A Novel Framework for Characterizing Spacetime Microstructure with Scaling

The Planck length is particularly significant, as it is the scale at which quantum gravitational effects are expected to become apparent, thus interactions require a working theory of quantum gravity to be studied. It is also a natural estimate for the fundamental string length scale and the characteristic size of compact extra dimensions bibitem1 . At the Planck scale, current models fail to describe the universe effectively, and a scientific model to explain the behavior of the physical universe has yet to be established. Our current understanding suggests that the universe began $\sim 10^{-43}$ seconds after the Big Bang. At this scale, the uncertainty principle reveals the quantum fluctuations of spacetime, which are integral to elucidating the original structure of the universe, the center of a black hole, and the structure of nature at a very short scale, in a framework of quantum gravity bibitem2 .

Many studies seek to address these challenges in quantum gravity, including two of the most potential of them, i.e., Loop Quantum Gravity (LQG) and String Theory. LQG is an approach to quantum gravity that quantizes spacetime itself by describing it in terms of discrete loops or spin networks bibitem2 ; Gambini ; Ashtekar ; Thiemann ; Pullin . It provides a non-perturbative and background-independent framework, where space is quantized, and the geometry emerges from the quantum states of the gravitational field. LQG introduces a new set of variables and applies loop quantization techniques, leading to finite areas and volumes, and avoiding singularities such as those in black holes Gambini2 . String Theory posits that the fundamental constituents of the universe are not point particles, but rather one-dimensional “strings” that vibrate at different frequencies to produce different particles. It provides a unified framework for all fundamental forces, including gravity, by incorporating the concept of extra dimensions and supersymmetry Zwiebach ; bibitem1 ; LeBihan . String Theory has successfully incorporated gravity into quantum mechanics, offering a potential solution to the problem of quantum gravity and explaining the existence of different particle types through string vibrations Vaid . Recent research in quantum gravity reveals various new approaches and theories Klauder ; Oriti ; Berenstein ; Pullin ; Shojai ; Carlip ; Harlow ; Pawlowski ; Braunstein ; Giesel ; King ; Bajardi ; Chakraborty ; Saueressig ; Bianchi . Although these studies have greatly improved our understanding of physics at the Planck scale, they are obviously not exhaustive.

Due to its incompleteness, quantum gravity imposes constraints on our understanding of spacetime at the smallest scales. This challenge has piqued the interest of many researchers who are drawn to the field in pursuit of unraveling its mysteries. At this scale, spacetime is hypothesized to deviate from the classical notion of a smooth continuum, potentially exhibiting a foam-like, fluctuating structure bibitem3 ; bibitem4 . This concept is rooted in the idea that the Heisenberg uncertainty principle could give rise to microscopic irregularities, as postulated by Wheeler. Additionally, Tryon’s hypothesis suggests that our universe may have originated from a quantum vacuum fluctuation, similar to the Big Bang scenario bibitem5 . Quantum fluctuations are not just theoretical curiosities but integral to the fabric of the universe’s microstructure. The significance of these fluctuations is underscored by ongoing research that highlights the necessity for a deeper exploration into the foundational characteristics of spacetime at the Planck scale. This quest for understanding is driven by the goal of reconciling the principles of quantum mechanics with the geometry of spacetime, with the ultimate goal of revealing the true nature of reality at its most fundamental level.

The Planck scale represents a frontier in theoretical physics, where quantum fluctuations dominate, and classical descriptions of spacetime become inadequate. The present work aims to explore the microstructure of spacetime at the Planck scale through micro-measurements of spacetime characterized by scaling functions. By examining the scaling behavior of micro-geometry, the research aims to provide a new perspective for exploring the nature of spacetime microstructures and offer valuable insights into fundamental spacetime properties. We develop a novel scaling-characterized metric tensor derived from the Lorentz scalar line element, enabling the analysis of spacetime microstructure. The differential operators are transformed into scaling form, providing a mathematical framework for examining fluctuation properties. The content is structured as follows: II. Micro-measuring principle and Scaling Characterization; III. Scaling Characterization in Metric; IV. The Lorentz Factor at the micro-scale; V. Micro Measurement in Spacetime with Schwarzschild Metric; VI. Scaling Geodesic Equations and Einstein Field Equations; VII. The Golden Ratio at the micro-scale; VIII. Scaling Klein-Gordon Equation and Dirac Equation; IX. Summary.

Heisenberg’s principle suggests that spacetime exhibits quantum fluctuations on a microscopic scale. These fluctuations result in uncertainties in the positions of points within spacetime coordinates and cause variations in microlengths. At the Planck scale, traditional concepts of spacetime geometry become inadequate due to quantum fluctuations. These fluctuations imply that spacetime is not a smooth manifold but has a more intricate structure.

We model the spacetime fluctuations using the differential $dx^{\alpha}$, representing spacetime microelements as microlength measurements relative to a reference length $dx^{\alpha}_{0}$. Here, $\alpha$ (equal to 0, 1, 2, 3, …) denotes the coordinate index in orthonormal frames. The microelements represent small segments of spacetime, making it a natural choice for modeling fluctuations at quantum scales, where spacetime might exhibit complex, non-continuous behavior. In quantum mechanics, uncertainty and fluctuations are inherent due to the Heisenberg uncertainty principle. Modeling spacetime at quantum scales involves accounting for these uncertainties, which microelements naturally accommodate by allowing for the representation of small changes and variations. Defining spacetime quantum fluctuations using differentials $dx^{\alpha}$ is a rational choice that fits well with current theoretical frameworks in physics. It allows for a coherent description of spacetime at very small scales, integrating principles of general relativity and quantum mechanics. This definition provides a useful starting point for exploring the quantum nature of spacetime.

The measurements of $dx^{\alpha}$ relative to $dx^{\alpha}_{0}$ is introduced by arbitrary scaling functions:

These scaling functions $L^{\alpha}(X^{\alpha}(\tau))$ suggest a way to quantify these quantum fluctuations, denoted by spacetime scales $X^{\alpha}$, and quantify how the microlength measurements deviate from their reference lengths due to quantum fluctuations. $dx^{\alpha}_{0}$ are micro reference lengths, serving as the baseline measurements in the absence of fluctuations. It establishes a well-defined reference scale that allows us to measure the deviations. To measure the micro proper length, scaling function $\tau$(X(s)) is introduced and defined as

in a similar way, where $d\lambda$ is the proper length microelement, $d\lambda_{0}$ is the reference length, and $s$ is defined as the proper time scale.

Consequently, the measurements of $dx^{\alpha}$ relative to proper length $d\lambda$ is given by

which involves choosing consistent reference lengths for $dx^{\alpha}_{0}$ = $d\lambda_{0}$. The Planck length is considered the optimal reference length as it represents the smallest measurable length scale in quantum gravity, making it a natural choice for investigating spacetime at this scale.

With these definitions, there are three types of spacetime micro measurements:

(a), When $L^{\alpha}=constant$ ($\tau=constant$), indicating that micro length measurements are conducted in a static mode (trivial measurement) without fluctuations, independent of $X^{\alpha}$($X$), indicating a perfectly stable spacetime region. When measurements are performed exclusively along a single dimension, the non-fluctuating mode can be represented as $L^{\alpha}=1$ as the $constant$ can be incorporated into the reference length through appropriate scaling.

(b), When $L^{\alpha}=K^{\alpha}_{1}X^{\alpha}+K^{\alpha}_{2}$ ($\tau=K_{1}X+K_{2}$), micro length measurements are generally performed in a linear fluctuating mode, where $K^{\alpha}(K)$ are coefficients. This suggests a linear relation between the microlength and their fluctuation. When measurements are performed exclusively along a single dimension, the linear fluctuating mode can be represented as $L^{\alpha}=X^{\alpha}$ as the coefficients can be incorporated into the reference length through translation and scaling.

(c), When $L^{\alpha}=L^{\alpha}(X^{\alpha})$ ($\tau=\tau(X)$) are nonlinear functions, micro length measurements occur in a nonlinear fluctuating mode.

Micro measurements might involve complex scaling functions, or even discrete rather than continuous measurements, leading to distinct spacetime structures. The exploration of these forms is crucial for a more comprehensive view of spacetime microstructure.

In this framework, spacetime fluctuation measurement can be described by the scaling function with non-trivial measurement. Additionally, the fluctuation induced by $X^{\alpha}$ signifies micro structures along the $\alpha$-axis. If the given $L^{\alpha}$ is the same for all axes, it indicates an isotropic structure.

Define $d\bar{x}^{\alpha}=\bar{L}^{\alpha}dx^{\alpha}_{0}=L^{\alpha}(\zeta^{\alpha}X^{\alpha})dx^{\alpha}_{0}$ with $\zeta^{\alpha}(\tau)$ being the $\alpha-axis$ rescaling factor of scale transformation. And also define $d\bar{\lambda}=\bar{\tau}d\lambda_{0}=\tau(\chi X)d\lambda_{0}$ with $\chi(s)$ being the rescaling scale transformation on $X(s)$. Then we can get

and the change of $r_{l^{\alpha}}$ relative to $r_{l}$ is

From $d(dx^{\alpha})=d\bar{x}^{\alpha}-dx^{\alpha}=(\frac{d\bar{x}^{\alpha}}{dx^{\alpha}}-1)dx^{\alpha}=(\frac{\bar{L}^{\alpha}}{L^{\alpha}}-1)dx^{\alpha}$ and $d(dx^{\alpha})=\frac{dL^{\alpha}}{dX^{\alpha}}dX^{\alpha}dx_{0}^{\alpha}$, we get

with

thus we have the first and second-order partial differential operators transformed in scaling form by

where

Eq. (8) and (9) connect with the differential operators to scaling functions, which are introduced to describe how the fluctuation of spacetime microelement can be adjusted according to a scaling factor $\zeta^{\nu}$. These equations introduce the factors $\hat{a}_{\nu}$ and $\hat{b}_{\nu}$ that modify the behavior of differential operators to account for changes in scale.

Similarly, introduce a scale transformation factor $\chi$ acting on $X(s)$. And from $d(d\lambda)=d\bar{\lambda}-d\lambda=(\frac{\bar{\tau}}{\tau}-1)d\lambda$ and $d(d\lambda)=\frac{d\tau}{dX}dXd\lambda_{0}$, we can get $\frac{dX}{d\lambda}=\hat{a}\frac{1}{d\lambda_{0}}$. Thus, we get the first and second-order differential operators transformed in scaling form by

where $\hat{a}=\frac{\bar{\tau}/\tau-1}{d\tau/dX}$ and $\hat{b}=-\frac{d\hat{a}}{dX}$.

Introducing the scale transformation $\bar{X}^{\alpha}=\zeta^{\alpha}X^{\alpha}$ with rescaling factor $\zeta^{\alpha}=\zeta^{\alpha}(X^{\alpha})$, we obtain $\frac{d\bar{X}^{\alpha}}{dX^{\alpha}}=\zeta^{\alpha}+X^{\alpha}\frac{d\zeta^{\alpha}}{dX^{\alpha}}$ and $\frac{dX^{\alpha}}{d\bar{X}^{\alpha}}=\frac{1}{\zeta^{\alpha}}$. Subsequently, this leads to the derivation of

Eq. (13) and (14) use the idea of scale transformations by focusing on partial derivatives of scaled coordinates. These equations describe how spacetime coordinates are rescaled using the factor $\zeta^{\alpha}$. This rescaling is crucial for analyzing how measurements adapt in transitioning between a linear and a nonlinear scaling regime when considering together Eq. (8) and (9), which will be further explained in the following sections VI-VIII.

Similarly, the transformation can be introduced for $\bar{X}=\chi X$ with rescaling factor $\chi=\chi(X)$. Thus $\frac{d\bar{X}}{dX}=\chi+X\frac{d\chi}{dX}$ and $\frac{dX}{d\bar{X}}=\frac{1}{\chi}$. Subsequently, we obtain

The introduction of these transformations establishes a mathematical framework for investigating the nature of spacetime at the micro-scale, which is further explored in the subsequent sections.

Micro-measuring principle at micro-scale: the distance between two infinitesimally close points in spacetime is denoted by $dx$, and its distance measurement requires definition through a comparison with a reference length. If $dx$ remains unchanging, the measured value only relies on the chosen reference length. We can establish a consistent spacetime measurement and facilitate comparisons of their respective lengths by selecting an appropriate fixed-length reference for measuring the distance. In cases where the micro distance $dx$ in spacetime undergoes fluctuations for some reason, the measurement of $dx$, relative to the reference length, is introduced as the scaling function. In the framework of quantum gravity theory, the reason for the fluctuations of spacetime microelements is considered to come from the quantum effects of spacetime at the microscopic scale, which can be inferred through the Heisenberg uncertainty principle. The measurement of microelements in quantum gravitational spacetime can be defined as fluctuating, including linear and nonlinear fluctuations. The spacetime microelement, defined by scaling functions, allows for the exploration of spacetime microstructure. This conceptualization enables the development of new mathematical tools and theories to investigate the nature of spacetime at the micro-scale.

The Lorentz scalar line element, also known as the spacetime invariant interval, is a fundamental concept in the theory of relativity. It measures the infinitesimal distance between two events in spacetime. It is significant due to its invariance under Lorentz transformations, meaning it remains constant for all observers, regardless of their relative motion. This invariance is a cornerstone of Einstein’s theory of relativity, ensuring that the laws of physics are the same in all inertial frames of reference.

The Lorentz scalar line element is crucial for maintaining the consistency of physical laws across all scales. It provides a foundational framework for understanding both macroscopic and microscopic phenomena, preserving spacetime’s geometric and causal properties across different observational perspectives. Kothawala Kothawala introduces the concept of a ”zero-point” length, suggesting that even at the smallest scales, the Lorentz scalar line element plays a significant role in defining the geometric properties of spacetime. As described by Volovik Volovik , it is linked to fundamental constants like the Planck length, acting as a natural ”zero-point” length. This connection is vital for exploring theories that aim to unify general relativity and quantum mechanics at the Planck scale. The Lorentz scalar line element helps characterize quantum fluctuations by providing a consistent metric that remains invariant under transformations, aiding in the interpretation of quantum behaviors.

The most general form of the Lorentz scalar line element is given by

where $g_{\alpha\beta}$ is the metric tensor, and the Einstein notation is used. The potential fluctuating nature of spacetime at small scales can be studied using the invariant interval. Utilizing the scaling representation defined in previous equations, we derive an equation for the metric tensor related to micro-measurements of spacetime fluctuations

with Einstein notation used. When $\tau=\tau_{C}$, $L^{\alpha}=L^{\alpha}_{C}$, and $L^{\beta}=L^{\beta}_{C}$ are $Constants$ respectively, we define a static measurement as

in a consistent choice of $d\lambda_{0}=dx^{\alpha}_{0}=dx^{\beta}_{0}$. This indeed corresponds to the definition of the classical Lorentz scalar line element. Consequently, for a non-static measurement, we have

with the consistent choice of $d\lambda_{0}=dx^{\alpha}_{0}=dx^{\beta}_{0}$.

Define the micro measurement of proper time to be $r_{t}=\frac{dt}{dt_{0}}=r_{t}(s)$ with reference time length $dt_{0}$. Here $s$ is the defined proper time scale. Introduce light speed measurement of $\frac{d\lambda}{dt}=r_{c}\frac{d\lambda_{0}}{dt_{0}}=r_{c}(\nu)\frac{d\lambda_{0}}{dt_{0}}$ from scale transformation of $\tau=r_{l}=r_{c}\cdot r_{t}$. $\nu=\nu(s)$ is defined as speed scale and a function of $s$. Thus we can get

with Einstein notation.

Define proper speed measurements from $r_{v^{\alpha}}=\frac{r_{l^{\alpha}}}{r_{t}}=\frac{L^{\alpha}}{r_{t}}=U^{\alpha}(V^{\alpha}(s))$. For space components, $r_{v^{i}}=U^{i}(V^{i}(s))=\frac{r_{l^{i}}}{r_{t}}=\frac{r_{l^{i}}}{r_{c}r_{T}}\frac{r_{c}r_{T}}{r_{t}}=\frac{v^{i}}{r_{c}}\frac{r_{c}r_{T}}{r_{t}}=\frac{v^{i}}{r_{c}}\frac{r_{l^{0}}}{r_{t}}=\frac{v^{i}}{r_{c}}r_{v^{0}}$, where $r_{c}r_{T}=r_{l^{0}}=r_{v^{0}}r_{t}$ and then the coordinate speed measurement $v^{i}=\frac{r_{l^{i}}}{r_{T}}$ with $r_{T}$ being the measurement of the coordinate time micro interval. Using Eq. (21), Thus we have

with Einstein notation.

Performing differentiation on both sides of Eq. (20) with respect to $s$, we aim to find how the metric tensor and scaling components evolve as time scale $s$ changes. Thus, we obtain

with Einstein notation. This equation governs how the metric tensor evolves under scaling transformations, showing the intricate dependencies between derivatives of scale variables and the metric. The terms represent a balance between metric changes and scaling influences, indicating that the metric must adjust to maintain consistency in spacetime geometry under scaling.

Under conditions of $L^{\alpha}\neq 0$, $L^{\beta}\neq 0$, and $d\tau/ds\neq 0$, we get equations for cases of $g_{\alpha\beta}\neq 0$

The conditions $L^{\alpha}\neq 0$ and $L^{\beta}\neq 0$ ensure that the microelements being measured are non-zero on the chosen scale. Such constraints help prevent ultraviolet (UV) divergence, ensuring the physical validity of the model bibitem1 ; bibitem2 . $d\tau/ds\neq 0$ ensures that $\tau(s)$ varies with $s$ in a non-trivial way. This condition ensures that $\tau(s)$ is not a constant function but varies with the parameter $s$. Physically, this means that the system is evolving or changing over time or space, representing a fluctuating mode in space or time scales. If $d\tau/ds=0$, $\tau(s)$ would be constant, implying no change or evolution, which would trivialize the analysis. These conditions ensure that the metric tensor and scaling behavior described by Eq. (24) are physically meaningful. They allow the equation to capture the dynamic, non-trivial structure of spacetime at microscopic scales, where quantum effects and scaling laws significantly influence spacetime geometry. After integral Eq. (24), we get

where $g_{\alpha\beta}=g_{\beta\alpha}$, and $\kappa_{\alpha\beta}$ are integral constant and satisfy the relation of

According to the equivalence principle of general relativity bibitem8 ; bibitem9 ; bibitem10 , the influence of a gravitational field is locally indistinguishable from an inertial frame, meaning the laws of physics reduce to those of special relativity in a sufficiently small local region. This principle allows the use of a local inertial frame where gravitational effects can be approximately neglected. Consequently, in such a locally inertial frame, the metric tensor $g_{\alpha\beta}$ can be approximated by the flat spacetime metric $\eta_{\alpha\beta}$. Therefore, in a local inertial frame or even in a flat spacetime, $\kappa_{\alpha\beta}$ in a D-dimensional flat spacetime with metric $g_{\alpha\beta}=\eta_{\alpha\beta}$ can be determined. Here, $\eta_{\alpha\beta}=(\pm,\mp 1,...,\mp 1)$ is the D-dimensional Minkowski spacetime metric for time-like and space-like interval bibitem11 .

In the realm of physics, special relativity bibitem12 ; bibitem13 provides a framework for understanding how time and space are perceived differently by observers in relative motion. The Lorentz factor is a key component of this theory, quantifying the degree of time dilation and length contraction experienced by objects moving at significant fractions of the speed of light. However, at the microscopic scale, where quantum effects play a crucial role, the influence of spacetime fluctuations on these relativistic effects becomes a fascinating subject of study Lieu1 ; Jacobson ; Abdo ; Zimdahl ; Maziashvili ; Lieu2 . This exploration seeks to extend the understanding of the Lorentz factor by incorporating the impact of quantum fluctuations, offering a nuanced view of how intrinsic measurements of space and time behave at the quantum level. By examining a scenario involving microscopic length measurements along a given spatial axis, we delve into the interplay between relativity and quantum mechanics, introducing scale parameters to capture the essence of intrinsic measurements in a D-dimensional space.

In special relativity, the Lorentz factor describes the relativistic effects on time and length measurements due to an object’s velocity. At the micro-scale, where quantum effects become significant, we explore how spacetime fluctuations influence this factor. We consider a scenario where a microscopic length is measured at a velocity $v^{1}$ along a given spatial axis $i$-axis (i=1, 2, 3, …, D-1, chose $i=1$). We introduce the scale parameter $\varsigma^{i}=\sigma^{i}/\tau$ with $\sigma^{i}$ representing the micro space length measurement along the i-axis in the rest frame, i.e., representing the intrinsic measurements. Let $\tau$ denote the micro proper length measurement. Thus, the micro intrinsic measurements in a D-dimensional space can be defined as:

in an instantaneous rest frame. For the relativistic length contraction effect, we have:

with $v^{1}=\frac{r_{l^{1}}}{r_{T}}=\frac{r_{v^{1}}\cdot r_{c}}{r_{v^{0}}}=\varsigma^{1}r_{c}/\gamma_{Ltz}^{2}$ and $v^{0}=r_{c}$. For the relativistic time dilation effect, we have:

Here, the Lorentz factor $\gamma_{Ltz}=1/\sqrt{1-\frac{[v^{1}]^{2}}{[v^{0}]^{2}}}$. Thus we derive equations

where $\vartheta^{1}=[\frac{r_{l^{1}}}{\tau}]^{2}=[\frac{r_{v^{1}}}{r_{c}}]^{2}$ and $\vartheta^{0}=[\frac{r_{l^{0}}}{\tau}]^{2}=[\frac{r_{v^{0}}}{r_{c}}]^{2}$. The solutions are:

From $(\vartheta^{1}_{2},\vartheta^{0}_{2})$ with considering positive solutions, The Lorentz factor is then given by:

The form and structure of Eq. (30) render it particularly suitable to be interpreted as an iterative equation, as it demonstrates how a system can approximate a state through recursive calculations under specific conditions. The value of $\vartheta_{0}$ depends on the preceding value of $\vartheta_{1}$, and conversely, the value of $\vartheta_{1}$ depends on the preceding value of $\vartheta_{0}$. This interdependence creates a recursive or cyclic relationship. Through repeated iterations, one can observe whether $\vartheta_{0}$ and $\vartheta_{1}$ converge towards a stable value, known as a fixed point. The two solutions in Eq. (31) represent fixed points, with the positive one being a stable point, as determined by analyzing the Jacobian matrix. It provides a framework for modeling the evolution of a system over time through successive calculations. This iterative process helps in approaching a system’s state under specific conditions, emphasizing the recursive relationships in spacetime dynamics and highlighting the mutual dependency between $\vartheta_{0}$ and $\vartheta_{1}$, which are scaling variables associated with spacetime measurements. This interdependency might illustrate the stability of these variables through iterative methods, offering potential insights into how spacetime could stabilize or reach fixed points when subjected to quantum fluctuations or scaling transformations.

The Lorentz factor can be obtained from a basic perspective. $\bar{\gamma}_{Ltz}$ is introduced to act as the Lorentz factor. Therefore, the micro space length measurement contracted by scale $\bar{\gamma}_{Ltz}$ gives $r_{l^{1}}=\sigma^{1}/\bar{\gamma}_{Ltz}=\varsigma^{1}\tau/\bar{\gamma}_{Ltz}$, while the micro time inflated by $\bar{\gamma}_{Ltz}$ gives $r_{T}=r_{l^{0}}/r_{c}=r_{t}\bar{\gamma}_{Ltz}$ (i.e. $r_{l^{0}}=r_{c}r_{T}=r_{c}r_{t}\bar{\gamma}_{Ltz}=\tau\bar{\gamma}_{Ltz}$). $v^{1}=\varsigma^{1}r_{c}/\bar{\gamma}_{Ltz}^{2}$. Since there is no contraction effect of the micro length along axis $i>1$, $r_{l}^{i>1}=\sigma^{i>1}=\varsigma^{i>1}\tau$. Using the D-dimension local flat spacetime matric $\eta_{\alpha\beta}$, we can get

From the constraint of $\Sigma_{\alpha,\beta}\kappa_{\alpha\beta}=1$, we get the constrained relation connecting the dimension and components of spacetime,

We call $\kappa_{\alpha\beta}$ the spacetime structure coefficients. Eq. (34) can be deemed as an equation of $\bar{\gamma}^{2}_{Ltz}$ and its solutions are

The limit of light speed needs $\bar{\gamma}^{2}_{Ltz}>0$ in Minkowski spacetime. The physical significance of this equation lies in its role in understanding how spacetime fluctuations at microscopic scales can be related to the relativistic effects of time dilation and length contraction. The Lorentz factor is a key component in the theory of relativity, used to describe how measurements of time and space differ for observers in different inertial frames. This equation demonstrates that the Lorentz factor depends on the scaling parameters $\varsigma^{k}$ which represent the fluctuations in spacetime geometry across different dimensions. The positive value of $\bar{\gamma}^{2}_{Ltz}$ is necessary to ensure the consistency of the model with the principles of relativity, particularly in preserving the limit on the speed of light. This relationship highlights how quantum fluctuations at the Planck scale can impact classical relativistic concepts, providing fundamental insight into the nature of spacetime.

Considering two limit situations for $v^{1}=0$ and $r_{c}$. If $v^{1}=0$, it corresponds to that $\varsigma^{1}=0$ and $\bar{\gamma}_{Ltz}^{2}=\sum_{k=2}^{D-1}[\varsigma^{k}]^{2}\pm 1$ or $\bar{\gamma}_{Ltz}^{2}=0$. Consequently, $\sigma^{1}=0$, signifying there is no spatial component in this direction. Additionally, we have $\sum_{k=2}^{D-1}[\sigma^{k}]^{2}=(\bar{\gamma}_{Ltz}^{2}\mp 1)\tau^{2}$ for the positive solution. If $v^{1}=r_{c}$, needing $\varsigma^{1}=\bar{\gamma}_{Ltz}^{2}$ and following with $\sigma^{1}=\bar{\gamma}_{Ltz}^{2}\tau^{2}$, then $\sum_{k=2}^{D-1}[\varsigma^{k}]^{2}=\mp 1$ and $\sum_{k=2}^{D-1}[\sigma^{k}]^{2}=\mp\tau^{2}$.

There is a generalization of Eq. (30). Based on Eq. (35), the generalized Lorentz inflation/contraction transformation becomes

where $\vartheta^{0}\vartheta^{1}=[\varsigma^{1}]^{2}$.

Comparing Eq. (35) to Eq. (32), when $\sum_{k=2}^{D-1}[\varsigma^{k}]^{2}$ equals 0 for time-like metric while 2 for space-like metric, we get positive $\bar{\gamma}_{Ltz}=\gamma_{Ltz}$. And then we have expressions

The speed $v^{i>2}$ can be defined as the speed of fluctuation in micro-length measurements ($\sigma^{i>2}$) associated with the fluctuation on $i=1$ axis with a speed of $v^{1}$ ($0\leq v^{1}\leq r_{c}$, TABLE 1 illustrates the relationships in the limiting situation for $v^{1}=0$ and $r_{c}$). This association is expressed via $v^{\alpha}$ being constrained by the relationship of

where $\sum_{k=2}^{D-1}[v^{k}]^{2}=\frac{2r_{c}^{2}}{\gamma_{Ltz}^{2}}=2(r_{c}^{2}-[v^{1}]^{2})$ for space-like metric and 0 for time-like metric.

The Schwarzschild metric describes the spacetime around a non-rotating, spherically symmetric black hole. If considering the black hole with Schwarzschild Metric bibitem14 , the space-like line element has the form of

in Schwarzschild coordinates, where $d\Omega^{2}=d\theta^{2}+sin^{2}(\theta)d\phi^{2}$. And $r_{s}=\frac{2GM}{c^{2}}$ is the Schwarzschild radius at where the event horizon is. $G$ is the gravitational constant, $M$ is the object mass, and $c$ is the speed of light. In the presence of gravitational fields, measurements of space and time are affected differently compared to special relativity. Specifically, gravitational fields can cause contraction effects in spatial measurements and dilation effects in time measurements. These effects arise due to the curvature of spacetime in general relativity. In regions with strong gravity, time appears to slow down (gravitational time dilation), while space can be warped, affecting perceived distances. These phenomena differ from the effects predicted by special relativity, which deals with time dilation and length contraction due to relative motion. However, in the absence of gravitational fields or in regions with weak gravity, the predictions of general relativity converge to those of special relativity. In such scenarios, spacetime is essentially flat, and the familiar Lorentz transformations apply, with the Lorentz factor $\bar{\gamma}_{Ltz}$ describing time dilation and length contraction.

Define a new factor $\gamma_{g}$ to replace $\bar{\gamma}_{Ltz}$ to act as the Lorentz factor in the spacetime of Schwarzschild Metric to measure the micro length contraction and micro time inflation effect. It can be seen as a correction factor with respect to $\bar{\gamma}_{Ltz}$ when measuring in a gravitational field. Using the constraint of $\Sigma_{\alpha,\beta}\kappa_{\alpha\beta}=1$, we get

When $M=0$ or $r\rightarrow\infty$, $\gamma^{2}_{g}=\bar{\gamma}^{2}_{Ltz}$. When $r=r_{s}$, $\gamma^{2}_{g}=\infty$. When $r=0$, $\gamma^{2}_{g}=0$. $\gamma^{2}_{g}$ is remaining positive when $\bar{\gamma}^{2}_{Ltz}<0$ at $r<r_{s}$ and $\bar{\gamma}^{2}_{Ltz}>0$ at $r>r_{s}$. It can be seen that $\gamma_{g}$ describes a dual effect of gravitational field in the general relativity frame and local motion in the special relativity frame. From this generalized Lorentz factor $\gamma_{g}$ in Schwarzschild Metric, we can still keep Eq. (34) remaining. Consequently, the physical laws satisfy the Lorentz transformation when the physical variables are defined to be scaled by $\gamma_{g}$.

There is also a generalization of Lorentz inflation/contraction transformation in Schwarzschild Metric. Namely,

where $\vartheta^{0}\vartheta^{1}=[\varsigma^{1}]^{2}(1-\frac{r_{s}}{r})^{-2}$.

The mass of a particle moving along $r$ scaled by $\gamma_{g}$ is expressed by

where $m^{2}_{g0}=(1-\frac{r_{s}}{r})^{-1}m^{2}_{0}$ is introduced as the measured mass of the particle at instantaneous rest in a gravitational field with Schwarzschild metric and $m_{0}$ is the mass of the particle at rest in a flat metric. When $r\rightarrow\infty$ or $M=0$, $m^{2}_{g0}=m^{2}_{0}$. When $r\rightarrow 0$, $m^{2}_{g0}=0$. Therefore, the energy-momentum relationship measured locally in the gravitational field with the Schwarzschild Metric becomes

where $E^{2}_{g}=\bar{\gamma}^{2}_{Ltz}(1-\frac{r_{s}}{r})^{-1}m^{2}_{0}c^{4}=\bar{\gamma}^{2}_{Ltz}m^{2}_{g0}c^{4}$. And $p^{2}_{g,1}=\bar{\gamma}^{2}_{Ltz}(1-\frac{r_{s}}{r})^{-1}m^{2}_{0}v^{2}_{g1}=\bar{\gamma}^{2}_{Ltz}m^{2}_{g0}v^{2}_{g,1}$ with $v_{g,1}=\frac{\varsigma^{1}r_{c}}{\gamma_{g}^{2}}$ and $p^{2}_{g,k>1}=\bar{\gamma}^{2}_{Ltz}m^{2}_{g0}v^{2}_{g,k>1}$ with $v_{g,k>1}=\frac{\varsigma^{k>1}r_{c}}{\gamma_{g}}$. Thus, $p^{2}_{g}=\sum_{k=1}^{D-1}p^{2}_{g,k}$. This energy-momentum relationship scaled by $(1-\frac{r_{s}}{r})$ on both sides can be back to the one in Minkowski spacetime. When $r\rightarrow\infty$ or $M=0$, the former is also back to the latter. When $r>>r_{s}$, we get

Geodesic equations bibitem8 ; bibitem9 ; bibitem10 ; Tsamparlis ; Hackmann are fundamental in the study of general relativity and differential geometry. They describe the path that an object follows under the influence of gravity alone, without any other forces acting on it. They are crucial for understanding the motion of objects in a gravitational field, serving as the foundation for analyzing how gravity shapes the trajectories of particles and light. They provide insights into both macroscopic phenomena, like the orbits of planets, and microscopic effects, such as the influence of gravity on quantum particles. As such, geodesics are integral to bridging the gap between classical and quantum descriptions of gravity.

The geodesic equations can be written in scaling form on a micro-scale via operators’ transformations. Firstly, by using Eq. (8), the Christoffel symbols can be represented in the form of

with Einstein notation, where $\Gamma^{\alpha}_{\beta\gamma}$ are defined as dimensionless scaling Christoffel symbols. Combining the Eq. (3) and Eq. (4), the scaling geodesic equations are obtained by

with Einstein notation. The equation demonstrates how local geodesic paths (which describe the shortest path between points in spacetime) evolve with scaling functions, offering insights into the complex structure of spacetime at microscopic scales.

In the framework of general relativity, gravity is conceptualized not as a force but as an outcome of the curvature of spacetime, with the stress-energy tensor serving as the source of this curvature. Within the geometric interpretation of gravity, geodesics are indicative of the curved geometry inherent in spacetime. All length-minimizing curves are geodesics, and all geodesics are length-minimizing, at least locally bibitem15 . The local geodesic equations can be viewed as a constraint on microscopic spacetime measurements at a local level. When representing the measurement of spacetime microelements through scaling functions, the geodesic equations represented by scales elucidate the restrictions on nontrivial fluctuations of scales.

The scaled Einstein Field Equations represented by scales $X^{\alpha}$ can be obtained in the same way. Which characterizes how the scales of micro geometric spacetime measurements are influenced by stress-energy tensor.

By considering a transformation of a linear mapping between coordinate $x^{\alpha}$ and scale $\bar{X}^{\alpha}$. The latter are new scales obtained by rescaling $\zeta^{\alpha}$ on $X^{\alpha}$. $\zeta^{\alpha}$ are the rescaling factors on differential operators, establishing a connection between coordinate $x^{\alpha}$ and scale $\bar{X}^{\alpha}$ and making the micro measurements being carried out in a linear way, i.e.,