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

1. Introduction

Classical General Relativity predicts singularities at black hole centers, and quantum gravity remains unresolved at Planck scales. Traditional approaches, such as string theory and loop quantum gravity, attempt to reconcile quantum mechanics and gravity but face conceptual and technical challenges.

Causal Lorentzian Theory (CLT) offers an alternative framework: it enforces causal field propagation, smooths point-like sources, and maintains classical energy conservation. CLT builds upon the velocity-dependent conformal Lorentz transformations (VDC-LT) [1,4,5], which generalize standard Lorentz transformations with a velocity-dependent conformal factor.

Recent theoretical and experimental proposals suggest that gravitational interactions can induce measurable phase shifts or correlations in quantum systems [8,9,10]. This motivates the study of gravitational phases and correlations in a semi-classical, causal background.

This paper explores CLT applications to:

Planck-scale black holes — regularized curvature and horizons.

Hawking radiation — causal, finite-energy emission.

Quantum particle correlations — gravitationally induced phase differences in multi-particle systems.

2. Velocity-Dependent Conformal Lorentz Transformations (VDC-LT)

The foundational transformation in CLT is:

$$x^{\mathrm{\text{'}}\mu }={\mathsf{\Lambda }}_{\nu }^{\mu }(v)x^{\nu }\mathsf{\Omega }(v),$$

where ${\mathsf{\Lambda }}_{\nu }^{\mu }(v)$ is the standard Lorentz matrix and $\mathsf{\Omega }(v)$ is a velocity-dependent conformal factor.

This factor accounts for:

Time contraction effects in moving systems

Relativistic mass scaling compatible with CLT

Causal propagation of fields at speed c

Using VDC-LT, CLT ensures causal field propagation, smooths classical singularities, and provides a semi-classical bridge between classical fields and quantum matter.

3. CLT Framework

3.1. Smoothed Mass Distribution

To remove singularities, the mass is distributed as a Gaussian:

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

$$m(r)=4\pi {\int }_0^r\rho (r^{\mathrm{\text{'}}})r^{\mathrm{\text{'}}2}d{r}^{\mathrm{\text{'}}}=M\left[\mathrm{e}\mathrm{r}\mathrm{f}(r/L)-{\displaystyle \frac{2r}{\sqrt{\pi }L}}{e}^{-r^2/L^2}\right],$$

where $L=l_p$ is the Planck length for Planck-scale black holes.

The metric reads:

$$d{s}^2=-\left(1−{\displaystyle \frac{2Gm(r)}{r}}\right)d{t}^2+{\left(1−{\displaystyle \frac{2Gm(r)}{r}}\right)}^{-1}d{r}^2+r^{2}d{\mathsf{\Omega }}^2.$$

The Kretschmann scalar is finite at $r=0$, removing the classical singularity.

3.2. Causal Hawking Radiation

The surface gravity is:

$$\kappa (r)={\displaystyle \frac{Gm(r)}{r^2}}-{\displaystyle \frac{G{m}^{\mathrm{\text{'}}}(r)}{r}}.$$

Using the semi-classical Hawking formula [11,12]:

$$P(r)={\displaystyle \frac{\mathrm{\hslash }\kappa (r{)}^2}{2\pi c^2}}.$$

In CLT, the flux propagates causally:

$$u(r,t)={\displaystyle \frac{P(r_h)}{4\pi r^2}}\delta \left(t−{\displaystyle \frac{r}{c}}\right),$$

ensuring smooth, finite emission. The black hole mass decreases as:

$${\displaystyle \frac{dM}{dt}}=-{\displaystyle \frac{P_{\mathrm{CLT}}}{c^2}}.$$

4. Micro-Scale Quantum Correlations

4.1. Single-Particle Phase Accumulation

A particle of mass $m$ at distance $r$ experiences a gravitational phase:

$$\varphi (t)={\displaystyle \frac{1}{\mathrm{\hslash }}}{\int }_0^{t}V(t^{\mathrm{\text{'}}})d{t}^{\mathrm{\text{'}}}=-{\displaystyle \frac{Gm}{\mathrm{\hslash }r}}{\int }_0^{t}M(t_r)d{t}^{\mathrm{\text{'}}},$$

with retarded time $t_r=t-r/c$.

4.2. Two-Particle and N-Particle Correlations

For two particles A and B:

$$\mathsf{\Delta }\varphi =({\varphi }_{A1}+{\varphi }_{B1})-({\varphi }_{A2}+{\varphi }_{B2})$$

$$={\displaystyle \frac{G}{\mathrm{\hslash }}}{\int }_0^{t}M(t^{\mathrm{\text{'}}})\left({\displaystyle \frac{m_A}{r_{A1}}}+{\displaystyle \frac{m_B}{r_{B1}}}-{\displaystyle \frac{m_A}{r_{A2}}}-{\displaystyle \frac{m_B}{r_{B2}}}\right)d{t}^{\mathrm{\text{'}}}.$$

For N particles, the correlation matrix is:

$$\mathsf{\Delta }{\varphi }_{ij}(t)={\displaystyle \frac{G{m}_i}{\mathrm{\hslash }}}{\int }_0^t\left[{\displaystyle \frac{M(t-r_{i1}/c)}{r_{i1}}}-{\displaystyle \frac{M(t-r_{i2}/c)}{r_{i2}}}\right]d{t}^{\mathrm{\text{'}}}.$$

This forms an N×N network of gravitationally induced phase correlations.

5. Experimental Relevance and Challenges

CLT predicts measurable phase differences in micro-scale quantum systems:

Levitated nanoparticles in interferometers [9]

Optomechanical micro-oscillators sensitive to gravitational correlations [8]

Gravitational Aharonov–Bohm-type experiments [10]

Challenges include:

Environmental decoherence

Extremely small phase measurements

Synchronization of N-particle systems

Control of classical gravitational backgrounds

6. Comparison with Existing Quantum Gravity Frameworks

CLT differs from string theory and loop quantum gravity by:

Enforcing causal propagation at speed c

Resolving singularities via smoothed mass distributions

Providing semi-classical predictivity for quantum phases

Serving as a classical baseline rather than a fully quantized spacetime

7. Numerical Examples

7.1. Single-Particle Phase Accumulation

Particle mass	Distance	Time	Phase
10−14 kg	10−12 m	1 s	1.38 × 1014 rad
10−14 kg	2 × 10−12 m	1 s	6.9 × 1013 rad
10−14 kg	1.5 × 10−12 m	1 s	9.2 × 1013 rad

7.2. Two-Particle Correlation

Particle	Branch 1	Branch 2	$\mathsf{\Delta }\varphi$
A	10−12 m	2 × 10−12 m	6.9 × 1013
B	1.5 × 10−12 m	2.5 × 10−12 m	3.7 × 1013
Total	—	—	1.24 × 1014

7.3. Three-Particle Correlation Matrix

	P1	P2	P3
P1	0	4.6 × 1013	6.9 × 1013
P2	−4.6 × 1013	0	2.3 × 1013
P3	−6.9 × 1013	−2.3 × 1013	0

8. Conclusions

CLT, built upon VDC-LT, provides a singularity-free, causal, and predictive classical framework for Planck-scale black holes and gravitationally induced quantum correlations. It offers measurable predictions for micro-scale experiments and serves as a bridge between classical fields and quantum matter. Future work includes larger N-particle simulations, experimental connections, and early-universe implications.