⚛️ general relativity

Uncertainty Principles and Non-local Black Holes

This paper argues that the Generalized and Extended Uncertainty Principles are effective descriptions of non-local gravitational interactions in Infinite Derivative Gravity, using their comparison with non-local black hole solutions to derive theoretical constraints on free parameters and establish universal laws for black hole physics beyond General Relativity.

Original authors: Salvatore Capozziello, Giuseppe Meluccio, Jonas R. Mureika

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Salvatore Capozziello, Giuseppe Meluccio, Jonas R. Mureika

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Fog and the Quantum Blur

Imagine the universe as a giant, cosmic game of billiards. For centuries, we've had two rulebooks for how the balls move. The first rulebook, called General Relativity, describes gravity as a smooth, curved trampoline made of space and time. It's perfect for big things like planets and stars. The second rulebook, Quantum Mechanics, describes the tiny, jittery world of atoms and particles, where things are fuzzy and you can't know exactly where a particle is and how fast it's going at the same time. This is the famous "Uncertainty Principle."

The trouble starts when these two rulebooks try to play together, especially in the most extreme places in the universe: black holes. These are cosmic vacuum cleaners so dense that they crush everything into a single, infinitely small point called a "singularity," where the math breaks down and the rules stop making sense. Scientists have been trying to write a new, unified rulebook—Quantum Gravity—to fix this. One promising idea is that gravity isn't perfectly smooth like a trampoline; instead, at the tiniest scales, it might be "fuzzy" or "smeared out," like a photograph that's slightly out of focus. This paper explores whether this cosmic fuzziness is the same thing as the quantum uncertainty we see in particles, and if so, what that means for the life and death of black holes.

The Paper's Big Idea: When Gravity Gets Fuzzy

In this study, physicists Salvatore Capozziello, Giuseppe Meluccio, and Jonas Mureika act like cosmic detectives trying to solve a mystery: Is the "fuzziness" of gravity (from a theory called Infinite Derivative Gravity) actually the same thing as the "fuzziness" of quantum mechanics (from the Generalized and Extended Uncertainty Principles)?

Think of it like this: Imagine you are trying to describe a blurry image. You could say, "The camera lens is dirty," which is a problem with the tool (gravity). Or, you could say, "The object itself is made of smoke," which is a problem with the thing being photographed (matter). This paper suggests that in the universe, these two explanations might actually be two sides of the same coin. The authors argue that the weird, fuzzy rules of quantum mechanics (which usually only apply to tiny particles) might actually be an "effective description" of a deeper, non-local reality where gravity itself is smeared out over space.

They looked at two different types of this "smearing":

The Tiny Scale (UV): This happens at incredibly small distances, near the Planck length ( $L_P$ ), where quantum effects dominate.
The Huge Scale (IR): This happens at massive distances, like the size of the entire universe (the Hubble length, $L_*$ ), where dark energy and dark matter might live.

The Detective Work: Matching the Potentials

To test their theory, the authors compared two different mathematical "potentials" (which describe how strong gravity is at a certain distance).

Potential A comes from the modified uncertainty principles (GUP and EUP), which assume the fuzziness is in the matter.
Potential B comes from the non-local gravity theory (IDG), which assumes the fuzziness is in the fabric of space itself.

The authors asked a simple question: "At what specific distance do these two different descriptions of gravity give the exact same answer?" By setting the equations equal to each other at these characteristic distances ( $L_{UV}$ and $L_{IR}$ ), they could solve for the "fuzziness" parameters ( $\beta$ and $\alpha$ ) that had previously been just free guesses.

The Surprising Findings

The results were quite specific and changed the picture of black holes in two major ways:

1. The "Fuzziness" Depends on Mass
The paper found that the amount of fuzziness isn't a constant number; it changes depending on how heavy the black hole is.

For the tiny scale (UV), the fuzziness parameter $\beta$ turns out to be negative and depends on the square of the mass ratio ( $m/M_P$ ).
For the huge scale (IR), the fuzziness parameter $\alpha$ is also negative and depends on the square of the inverse mass ratio ( $M_P/m$ ).
This suggests that the "rules" of quantum uncertainty might actually break the Equivalence Principle (the idea that all objects fall the same way) at these specific scales, because the fuzziness changes based on the object's weight.

2. Black Holes Get Smaller and Hotter
Because of this new "smearing" effect, the authors calculated that black holes look different than Einstein predicted:

Smaller Horizons: The event horizon (the point of no return) shrinks.
- In the tiny-scale (UV) scenario, the horizon becomes about 0.5 times the standard Schwarzschild radius ( $R_S$ ).
- In the huge-scale (IR) scenario, the horizon shrinks even more, to about 0.1 times the standard radius.
Hotter Evaporation: Black holes don't just sit there; they slowly evaporate. The paper suggests that because the horizon is smaller, these black holes would evaporate much faster and hotter than standard theory predicts.
- In the UV case, the temperature would be about 2 times the standard Hawking temperature ( $T_H$ ).
- In the IR case, the temperature would be a scorching 10 times the standard Hawking temperature.

What This Means (and What It Doesn't)

The authors are careful to note that they haven't proved this is how the universe works. Instead, they suggest that if non-local gravity is real, then the Generalized and Extended Uncertainty Principles are likely the "shadow" or the "effective description" of that reality.

They explicitly rule out the idea that these effects are just random or that the parameters are arbitrary; their math forces specific values based on the mass of the black hole. However, they also admit that this is currently a theoretical exercise based on black hole physics. They haven't measured this in a lab or observed it in a telescope yet. The paper concludes that while these findings offer a "universal law" for black holes beyond General Relativity, we need to do more work to see if these ideas hold up in other physical scenarios or if we can spot these "hotter, smaller" black holes in the real sky.

In short, the paper proposes a beautiful link: the "fuzziness" of the quantum world might just be our way of seeing the "smear" of a non-local universe, and if we're right, black holes are smaller, hotter, and more mysterious than we ever imagined.

=== DRAFT ===
Technical Summary: Uncertainty Principles and Non-local Black Holes

Problem Statement
Black holes serve as a primary theoretical testing ground for deviations from both Quantum Mechanics (QM) and General Relativity (GR). While General Relativity predicts singularities and standard thermodynamics, Quantum Gravity (QG) research seeks to resolve these shortcomings. A promising avenue involves non-locality, specifically Infinite Derivative Gravity (IDG), which offers singularity-free, unitary, and renormalizable QG theories. Concurrently, modifications to the Heisenberg Uncertainty Principle, namely the Generalized Uncertainty Principle (GUP) for UV scales and the Extended Uncertainty Principle (EUP) for IR scales, have been proposed to describe quantum gravitational effects. The central problem addressed is the lack of a unified theoretical framework connecting these modified uncertainty principles with non-local gravitational theories. Specifically, it remains unclear whether GUP and EUP are fundamental laws or merely effective descriptions arising from the non-local features of gravity.

Methodology
The authors investigate static black hole solutions within the framework of IDG, focusing on two distinct regimes:

UV Regime (GUP): The study considers IDG theories involving analytic functions of the d'Alembert operator ( $\square$ ) that resolve UV singularities. The authors derive the Newtonian potential for a static point-like source in the weak-field limit, which results in a "smeared" source distribution characterized by a non-local length scale $L_{UV}$ . This potential, $\Phi_{NL}(r)$ , is compared against the potential derived from GUP-corrected gravity, $\Phi_{GUP}(r)$ .
IR Regime (EUP): Similarly, the study examines IDG theories involving inverse d'Alembert operators ( $\square^{-1}$ ) relevant to IR corrections (e.g., dark energy/matter). The authors derive the Newtonian potential $\Phi_{NL}(r)$ for a static source with a characteristic IR length scale $L_{IR}$ , involving generalized hypergeometric functions. This is compared against the potential derived from EUP-corrected gravity, $\Phi_{EUP}(r)$ .

The core methodological approach involves equating the non-local gravitational potentials ( $\Phi_{NL}$ ) with the modified uncertainty principle potentials ( $\Phi_{GUP}$ or $\Phi_{EUP}$ ) at their respective characteristic length scales ( $L_{UV}$ and $L_{IR}$ ). By solving the equation $\Phi_{mod}(L) = \Phi_{NL}(L)$ , the authors determine the values of the dimensionless parameters $\beta$ (for GUP) and $\alpha$ (for EUP) that would make the two descriptions equivalent.

Key Contributions and Results
The paper argues that GUP and EUP can be viewed as effective descriptions of non-local gravity at specific scales. The comparison yields the following specific results:

UV/GUP Connection:
- By matching the GUP potential to the non-local potential at $L_{UV}$ , the authors derive a specific form for the GUP parameter $\beta$ . The result indicates that $\beta$ is negative and depends quadratically on the ratio of the black hole mass to the Planck mass ( $\beta \propto -m^2/M_P^2$ ).
- This specific mass dependence of $\beta$ cancels the explicit mass dependence in the GUP-inspired metric correction term, leading to a universal prediction for the event horizon radius: $R_{EH} \approx 0.5 R_S$ (where $R_S$ is the standard Schwarzschild radius).
- Consequently, the black hole evaporation temperature is predicted to be approximately twice the standard Hawking temperature ( $T_e \approx 2 T_H$ ).
IR/EUP Connection:
- By matching the EUP potential to the non-local potential at $L_{IR}$ , the authors derive the EUP parameter $\alpha$ . Similar to the UV case, $\alpha$ is found to be negative and depends quadratically on the inverse mass ratio ( $\alpha \propto -M_P^2/m^2$ ) and the ratio of the characteristic scales.
- This leads to a universal prediction for the event horizon radius in the IR regime: $R_{EH} \approx 0.1 R_S$ .
- The evaporation temperature is predicted to be approximately ten times the standard Hawking temperature ( $T_e \approx 10 T_H$ ).
Theoretical Constraints:
- The analysis provides theoretical constraints on the otherwise free parameters $\beta$ and $\alpha$ , linking them directly to the mass of the source and the fundamental scales of the non-local theory.
- The results suggest that the "fuzziness" introduced by GUP/EUP is an effective manifestation of the non-local "smearing" of matter sources in IDG.

Significance and Claims
The paper claims that the connection between non-locality and modified uncertainty principles arises from the dynamics of gravitational interactions. While GUP/EUP modify the behavior of matter fields assuming standard GR, and non-local gravity modifies the spacetime curvature assuming standard QM, the field equations allow these effects to be attributed to either sector. The authors posit that non-local gravity offers a more fundamental reformulation, which can be approximated by GUP or EUP in appropriate regimes.

The significance of these findings lies in the prediction of universal deviations from GR for black hole physics beyond the standard model:

Reduced Event Horizons: Both UV and IR non-local effects predict event horizons significantly smaller than the Schwarzschild radius.
Enhanced Evaporation: Black holes in these frameworks are predicted to evaporate at temperatures significantly higher than the Hawking prediction.
Equivalence Principle Violation: The derived expressions for $\beta$ and $\alpha$ imply a mass dependence that would suggest a potential violation of the equivalence principle at the quantum level, as the fundamental commutation relations become mass-dependent.

The authors maintain a modest scope, noting that these results are derived specifically within the framework of black hole physics and that the domain of validity for these mass-dependent parameters requires further investigation in different physical scenarios. They do not claim to have experimentally verified these deviations but rather provide theoretical constraints and universal laws for black hole physics in the context of non-local gravity.

The Cosmic Fog and the Quantum Blur

The Paper's Big Idea: When Gravity Gets Fuzzy

The Detective Work: Matching the Potentials

The Surprising Findings

What This Means (and What It Doesn't)

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