Scalar Metrics Tensor Gradation Rank-n Quantum Gravity Physics

Abbreviations

GR: General Relativity; QR: Quantum Relativity; LQG: Loop Quantum Gravity; GUTs: Grand Unified Theories; LHC: Large Hadron Collider; CMBR: Cosmic Microwave Background Radiation; QFT: Quantum Field Theory; TEGS: Teknet Earth Global Symposia.

Introduction

The quest for unifying the general relativity (GR) as well as the quantum mechanics (QM) has led to various theoretical frameworks, including string theory and loop quantum gravity. However, a complete theory of quantum gravity remains elusive. Emergent theories like Loop Quantum Gravity (LQG) propose that spacetime is nothing but quantized at the smallest scales, with a suggestion that both the gravity and spacetime geometry can be emerging from the quantum states of the gravitational field as emergent properties. So, one can visualize the space –time through LQG as a network of loops, known as spin networks, where whose edges represent the quantum states of the gravitational field, while nodes represent as an interaction between these states. This discrete structure , therefore, helps us to avoid the crisis of infinities that arise in classical gravity theories. Recent researches on LQG have explored the concept of boundary maps, where the quantum states of the bulk geometry are represented as wave-functions on the boundary. This approach helps to understand the relationship between the bulk and the boundary in quantum gravity. Additionally, matrix models offer an alternative approach in to describing the quantum spacetime. The use of matrices in these models in quantifying the degrees of freedom of spacetime provides a different perspective related to quantum gravity. For example , certain solutions in matrix models suggest a quantized cosmological spacetime, where the fluctuations of these matrices can give rise to gravitational phenomena. The idea that the bulk properties of spacetime can be understood through its boundary can be treated as is a significant concept in both LQG and matrix models. This correspondence is crucial for understanding how quantum states of geometry relate to the observable phenomena. Both the LQG and matrix models suggest that spacetime and gravity might be those emergent properties which are arising from the fundamental quantum structures little bit more. Please see extensive references as listed at the end of this article that will explore further on what is the sized of the literature associated with this present paper [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76].

The author has considered advancing quantum gravity with gravity and tensor time metrics [40, 41]. Emergent theories like Loop Quantum Gravity, which states the quantization of Spacetime as discrete structure at the smallest scales, suggests that gravity and spacetime geometry emerges from the quantum states of the gravitational field as emergent properties The Aspects of Underlining Spacetime maybe a Result of: (i) mixing of the domains spatial temporal; (ii) fixing of the metrics equations with c2dt2. We propose in this paper a method to quantize the gravitational field by examining the gradation of rank tensors within a framework of metric wavefunction.

The purpose of this study is to address the fundamental issues of spacetime quantization and to explore the potential for the unified description of gravity and quantum phenomena. The gradation of time tensors from rank-6 to rank-1 vectors in spacetime presents a novel approach to unifying General Relativity (GR) and Quantum Relativity (QR). In this study we examine:

• The metrics that affect time wavefunctions through Hamiltonian action, and
• Then compares them to those influencing gravitational gradients.
• Finally, an approach of tensor gradation within a framework of metric wavefunction for expressing the existing quantum gravity and the emergent theories that are given below through highlighting unique aspects as well as common goals.

**Results with Mathematical Derivations, as well as Discussions and Experimental Validations**

Results: Logical approach having Quantum Gravity general analysis with idea of $g_{\mu\nu} |\Psi\rangle$, where $g_{\mu\nu} \equiv$ gravity metric; $\Psi \equiv$ wavefunction quanta was considered first. Another way was to use the metric tensor ($g_{\mu\nu}$) as a quantum field. The wavefunction ($\Psi$) of the gravitational field can be expressed as a functional of the metric tensor: $\Psi[g_{\mu\nu}]$. This wavefunction describes the probabilistic distribution of different metric configurations. Quantization of the rank tensors applying the principles of QFT (quantum field theory) [49] to the operator metric tensor, treating it as an operator with associated creation and annihilation operators to give equation metric relationship: $\hat{g}{\mu\nu} = \left[ d^3 k \left( a{\mu\nu} (k) e^{ikx} + a_{\mu\nu} ^\dagger (k) e^{-ikx} \right) \right],$ where $a_{\mu\nu} (k)$ and $da_{\mu\nu} ^\dagger (k)$ are the annihilation and creation operators, respectively. The wavefunctionality of the metric field is extended to account for tensor gradation: $\Psi \left[ g_{\mu\nu} \dots g_{\mu\nu} \right]$.

This wavefunction describes the quantum states of the metric field with varying tensor ranks, providing a comprehensive framework for spacetime quantization. However, this simple approach, even with Schwarzschild-like metric wasn't enough to quantify quantum gravity. One of the reasons addresses the question of domains. Dimensions, coordinates, and the units problem underlining spacetime maybe a result of: (i) mixing of the spatial temporal domains; (ii) fixing of the metrics equations with $c^2 dt^2$ to force-fit matching units, requiring to have constant speed of light criteria. Besides questioning of the units as well as unitarization like the $c^2 = 1$ and other constants like h/c taken as 1, incorporation of units with appropriate dimensions have posed unresolved problem to accommodate quantum with general relativity, apart from other aspects. Hence, the author had novel approach to solution by converting everything parametrically to one domain, for example, onto time domain by operator calculus algebra with general transforms, formalism of which had been already presented on many peer-reviewed publications [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. This resulted eventually to the approach of Rank-n tensor time PHYSICS [41]. The concept of wavefunctionality is extended to the metric field, where the wavefunction describes probabilistic distribution of metric configurations. This framework provides a natural way to incorporate quantum fluctuations of the metric, leading to a more comprehensive understanding of spacetime dynamics.

The author has previous results with concept that the universe operates akin to a "black box," as revealed by the graphical general transform equation, where inverse transform giving four-vector time matrix interactively coupling commutes with gravity matrix analogous to Schwarzschild metrics. The following methodologies resolve problems using techniques, such as: (i) Metrics guiding time tensor collapse interactively coupled to the gravitational gradient, with their own stochastic metrics determining the scalar spacetime event matrix. (ii) For example, front-loading "quantum foam" time tensor wavefunctions which works physically interactively mathematical equivalent to curl (iii) and couple with the gravitational "vacuum bath" gradient, acting back in time event matrix sequences. (iv) Metrics gauging both create ongoing present situational states. These aspects appear in recent peer-reviewed publications [35, 36, 37, 38, 39, 40, 41, 42, 43].

Proof of the concept Rank-4 Tensor Time Matrix [41,43]

• Rank-4 tensor time,

4 4 T S H g g p p p p x x x x µναβ µναβ µναβ µ ν α β µ ν α β ∂ ∂ = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

1 gµναβ −    

• Manipulating,

4 4 S H T p p p p x x x x µναβ µ ν α β µ ν α β $$ 1 = \frac {\partial^ {4} S}{\partial x ^ {\mu} \partial x ^ {\nu} \partial x ^ {\alpha} \partial x ^ {\beta}} = \frac {\partial^ {4} H}{\partial p _ {\mu} \partial p _ {\nu} \partial p _ {\alpha} \partial p} $$ • Hence: Action, { } $$ S = \int \left\{\left[ g _ {\mu \nu \alpha \beta} \right] ^ {- 1} T _ {\mu \nu \alpha \beta} \partial x ^ {\mu} \partial x ^ {\nu} \partial x ^ {\alpha} \partial x \right. $$

1 g T x x x S x µ ν α β µναβ µναβ • Hamiltonian, { }

1 g T p p p p H µναβ µναβ µ ν α β −   =  ∂ ∂ ∂ ∂  ∫

where p: momentum; x’s are the space-time coordinates; {μ, ν, α, β} = 0,1,2,3; and the metric gμναβ converts action derivative to appropriate domain of time [41, 43].

Since, $i\hbar \frac{d}{dt}|\Psi\rangle = \hat{H}|\Psi\rangle$ [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation] to get the equation: $i\hbar \frac{d}{dt}\Psi(r,t) = -\frac{\hbar^2}{2m} \nabla^2\Psi(r,t) + V(r)\Psi(r,t)$

The momentum-space counterpart involves the Fourier transforms of the wave function and the potential: $i\hbar \frac{\partial}{\partial t}\bar{\Psi}(p,t) = -\frac{p^2}{2m}\bar{\Psi}(p,t) + (2\pi\hbar)^{-3/2} \int d^3 p \bar{V}(p-p')\bar{\Psi}(p,t)$

The functions $\Psi(r,t)$ and $\bar{\Psi}(p,t)$ are derived from $|\Psi(t)\rangle$ by $\Psi(r,t) = \langle r|\Psi(t)\rangle$, $\Psi(p,t) = \langle p|\Psi(t)\rangle$ where $|r\rangle$ and $|p\rangle$ do not belong to the Hilbert space itself but have well-defined inner products with all elements of that space.

Our results indicate that the immediate past metrics affecting time wavefunctions differ significantly from those affecting gravitational gradients. The gravitational gradient curling tensor time depends on the immediate future of the action that generated the time wavefunction, transforming it into a scalar spacetime event that advances the timeline. This separation of metrics allows for the empirical prediction of future time events. Schrödinger equation, which had been shown above, the momentum-space counterpart involves the Fourier transforms of the wave function and the potential.

Matching Tensor Ranks [41, 43]: To match the rank of the tensor time with that of the gravitational gradient, we use the time matrix in a compact form of the Kronecker delta and the Levi-Civita symbol: $T_{uvv} = g_E \delta_{uvv} \delta_{uvf} E + g_L \int_{uvv} L + g_P \delta_{uvf} \delta_{vf} P + g_S \delta_{uvf} \delta_{vf} S$, where $E$ is the energy, $L$ is the angular momentum, $P$ is the momentum, and $S$ is the entropy. These four fundamental informational observables characterize the physical quantifiability of any system in any domain of reality. This results in mostly rank-2 matrices, except for one rank-4 matrix, which can be related to a 4D hyperspace matrix and then flattened using IT techniques like factorizing [41, 43].

The author Iyer R [41, 43] has rationalized elsewhere as to why Rank4 matrix time needs extension to Rank6 tensor time to quantify complex phenomena that involve translational domain $\{t(x), t(y), t(z)\}$ and spherical rotations $\{t(\theta), t(\varphi), t(Y)\}$ in different realities, such as quantum entanglement, wormholes, and parallel universes; additionally, boosts, Euler rotations, and/or chirality would be addressed. Rank 6 $t_{xyz\theta\varphi}$ in Equation (6) notation, $T_{ijklm}$ in suggestive physics notation may be written as: $T_{abcdef}$ or $t_{abcdef}$. The rank-6 tensor time matrix can be used to describe the transformations of rotational parameters in different domains of reality, such as:

• Quantum Space: the domain of reality where space fields conceal time, and quantum entanglement phenomena reveal the nonlocality of time probability aspects;
• Mesoscopic Environment: the domain of reality where time coexists with entropic matter space fields, and neutrino oscillations and gravitational lensing indicate the influence of cosmic gravity;
• Astrophysical Field: the domain of reality where time is curved with space sense containment extending to infinite vacuum, and Hawking radiation reflects the quantum as well as thermal effects of black holes with the causality of negative energy.

The rank-6 tensor time matrix was written as [41, 43]: $T_{ijklm} = g_{ijklm} \frac{\partial \theta}{\partial t} \frac{\partial \phi}{\partial t} \frac{\partial \psi}{\partial t} \frac{\partial \chi}{\partial t}$, where $\theta_p$, $\phi_k$, $\psi_m$, and $\chi_n$ are the Euler angles of rotation in the i-th, k-th, m-th, and n-th domain of reality, respectively, and $t_p$, $t_k$, $t_m$, and $t_n$ are the time coordinates in the j-th, l-th, n-th, and p-th domain of reality, respectively; metric $g_{ijklm}$ converts angular velocity to appropriate time domain. A rank-6 tensor time matrix has 64 components, corresponding to the 64 possible combinations of the six indices $i, j, k, l, m$, and $n$. These equations represent the "black box" universe, where the rank-6 tensor time matrix is the key to unlock the secrets of reality [41, 43]. We can manipulate this to obtain the scalarized form by integrating over the appropriate variables to convert to parameter of action $S(\theta_i\phi_k\psi_m,\text{and}\chi_n) = \int [g_{ijklm}]^{-1} T_{ijklm} \partial_t \partial_t \partial_t \partial_t p$.

Having Euler angles expressed as a function of time R.H.S. can be evaluated to proceed to equivalent Kronecker delta Levi Civita form and then use flattening matrix factorization obtaining gaging rank of a rank-6 tensor time matrix to minimum number of rank-1 tensors written as the outer product of six vectors, one for each index like $(u_1 \otimes u_2 \otimes u_3 \otimes u_4 \otimes u_5)$ with components, for example, like $(u_1 \otimes u_2 \otimes u_3 \otimes u_4 \otimes u_5)_{ijklm} = u_1 u_2 j u_3 k u_4 u_5 m_6$ where $u_1, u_2, u_3, u_4, u_5, u_6$ are vectors (that can be added together to obtain the given tensor) [41, 43].

Applying this technique, we can obtain quantum domain space rank-6 tensor time factorizing: $T_{ijklm} = \vec{t}_1 \otimes \vec{t}_2 \otimes \vec{t}_3 \otimes \vec{t}_4 \otimes \vec{t}_5 \otimes \vec{t}_6$ in vector time multiplication. We will apply these techniques repeatedly achieving formalisms scalarizing time and the gravity metrics.

**Results of Mathematical Physics Derivations**

Scalarizing Time: After flattening of tensor to vector operator like-time matrix equates:

$$\hat{i}_{ijklm} = \vec{t}_1 \otimes \vec{t}_2 \otimes \vec{t}_3 \otimes \vec{t}_4 \otimes \vec{t}_5 = \langle \vec{t}_q \rangle, [q] \{1, 2, 3, 4, 5, 6\}$$

multiplicative vectors. Sense-fields-effect mesoscopic

coupling with gravitational gradient w G ρ

will generate gravitational fields wavefunction, € €S w g G = Ψ ρ

to energize giving momentum interactively to density time gravity Schwarzchild-like expanded Matrix, |Gw><tq| = Gij, having (i, j = 1, 2, 3, 4) producing € € q S w q g t G t = Ψ ρ ρ ρ =tqg, having t = scalar time; “qg” representing quantum gravity. Thus, scalarizing tensor time with matrix factorization mechanism-like magic square prime factorizing vacuum (earlier publication) absolute matrix to originate gravity- time matrix is possible.

We can generalize metrics technique applied to scalarize time to gravity-time metrics. Achieving that will involve the following sequences: Rank 6-time metrics, written in the general form like ˆ T ijklmn G is matrix factorizable to Rank 1 time metrics of the kind t q G ρ “bra” matrix that has capability to have interactive coupling with gravity gradient, g w G ρ “ket” matrix, similar to what was shown earlier with

Schwarzschild-like metrics analogue, to obtain configuration

of density matrix quantification gravity-time,

ˆ gt v Gµ   

, that is

written to be:

$$ \left[ ^ {g t} \hat {G} _ {\mu v} \right] = \left| ^ {g} \vec {G} _ {w} \right\rangle \left\langle^ {t} \vec {G} _ {q} \right| $$

, whereby {µ, ν} = 0, 1, 2, 3;

this matrix will have a fully Euclidean type expanded matrix G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G $$ \left[ \begin{array}{c c c c} G _ {0 0} & G _ {0 1} & G _ {0 2} & G _ {0 3} \\ G _ {1 0} & G _ {1 1} & G _ {1 2} & G _ {1 3} \\ G _ {2 0} & G _ {2 1} & G _ {2 2} & G _ {2 3} \\ G _ {3 0} & G _ {3 1} & G _ {3 2} & G _ {3 3} \end{array} \right] = \left[ \begin{array}{c c c c} G _ {0 0} & G _ {0 x} & G _ {0 y} & G _ {0 z} \\ G _ {x 0} & G _ {x x} & G _ {x y} & G _ {x z} \\ G _ {y 0} & G _ {y x} & G _ {y y} & G _ {y z} \\ G _ {z 0} & G _ {z x} & G _ {z y} & G _ {z z} \end{array} \right] $$

which represents expansion from 2x2 matrix with zero

point or origin in the matrices as G00 then Cartesian

coordinates (x, y, z) so that whole matrix has {µ, ν} = 0, x,

y, z to account for 4D type space time gravity; note that

can represent time to spatial coordinate with Minkowski

spacetime which becomes like Schwarzschild metrics {c2dt2}

00 0 0 0 00 01 02 03

x y z

0 10 11 12 13 x xx xy xz

0 20 21 22 23 y yx yy yz

of the form

0 30 31 32 33 z zx zy zz

2 2 2 2 2 2 2 2 2

2 1 sin 2 1 GM dr ds c dt r d d GM c r c r $$ = - c ^ {2} \left(1 - \frac {2 G M}{c ^ {2} r}\right) d t ^ {2} + \frac {d r ^ {2}}{1 - \frac {2 G M}{c ^ {2} r}} + r ^ {2} \left(d \theta^ {2} + \sin^ {2} \theta d \phi^ {2}\right) $$ ( )

2 , Schwarzschild K [64], Einstein A [19, 20] discussed more below.

In the 2x2 reduced matrix form it conforms consistently with Iyer R, et al. [32] point PHYSICS Helmholtz [31] decomposition 2x2 matrix fields, discussed earlier:

ˆ ˆ , ˆ ˆ

  =      quantifying zero point and microblackhole, which is like a point singularity within the Schwarzschild metrics above, demonstrating quantifying power characterizing compact astrophysical domain temporal spatial fields that has gravity-time incorporated, extended beyond Loop Quantum Gravity Theory [63].

v r v g v r g v v v r

µ µ µ µ µ µ ε ε ε ε ε ε ,

g, Above 4x4 matrix,

$$ \left[ ^ {g t} \hat {G} _ {\mu v} \right] = \left| ^ {g} \vec {G} _ {w} \right\rangle \left\langle^ {t} \vec {G} _ {q} \right| $$ is mesoscopic representation intermediary between astrophysical and quantum spacetime of the universe. We can think compactly in terms of schematics: Gravity metrics with time metrics will proceed to determine future event matrix, having algorithm quantifiable foregoing PHYSICS stochastically moving progressively. Similar to explanations within Introduction section on physics literature, we can utilize tensor rank gradation by matrix factorization to convert higher rank like rank6 to rank1 vector or even to rank0 scalar parameters, formalizing like techniques enunciated below, summarizing gist with schematics results, showing how rank of tensor is critical to define different domains of reality.

Quantum domain space rank-6 tensor time =

1 2 3 4 5 6 t t t t t t ⊗ ⊗ ⊗ ⊗ ⊗ υρ υρ υρ υρ υρ υρ

like monopoles Mesoscopic environment spacetime = [oscillation]{rank4, rank2] gravity-time metrics Astrophysical domain temporal spatial fields = rank2 Gij= |Gw><tq|| like gravitons Graphically this may have Plotting Schematics like:

Click to enlarge

shows how the different domains of reality can be represented quantitatively schematically to show rank gradation with astrophysical domain representation possibly with rank 2 temporal spatial fields like Schwarzschild metrics. When we proceed to mesoscopic environment, we require oscillatory gravity-time metrics between rank4 and rank 2 spacetime which is like Kronecker delta and the Levi-Civita symbol:

$$ T _ {\mu \nu \alpha \beta} = g _ {E} \delta_ {\mu \nu} \delta_ {\alpha \beta} E + g _ {L} \in_ {\mu \nu \alpha \beta} L + g _ {P} \delta_ {\mu \alpha} \delta_ {\nu \beta} P + g _ {S} \delta_ {\mu \beta} \delta_ {\nu a} S $$

discussed above having mixed ranks of 2 and 4, capable of

incorporating energy, momentum, angular momentum and

entropy aspects that are typical of material-environment

mesoscopic systems (Figure 3). Applying transformation

with Euclidean to Riemann geometry, like indicated by the

SCHEMATIC PHYSICS below, we can effectively convert

rank4, for example, ϵ_μναβ to the rank2 equivalent. However, quantum regime requires much higher rank, i.e. rank6 tensor since entities are essentially stripped to elemental form, causing field polarization, for example, polarization with vacuum energy. Essentially this will reflect onto geometric effect having eigen translational and rotational directions pulled-out or dissociated like algebra form, possibly non- Abelian or multi-directional dimensional aspects evolving to models of the String theoretic extra-dimensionality.

Using the above tensor rank factorization techniques, these tensors may be converted to rank1 vectors, which are possibly scalarized like above time metrics.

Overall physics graphically schematically shown elsewhere [41, 43] is brought to attention here to get picture how origin of universe from “quagmire” noisy superluminal phase, that will have only imaginary spacetime, unobservable dark matter energy form having only wavefunctions that are like rank6 tensors or only random sense fields. These aspects will be considered more in the subsequent publications. Here, only schematics will be given below to illustrate that (Figure 4).

Click to enlarge

Referring to simplified schematics explaining physics picture-wise above, the following aspects may highlight key points, taking away to proceed to better understanding that.

Point “Singularity” and Faster than Light Superluminal “Quagmire” Related to Tensors: Transverse waves curled up from longitudinal waves get activated by random distortions of superluminal “quagmire” to generate zero-point- microblackhole Calibu-Yau 6-fold brane string theory like manifolds. With true and false vacuum creations, sufficient energy will be available to activate Hod-PDP mechanism originating particle spectra. Vacuum interphase having light mediating popup subluminal phase particle to matter energy signals with evaporation process stabilizing real matter universe will also have condensation process matter to entropy symmetry. Schematic flowchart picturing rank-n tensor expanding and collapsing to more stable rank2 4x4 matrix form mesoscopic scale have been sketched to depict quantum, mesoscopic, astrophysical levels. References listed below with the author’s earlier publications also having more references explain in detail these aspects further.

The faster than light speed maybe happening primarily with higher rank tensors at quantum level wavefunction occurring at superluminal phase with phenomena of entanglement superposition processes making instantaneity till wavefunction becomes collapsed to equivalent scalar parametric metric fields.

Gradation of tensors’ rank provide through sense-given domains of reality within existential spatial temporal gauge fields having going up the rank in the quantum level, while mesoscopic to astrophysical levels, the rank goes towards scalar observables. They will have consonance with Table of Realities within advanced Discontinuum PHYSICS [71], where gradation scales argumentation deeply logically considered having specific numbers. Here, mathematical algebra with matrix metrics have been quantitatively algorithmized tensors to scalars, with questions regarding measurements, physics having dark matter energy, quantum superpositions entanglements, mesoscopic existence, point singularities, blackhole, exotic systems, physics proposing multiverse, higher dimensional strings, M-theory with branes [72], wormholes, superluminal communication, operator universal matrix simulation “mind over matter” alternate universes or multiverse only touched perhaps to hint that these are developable from these formalisms, that will be the subject of further enquiry. Here, gravity-time interactive coupling aspects will be quantified further analyzed below to advance formalism towards unified PHYSICS with discretized spacetime having quantum ensembles, pointing M-branes oscillations of gravity-time, following literature enumerated earlier in Introduction of this paper.

We will proceed now to incorporate gravity into time metrics, like analysis extended to scalarize gravity gradient metrics to rank1, noting General Relativity principle of stress-tensor that quantifies mass tensor, discussed more below. Before advancing scalarization of gravity gradient, the following will have to be considered specifically contrasting with time metrics scalarization technique.

Literature Discussion Highlighting Key Aspects to Consider below: Einstein’s Field Equations describe the gravitational interaction in terms of the curvature of spacetime [20, 64]. They are given by (Figure 5):

Click to enlarge

1 2 G R g R µν µν µν = − , where: Gμν the Einstein spacetime curvature tensor combines Rμν the Ricci tensor which gives curvature of spacetime on a pseudo-Riemannian manifold [20], gμν metric tensor, and R, scalar Ricci curvature. Λgμν is the curvature spacetime cosmological constant; G is the Einstein gravitational constant; Tμν is the stress-energy tensor, describing the distribution of local energy, momentum, and stress; and then c is the speed of light in vacuum. The Schwarzschild’s metric, describing gravitational field outside a spherically symmetric mass, such as non-rotating black hole is given by having Ricci-flat spacetime, Rμν = 0 (vacuum), written in calculus form:

2 2 2 2 2 2 2 2 2

2 1 sin 2 1 GM dr ds c dt r d d GM c r c r $$ ^ {2} = - c ^ {2} \left(1 - \frac {2 G M}{c ^ {2} r}\right) d t ^ {2} + \frac {d r ^ {2}}{1 - \frac {2 G M}{c ^ {2} r}} + r ^ {2} \left(d \theta^ {2} + \sin^ {2} \theta d \phi^ {2}\right) $$ ( )

2 where: s is spatial distance (distance from the center of a massive object, like a black hole or massive star); t is time coordinate (observer experienced time far away from the massive object, asymptotically flat region; for an observer at rest relative to the massive object, t is proper time measured by a clock at that location); M is mass of the central object; c is speed of light in vacuum; r is radial coordinate; {θ, ϕ} are angular coordinates; also, ds is spacetime distance; and dr is spatial distance. The author will refer to Standard PHYSICS text on General Relativity to get in-depth knowledge with extensive literature.

Gradation of Time Tensor: Rank6 to rank1 vector spacetime – General Relativity PHYSICS rank1 vector spacetime-like to rank6 time tensor – Quantum Relativity PHYSICS Metrics that affect time wavefunctions by Hamiltonian (action) via Rank6 time tensor per algorithm immediate past is different from metrics that affect gravitational gradient, since gravity is more localized phenomena stemming out of curvature of spacetime to produce unidirectional effect (these aspects will be considered more in detail at later publications). Hence, gravitational gradient that curls tensor time will depend on immediate future of the action that generated time wavefunction to make it scalar spacetime event to advance timeline. Prediction of the future events will then be highly complex, particularly metrics that decide tensor time wavefunction vary from metrics operating gravitational gradient. These scenarios are based on interpreting algorithmic equations developed by my ongoing rank-$n$ tensor time conjectures. By making the separation of the two metrics, assumed to vary independently much like separation of the space and time for solution of quantum wavefunction solutions, empirical prediction of future time event matrix is possible. Applying data statistics of possible event matric states, extreme value statistical projection as well as expectation value may help to determine event path "timeloop" and/or trajectory of most possible states of time event sequential matrix ongoing. Because of the statistical nature of quantum probabilistic wavefunctions, metric stochastics would be key to predictive determinant. These points will elaborate at later publications. Here, in this paper it suffices to mention contrast. A formalism that will help to scalarize gravity to enable measurable observables will be advanced below ensuring quantum ensembles.

Scalarizing Gravity: Like scalarizing tensor time, mass tensor $\langle \vec{m}w \rangle$, that is like stress-tensor with General Relativity interacts with gravitational fields wavefunction $\left| \Psi{kg} \right\rangle$ to produce $\langle \vec{m}w \| \Psi{kg} \rangle = m_{wg}$, having $m =$ scalar mass; "wg" will represent weight with gravity. Hence, $m_{wg}$ will be gravitational mass, while $\langle \vec{m}w \rangle =$ inertial mass from factorization of tensor mass energy like mass matrix $\dot{m}{affy} = \vec{m}1 \otimes \vec{m}_2 \otimes \vec{m}_3 \otimes \vec{m}_4 = \langle \vec{m}_w \rangle$, where $\{w\} = 1, 2, 3, 4$ representative of like foregoing analysis with a 4D toroidal space to Euclidean Cartesian coordinates. Mass tensor $\dot{m}{affy}$ will be like the General Relativity stress-energy tensor $T_{uv}$ describing distributions - local energy, momentum, and stress to create gravitational interaction in terms of the curvature of spacetime.

**Experimental Validations**

To validate the theoretical framework presented, we propose the following experimental approaches [1, 4, 5, 17, 21, 65, 72].

Astrophysics

Gravitational Wave Detections: Quantum Signatures in Gravitational Waves: Next-generation gravitational wave detectors, such as the Einstein Telescope and Cosmic Explorer, could potentially detect quantum gravitational signatures. These detectors might observe deviations from classical predictions, indicating the presence of quantum effects in gravitational waves. Primordial Gravitational Waves: The approach could provide new predictions for the power spectrum of primordial gravitational waves, which can be tested through B-mode polarization measurements of the Cosmic Microwave Background (CMB).

Conclusions

In this paper, we have explored the gradation of time tensors from rank-6 to rank-1 vectors in spacetime, providing a novel approach to unifying General Relativity (GR) and Quantum Relativity (QR). By examining the metrics that affect time wavefunctions through Hamiltonian action as well as comparing them to those influencing gravitational gradients, we have proposed a method for empirical prediction of future time events. Quantum domains might cause fields polarization to the rank6 tensor time, while mesoscopic environment would manifest rank mixing 2 and 4. Astrophysical regions would demonstrate a rank2 tensor, analogous to Schwarzschild metrics.

While the tensor gradation approach shares common goals with existing quantum gravity theories, such as the quantization of spacetime and the unification of general relativity and quantum mechanics, it offers unique insights through the gradation of rank tensors and the concept of metric wavefunctionality. These features provide a fresh perspective on the discrete structure of spacetime and the nature of gravitational interactions at the quantum level [28, 52, 59, 74].

Tensor gradation approach within a metric wavefunction framework shares many conceptual similarities with emergent spacetime theories. Both of these perspectives would in general emphasize further the non-fundamental quantized nature of spacetime and gravity, proposing that these arise from deeper quantum phenomena. By exploring the gradation of tensors and the probabilistic nature of metric fields, the current approach provides a novel and complementary perspective on the emergence of spacetime with having quantum ensembles.

Our results demonstrate that scalarizing tensor time and gravity through matrix factorization mechanisms can lead to the formation of a gravity-time matrix. This process involves the interactive sense-fields-effect mesoscopic coupling with gravitational gradients, generating gravitational fields wavefunctions. The scalarizations of tensor time and mass provides a framework for understanding the classical scalar equation of motion under gravitational fields, linking scalar gravitational mass and scalar quantum gravity time operators with gravitational fields.

We have also discussed the implications of Einstein’s Field Equations and the Schwarzschild metrics to explain the gravitational interaction in terms of spacetime curvature. The proposed many experimental approaches, including high-energy particle collisions, gravitational wave observations, quantum entanglement experiments, astrophysical observations, and laboratory simulations, aim to validate the theoretical framework of rank tensor gradation and metric wavefunctionality. The experimental implications of the tensor gradation approach are vast and varied, spanning gravitational wave detection, quantum entanglement experiments, high-energy particle collisions, precision measurements, and cosmological observations. By designing as well as conducting experiments in these areas, researchers can test the predictions of the tensor gradation approach and potentially uncover new aspects of quantum gravity.

The implications of tensor gradation extend far beyond theoretical physics, potentially influencing a wide range of scientific and technological fields. By providing a deeper understanding of the quantum structure of spacetime, tensor gradation could lead to significant advancements in quantum computing, cosmology, particle physics, metrology, and more. As research in this area progresses, we can expect to see new applications and innovations that leverage the unique properties of quantized spacetime. By combining these theoretical insights with experimental validation, we hope to provide a deeper understanding of quantum gravity and its implications for spacetime. This work lays the foundation for future research in the field, potentially leading to new discoveries and advancements in our understanding of the universe.

Gradation of Time Tensors: The paper explores the transition of time tensors from rank-6 to rank-1 vectors, aiming to bridge General Relativity (GR) and Quantum Relativity (QR). Metric Differences: It highlights the differences between metrics affecting time wavefunctions via Hamiltonian action and those influencing gravitational gradients. Scalarization Process: The process of scalarizing tensor time and gravity through matrix factorization mechanisms is detailed, leading to the formation of a gravity-time matrix. Gravitational Fields Wavefunction: The interaction of sense-fields-effect mesoscopic coupling with gravitational gradients generates gravitational fields wavefunctions. Classical Scalar Equation of Motion: The scalarization of tensor time and mass provides a framework for understanding the classical scalar equation of motion under gravitational fields. Einstein’s Field Equations: The paper discusses the implications of Einstein’s Field Equations and the Schwarzschild metric in describing gravitational interactions in terms of spacetime curvature [19, 20, 64]. Experimental Validation: Proposed experimental approaches include high-energy particle collisions, gravitational wave observations, quantum entanglement experiments, astrophysical observations, and laboratory simulations to validate the theoretical framework. Future Research: The work lays the foundation for future research in quantum gravity, potentially leading to new discoveries and advancements in our understanding of the universe.

Acknowledgement

Engineering Inc. International Operational Teknet Earth Global has provided a platform launching wonderful ongoing projects that will be most useful to future human progress. Scientists worldwide specifically have contributed to the success of RESEARCHGATE forums as well as Virtual Google Meetings posted on YouTube as well per TEKNET EARTH GLOBAL SYMPOSIA (TEGS) website: https://www. youtube.com/channel/UCdUnenH0oEFiSxivgVqLYw that has successfully promoted peer-reviewed publications with ongoing project work. It is with great honor and gratitude that the author would be liking to thank collaborative international physicists ‘scientists starting with Dr. Emmanouil Markoulakis, Experimental Physicist with Hellenic Mediterranean University, Greece in mutually coauthored peer publications of many ansatzes breakthrough sciences to explore and successfully pursue quantum astrophysics. The author would too like to thank and be always grateful to Mr. Christopher O’Neill, IT Physicist with Cataphysics Group, Ireland for peer coauthored papers publications, professional graphics, and expert comments with collaborative evaluator feedback on the key contents, especially with TEKNET conference sessions discussing concepts and the graphics suggestions appreciatively exceptionally mutual project workouts ongoing. With highly engaging fruitful debates as well as discussions, the author extends profoundly high appreciation to project collaboratively engaging physicists Drs. Manuel Malaver, John Hodge, Emory Taylor, Wenzhong Zhang, Andreas Gimsa, Christian Wolf, Gerd Pommerenke, and other participating scientists. The author would be always indebted to many upcoming progressive outstanding journals who have promoted publications with excellent peer reviews of our papers’ articles.

Conflict of interest: The author declares that there is no conflict of interest regarding the publication of this article.

Funding Source: The author declares that the funding is done by self only.