Black holes are among the most mysterious objects in the universe. They are regions of space where gravity becomes so strong that nothing—not even light—can escape. For decades, scientists have been trying to understand not only how black holes work, but also how they fit into the laws of physics that govern the universe.
One of the biggest discoveries in modern science is that black holes behave like thermodynamic systems. Just like a hot cup of coffee has temperature and energy, black holes also have temperature, entropy, and other thermodynamic properties. A new study by Fu and his team has taken an important step toward understanding how these properties behave when a black hole is changing and growing.
Their research suggests that the entropy of a dynamical black hole is closely linked to the area of its apparent horizon, a boundary that may be more important than the famous event horizon when a black hole is evolving.
What Is Black Hole Entropy?
Entropy is a measure of disorder or hidden information in a system. In everyday life, entropy explains why ice melts, why heat flows from hot objects to cold ones, and why natural processes tend to move toward greater disorder.
For black holes, entropy is even more fascinating. Scientists believe it contains information about the microscopic structure of spacetime itself. Understanding black hole entropy could help answer some of the biggest questions in physics, such as:
What is the true nature of gravity?
How does quantum mechanics work with gravity?
What happens to information that falls into a black hole?
How can we develop a theory of quantum gravity?
Because of this, black hole entropy is one of the most important topics in theoretical physics.
The Beginning of the Story
In the 1970s, physicist Jacob Bekenstein proposed a revolutionary idea. He suggested that the entropy of a black hole is proportional to the area of its event horizon.
At first, this idea seemed strange. Why should a black hole's entropy depend on its surface area instead of its volume?
Soon afterward, Stephen Hawking discovered that black holes emit a tiny amount of radiation due to quantum effects. This radiation, now called Hawking radiation, showed that black holes have a temperature.
Together, these discoveries led to the famous conclusion that a black hole's entropy is directly related to the area of its horizon.
For stationary black holes—those that remain unchanged over time—this relationship is well understood. But real black holes are rarely perfectly still.
Why Dynamical Black Holes Are Different
In reality, black holes constantly interact with their surroundings. They absorb gas, dust, stars, and even merge with other black holes. As a result, they grow and change over time.
Scientists call these changing objects dynamical black holes.
When a black hole evolves, defining its entropy becomes much more difficult. The traditional event horizon depends on the entire future history of the universe, making it difficult to use in situations where the black hole is actively changing.
This has led researchers to explore another important boundary known as the apparent horizon.
Unlike the event horizon, the apparent horizon can be identified using information available at the present moment. It is a local boundary that changes as the black hole evolves.
Many physicists now believe that the apparent horizon may provide a better description of black hole entropy in dynamical situations.
What Did Fu and His Team Study?
Fu and his colleagues wanted to investigate whether the entropy-area relationship remains true when a black hole changes over time.
To do this, they studied a black hole that is initially stable and spherically symmetric. They then introduced small disturbances and calculated how the black hole responds.
Instead of looking only at first-order changes, which are relatively simple, they analyzed second-order perturbations. These calculations capture more detailed and realistic changes in the black hole's geometry.
The researchers used a mathematical framework called Gaussian null coordinates, which is particularly useful for studying horizons.
Using this method, they carefully tracked how the apparent horizon changes when matter and energy fall into the black hole.
A New Way to Connect Energy and Entropy
An important part of the study involved understanding how energy entering the black hole affects its entropy.
The researchers used a powerful mathematical approach called the covariant phase space formalism, which helps physicists study gravitational systems in a consistent way.
However, previous methods did not fully include the effects of matter falling into the black hole.
To solve this problem, Fu and his team introduced a modified version of something called canonical energy. This new quantity includes contributions from external matter fields and allows scientists to accurately track the flow of energy into the black hole.
Using this approach, they derived a balance law connecting:
Energy entering the black hole
Changes in the horizon geometry
Variations in entropy
This balance law provides a deeper understanding of how black holes evolve thermodynamically.
The Main Discovery
After performing their calculations, the researchers found something remarkable.
When the Null Energy Condition is satisfied—a common assumption in classical physics stating that matter carries positive energy—the complicated entropy formula becomes much simpler.
All additional correction terms disappear.
As a result, the entropy of the dynamical black hole turns out to be exactly proportional to the area of the apparent horizon, even when second-order effects are included.
This means that the famous relationship between entropy and area remains valid beyond the simplest approximations.
In other words, the apparent horizon appears to be the correct place to measure entropy when a black hole is actively evolving.
The Second Law Still Works
Another important result concerns the second law of thermodynamics.
According to this law, entropy cannot decrease in an isolated system. It generally increases over time.
Fu and his team showed that their definition of black hole entropy also follows this rule.
As matter and energy fall into the black hole, the entropy associated with the apparent horizon increases.
This confirms that dynamical black holes behave consistently with the fundamental laws of thermodynamics.
Questions That Remain
Although the results are encouraging, several mysteries remain.
One major question involves situations where the Null Energy Condition is violated.
In quantum physics, certain effects can produce negative energy densities. Examples include quantum vacuum fluctuations and some aspects of Hawking radiation.
In such cases, it is not yet known whether entropy will still be exactly proportional to the apparent horizon area.
Scientists will need further research to answer this question.
Another challenge is determining whether the same results apply to more complex black holes, such as:
Rotating black holes
Black holes in curved cosmological environments
Higher-dimensional black holes
Alternative theories of gravity
These systems are much more complicated and could reveal new aspects of black hole thermodynamics.
Looking Toward the Future
The work of Fu and his collaborators provides strong evidence that the apparent horizon plays a special role in dynamical black holes.
Their study shows that even when a black hole changes and grows, its entropy remains closely connected to the area of the apparent horizon. The results also support the second law of thermodynamics and strengthen the link between gravity, energy, and information.
Most importantly, this research brings scientists one step closer to understanding the deep connection between black holes and the fundamental structure of the universe. As physicists continue exploring the mysteries of entropy and spacetime, studies like this may eventually help reveal the long-sought theory of quantum gravity and unlock some of the universe's greatest secrets.
Reference: Wen-Tao Fu, Ming-Fei Ji, Yu-Sen Zhou, Li-Ming Cao, "The entropy of black hole under second-order deviation from equilibrium", Arxiv, 2026. https://arxiv.org/abs/2606.16757
Technical Terms
1. Black Hole
A black hole is a region in space where gravity is so strong that nothing can escape from it, not even light. It forms when a very massive star collapses.
2. Entropy
Entropy is a measure of disorder or missing information in a system.
High entropy = more disorder, more randomness
Low entropy = more order, more structure
For black holes, entropy is related to how much hidden information is stored inside them.
3. Event Horizon
The event horizon is the “point of no return” around a black hole.
Once anything crosses this boundary, it can never come back out. Even light cannot escape from inside it.
4. Apparent Horizon
The apparent horizon is a more “local” boundary of a black hole.
It changes with time
It depends only on what is happening right now
It is easier to study in evolving (changing) black holes
Many scientists think it is more useful than the event horizon for dynamic situations.
5. Dynamical Black Hole
A dynamical black hole is a black hole that is changing over time.
This can happen when it:
Eats matter (gas, stars)
Grows in size
Merges with another black hole
It is not in a fixed, stable state.
6. Perturbation
A perturbation is a small disturbance or change in a system.
Example:
Dropping a small amount of matter into a black hole creates a “perturbation”
Scientists study how systems respond to these small changes.
7. Second-Order Perturbation
This means studying not just the first small change, but also the next level of effects caused by that change.
First-order = direct effect
Second-order = deeper, more detailed effects (like ripple effects of a ripple)
It gives a more accurate picture of reality.
8. Gaussian Null Coordinates
This is a special mathematical tool used to describe space near a black hole horizon.
In simple terms:
It is a smart way to “map” the area around a black hole so scientists can calculate things more easily.
9. Covariant Phase Space Formalism
This is a powerful method used in theoretical physics to study gravity.
Simple meaning:
It is a mathematical framework that helps physicists track energy, motion, and changes in spacetime in a consistent way.
10. Canonical Energy
This is a measure of energy used in advanced physics to understand how a system changes.
In this study, it helps track how energy enters a black hole and affects its structure.
11. Energy Flux
Energy flux means the flow of energy into or out of a system.
For black holes:
It usually means energy (matter, radiation) falling into the black hole.
12. Null Energy Condition (NEC)
This is an assumption in physics that says:
Energy measured along light-like paths is always positive or zero.
Simple idea:
It means matter behaves normally and does not have “negative energy.”
Some quantum effects can violate this condition.
13. Raychaudhuri Equation
This is a key equation in general relativity.
Simple meaning:
It describes how bundles of light rays move and focus in curved spacetime.
It helps explain how horizons grow or shrink.
14. Balance Law
A balance law is like a “bookkeeping rule.”
It connects:
Energy entering the black hole
Change in its structure
Change in entropy
It ensures everything is conserved and consistent.
15. Second Law of Thermodynamics (in Black Holes)
In simple terms:
The entropy of a black hole never decreases in normal conditions.
It is similar to how disorder in the universe generally increases over time.
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