been chosen. In general, the true Hamiltonian will be time dependent even for an isolated gravitational field”. This is why Feynman used the field theory approach to the gravity theory as a second rank tensor in Minkowski space (see his book Feynman R., Morinigo F., Wagner W., (1995), Feynman Lectures on Gravitation, Addison-Wesley Publ. Comp.)

The dependence of physical quantities on coordinate systems in geometrical GR is in contradiction with independence of physics on coordinate systems. The confusion between coordinate and reference systems is an unsolved conceptual problem in GR. In cosmology it appears in the static and non-static forms of De Sitter metric.