"However, something very different happens with the energy-momentum of gravity itself ... "
Roger Penrose, "The Road to Reality" (2004) p. 456

To appear in my book Super Cosmos that I am keeping from release until after the election is decided.

On Oct 26, 2004, at 5:57 PM, Jack Sarfatti wrote:

OK now I recall all that.

Will get back to you tomorrow.

NOTE they are using NON-COVARIANT divergences appropriate ONLY to flat space-time where you CAN do integrals and preserve the Poincare group tensor properties after the integration, which you cannot do with Diff(4)!

This is the source of the pseudo-problem and WHY Yilmaz introduced that second flat space-time metric that Hal Puthoff loves so much.

On Oct 26, 2004, at 5:37 PM, iksnileiz@earthlink.net wrote:

Here are scans of Pauli's "Theory of Relativity", Section 61, pp 175-178:

OK I am in my office with all the relevant books. Also have Penrose's latest.

CAPS below are mine not in the original quotes.

Note 1 of several over an extended period of time for this allegedly still not settled problem in general relativity.

1.1 Wheeler in "Geometrodynamics" (Acad Press 1962) this is 40 years after Pauli's unripe immature though genius lectures in GR's infancy.

"energy not defined for a closed universe" pp 64,66

Paul have you confused "closed system" in sense of thermodynamics with closed universe, i.e. k = + 1 in FRW metric?

As distinct from a linear theory with only linear coordinate transformations like 1905 global special relativity that is violated in general relativity i.e. reduced from the admissibility of Global Inertial Frames GIFs to only Local Inertial and Non-Inertial Frames (LIF & LNIF):

"Geometrodynamics differs in character even more decisively than previously suspected from any linearized version of field theory. Being so different, Einstein's general theory of relativity has to be discussed on its own terms, in the light of the most general principles - not in terms of the workaday tools and concepts used for a linear theory.

1.1.1 The total energy is not a defined quantity for a closed universe. The usual derivation of the law of conservation of energy considers a 3-space and in this 3-space considers a 2-surface which contains all relevant parts of the system in question. However, in a closed universe, surfaces drawn at greater and greater distances from a given point ultimately have to contract and shrink to nothingness in some antipodal region of space. The law of conservation of energy reduces to the trivial entity 0 = 0 ...

1.1.2 total energy [is] a defined quantity [only for] a linearized field [where it can be represented] as the sum ... from individual [normal] modes of excitation [Fourier analysis]. In contrast, there is not the least evidence that a geometrodynamical universe naturally admits any description in terms of normal modes of excitation. No total energy, no normal modes are two novel features of classical geometrodynamics. There is a third: There existg situations in which there are pairs of points in space-time which cannot be connected by a geodesic ... the Schwarzschild metric furnishes the best studied example ..."

1.2 Roger Penrose "The Road to Reality" (2004) 19.5 The Energy-Momentum Tensor

1.2.1 "[What is] the energy density of a field?, this density being the source of gravity ... such as Maxwell's ... it does so via a tensor quantity ... This is a symmetric [02]-valence tensor which satisfies a [local] conservation equation [COVARIANT 4-divergence of energy-momentum tensor vanishes] ... The quantity T^ab collects together all the different densities and fluxes of the energy and momentum in the fields and particles ... in a standard [flat] Minkowski coordinate system, the covector T^0b defines the density of 4-momentum, and the three co-vectors T^1b, T^2b, T^3b, provide the flux of 4-momentum in the three independent spatial directions ... T00 measures the energy density, and T11, T22, T33 measure the pressure, in the three directions of the spatial coordinate axes."

[Note by Jack on micro-quantum random zero point exotic vacuum energy-momentum tensor (AKA stress-energy density tensor true for all quantum fields of any spin;

Trace{Tab(zpf)} = T00 + T11 + T22 + T33 = 0

P(zpf) = T11 + T22 + T33

Energy Density (zpf) = T00

w(zpf) = P(zpf)/[Energy Density (zpf)] = -1 ]

1.2.2 "The energy momentum tensor of the electromagnetic field [is]

Tab(EM) = (1/8pi)(FacF^cb + *Fac*F^cb)

Fac = Aa,c - Ac,a

i.e. 4D curl of EM 4-potential

Cut to the quick with the bottom line

1.2.3 "However, something very different happens with the energy-momentum of gravity itself ... when gravity is absent space-time is flat (i.e. Minkowski space), and we can use flat (Minkowskian) coordinates. Then each of the four vectors T^a0, T^a1, T^a2, T^a3 INDIVIDUALLY satisfies exactly the same conservation equation as does the vector J^1 [4-Divergence of T^a0 = 0 etc., for T^a1, T^a2, T^a3 analogous to 4 divergence of J^a = 0]. with the implication that there is an INTEGRAL CONSERVATION LAW exactly analogous to that of charge ... for each of the four 0-components [T^a0] of energy-momentum separately. Thus total mass is conserved [i.e. 3D space integral of T^00], and so are the 3 components of total momentum [i.e. 3D space integrals of T10, T20, T30]. But recall the discussion [Ch 17] of Einstein's equivalence principle, and of why this leads us to a curved space-time. Thus, when gravity is present [e.g. in a non-geodesic LNIF where "gravity force" = "g-force" ~ "connection field" = "weight" is measured - this is not same as operational measurement of the curvature tidal force that CAN BE DONE in an LIF, do not confuse the two], we must take into account that the [LNIF covariant derivative ;u] is no longer simply [the LIF ordinary partial derivative ,u] but (14.3) there are extra [connection fields that are not homogeneously multi-linear transforming Diff(4) local tensors] that confuse the very meaning of [4-divergence of T^a0] AND WHICH CERTAINLY PREVENTS US FROM DERIVING AN INTEGRAL CONSERVATION LAW FOR ENERGY AND MOMENTUM just from our [local] conservation law [Covariant 4-divergence of the energy-momentum tensor vanishes]. The problem can be phrased as the fact that the EXTRA INDEX b in Tab PREVENTS IT FROM BEING THE DUAL OF A 3-form, AND WE CANNOT WRITE A COORDINATE-INDEPENDENT FORMULATION of a 'conservation equation'(like the vanishing exterior derivative of the 3-form *J [in flat space-time Maxwell EM field theory] in d*J = 0."

[Note from Jack - so this a relation between curvature and topology. In order to get a global conservation law, topology is a new ingredient in curved manifolds. The topology of globally flat Minkowski space-time is trivial and we were lucky. Pauli and Einstein did not know this in 1921! Cartan did not do the relevant math until later! You need a closed 3-form to make a DeRham integral conservation law on the curved manifold which is generally not topologically trivial as the exact vacuum solutions of GR show!]

"We seem to have lost these most crucial conservation laws of physics, the laws of conservation of energy and momentum."

Note Paul, that so far this is general for any Tab not only tab(pure gravity vacuum).

Next time "The Fix" i.e. Killing vector fields for isometries in curved manifolds.