Causal Lorentzian Theory (CLT) Applied to Planck-Scale Black Holes and Micro-Scale Quantum Correlations

1. Introduction

General Relativity (GR) successfully describes gravitation in weak-field and astrophysical regimes, yet it generically predicts curvature singularities and event horizons in strong-field scenarios. At Planck scales, the classical theory loses predictive power, while a fully consistent quantum theory of gravity remains elusive.

Most approaches attempt to quantize spacetime itself, leading to conceptual difficulties such as nonlocality, loss of causality, and ambiguities in energy localization. An alternative strategy is to preserve flat spacetime while modifying the physical description of gravitation.

Causal Lorentzian Theory (CLT) adopts this approach. Gravity is described as a physical field propagating causally at speed $c$, carrying local, positive-definite energy. Gravitational effects arise from conformal time dilation, not spatial curvature. This permits a consistent treatment of strong fields, removes singularities, and allows controlled semi-classical coupling to quantum matter.

Recent experimental advances motivate the study of gravitationally induced quantum phases and correlations in micro-scale systems. CLT provides a natural framework for such investigations.

This paper applies CLT to:

• Planck-scale compact objects,
• finite horizon-free gravitational energy emission,
• gravitationally induced quantum phase correlations in multi-particle systems.

2. Velocity-Dependent Conformal Lorentz Transformations

The kinematical foundation of CLT is the velocity-dependent conformal Lorentz transformation

$$x^{\text{'}\mu }=\mathsf{\Omega }\left(v^2\right){\mathsf{\Lambda }}_v^{\mu }\left(v\right)x^v$$

Where ${\mathsf{\Lambda }}_v^{\mu }\left(v\right)$is the standard Lorentz transformation and $\mathsf{\Omega }$ is a conformal factor determined by gravitational field energy.

Unlike GR, spacetime remains Minkowskian. The conformal factor modifies physical clocks and rulers, producing gravitational time dilation while preserving Lorentz invariance and causality.

3. CLT Framework for Planck-Scale Compact Objects

3.1. Smoothed Mass Distribution

To eliminate point singularities, matter sources are modeled by smooth distributions. We adopt a Gaussian profile

$$\rho \left(r\right)={\displaystyle \frac{3M}{4\pi L^3}}{e}^{-r^2/L^2},$$

with enclosed mass

$$m\left(r\right)=M\left[\mathrm{e}\mathrm{r}\mathrm{f}\left({\displaystyle \frac{r}{L}}\right)−{\displaystyle \frac{2r}{\sqrt{\pi }L}}{e}^{-r^2/L^2}\right].$$

For Planck-scale objects, $L={\mathcal{l}}_p$.

This regularization ensures finite gravitational fields and finite field energy everywhere.

3.2. Conformal Gravitational Field (No Curved Space)

In CLT, gravity enters through a conformal scaling of flat spacetime:

$$d{s}^2={\mathsf{\Omega }}^2\left(r\right)\left(c^{2}d{t}^2−d{r}^2−r^{2}d{\mathsf{\Omega }}^2\right),$$

We define the inverse conformal factor as

$${\mathsf{\Omega }}^{-1}\left(r\right)=1-{\displaystyle \frac{GM}{c^{2}r}}.$$

This choice ensures agreement with the standard weak-field gravitational time dilation at first order.

Squaring immediately gives

$${\mathsf{\Omega }}^{-2}\left(r\right)={\left(1−{\displaystyle \frac{GM}{c^{2}r}}\right)}^2=1-{\displaystyle \frac{2GM}{c^{2}r}}+{\displaystyle \frac{G^2{M}^2}{c^4{r}^2}}.$$

Thus, the metric reproduces the correct Newtonian limit while providing a well-defined second-order correction in strong gravitational fields, without invoking spacetime curvature or horizons.

There is:

• no spatial curvature,
• no event horizon,
• no curvature singularity.

Time dilation remains finite for all $r$.

4. Horizon-Free Gravitational Energy Emission

CLT does not admit Hawking radiation in the thermodynamic sense, as event horizons do not exist. Instead, compact objects emit a finite, causal gravitational energy flux arising from time-dependent gravitational field energy.

The effective potential is

$${\mathsf{\Phi }}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(r\right)=-{\displaystyle \frac{Gm\left(r\right)}{r}}.$$

A representative expression for the emitted power is

$$P_{\mathrm{C}\mathrm{L}\mathrm{T}}\propto {\displaystyle \frac{\mathrm{\hslash }}{c^2}}{\mid \nabla {\mathsf{\Phi }}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(r\right)\mid }^2,$$

which remains finite everywhere. Energy propagates outward at speed $c$, ensuring strict causality.

This horizon-free emission plays the physical role attributed to Hawking radiation in GR while avoiding divergences and information-loss paradoxes.

5. Gravitationally Induced Quantum Phase Accumulation

5.1. Single-Particle Phase

In CLT, quantum phase accumulation arises from conformal time dilation:

$$\varphi ={\displaystyle \frac{m{c}^2}{\mathrm{\hslash }}}\int \left[\mathsf{\Omega }\left(r\right(t\left)\right)-1\right]dt.$$

Absolute phases may be large, but they are not observable. Only relative phase differences modulo $2\pi$ have physical meaning.

5.2. Two-Particle and Multi-Particle Correlations

For a particle traversing two branches,

$$\mathsf{\Delta }\varphi ={\displaystyle \frac{m{c}^2}{\mathrm{\hslash }}}\int \left[\mathsf{\Omega }\right(r_1\left(t\right))-\mathsf{\Omega }(r_2\left(t\right)\left)\right]dt.$$

For $N$ particles, CLT predicts a causal phase-correlation network

$$\mathsf{\Delta }{\varphi }_{ij}={\displaystyle \frac{m_i{c}^2}{\mathrm{\hslash }}}\int \left[\mathsf{\Omega }\right(r_{ij}^{\left(1\right)}(t-r/c))-\mathsf{\Omega }(r_{ij}^{\left(2\right)}(t-r/c)\left)\right]dt.$$

These correlations arise from shared gravitational field energy and do not require quantization of gravity.

6. Comparison with General Relativity

CLT and GR agree in the weak-field regime but diverge fundamentally in strong-field and micro-scale contexts. CLT preserves flat spacetime, enforces explicit causality, localizes gravitational energy, and remains finite at all scales. A conceptual comparison is summarized in Table 1.

Caption: Conceptual and qualitative distinctions between General Relativity (GR) and Causal Lorentzian Theory (CLT). Both theories agree in the weak-field regime, while differing fundamentally in their treatment of strong fields, causality, and gravitational energy.

7. Experimental Relevance

CLT predicts observable relative phase shifts in micro-scale quantum systems. Order-of-magnitude estimates indicate potential detectability in:

• levitated nanoparticle interferometry,
• atom interferometers,
• optomechanical oscillator networks.

Absolute phases are unobservable; experimental signatures appear as fringe displacements or correlated phase noise.

Table 2. Experimental-Scale Predictions of Gravitational Phase Shifts in CLT. 

Experimental system	$\mathbf{Test}\mathbf{mass}\mathit{m}$	$\mathbf{Path}\mathbf{separation}\mathsf{\Delta }\mathit{r}$	$\mathbf{Interaction}\mathbf{time}\mathit{t}$	$\mathbf{Predicted}\mathbf{relative}\mathbf{phase}\mathbf{shift}\mathsf{\Delta }\mathit{\varphi }$
Levitated nanoparticle interferometer	${10}^{-17}\mathrm{k}\mathrm{g}$	$1\mu \mathrm{m}$	$1\mathrm{s}$	${10}^{-6}\text{–}{10}^{-3}\mathrm{r}\mathrm{a}\mathrm{d}$
Atom interferometer	${10}^{-25}\mathrm{k}\mathrm{g}$	$10\mu \mathrm{m}$	$0.1\mathrm{s}$	$\lesssim {10}^{-6}\mathrm{r}\mathrm{a}\mathrm{d}$
Optomechanical micro-oscillator pair	${10}^{-14}\mathrm{k}\mathrm{g}$	$0.1\mu \mathrm{m}$	$1\mathrm{s}$	$\lesssim {10}^{-2}\mathrm{r}\mathrm{a}\mathrm{d}$
Three-particle configuration	comparable	comparable	$1\mathrm{s}$	Correlated phase signal
$N$-particle network	variable	variable	variable	Correlation scaling $\propto N(N-1)$

Table 2. Experimental-Scale Predictions of Gravitational Phase Shifts in CLT. 

Experimental system	$\mathbf{Test}\mathbf{mass}\mathit{m}$	$\mathbf{Path}\mathbf{separation}\mathsf{\Delta }\mathit{r}$	$\mathbf{Interaction}\mathbf{time}\mathit{t}$	$\mathbf{Predicted}\mathbf{relative}\mathbf{phase}\mathbf{shift}\mathsf{\Delta }\mathit{\varphi }$
Levitated nanoparticle interferometer	${10}^{-17}\mathrm{k}\mathrm{g}$	$1\mu \mathrm{m}$	$1\mathrm{s}$	${10}^{-6}\text{–}{10}^{-3}\mathrm{r}\mathrm{a}\mathrm{d}$
Atom interferometer	${10}^{-25}\mathrm{k}\mathrm{g}$	$10\mu \mathrm{m}$	$0.1\mathrm{s}$	$\lesssim {10}^{-6}\mathrm{r}\mathrm{a}\mathrm{d}$
Optomechanical micro-oscillator pair	${10}^{-14}\mathrm{k}\mathrm{g}$	$0.1\mu \mathrm{m}$	$1\mathrm{s}$	$\lesssim {10}^{-2}\mathrm{r}\mathrm{a}\mathrm{d}$
Three-particle configuration	comparable	comparable	$1\mathrm{s}$	Correlated phase signal
$N$-particle network	variable	variable	variable	Correlation scaling $\propto N(N-1)$

Caption: Order-of-magnitude estimates of observable relative gravitational phase shifts predicted by Causal Lorentzian Theory (CLT) for experimentally accessible systems. Absolute gravitational phases are unobservable; experimental signatures appear as relative phase differences, fringe displacements, or correlated phase noise. Values shown assume optimized isolation and coherence.

8. Conclusions

Causal Lorentzian Theory provides a consistent, singularity-free, causal framework for gravitation at Planck scales and its interaction with quantum matter. By replacing spacetime curvature with conformal time dilation, CLT eliminates horizons and divergences while remaining predictive and experimentally accessible. The theory naturally produces gravitationally induced quantum phase correlations without quantizing gravity, offering a viable semi-classical bridge between gravitational and quantum phenomena.

Future work will extend numerical modeling, explore larger multi-particle networks, and refine experimental proposals.