⚛️ general relativity

EGUP effects on the thermodynamic properties of the Kerr-Newman black hole surrounded by quintessence

This paper investigates how the Extended Generalized Uncertainty Principle (EGUP) influences the thermodynamic properties, stability, and remnant characteristics of a Kerr-Newman black hole surrounded by quintessence, while comparing these effects with those derived from the Generalized Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP) to illustrate variations across different cosmic eras.

Original authors: Aheibam Boycha Meitei, Yenshembam Priyobarta Singh, T. Ibungochouba Singh, Irom Ablu Meitei, Kangujam Yugindro Singh

Published 2026-06-24

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CC BY 4.0

Original authors: Aheibam Boycha Meitei, Yenshembam Priyobarta Singh, T. Ibungochouba Singh, Irom Ablu Meitei, Kangujam Yugindro Singh

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

Imagine the universe as a giant, complex machine. At its smallest scales, this machine isn't smooth and predictable like a clock; instead, it's fuzzy and chaotic, like a foam of bubbles. This is what physicists call "quantum foam."

This paper is a theoretical investigation into how this fuzzy, chaotic nature of space affects the behavior of a very specific type of cosmic monster: a Kerr-Newman black hole. But this isn't just any black hole; it's a spinning, electrically charged one, and it's currently sitting in a sea of "quintessence"—a mysterious, invisible energy that is pushing the universe apart (like a cosmic wind).

The authors want to know: How does the "fuzziness" of space change the temperature, stability, and life cycle of this black hole?

To answer this, they use three different "rulers" to measure the universe, which act like different sets of rules for how things can move and exist:

The Standard Ruler (HUP): The classic rules of physics we learned in school.
The "Too Small" Ruler (GUP): A rule that says, "You can't get infinitely small. There is a minimum size (the Planck length) below which you can't go." This is like trying to zoom in on a digital photo until you hit the pixel limit; you can't see anything smaller than a single pixel.
The "Too Big" Ruler (EUP): A rule that says, "Just as there's a limit to how small things can be, there might also be a limit to how big the universe is." It's like saying the universe has a maximum zoom-out level.
The "All-in-One" Ruler (EGUP): The authors combine the "Too Small" and "Too Big" rules into one super-rule called the Extended Generalized Uncertainty Principle (EGUP). This is the main tool they use for their study.

The Main Findings, Explained Simply

Here is what happens to the black hole when they apply these new, fuzzy rules:

1. The Temperature of the Black Hole
Black holes aren't actually black; they glow with a faint heat called "Hawking radiation." The authors found that when you use the EGUP (the combined rule), the temperature of the black hole changes.

The Analogy: Imagine the black hole is a campfire. The standard rules say it burns at a steady rate. The new rules say that because space is "fuzzy," the fire behaves differently.
The Result: The black hole has a "safe zone" for its size. If it gets too small, the math breaks down unless it stops shrinking at a specific point. This suggests the black hole doesn't vanish completely into nothingness; instead, it leaves behind a tiny, stable "remnant" (like a glowing ember that never goes out).

2. Stability and Phase Transitions (The "Mood Swings")
The paper looks at "heat capacity," which is basically how stable the black hole is.

The Analogy: Think of the black hole as a person. Sometimes they are calm and stable (positive heat capacity). Sometimes they are moody and unstable (negative heat capacity).
The Result:
- Under the EGUP and EUP (the rules that include the "universe size" limit), the black hole has two distinct "mood swings" or tipping points. It can be stable when it's very small, unstable in the middle, and stable again when it's very large. It's like a person who is calm as a baby, chaotic as a teenager, and calm again as an adult.
- Under the GUP (only the "too small" rule), the black hole only has one mood swing. It's a simpler, less dramatic change.

3. The "Swallowtail" Shape
When the authors plotted the "Gibbs Free Energy" (a measure of how much the black hole wants to change its state), they saw a specific shape for the EGUP and EUP cases.

The Analogy: Imagine a bird's tail feathers. The graph looks like a "swallowtail." This shape tells physicists that the black hole can suddenly jump from being a "small black hole" to a "large black hole" in a dramatic event (a first-order phase transition).
The Result: This dramatic jump only happens when the universe's "size limit" (EUP) is taken into account. If you only look at the "too small" limit (GUP), this dramatic jump disappears.

4. Entropy (The Messiness)
Entropy is a measure of disorder or "messiness."

The Result: When the correction parameters (the strength of the fuzzy rules) are small, the black hole gets messier as it grows, which is normal. However, if the fuzzy rules are very strong, the messiness (entropy) rises to a peak and then starts to decrease, eventually becoming negative.
The Warning: A negative entropy is a sign of thermodynamic chaos. It suggests that under extreme fuzzy conditions, the black hole becomes unstable and breaks the usual laws of physics. Interestingly, the "Too Small" rule (GUP) keeps the entropy positive and stable, while the "All-in-One" rule (EGUP) can lead to this instability.

5. Pressure
The authors also looked at the pressure exerted by the "quintessence" (the cosmic wind) around the black hole.

The Result: The pressure drops as the black hole gets bigger. However, the "fuzziness" of space (the EGUP corrections) changes how strong this pressure is, especially for larger black holes.

The Bottom Line

This paper is a mathematical exploration of how the universe's fundamental "pixelation" (both at the smallest and largest scales) changes the life story of a spinning, charged black hole.

Without these rules: The black hole evaporates and disappears.
With these rules: The black hole might stop shrinking at a tiny size, leaving a stable remnant. It also suggests that the black hole can undergo complex "mood swings" (phase transitions) that wouldn't happen if we only looked at the smallest scales.

The authors conclude that the "Extended" rules (EGUP) paint a much more complex and interesting picture of black hole thermodynamics than the standard rules, revealing that the universe's size limits play a crucial role in how these cosmic monsters behave.

Technical Summary: EGUP Effects on the Thermodynamic Properties of the Kerr-Newman Black Hole Surrounded by Quintessence

Problem Statement
This paper investigates the thermodynamic behavior of a Kerr-Newman black hole surrounded by a quintessence field (KNBHQ) under the influence of the Extended Generalized Uncertainty Principle (EGUP). While the Generalized Uncertainty Principle (GUP) addresses the existence of a minimum observable length (Planck scale) relevant to the early universe, and the Extended Uncertainty Principle (EUP) addresses a maximum fundamental length scale relevant to the later universe, the authors aim to unify these effects. The study specifically examines how the integration of GUP and EUP into the EGUP framework modifies key thermodynamic quantities—Hawking temperature, heat capacity, Gibbs free energy, entropy, and pressure—and how these modifications influence the stability and phase transitions of the black hole in the presence of dark energy (quintessence).

Methodology
The authors employ a theoretical framework based on natural units ( $\hbar = c = G = 1$ ). The methodology proceeds as follows:

Theoretical Formulation: The study begins by deriving the EGUP commutation relation, which incorporates both a momentum-dependent term (GUP) and a position-dependent term (EUP). This leads to modified uncertainty relations for position ( $\Delta x$ ) and momentum ( $\Delta p$ ).
Metric and Horizon: The KNBHQ metric in the presence of a quintessence field is defined using Boyer-Lindquist coordinates. The event horizon radius ( $r_h$ ) is determined by the roots of the metric function $\Delta_r = 0$ .
Thermodynamic Derivation:
- Temperature: The Hawking temperature is derived using the first law of black hole mechanics and the modified uncertainty principle. The authors utilize the relation $dA/dS \propto \Delta x \Delta p$ to incorporate quantum corrections into the temperature formula ( $T_{EGUP}$ ).
- Heat Capacity: Calculated via $C = dM/dT$, analyzing the stability of the black hole based on the sign of the heat capacity.
- Gibbs Free Energy: Derived in the canonical ensemble ($G = M - TS$) to analyze global stability and phase transitions.
- Entropy and Pressure: Entropy is calculated by integrating the modified area-entropy relation, and pressure is derived from the quintessence equation of state, adjusted for the EGUP corrections.
Comparative Analysis: The results for EGUP are systematically compared against those derived from GUP alone ( $\eta=0$ ), EUP alone ( $\beta=0$ ), and the standard Heisenberg Uncertainty Principle (HUP). The analysis focuses on two specific quintessence equation of state parameters: $\omega = -1/3$ and $\omega = -2/3$ .

Key Contributions and Results

Modified Hawking Temperature: The study derives explicit expressions for $T_{EGUP}$ , $T_{GUP}$ , and $T_{EUP}$ . The analysis reveals that for $\omega = -1/3$ , the modified temperatures yield positive real values for $r_h \geq 0.330$ . For $\omega = -2/3$ , positive real values exist within the interval $0.320 < r_h < 5$ . The EGUP and EUP corrections significantly influence the temperature, whereas the GUP correction's effect is noted as negligible in comparison.
Heat Capacity and Phase Transitions:
- For $\omega = -1/3$ , the EGUP and EUP corrected heat capacities exhibit two distinct divergence points, indicating two second-order phase transitions. This divides the system into three regimes: stable small black holes, an unstable intermediate phase (negative heat capacity), and stable large black holes.
- In contrast, the GUP corrected heat capacity shows only a single divergence point, corresponding to a single second-order phase transition.
- For $\omega = -2/3$ , all three models (EGUP, EUP, GUP) exhibit only a single phase transition.
- The phase transition points shift to lower event horizon radii as the correction parameters ( $\eta, \beta$ ) increase in the EGUP and GUP models.
Black Hole Remnants: The vanishing of heat capacity implies the formation of a black hole remnant. The authors derive specific expressions for the remnant mass ( $M_{rem}$ ) and remnant temperature ( $T_{rem}$ ) for the EGUP and GUP cases. Notably, in the EUP scenario, no remnant temperature or mass dependent on the parameter $\eta$ is observed.
Gibbs Free Energy and Stability:
- The Gibbs free energy ( $G$ ) analysis reveals a characteristic swallowtail structure for EGUP and EUP when $\omega = -1/3$ , signaling a first-order phase transition between small and large black hole phases, with cusps marking second-order transitions.
- For $\omega = -2/3$ and the GUP case, the swallowtail structure is absent, showing only a single cusp, consistent with a purely second-order phase transition.
- The study notes that EUP corrections indicate strong instability for small black holes (sharp divergence of $G$ at small $r_h$ ), while EGUP and EUP favor large black holes at high $r_h$ .
Entropy and Pressure:
- The corrected entropies for EGUP, EUP, and GUP are found to be independent of the quintessence parameters ( $\omega, \alpha$ ).
- Under large correction parameters, entropy for EGUP and EUP rises to a peak and then decreases, potentially becoming negative (indicating instability), whereas GUP-corrected entropy remains positive and increasing.
- Pressure decreases as the event horizon radius increases. The correction parameters ( $\eta, \beta$ ) lower the pressure, with EGUP and EUP effects being more significant for larger black holes, while GUP effects are stronger for smaller black holes.

Significance and Claims
The paper claims that the integration of the Extended Generalized Uncertainty Principle provides a more comprehensive framework for understanding black hole thermodynamics across different cosmic eras (early vs. late universe). The primary significance lies in demonstrating that EGUP and EUP corrections induce multiple phase transitions and alter the stability regions of Kerr-Newman black holes in a way that GUP alone does not. Specifically, the presence of two divergence points in the heat capacity for certain quintessence parameters suggests a richer thermodynamic phase structure than previously modeled by GUP alone. The authors conclude that quantum gravity corrections significantly modify the stability of black holes and shift phase transitions to higher temperatures, with the EGUP framework offering a unified view of minimum and maximum length scale effects on black hole evaporation and remnant formation.

The Main Findings, Explained Simply

The Bottom Line

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