solution concept of LQT is basically about "A
spin network represents a .quantum state of a gravitational field: a
quantum state of space; it is characterized by a volume "v" for every
node and a half-integer "j" for every line. ... The crucial difference
between photons (the quanta of the electromagnetic field) and the nodes
of the graph (the quanta of gravity) is, that photons exists in space,
whereas the quanta of gravity constitute space themselves ((RoC2) p.
148)" The proposed distributional quantum
state H(-1/2) with corresponding inner product admits and requires With respect to the two parameters characterizing a spin network we refer to a corresponding wavelet properties (BrK6): The ( can
be compared with each other. We note that for
a convenient choice of the two wavelet functions the two
wavelet (optics) functionsGibbs phenomenon
disappears (see also (RoC) 5.5, "Complements").
(CiI) 2.8: , basically as the (CiI) 3.4.covariant divergence of the Einstein tensor is zero. At the same point in time there are multiple tests of the geometrical structure and of the geodesic equation of motion, e.g. gravitational deflection and delay of electromagnetic waves, de Sitter and Lense-Thirring effect, perihelion advance of Mercury, Lunar Laser Ranging with its relativistic parameters: time dilation or gravitational redshift, periastron advance, time delay in propagation of pulse, and rate of change of orbital period,(CiI) 3.5: " With respect to the overall conceptual idea of this homepage a Hilbert space based geometrodynamics is built on "space-time states", which are represented by elements of H(-1/2), while their corresponding "space-time energy" elements are represented by the corresponding "dual" (wavelets) elements in H(1/2). We emphasis that if ((u,v)) denotes the inner product of H(-1/2) the following relationships hold true: ((div(u),v)) ~ (u,v) ~ ((u,grad(v))).
The LQT is a modern version of the The C.
Rovelli) is the choice of a different algebra of basic field functions: a
noncanonical algebra based on the holonomics of the gravitational connections
((RoC) 1.2.1). The holonomy (or the "Wilson loop") is the matrix of
the parallel transport along a closed curve and spacetime itself
is formed by loop-like states. Therefore the position of a loop
state is relevant only with respect to other loops, and not with respect to the
background. The state space of the theory is a separable Hilbert
space spanned by loop states, admitting an orthogonal basis of spin network
states, which are formed by finite linear combinations of loop states,
and are defined precisely as the spin network states of a lattice Yang-Mills
theory."
We briefly sketch the central (RoC1): "
The three conceptual elements of the quantum mechanics
(remaining in LQT) are (RoC1): "
(2)
Indeterminacy: the future is not determined unequivocally by the past.
Even the more rigid regularities we see are, ultimately, statistical(3) Relationality: the events of nature are always interactions. All events of a system occur in relation to another system (i.e. it is about relations of
physical variables resp. phenomena).
In quantum
mechanics " is the standard L(2) framework of probability theory, statistical analysis and quantum mechanics. Some related aspects related to Schrödinger's view on statistical thermodynamics are given in (BrK7) p. 25. In other words, the physical (H(0), state resp. H(1), energy Hilbert) spaces are made of quanta (H(-1/2), state resp. H(1/2) energy Hilbert) spaces.An analogue situation regarding the compactly embedded Hilbert space H(a) into H(b) for a>b, is given by the rational numbers
(~H(0), H(1)) as subset of the real or hyperreal numbers (~H(-1/2), H(1/2)): the
rational numbers are embedded into the From a purely mathematical point of view the baseline of all mathematical models are "axioms"; the very first one to be mentioned in the context of the above is the " The last section of (RoC1) is related to philosophical aspects (including the words of Anaximander, (HeM) "Der Spruch des Anaximander"). From (HeM), Die Zeit des Weltbildes, 72) we recall the following: " From the below we quote: The Offenbar
haben wir beim Energieprinzip eine typische Entwicklung vor uns: wenn
das Prinzip der reflektierenden Urteilskraft mit einer seiner Maximen
vollen Erfolg gehabt hat, rückt sein Ergebnis aus dem Reich der Vernunft
im Kantischen Sinne, zu welchem die reflektierende Urteilskraft gehört,
in die Sphäre des Verstandes herab und wird zum allgemeinen Naturgesetz
(".law of nature)The C. Rovelli) is the choice of a different algebra of basic field functions: a
noncanonical algebra based on the holonomics of the gravitational
connections ((RoC) 1.2.1). The holonomy (or the "Wilson
loop") is the matrix of the parallel transport along a closed curve and spacetime itself is formed by loop-like states.
Therefore the position of a loop state is relevant only with respect to
other loops, and not with respect to the background. The state
space of the theory is a separable Hilbert space spanned by loop states,
admitting an orthogonal basis of spin network states, which are formed
by finite linear combinations of loop states, and are defined precisely
as the spin network states of a lattice Yang-Mills theory."
The central mathematical concepts of the GRT are differentiable manifolds, affine connexions with the underlying covariant derivative definition on corrresponding tangential (linear) vector spaces. Already the "differentiability" condition is w/o any physical justification. The only "affine" connexion concept and its corresponding locally defined metrical space framework jeopardizes a truly infinitesimal geometry, which is compatible with the Hilbert space framework of the quantum theory and the proposed distributional Hilbert space concept in (BrK). In sync with the above we propose a generalized Gateaux differential operator: let H(1/2) = H(1) + H(*) denote the orthogonal decomposition of the alternatively proposed "energy/momentum/velocity" Hilbert space, whereby H(1) denotes the (compactly embedded) standard energy space with its inner product, the Dirichlet integral; "lim" denotes the limes for t --> 0 for real t. Then for x,y ex H(1/2) the operator VF(x,y) is defined by VF(x,y):=lim((F(x+t*y)-F(x))/t), whereby the limes is understood in a weak H(-1/2) sense. The operator is homogeneous in y; however, it is not always a linear operator in y ((VaM) 3.1). The main tools used in geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing space-time. A Lorentz manifold L is likewise equipped with a metric tensor g, which is a nondegerated symmetric bilinear form on the tangential space at each point p of L. The Minkowski metric is the metric tensor of the (flat space-time) Minkowski space. The least action principle can refer to the family of variational principles. The most popular among these is Hamilton's principle of least action. It states that the action is stationary under all path variations q(t) that vanishes at the end points of the path. It does not not strictly imply a minimization of the action. The Offenbar haben wir beim Energieprinzip eine typische Entwicklung vor uns: wenn das Prinzip der reflektierenden Urteilskraft mit einer seiner Maximen vollen Erfolg gehabt hat, rückt sein Ergebnis aus dem Reich der Vernunft im Kantischen Sinne, zu welchem die reflektierende Urteilskraft gehört, in die Sphäre des Verstandes herab und wird zum allgemeinen Naturgesetz (".law of nature)The The Einstein-Hilbert functional is an ) plus a divergence integral, that is an integral whose integrand is of the form div(" This is where a alternative field-action functional of gravitation in a alternative framework (as proposed above) can be defined, based on a w). Hence for the corresponding variations of theh potential functions g(i,k) the variations of both funtionals are identical; therefore the replacement of the physically required scalar density G by the integrand of the W(g) is justified (as the essential feature of the Hamilton's principle is fulfilled with W(g))." in a "scalar density" functionPlemelj" (Stieltjes integral) sense.The electromagnetic field is built up from the co-efficients of an invariant - replacing of the mathematical " - replacing the manifold concept by a (semi) Hilbert
space-based concept, where a non-linear invariant integral functional F(V(g)) is defined by a
distributional (semi-) inner product, which is equivalent to a corresponding
functional F(R(g)) of a related inner product (where R denotes the Riesz
operators (which commute with translations & homothesis having nice
properties relative to rotations)) plus a (non-linear) compact disturbance term; the concept enables variational methods of nonlinear operators based on Stieltjes and curvilinear integrals (VaM). The Yang-Mills functional is of similar structure than the Maxwell functional regarding the underlying constant fundamental tensor. The field has the property of being self-interacting and equations of motions that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. The To merge two inconstent theories requires changes on both sides. In the above case this is about a newly proposed common "mass/substance element" concept, alternatively to the today's "mass density" concept, while, at the same time, the linear algebra tensor tool (e.g. a "density" tensor) describing classical PDE systems is replaced by non-linear operator equations defined by weak (variational) functional systems. Those (weak) equations provides the mathematical model of physical phenomena, while its correspondin classical PDE systems (requiring purely mathematical additional regularity assumptions) are interpreted as approximation solutions, only.
As a shortcut reference to geometrodynamics is
given by (WhJ). For a review of discoveries in the nonlinear dynamics of curved spacetime, we refer to ((ScM). An introduction to the foundations and tests of
gravitation and geometrodynamics or the meaning and origin of inertia in
Einstein theory is provided in (CiI). In ((CiI) 4.6) the Gödel model universe is discussed, which is a four-dimensional model universe, homogeneous both in space and time, which admits the whole four-dimensional simply transitive group of isometries, in other words, a space-time that admits all four "simple translations" as independent Killing vectors. As the Gödel model universe is homogeneous both in space and time it is stationary. In other words, in this model the cosmological fluid is characterized by zero expansion and zero shear. Thus the Gödel model runs into difficulty with the expansion of the universe. " The related clarifcations regarding the distortion tensor or gravitomagnetic field is provided in ((CiI) §5.2.6, § 5.2.7). The Laplacian equation for the gravitomagnetic vector
potential W, in terms of the current J of mass-energy is discussed in
((CiI) 5.3). The Neumann problem and its related integral equations with double layer potential leads to the Prandtl operator, defining a well posed integral equation in case of domain H(1/2) with range H(-1/2) ((LiI) theorem 4.3.2). whereby (((*,*))) defines the H(-1) inner product and
((*,*)) defines the H(-1/2) inner product of the corresponding Hilbert
scales building on the eigen-pair solutions of the Prandtl operator
equation with domain H(1/2). The proposed alternative Hilbert space based framework provides
also a "variational wave equation/ function" based approach of the "evolution of
geometric structures on 3-manifolds" in the context of Thurston's " "T If we think these ideas consistently through to the end we must expect the whole inertia, that is, the whole g(i,k)-field, to be determined by the matter of the universe, and not mainly by the boundary conditions at infinity.
The wave equation can be derived from the Maxwell equations by applying the rot-operator. It results into the "light" phenomenon. A similar tranformation is not possible for Einstein equations, which results into the "gravitation" phnomenon. The "approximation" approach is about the split g(i,k)=m(i,k)+h(i,k), where m(i,k) denotes the flat Minkowski metric. The perturbance term h(i,k) admits a retarded (only) potential representation, representing a gravitational perturbance propagating at the speed of light ((CiI) 2.10). An alternative splitting with defined distortion tensor enabling an analogue approach with electrodynamics is provided in ((CiI) 5.2.7). In ((CiI) (2.7.10)) an „ effective energy-momentum tensor for matter, fields and gravity field". As the Einstein (gravity) tensor is derived from the condition of a divergence-free energy-momentum tensor, this results to an alternative Einstein tensor. The additional term of this alternative Einstein tensor could be interpreted as "cosmologic term", not to ensure a static state of the universe (which is not the case due to Hubbles observations), but to model the "vacuum energy" properly. This then would also be in sync with the physical interpretation of the corresponding term in the modified Maxwell equations with its underlying split of divergence-free and rotation-free tensors. At the same point in time the approach avoids the affine connexion concept and the "differentiable" manifolds regularity requirement, which is w/o any physical justification.There are eight 3-dimensional geometries in the context of "nice"
metrics. The nicest metrics are those with a constant curvature, but
there are other ones. Their classification in dimension three is due to
Thurston (ScP). In (GrJ) philosophical aspects of the geometrodynamics are considered. We quote from the cover letter summary: The above
In (CoR) there is a
We mention that the existing
electromagnetic phenomena on earth are the result of plasma physics phenomena
underneath the earth crust. Those “activities” are all triggered by gravitation "forces". The above (distributional) Hilbert space based
alternative geometrodynamic modelling framework provides an alternative
approach to " ((RoC) 1.2.1). The holonomy (or the "Wilson loop") is the matrix of the parallel transport along a closed curve. ... In LQT, the holonomy becomes a quantum operator that creates "loop states" (to overcome the issue of current dynamics model of coupled gravity + matter system, simply defined by adding the terms defining the matter dynamics to the gravitational relativistic hamiltonian ((RoC) 7.3)). ..." The proposed distributional quantum state H(-1/2) above admits and requires infinite linear combinations of those "loop states" (which we call "quantum fluid" state), i.e. overcomes the Space-time itself is formed by loop-like states. Therefore the position of a loop state is relevant only with respect to other loops, and not with respect to the background. ... The state space of the theory is a separable Hilbert space spanned by loop states, admitting an orthogonal basis of spin network states, which are formed by finite linear combinations of loop states, and are defined precisely as the spin network states of a lattice Yang-Mills theory.current challenge of LQT defining the scalar product of the spin network state Hilbert space ((RoC) 7.2.3). The physical space is a quantum superposition of "" in LQT corresponds to an orthogonal projection of H(-1/2) onto H(0). This othogonal projection can be interpreted as a general model for a "spin networksspontaneous symmetry break down".
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