Static Spherically Symmetric Solutions in Nonconservative Unimodular Gravity

In the vast architecture of the universe, gravity is the force that sculpts space and time. For over a century, physicists have relied on a specific set of rules, known as general relativity, to describe how massive objects like stars and black holes bend the fabric of reality. These rules work with stunning precision, predicting the orbits of planets and the collision of distant galaxies. However, when scientists look closely at the edges of these theories, particularly around the most extreme objects in the cosmos, they begin to wonder if the rules might need a slight adjustment. One such adjustment is a theory called unimodular gravity. It is a variation of the standard rules that treats the volume of space differently, allowing for a subtle freedom in how energy and matter are accounted for. In this framework, the total amount of energy in a region is not strictly fixed by the geometry of space alone; instead, there is a choice to be made about how much of the observed effect comes from actual matter and how much comes from the vacuum of space itself. This freedom creates a puzzle: if we see a specific shape of space around a black hole, can we be sure what is causing it?

A team of researchers has now tackled this puzzle by examining a specific, slightly deformed version of a black hole. They focused on a mathematical description of space that includes a small, unusual term that drops off very quickly as you move away from the center. In the standard view, this term would be caused by a specific type of matter or charge. But in the flexible world of unimodular gravity, that same term could be interpreted in two very different ways. The researchers set out to determine exactly what this term represents and whether the different interpretations lead to different physical realities. They discovered that the answer depends entirely on how one chooses to split the effects between matter and the vacuum. Crucially, they found that while some conclusions about the black hole remain the same regardless of this choice, others change completely. This means that certain ideas about what black holes are made of, which were previously thought to be impossible, might actually be possible if the universe operates under these specific rules.

The study centers on a black hole that is not perfectly simple but has a slight twist in its structure, described by a term that falls off as the cube of the distance. In the standard theory of gravity, this twist would be a direct fingerprint of a specific kind of matter. However, in unimodular gravity, the equations allow for a "reconstruction freedom." This means that for the exact same shape of space, one can choose to say the twist is caused by ordinary matter, or one can choose to say it is caused by a non-conservative exchange of energy between matter and the vacuum. The researchers mapped out every possible way to make this choice. They found that the twist is the key that unlocks the non-conservative part of the theory. If the twist is zero, the theory behaves exactly like standard gravity. But if the twist is present, it acts as a dial that controls how much energy is being exchanged between the matter and the empty space around it.

One of the most significant findings concerns the rules that matter must obey to be considered "normal." Physicists use a set of conditions, known as energy conditions, to decide if a source of gravity is made of ordinary stuff or something exotic and strange. The researchers showed that some of these rules are robust, meaning they hold true no matter how you choose to split the effects between matter and vacuum. These are the rules that apply to light moving through space. However, other rules, which apply to the density and pressure of matter, are fragile. They change depending on the choice made. This distinction is vital because it means that a claim that a black hole must contain "exotic" matter to exist might be an illusion created by a specific choice of how to interpret the data. If one chooses the most natural split, where the matter has no net trace, the black hole can be supported by ordinary electromagnetic fields, just like a standard charged particle, without needing any strange, exotic ingredients.

The team also looked at the observable consequences of this twist. They calculated how light would bend around such a black hole and how stable orbits for stars and gas would behave. They found that the twist affects the position of the photon sphere, the region where light orbits the black hole, and the innermost stable orbit where matter can circle without falling in. By measuring the shifts in these positions, astronomers could, in principle, detect the presence of this twist. The researchers identified a specific combination of measurements that would isolate the twist from other factors, such as electric charge. This combination would act as a clean test for the non-conservative sector of the theory. However, they also noted that current telescopes are not yet precise enough to detect this effect. The signal is too small, and the measurements required are far beyond our current capabilities. Furthermore, the rotation of real black holes would complicate these measurements, meaning this test cannot be applied to the spinning black holes we observe in the sky until the theory is extended to include rotation.

Perhaps the most surprising result involves the existence of multiple rings of light around a black hole. In standard gravity, if a black hole has three distinct rings of light outside its event horizon, it usually implies that the matter surrounding it violates the rules of normal physics. The researchers found that in their framework, this is not necessarily true. By choosing the optimal split between matter and vacuum, it is possible to have a black hole with three light rings while still satisfying all the rules of normal matter. This does not mean such black holes are common; the mathematical conditions required for them are extremely specific and correspond to objects that are more highly charged than any known black hole. Nevertheless, the result proves that the barrier preventing such configurations is not a fundamental law of geometry, but rather a consequence of how we choose to interpret the source of gravity.

The study also addressed a potential objection: could this unusual term simply be an artifact of how we measure distance? The researchers demonstrated that the term is tied to a fundamental property of space called curvature, which cannot be removed by changing coordinates. It is a real, physical feature. They also explored what happens if a cosmological constant, a term representing the energy of empty space, is included. They found that while this constant changes the location of the event horizon, it does not change the fundamental nature of the twist or the matter required to support it. The twist remains the carrier of the non-conservative sector, and the ability to satisfy normal matter conditions remains intact.

In the end, this work provides a clear map of the possibilities within unimodular gravity. It shows that the universe could contain black holes with this specific twist, supported by ordinary matter, if we allow for a non-conservative exchange of energy. It clarifies which statements about black holes are absolute truths of geometry and which are merely artifacts of a specific interpretation. While we cannot yet measure these effects, the theoretical groundwork is laid. The researchers have shown that the door to these exotic possibilities is not locked by the laws of physics, but by the precision of our instruments and the specific choices we make in how we define the contents of the universe. The path forward requires building the tools to measure these subtle shifts and extending the theory to the rotating black holes that populate our sky, but the theoretical landscape has been significantly clarified.