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Subject: Cambridge University physicists say that nonlocality of gravity field energy still stands BG attempt fails and is trashed.

B-G do not use tetrads and in the end their theory fails - excess formal baggage like Yilmaz's failed attempt - another bites the dust.

I suspect any theory needing a second shadow connection in addition to the Levi-Civita connection will fail, i.e. not be Popper falsifiable, but only an artifact of a redundant mathematical extension not demanded by experiment.

The physical significance of the Babak-Grishchuk
gravitational energy-momentum tensor
Luke M. Butcher,∗ Anthony Lasenby, and Michael Hobson
Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK.
(Dated: November 4, 2010)

"We examine the claim of Babak and Grishchuk [1] to have solved the problem of localising the energy and momentum of the gravitational field. After summarising Grishchuk’s flat-space formulation of gravity, we demonstrate its equivalence to General Relativity at the level of the action. Two important transformations are described (diffeomorphisms applied to all fields, and diffeomorphisms applied to the flat-space metric alone) and we argue that both should be considered gauge transformations: they alter the mathematical representation of a physical system, but not the system itself. By examining the transformation properties of the Babak-Grishchuk gravitational energy-momentum tensor under these gauge transformations (infinitesimal and finite) we conclude that this
object has no physical significance."

However, this spacetime coordinate gauge transformation is unlike the internal symmetry redundant gauge transformations of the U1 SU2 SU3 electro-weak-strong forces because the equivalence organizing principle makes the former physical in terms of the locally coincident detectors measuring the same events (which can be distant on or inside their light cones). I am not aware of any detector configurations that can distinguish A from A' = A + df where d^2 = 0 in the U1 SU2 SU3 cases. In contrast:

We have three classes of local spacetime transformations for COINCIDENT detectors:

1) Local Lorentz subgroup O1,3(x) LIF <---> LIF'

LIFs have no angular momentum about their Centers Of Mass (COM) that move on timelike zero g-force geodesics.

2) Einstein's 1916 "General Coordinate Transformations" (GCT), i.e., the local translation subgroup T4(x)

LNIF <---> LNIF'    each LNIF detector is not on a timelike geodesic and may or may not have angular momentum about their COMs.

A non-gravity force is needed to push the detector off the objective invariant local geodesic whose global pattern with its neighbors defines the real T4(x) tensor gravity field.

3) Finally the local Tetrad subgroup

LIF <----> LNIF

Indeed I suspect that we can make a commutative diagram in the sense of abstract algebra

LNIF <---> LNIF'
  |                 |
LIF   <---> LIF

This argues against extra-dimensional Kaluza-Klein theories, but not against fiber-bundle theories connecting internal and external symmetries.
Note that supersymmetry is the square root of T4.

Back to the Cambridge physicist's discussion of BG:

"Despite the central role played by the energy-
momentum tensor of matter in General Relativity, there
is no widely accepted way to localise the energy and mo-
mentum of the gravitational field itself. In the place
of a genuine solution to this problem, we are forced to
make do with an over-abundance of energy-momentum
pseudotensors, objects designed to display some or other
property befitting a measure of gravitational energy-
momentum, but whose coordinate dependence renders
them of little physical significance beyond giving the cor-
rect integrals at infinity in asymptotically flat spacetimes.
Even for weak gravitational waves, the best measures at
our disposal only become meaningful once we have aver-
aged over many wavelengths.

The canonical response to the gravitational energy-
momentum problem is to dismiss it as “looking for the
right answer to the wrong question”[2]; but while the
well-known argument presented by Misner, Thorne and
Wheeler is certainly compelling, it is far from watertight.
They remind us that the equivalence principle ensures
that all “gravitational fields" (i.e. Levi-Civita connection)
can be made to vanish at a point by a suitable choice of coordinates, and
conclude that because gravity is locally zero, there can
be no energy density associated with it.

However, this argument fails to consider tensors containing second deriva-
tives of the metric, which unlike (Levi-Civita connection)
cannot be made to vanish by choice of coordinates, and really do reflect the
local curvature of spacetime: for example, the Riemann
tensor can be used to construct objects such as the Bel-
Robinson tensor [3]. Misner, Thorne and Wheeler also
point out that, while the matter energy-momentum tensor derives its physical significance by curving space, a similar tensor for gravity would not be a source term for
the field equations. However, this stance is based around
a prejudice for writing the Einstein field equations as
Gab = kTab with gravity on the left and matter on the
right; there is nothing to stop us splitting up Gab in a
covariant fashion, grouping one part with Tab, and interpreting this as the total energy-momentum source, taking the remainder of Gab to be the gravitational ‘response’.
Despite these reservations, the argument in [2] remains
vindicated as yet by the failure of these escape-routes to
yield anything which can be physically interpreted as an
energy-momentum tensor.

It might appear that the only straightforward solution
to the problem is to extend the definition of the matter
energy-momentum tensor Tab (a functional derivative of
the matter Lagrangian with respect to the metric) to the
gravitational field, and conclude that the gravitational
energy-momentum tensor is −Gab/k, where  k= 8piG/c4.
The Einstein field equations could then be interpreted
as a constraint that everywhere sets to zero the sum
of gravitational and matter energy-momentum. While
one might claim this simple idea conveys some impor-
tant physical insight, it suffers from numerous problems.
Firstly, −Gab/k lacks the analytical power one expects
from an energy-momentum tensor: the ability to split
the set of all physical systems at a particular time into
classes of different total energy and momenta, so that
conservation laws alone can reveal that two particular
systems could never be part of the same spacetime. Sec-
ondly, it leads us to conclude that the gravitational field
only has energy where matter is also present, precluding
the use of this prescription to describe the energetics of
gravitational waves, or define a gravitational tension in
the vacuum between massive bodies. Thirdly, the energy-
momentum tensors for gravity and matter are conserved
separately (∇aGab = 0 and ∇aTab = 0) so that although
there is a delicate balance that keeps their sum zero, it
is not the case that energy or momentum simply ‘flows’
between gravity and matter, as ∇a(Tab − Gab/k) = 0
alone would imply. Lastly, we note that the conserva-
tion law ∇aGab = 0 actually tells us nothing at all about
the gravitational field; it is satisfied identically, without
any need for the equations of motion to hold. Because of
these drawbacks, if we are to regard −Gab/k as a solu-
tion to the gravitational energy-momentum problem, we
consider it rather a trivial one. Clearly, the reason for
this triviality is that we have over-worked the metric: we
cannot use the functional derivative with respect to a dy-
namical field as a way of defining the energy-momentum
tensor for that same field, as we will only end up writing
down the equations of motion twice. This line of rea-
soning leads us to consider that one method of attack
for this problem may be to separate the two roles played
by gab in General Relativity, that of dynamic field and
spacetime metric. 

In [4], Grishchuk develops a “field-theoretical” approach to gravitation, which expresses the physical content of General Relativity (GR) in terms of a dynamical
symmetric tensor field in flat Minkowski spacetime. Although this formulation has been carefully designed to agree with the empirical predictions of GR, in [1] Babak and Grishchuk claim that the flat-space approach allows them to define a unique, symmetric, and non-trivial energy-momentum tensor for the gravitational field.

We cannot fault Grishchuk’s formulation of gravitational dynamics within the realm of General Relativity, as agreement over predictions of ‘geometrical phenomena’ (as they would be interpreted in GR) has been achieved by design.6 However, in comparison with General Relativity, the flat-space theory possesses additional mathematical structure: two tensors hab and &ab fulfil the role played by gab alone. This extra structure endows the flat-space theory with an increased range of expres-sion, making possible the definition of tensors that cannot be constructed within the framework of GR. As we shall show, the gravitational energy-momentum tensor is one of these ‘non-GR’ quantities.7 We investigate here
whether tab (or any non-GR quantity) can be physically significant, or whether it can only ever be interpreted as an artefact of the mathematics.

Besides allowing us to interpret gravity as a force-field on flat space, the presence of &ab has had the important side-effect of increasing the space of gauge transformations of the theory. The core reason for this is that the flatness constraint (2) is not enough to define a unique
&ab for a given gab, a tensor which, through the correspondence with GR, can be used alone to construct the observable predictions of the theory. In this section we examine two transformations and justify their status as gauge transformations, i.e. that they alter the mathematical representation of a physical system, but not the system itself.

Clearly, no ‘geometric’ measurements can ever reveal which &ab is hidden beneath the gab metric, because ‘geometric’ phenomena are invariant under the &-transformation. The only possibility of revealing &ab empirically would be if we could directly measure a non-GR
tensor like tab. However, to assume that such a measurement could be carried out would make our logic circular, as for that to be possible the tensor would certainly need to be physically meaningful, and it is the truth of precisely this assertion that we have been trying to determine!


Thus we must finally conclude that the &-transformation (10) is a gauge transformation of
B &Grishchuk’s formalism, and that not only is the flat metric &ab unobservable it is impossible to define a ‘canonical’ choice of &ab in a diffeomorphism gauge covariant, systematic, and natural fashion. ...

Of course, the expected form of these invariants rather depends on what one supposes the physical content of tab to be. If it is, indeed, an energy momentum tensor, then an observer with 4-velocity ua would expect to ‘find’ some energy density

or possibly

It is easy to check that neither of these quantities are invariant under a  &-transformation,
despite the fact that we were forced to conclude that these transformations do not alter whatsoever the physical system we are examining. From this we deduce that, whatever physical meaning tab may have, since it cannot define a meaningful energy-density in the standard way, it is definitely not an energy-momentum tensor. ...

The formulation of gravity presented in [4] succeeds in
recasting General Relativity as a flat-space theory of a
symmetric tensor field. While we do not find fault with
the formalism itself, we assert that care must by taken
in its interpretation, as we believe we have demonstrated
that only those quantities which can be defined solely in
terms of GR tensors are of any physical importance. The
physically insignificant content of the flat-space formalism is a consequence of an unmeasurable field &ab, which is not uniquely determined by the requirement that it be
a flat metric tensor. ...

Accepting that &-transformations and &-fixed transformation are maps between different mathematical representations of the same physical system, we conclude that the exotic gauge transformation properties of tab cannot allow us to interpret this tensor as a local measure of the energy and momentum content of the gravitational field. Although tab is a perfectly legitimate mathematical construction, its dependence on the unmeasurable and non-unique tensor &ab renders it ill-defined, and devoid of physical meaning.

On Nov 27, 2010, at 10:20 AM, JACK SARFATTI wrote:

Thanks these are useful references on the nonlocality of gravity field energy

5. Luke M. Butcher, Anthony Lasenby, and Michael Hobson, The physical significance of the Babak-Grishchuk gravitational energy-momentum tensor, arXiv:0807.0112v1 [gr-qc]

"The canonical response to the gravitational energy-momentum problem is to dismiss it as “looking for the right answer to the wrong question” [2]; but while the well-known argument presented by Misner, Thorne and Wheeler is certainly compelling, it is far from watertight.

"They remind us that the equivalence principle ensures that all “gravitational fields” [X] can be made to vanish at a point by a suitable choice of coordinates, and conclude that because gravity is locally zero, there can be no energy density associated with it. However, this argument fails to consider tensors containing second derivatives of the metric, which unlike [X] cannot be made to vanish by choice of coordinates, and really do reflect the local curvature of spacetime: for example, the Riemann tensor can be used to construct objects such as the Bel-Robinson tensor.

"Despite these reservations, the argument in [2] remains vindicated as yet by the failure of these escape-routes to yield anything which can be physically interpreted as an energy-momentum tensor."

6. Luke M. Butcher, Michael Hobson, and Anthony Lasenby, Localising the Energy and Momentum of Linear Gravity, arXiv:1008.4061v2 [gr-qc]; Phys. Rev. D 82, 104040 (2010)

On Nov 27, 2010, at 5:36 AM, Dimi Chakalov wrote:

Thank you very much. An alternative proposal can be read at


All the best,


On Sat, Nov 27, 2010 at 12:09 PM, Carlos Castro

Dear Colleagues :

Since Exceptional algebras like E_8 were discovered thanks to the Octonions.

you might be interested in the article

"Nonassociative Octonionic Ternary Gauge Field Theories"

that was submitted to the J. Phys. A : Math and Theor.

Best wishes


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