Displacement of Geometric Levels in an Extended Scalar Field via Point Mass Interaction

1. Introduction

One of the deepest and most persistent challenges in contemporary physics lies in the unification of quantum mechanics with General Relativity, a problem that has driven the search for alternative frameworks under the concept of emergent gravity [1,2]. In recent decades, the scientific literature has dedicated considerable effort to understanding spacetime not as a fundamental and predefined entity, but as a macroscopic structure that emerges from underlying microscopic degrees of freedom, emulating the phenomena observed in condensed matter systems [3]. Within this paradigm, extended scalar fields have gained renewed prominence. Although in standard cosmology scalar fields are conventionally employed to model inflation or dark energy [4], recent theoretical developments suggest that quantum vacuum fluctuations and scale perturbations in these fields can induce true metric deformations in the spacetime structure [5,6,7].

However, the formal description of localized massive objects within these emergent approaches usually faces the problem of gravitational singularities and the loss of consistency in the weak-field limit [8]. Previous studies, such as those presented in the literature on semiclassical gravity, attempt to mitigate these divergences by introducing higher-order corrections or effective field theories [9,10], but the quantum-mechanical origin of a point mass without resorting to spacetime singularities remains an area of intense debate. In this context, the concept of geometric level displacements induced by topological or perturbative interactions offers an alternative and promising route [11], allowing a connection between the physics of bound states in the vacuum and the geometric properties of the background [12,13].

Taking as a starting point and a natural extension the foundations established in recent literature, particularly in the framework of modified extended scalar fields as proposed in Ref. [14], this work aims to explicitly investigate the connection between geometric level displacements induced by scale perturbations and the resulting metric deformations under the presence of interactions with point masses. The main objective of this paper is to propose and develop a rigorous perturbative mechanism that, by modifying the ground state of the system, is capable of naturally modeling the anisotropy between the temporal and spatial sectors of the emergent spacetime.

Throughout this manuscript, structured to guide the reader through the different phases of the formalism, we will demonstrate how this approach reveals an internal compatibility condition. This condition not only reproduces exactly the metric reciprocity relation, but also gives rise to a localized point mass of a purely emergent character [15]. After offering a detailed review of the extended scalar field in Sec. 2, we address the core of the research in Sec. 3, which includes the perturbative calculation via ground state modification and the characterization of the point mass effects. Finally, it will be verified that in the long distance regime the obtained metric deformation consistently converges with the predictions of General Relativity in its weak-field limit, with the remarkable advantage that the resulting local deformation is entirely free of singularities, thus establishing a regular emergent gravitational background derived directly from vacuum collapse [16,17].

2. Review of Extended Scalar Field

Consider an extended scalar field $\tilde{\chi }\left(\xi \right)$ depending on the dimensionless auxiliary $\xi$-space, whose Lagrangian can be written as proposed in Ref. [14]

$$\tilde{\mathcal{L}}\left(\xi \right)={\displaystyle \frac{1}{2{\delta }_{\mathrm{t}}^2}}{\left({\partial }_{{\xi }^0}\tilde{\chi }\right)}^2-{\displaystyle \frac{1}{2{\delta }_{\mathrm{s}}^2}}{\left({\nabla }_{\xi }\tilde{\chi }\right)}^2-{\displaystyle \frac{1}{2}}{m}^2{\tilde{\chi }}^2-{\displaystyle \frac{1}{2}}\lambda {\tilde{\chi }}_0^2{\tilde{\chi }}^2.$$

(1)

The mapping from the physical x-spacetime to the dimensionless $\xi$-space is expressed as

$${\xi }^0=(1/{\delta }_{\mathrm{t}})\Delta x^0,{\xi }^i=(1/{\delta }_{\mathrm{s}})\Delta x^i,$$

(2)

where $\Delta x^{\mu }=x^{\mu }-x^{\prime \mu }$ denotes a variation of the coordinate $x^{\mu }$ pivoting around a control point $x^{\prime \mu }$, with characteristic size ${\delta }_{\mathrm{t}}$ and ${\delta }_{\mathrm{s}}$ for the temporal and spatial sectors, respectively. Under this mapping, the field is transformed as $\chi \left(x\right)=\tilde{\chi }\left(\xi \right)$. Furthermore, the specific coupling ${\tilde{\chi }}_0^2{\tilde{\chi }}^2$ represents a self-interaction of the field with its ground state, which acts to organize the field excitations. Consequently, the resulting Klein-Gordon equation reads

$${\displaystyle \frac{1}{{\delta }_{\mathrm{t}}^2}}{\partial }_{{\xi }^0}^2\tilde{\chi }-{\displaystyle \frac{1}{{\delta }_{\mathrm{s}}^2}}{\nabla }_{\xi }^2\tilde{\chi }+m^2\tilde{\chi }+\lambda {\tilde{\chi }}_0^{\prime 2}\tilde{\chi }=0,$$

(3)

where the following approximation function is assumed for ${\tilde{\chi }}_0^{\prime }$ in terms of ${\left|\xi \right|}^2={\left({\xi }^0\right)}^2+{\xi }^i{\xi }^i$

$${\tilde{\chi }}_0^{\prime }={\upsilon }_0\left(1-{\displaystyle \frac{1}{2}}{\left|\xi \right|}^2\right).$$

(4)

By retaining terms up to quadratic order and neglecting higher-order contributions $\mathcal{O}\left(\right|\xi {|}^4)$, the ansatz yields the quadratic form ${\tilde{\chi }}_0^{\prime 2}\approx {\upsilon }_0^2{(1-|\xi |}^2)$. This is compatible with the ground state solution ${\tilde{\chi }}_0={\upsilon }_0{e}^{-{\displaystyle \frac{1}{2}}{\left|\xi \right|}^2}$, which, under the same approximation, reduces to ${\tilde{\chi }}_0\equiv {\tilde{\chi }}_0^{\prime }$. Similarly, the general solution of equation (3) can be expressed as

$${\tilde{\chi }}_n=\sum C_n{H}_n\left(\xi \right)e^{-1/{2\left|\xi \right|}^2},$$

(5)

with $H_n\left(\xi \right)={\prod }_{\mu =1}^4{H}_n\left({\xi }^{\mu }\right)$, where $H_n\left(\xi \right)$ are the Hermite polynomials. The amplitude $C_0$ of the ground state must necessarily coincide with the value ${\upsilon }_0$ defined in the ansatz. Formally, we proceed using the mathematical method of separation of variables. The physical justification for imposing a vanishing quantum separation constant corresponds to strict conservation of the system’s global internal energy. Since there is no energetic coupling constant, the perturbative balance does not alter conventional energy states but instead shifts the constraints directly toward the scales of the system. In this way, the quantization conditions are transferred from energy levels to geometric levels, forcing the temporal and spatial sectors to self-compensate independently through the dynamics of their own scales. This solution is viable only if the geometric scales in both the temporal sector ${\delta }_{\mathrm{t}}$ and the spatial sector ${\delta }_{\mathrm{s}}$ are governed by a single amplitude $\delta$, defined as

$${\delta }_{\mathrm{t}}=\delta ,\mathrm{and}{\delta }_{\mathrm{s}}=i\delta .$$

(6)

In addition, two quantum constraints are required as a consequence of the cancellation of independent and quadratic terms in $\xi$ by substituting the solution (5) into the equation (3). The first constraint takes the form

$$\lambda {\delta }^2{\upsilon }_0^2=1,$$

(7)

which arises from the condition that the amplitude $C_0$ is predefined by ${\upsilon }_0$. The second constraint depends on the excitation level and is given by

$${\delta }^2{m}^2=(2n_{\mathrm{t}}+1)+3(2n_{\mathrm{s}}+1)-1,$$

(8)

where $n_{\mathrm{t}}=n_0$ and $n_{\mathrm{s}}=n_1=n_2=n_3$ are non-negative integers ($n_{\mathrm{t}},n_{\mathrm{s}}\in \{0,1,2,\dots \}$), with $n_{\mathrm{s}}$ assumed equal for simplicity to ensure homogeneity in the spatial coordinates. Under certain conditions, when m relaxes and becomes fixed, the field excitations can induce a geometric background characterized by the linear relationship between $\delta$ and the discrete integer $n_{\mathrm{s}}$ [14].

Considering the transformation proposed in (2) and the condition given in (6), it is possible to relate the relativistic interval, defined as $s^2=\Delta x_{\mu }\Delta x^{\mu }$, to the dimensionless Euclidean distance ${\left|\xi \right|}^2$ via

$$s^2={\delta }^2{\left|\xi \right|}^2.$$

(9)

This relation is controlled by the parameter $\delta$, which can be interpreted as an emergent geometric structure from the extended field $\tilde{\chi }$, assuming the dimensionless auxiliary $\xi$-space as primary.

3. Geometric Level Displacement Due to Interaction with a Point Mass

Unlike the analysis presented in Sec. 2, a point mass M interacting with the extended field $\tilde{\chi }$ is expected to produce an anisotropic perturbation of the geometric scales. Consequently, the relation (6) is no longer satisfied, leading to ${\delta }_{\mathrm{t}}\ne {\delta }_{\mathrm{s}}$ in general. To this end, the ground state amplitude ${\upsilon }_0$ must be split into two modes: ${\upsilon }_{0\mathrm{t}}$ for the temporal sector and ${\upsilon }_{0\mathrm{s}}$ for the spatial sector. Accordingly, we propose the following relation

$${\upsilon }_0^2={\upsilon }_{0\mathrm{t}}^2+{\upsilon }_{0\mathrm{s}}^2.$$

(10)

To calculate this perturbation, the definition of first-order unitary variations for several variables is employed. The perturbative form of the ground state amplitude reads

$${\upsilon }_{0\mathrm{t}}={\displaystyle \frac{{\upsilon }_0}{\sqrt{2}}}(1+{\eta }_{\mathrm{t}}),{\upsilon }_{0\mathrm{s}}={\displaystyle \frac{{\upsilon }_0}{\sqrt{2}}}(1+{\eta }_{\mathrm{s}}).$$

(11)

Suppose that the modification of the ground state ${\tilde{\chi }}_0^{\prime }$ caused by these perturbations does not alter the vacuum energy density ${\rho }_{\mathrm{vac}}$, namely, ${\rho }_{\mathrm{vac}}\propto {\upsilon }_0^2=\mathrm{const}.$, according to Ref. [14]. Then, substituting relations (11) into (10) and applying this condition, we obtain

$${\eta }_{\mathrm{t}}=-{\eta }_{\mathrm{s}}.$$

(12)

Similarly, for the geometric scales where the perturbation breaks the isotropy between the temporal and spatial sectors, it follows that

$${\delta }_{\mathrm{t}}=\delta (1+{ϵ}_{\mathrm{t}}),{\delta }_{\mathrm{s}}=i\delta (1+{ϵ}_{\mathrm{s}}).$$

(13)

The perturbed bare mass $m^{\prime }$ is given by

$$m^{\prime }=m(1+\mu ).$$

(14)

Based on these definitions and within a perturbative framework, the unitary variations of all variables satisfy $|{\eta }_{\mathrm{t}}|,|{\eta }_{\mathrm{s}}|,|{ϵ}_{\mathrm{t}}|,|{ϵ}_{\mathrm{s}}|,|\mu |\ll 1$.

3.1. Point Mass Effects and Internal Compatibility

The ansatz quadratic form ${\tilde{\chi }}_0^{\prime 2}\approx {\upsilon }_0^2{(1-|\xi |}^2)$ is insufficient to represent a field $\tilde{\chi }$ with non-isotropic geometric scales ${\delta }_{\mathrm{t}}\ne {\delta }_{\mathrm{s}}$. Consequently, a generalized form of ansatz becomes necessary and is given by

$${\tilde{\chi }}_{0\mathrm{g}}^2={\upsilon }_{0\mathrm{t}}^2{[1-{\left({\xi }^0\right)}^2]}^2+{\upsilon }_{0\mathrm{s}}^2{[1-{\xi }^i{\xi }^i]}^2.$$

(15)

Using the relations (10) and (11) and retaining terms up to the quadratic order and neglecting higher-order contributions of $\mathcal{O}\left(\right|\xi {|}^4)$, leads to this other equivalent expression

$${\tilde{\chi }}_{0\mathrm{g}}^2={\tilde{\chi }}_0^{\prime 2}-2{\upsilon }_0^2[{\eta }_{\mathrm{t}}{\left({\xi }^0\right)}^2+{\eta }_{\mathrm{s}}{\xi }^i{\xi }^i].$$

(16)

A perturbation of the ground state is conveniently defined as ${\tilde{\chi }}_{0\mathrm{p}}^2=2{\upsilon }_0^2[{\eta }_{\mathrm{t}}{\left({\xi }^0\right)}^2+{\eta }_{\mathrm{s}}{\xi }^i{\xi }^i]$. This expression reflects the internal compatibility of the system, as it represents a perturbative contribution independent of the action mechanism.

Complementary to the above, an equivalent approach is to assume that the ground state is modified due to a perturbation. Then, the generalized ansatz for this configuration can be written as ${\tilde{\chi }}_0^{\prime 2}{\tilde{h}}_0^2(\xi ,\alpha ,\beta )$. Here, $\alpha$ is the parameter that controls the perturbation and ${\tilde{h}}_0^2$ is a function acting as a modification mechanism for the vacuum state. We propose the explicit form of this function as follows

$${\tilde{h}}_0^2(\xi ,\alpha ,\beta )=1+\alpha +(\alpha +{\beta }_{\mathrm{t}}){\left({\xi }^0\right)}^2+(\alpha +{\beta }_{\mathrm{s}}){\xi }^i{\xi }^i.$$

(17)

Expanding the product ${\tilde{\chi }}_0^{\prime 2}{\tilde{h}}_0^2(\xi ,\alpha ,\beta )$ and comparing it with ${\tilde{\chi }}_{0\mathrm{g}}^2$ in the limit where $\alpha$ approaches zero, yields the equivalences ${\beta }_{\mathrm{t}}\equiv -2{\eta }_{\mathrm{t}}$ and ${\beta }_{\mathrm{s}}\equiv -2{\eta }_{\mathrm{s}}$, thereby ensuring internal compatibility. Assuming that the perturbation is produced by a mass M that interacts with the field $\tilde{\chi }$ via a coupling constant $\kappa$, the parameter $\alpha$ can be identified as $\alpha \equiv \kappa M$. Consequently, all of this determines the relation

$${\tilde{\chi }}_0^{\prime 2}{\tilde{h}}_0^2(\xi ,\kappa )={\tilde{\chi }}_0^{\prime 2}-{\tilde{\chi }}_{0\mathrm{p}}^2+{\upsilon }_0^2\kappa M.$$

(18)

In summary, the ground state ${\tilde{\chi }}_0^{\prime 2}$ has been disturbed by two contributions: one related to internal compatibility ${\tilde{\chi }}_{0\mathrm{p}}^2$ and the other to point action ${\upsilon }_0^2\kappa M$. This implies that the global vacuum modification produced by the parameter $\alpha$ leads to a spontaneous localization of the field into a pointlike configuration [18].

3.2. Perturbative Calculation via Ground State Modification

We can introduce the product ${\tilde{\chi }}_0^{\prime 2}{\tilde{h}}_0^2(\xi ,\kappa )$ into the Lagrangian (1) and considering the equality (18) to obtain a perturbed Klein-Gordon equation similar to (3), yields the following result

$${\displaystyle \frac{1}{{\left[\delta (1+{ϵ}_{\mathrm{t}})\right]}^2}}{\partial }_{{\xi }^0}^2\tilde{\chi }-{\displaystyle \frac{1}{{\left[i\delta (1+{ϵ}_{\mathrm{s}})\right]}^2}}{\nabla }_{\xi }^2\tilde{\chi }+{\left[m(1+\mu )\right]}^2\tilde{\chi }+\lambda {\tilde{\chi }}_0^{\prime 2}\tilde{\chi }-\lambda {\tilde{\chi }}_{0\mathrm{p}}^2\tilde{\chi }+\lambda {\upsilon }_0^2\kappa M\tilde{\chi }=0.$$

(19)

The relations (13) for ${\delta }_{\mathrm{t}}$ and ${\delta }_{\mathrm{s}}$ were used along with the relation (14) for the bare mass. Then, this equation contains all unknown unitary variations, primarily those related to the geometric scales ${ϵ}_{\mathrm{t}}$ and ${ϵ}_{\mathrm{s}}$, and secondarily the variation corresponding to the bare mass $\mu$. These unitary perturbations must be related to the internal compatibility conditions ${\beta }_{\mathrm{t}}$ and ${\beta }_{\mathrm{s}}$, implicitly contained in ${\tilde{\chi }}_{0\mathrm{p}}^2$, as well as to the interaction term explicitly denoted by $\kappa M$. Importantly, determining the geometric level displacements is crucial, as they are formulated in analogy with the energy level shifts of the standard quantum perturbation theory, although in our framework they manifest as metric deformations.

The procedure for solving (19) is straightforward, since the perturbative term of the ground state ${\tilde{\chi }}_{0\mathrm{p}}^2$ exhibits the same quadratic dependence on $\xi$ as the unperturbed field ${\tilde{\chi }}_0^{\prime 2}$. Consequently, for this case we can propose a general solution analogous to the expression (5). Substituting this solution into the Klein-Gordon equation and applying the quantum constraints (7) and (8) eliminates the zeroth-order terms, leaving an equation with the following perturbative contributions

$$-2{ϵ}_{\mathrm{t}}{\partial }_{{\xi }^0}^2\tilde{\chi }-2{ϵ}_{\mathrm{s}}{\nabla }_{\xi }^2\tilde{\chi }+2{\delta }^2{m}^2\mu \tilde{\chi }-2[{\eta }_{\mathrm{t}}{\left({\xi }^0\right)}^2+{\eta }_{\mathrm{s}}{\xi }^i{\xi }^i]\tilde{\chi }+\kappa M\tilde{\chi }=0.$$

(20)

Following the physical rationale established in Sec. 2, the variables are separated by enforcing a vanishing quantum separation constant, consistent with the strict conservation of the system’s global internal energy under perturbative corrections. Since no external energy exchange is permitted, the temporal and spatial sectors must self-compensate independently. Consequently, we propose the decoupled general solution

$$\tilde{\chi }=T\left({\xi }^0\right)S\left({\xi }^i\right),$$

(21)

$$-{ϵ}_{\mathrm{t}}{\displaystyle \frac{1}{T}}{\partial }_{{\xi }^0}^{2}T+{\delta }^2{m}^2\mu -{\eta }_{\mathrm{t}}{\left({\xi }^0\right)}^2=0,$$

(22)

$$-2{ϵ}_{\mathrm{s}}{\displaystyle \frac{1}{S}}{\nabla }_{\xi }^{2}S-2{\eta }_{\mathrm{s}}{\xi }^i{\xi }^i+\kappa M=0.$$

(23)

From this point forward, the same procedure as in Sec. 2 is followed to obtain the solutions for $T\left({\xi }^0\right)$ and $S\left({\xi }^i\right)$. This involves seeking constraint conditions that relate the perturbed unit variations to the quadratic and independent terms in $\xi$, bearing in mind that the proposed solutions are of the form $T\left({\xi }^0\right)\propto H_n\left({\xi }^0\right)e^{-1/2{\left({\xi }^0\right)}^2}$ and $S\left({\xi }^i\right)\propto H_n\left({\xi }^i\right)e^{-1/2{\xi }^i{\xi }^i}$. In the first case, cancellation of the quadratic terms of equations (22) and (23) yields, respectively,

$${ϵ}_{\mathrm{t}}=-{\eta }_{\mathrm{t}},{ϵ}_{\mathrm{s}}=-{\eta }_{\mathrm{s}}.$$

(24)

As a result of this, the relation (12) gives rise to the important condition of internal compatibility relating the perturbative unit variation of the temporal sector to the spatial sector, as follows

$${ϵ}_{\mathrm{t}}=-{ϵ}_{\mathrm{s}}.$$

(25)

All of this has interesting significance in the context of the metric tensor $g_{\mu \nu }$ in General Relativity. Specifically, in the weak-field approximation of this theory, where the perturbed diagonal components of the metric tensor are given by $g_{00}+h_{00}$ and $g_{ii}+h_{ii}$, a symmetry relation of the type $h_{00}=-h_{ii}$ is verified [19,20]. We propose its equivalent for the case of the perturbed geometric scales given in (25), although in this context the relation is motivated by a different condition, namely the constancy of the vacuum energy density under ground state perturbation.

In the second case, independent term cancellation in the temporal sector of (22) yields

$$(2n_{\mathrm{t}}+1){ϵ}_{\mathrm{t}}+{\delta }^2{m}^2\mu =0.$$

(26)

Assuming that the system is under stationary conditions, so that $n_{\mathrm{t}}>>n_{\mathrm{s}}$, therefore, according to (8) we can write ${\delta }^2{m}^2=(2n_{\mathrm{t}}+1)$ and the following is obtained

$$\mu =-{ϵ}_{\mathrm{t}}.$$

(27)

A modification in the bare mass of the field can occur as a consequence of variations in the geometric scale. This would be consistent with the conservation of ${\delta }^2{m}^2$ in the system. On the other hand, independent term cancellation in the spatial sector of (23) gives

$$6(2n_{\mathrm{s}}+1){ϵ}_{\mathrm{s}}+\kappa M=0,$$

(28)

from which it follows that

$${ϵ}_{\mathrm{s}}=-{\displaystyle \frac{\kappa M}{6(2n_{\mathrm{s}}+1)}}.$$

(29)

Finally, a unit variation ${ϵ}_{\mathrm{s}}$ on the geometric scale ${\delta }_{\mathrm{s}}$ is found to be directly proportional to the interaction constant $\kappa$ of the mass M with the field $\tilde{\chi }$, and inversely proportional to the excitation number $n_{\mathrm{s}}$ in the large-$n_{\mathrm{s}}$ limit. Note that for $n_{\mathrm{s}}=0$, no geometric singularity appears. Suppose that the mass M is located at the level $n_{\mathrm{s}}=0$, its effect on the geometric scale decreases as $n_{\mathrm{s}}$ takes on larger values. Furthermore, by identifying $n_{\mathrm{s}}$ with geometric levels via a relation such as $n_{\mathrm{s}}\propto R$, it could be interpreted that the interaction of a mass with the field modifies the geometric background, with its effect decreasing as a function of the distance R in the physical x-space.

3.3. Deformed Metric of Emergent Flat Geometry

Based on the results of the previous section, the effect of the perturbation on the emergent metric defined in (9) can be verified. Using the identities (2), the metric components can be written as

$${(1/{\delta }_{\mathrm{t}})}^2{(\Delta x^0)}^2+{(1/{\delta }_{\mathrm{s}})}^2\Delta x^i\Delta x^i={\left|\xi \right|}^2,$$

(30)

introducing the perturbative relations (13) for the geometric scales ${\delta }_{\mathrm{t}}$ and ${\delta }_{\mathrm{s}}$, while taking into account the result (25), yields

$$(1+2{ϵ}_{\mathrm{s}}){(\Delta x^0)}^2-(1-2{ϵ}_{\mathrm{s}})\Delta x^i\Delta x^i={\delta }^2{\left|\xi \right|}^2.$$

(31)

Substituting the value of ${ϵ}_{\mathrm{s}}$ given in (29) leads to the expression

$$(1-{\displaystyle \frac{\kappa M}{3(2n_{\mathrm{s}}+1)}}){(\Delta x^0)}^2-(1+{\displaystyle \frac{\kappa M}{3(2n_{\mathrm{s}}+1)}})\Delta x^i\Delta x^i={\delta }^2{\left|\xi \right|}^2.$$

(32)

Consequently, we can interpret that the geometric level displacements in the $\xi$-space, calculated using perturbative methods, give rise to deformations in the temporal and spatial sectors of the physical x-space. Assuming $n_s$ is related to some type of distance R, such that $n_s{\ell }_{\mathrm{p}}\to R$, where ${\ell }_{\mathrm{p}}$ is the Planck length, the deformation factor $\kappa M/3(2n_{\mathrm{s}}+1)$ can be written as $\kappa M{\ell }_{\mathrm{p}}/6R$ in the large-R limit [21]. Knowing that this factor in the weak-field approximation of General Relativity for the case of a localized mass is equal to $2GM/R$, it follows that the free parameter $\kappa$ must have a value of $\kappa =12G/{\ell }_{\mathrm{p}}$, in natural units $\hslash =c=1$. Therefore, using a deformation factor $4GM/(2R+{\ell }_{\mathrm{p}})$ in (32) for large distances R, it is possible to derive contractions and dilations of time and space intervals compatible with those obtained locally by the weak-field approximation of Einstein’s gravity for the case of a spacetime deformed by a localized mass. This framework suggests that the emergent point mass originates from a local vacuum collapse. Although this mass arises intrinsically from the perturbation configuration, it effectively behaves as an external and localized mass sourcing the gravitational field in the GR description. Additionally, at the limit $R=0$, the local metric deformation remains strictly finite and free of singularities [22,23], taking the maximum bounded value of $4GM/{\ell }_{\mathrm{p}}$. This regular behavior is a direct consequence of the underlying quantum structure of the extended field $\tilde{\chi }$, where the geometric displacements are governed by the discrete pattern $2n_{\mathrm{s}}+1$, which naturally prevents structural collapses by remaining non-zero at the ground state $n_{\mathrm{s}}=0$.

4. Conclusions

Based on the premises established in Ref. [14], a perturbative calculation within an extended scalar field framework $\tilde{\chi }$ was developed to map geometric level displacements induced by scale perturbations directly to macroscopic metric deformations, naturally breaking anisotropy between the temporal and spatial sectors.

A mechanism for modifying the vacuum state ${\tilde{\chi }}_0$ has been proposed via a specific modification function ${\tilde{h}}_0^2(\xi ,\kappa )$. Crucially, this vacuum modification reveals an internal compatibility condition equivalent to the metric reciprocity relation ${ϵ}_{\mathrm{t}}=-{ϵ}_{\mathrm{s}}$, while providing a fundamental physical mechanism where a localized point mass emerges intrinsically from the self-compensating dynamics and scale variations of the vacuum state itself.

In the long-distance regime $2R\gg {\ell }_{\mathrm{p}}$, the resulting metric deformation rigorously converges with the predictions of weak-field General Relativity, demonstrating that the emergent point mass behaves identically to a conventional localized gravitational source generating Einstein’s fields.

Finally, the most remarkable feature of this framework is that the local deformation factor $1\pm [4GM/(2R+{\ell }_{\mathrm{p}})]$ is completely free of singularities throughout the spacetime domain. At the ultraviolet limit $R=0$, the deformation factor remains strictly finite and saturates at a bounded value of $4GM/{\ell }_{\mathrm{p}}$. This regular behavior is a direct consequence of the underlying quantum structure of the extended field, where geometric level displacements are governed by the discrete pattern $2n_{\mathrm{s}}+1$, which remains non-zero even at its lowest possible ground state $n_{\mathrm{s}}=0$. Consequently, this work offers a promising singularity-free alternative for the origin of localized mass backgrounds derived directly from local vacuum collapse (18).