Thermodynamic stability and geometric thermodynamics of charged black holes in Rastall-massive gravity under quintessence

Imagine the universe as a giant, cosmic stage where gravity is the director. For decades, we've watched the show using a script written by Albert Einstein, known as General Relativity. It's a brilliant script that explains how massive objects like stars bend space and time, creating the "gravity" we feel. But sometimes, the universe throws plot twists that the old script can't quite explain, like why the universe is speeding up its expansion. To fix the script, scientists have started writing "sequels" and "spin-offs"—new theories of gravity that tweak the rules. Two of the most popular new drafts are "Rastall gravity," which changes how energy and matter talk to geometry, and "massive gravity," which gives the invisible messenger of gravity (the graviton) a tiny bit of weight.

In this cosmic theater, there are also mysterious actors called "black holes." These are regions where gravity is so strong that not even light can escape. But black holes aren't just dark pits; they have a temperature and an entropy, acting like hot, messy thermodynamic systems (think of them as cosmic steam engines). Recently, scientists have been asking: What happens if we put a black hole in a room filled with "quintessence"? Quintessence is a mysterious, ghostly energy field that pushes the universe apart, acting like a negative pressure. The big question is: How do these new gravity rules and this ghostly energy change the way black holes behave, heat up, and even change phases? It's like asking how a steam engine would run if you changed the fuel and the pressure gauge at the same time.

This paper dives into that exact question, but with a twist. The authors, Mohamed Chabab, Samir Iraoui, and Hicham Sriba, create a brand-new mathematical model of a charged black hole. They combine Rastall gravity, massive gravity, and a surrounding field of quintessence. Their main discovery is that this complex mix of forces creates a black hole that behaves remarkably like a Van der Waals fluid—the same type of physics that describes how water turns into steam or how gases condense into liquids.

Here is the fun part: The authors suggest that the "quintessence" surrounding the black hole acts like a thermodynamic pressure. Just as you can squeeze a gas to turn it into a liquid, they found that changing the strength of this surrounding field causes the black hole to jump between a "small" phase and a "large" phase. They mapped out these transitions and found a critical point where the black hole is on the edge of changing states, much like water at its boiling point.

But the paper doesn't stop at math equations. The authors propose a clever way to "see" these invisible thermodynamic changes. They looked at the paths of light (photons) swirling around the black hole. In the chaotic zone just outside the event horizon, there is a "photon sphere" where light can orbit the black hole in unstable circles. The authors found that the size of this photon sphere and the angle at which light gets captured (the impact parameter) change in a way that perfectly mirrors the black hole's thermodynamic phase transitions. It's as if the black hole's "thermostat" is written in the way light dances around it. If we could observe these light orbits with future telescopes, we might be able to detect these phase changes without ever touching the black hole.

To double-check their work, the authors used a tool called "geometrothermodynamics." This is like drawing a map of the black hole's thermodynamic landscape. They found that the "mountains" and "valleys" on this map (specifically, the points where the map's curvature blows up) line up exactly with the points where the black hole's heat capacity becomes infinite. This confirms that their thermodynamic findings are solid and consistent with the geometry of the system.

In short, the paper suggests that by mixing Rastall gravity, massive gravity, and quintessence, we get a black hole that acts like a cosmic fluid, switching between small and large states. While they haven't observed this in the real sky yet, their calculations show that the way light orbits these black holes could serve as a hidden signature, allowing us to spot these thermodynamic shifts from afar. The results are consistent with known physics for charged black holes in other theories, but this new framework offers a unified way to understand how these different gravity theories and energy fields interact.