**Analyzing
and testing the GTR2 proposal**

In that document (EAN 133-2, first part – actually in revision – a version in French language is in preparation), we aim to test the
consequences of the GTR2 proposal ([a] and [b]). This approach reproduces the
construction of Einstein’s theory of relativity. I.e.: it studies the
variations of four basis vectors generating a space {E(4, R), Ä_{T}}
where Ä_{T}
denotes a standard tensor product deformed by some cube T of 64 scalars
arbitrarily chosen in R in pushing the precision of calculations at least until
the second order. The various possible T-cubes play a role which is like the
one which is played by Christoffel’s cube in the precise sense that they allow
the construction of a Riemann-Christoffel’s-like tensor. Nevertheless, these
cubes T cannot be systematically identified with the Christoffel’s cube of the
local Levi-Civita connection. In general, it is more convenient to interpret
them within a context already studied by E. Cartan in his document [01] (See
the French part of that exploration: GTR2-validation).

Within the GTR2 proposal, the coherence of the
geometric structure imposes the existence of bizarre electromagnetic fields
(the GTR2-EM-fields) living no imprint on the energy-impulse tensor but
eventually one on the Ricci tensor. This mathematical fact carries the following important
message: the GTR2-EM-fields and the gravitational fields are untimely related
to each other. The document called "The flow of time" [c] analyzes a
first and semi-classical consequence of that fact [c].

In the latter, we have understood how the GTR2
calculations may bring some elements enlightening the end of the life of a
black hole. In resume, we can say that an extreme pressure due to the
gravitation is forcing the ultimate constituents of the matter (quarks) to
organize themselves inside a reduced four-dimensional volume in such a way that
they form a neutral ball. For the event which has been detected by the LIGO
experiment (r_{S} = 180 km), this contraction should result into a
spherical ball with ratio r_{S}/r = 0,3, hence with a radius of about
600 km. Curiously, this ratio can also be obtained after the contraction of an
object like our Sun and ending as a neutron star [02]. On one side, the value
of the ratio is encouraging; on the other side, it tells us that "all ways
are ending in Rome" in that sense that several histories may end with
similar objects. A given scenario probably depends on the initial, chemical,
mechanical and dynamical conditions. The semi-classical scenario proposed in
"the flow of time" does not tell so much on the details. We must ask
the GTR2 properties again to get more information.

The GTR2 proposal brings only six EM-like
fields. This indication, when confronted with the vanishing of the
energy-impulse tensor related to the global effect of these six EM-fields,
strongly suggests that the edges of tetrahedrons may represent these fields. This
suggestion is a first possible link with usual or unusual considerations
developed within diverse theories of quantum gravity (e.g.: [03], [04]). We are
legitimate to ask what (and how) the edges of the figurative tetrahedra
represent: energies? The invariant quantities of the six involved EM fields?
The answers are still open.

Within the GTR2 paradigm, we are forced to
work with the following reduction of Einstein’s field equation: [R_{lm}]
= (½. R – L).
[g_{lm}].
With other words, the components of the Ricci tensor (with or without
cosmological constant) must be proportional to the components of the metric. Since
a vanishing metric would be meaningless, a vanishing Ricci tensor is compatible
with a GTR2 universe only if the cosmological constant vanishes. It is, in that
particular case, a tautology since (i) a vanishing Ricci tensor induces a
vanishing Ricci scalar and (ii) the energy-impulse tensor of the global EM
field always vanishes. When the Ricci tensor vanishes, a confrontation with the
Petrov’s classification demonstrates that the nine components of the M(3, C)
matrix representing the four-dimensional Riemann’s curvature tensor are linear
combinations of components of the six electrical-fields.

In all other cases (the Ricci tensor is not
null), this proportionality also justifies the fact that we must look for a
link between the metric and the Riemann’s curvature tensor. The traditional way
of finding that link has been given in Christoffel’s work [05]. Since the GTR2
paradigm does not necessarily coincide with the context of Einstein’s theory,
the link we are looking for should preferably not depend on the Christoffel’s
cube; at least in a first step. The natural alternative to the historical one
consists in supposing that the metric is slightly unstable and -in some ad hoc manner-
connected to the average Minkowski’ metric. This vision is coherent with the
point of view exposed in [c].

In doing so, we can recover Einstein’ field
equation, but the specific expression for the energy-impulse tensor resulting
from the procedure (although it must vanish within the GTR2 context) must yet
be rechecked (recalculated precisely for verification).

Another crucial characteristic of the GTR2
proposal is that the existence of GTR2 EM-fields is warrantied only if: (i) we
postulate that all solutions of the GTR (inclusively those of the GTR2) implicitly
contain the proof for the existence of Ä_{T}(d**x**, **…**) deformed tensor products; (ii) each
of these products is decomposed in the following way, at least approximately: |Ä_{T}(d**x**, **…**)> = [_{lm}P].
|**…**> + |_{lm}**z**>; and
(iii) each infinitesimal variation of a given component of the metric, dg_{lm},
is not identifiable with a classical Taylor’s development of that component but,
in opposition, is identifiable with: |_{T}F(d**x**) – [_{lm}P]|
– |_{lm}P|
= dg_{lm}.
For now, further mathematical explorations are needed to better understand the
links between these GTR2 EM-fields and four-dimensional volumes.

© Thierry PERIAT, 30 January 2019.

[a] PERIAT, T.: Foundations of
the GTR2, projective variations of the local tetrad until the second order and
links with the A. Einstein’s theory of relativity. ISBN 9782-36923-091-5, EAN
9782369230915.

[b] PERIAT, T.: GTR2, Riemann tensor
and electromagnetic fields; ISBN-978-2-36923-131-8, EAN 9782369231318,

[c] PERIAT, T.: GTR2, the flow of time
(Do black holes end as neutrons stars?); ISBN-978-2-36923-145-5, EAN 9782369231455,
January 2019,

**Bibliography**

[01] Cartan, E.: Les espaces
métriques fondés sur la notion de d’aire dans “Actualités scientifiques et industrielles”,
numéro 72, exposés de géométrie publiés sous la direction de monsieur Elie
Cartan, membre de l’institut et professeur à la Sorbonne; Paris, Hermann et
Cie, éditeurs, 1933.

[02] Fliessbach, T.: Allgemeine
Relativitätstheorie, 4. Auflage, Spektrum Lehrbuch, © 2003, 1998, 1995,
Spektrum Akademischer Verlag, Heidelberg, Berlin, ISBN 3-8274-1356-7.

[03] Statistical equilibrium of
tetrahedra from maximum entropy principle; arXiv:1811.00532v1 [gr-qc] 1
November 2018.

[04] Rovelli, C. and Vidotto, F.: Covariant Loop Quantum
Gravity, an elementary introduction to quantum gravity and spinfoam theory;
ISBN 9781107069626, December 2014.

[05] Christoffel, E. B.: Über die Transformation der
homogenen Differentiale Ausdrücke zweiten Graden; Journal für die reine und
angewandte Mathematik, pp. 46-70, 3 Januar 1869. This document can be studied at the University of Göttingen (extern link in English – the
University is in Germany).