The Physics of Inequalities: Energy Conditions in General Relativity

Like the second law of thermodynamics, there are natural laws in physics where inequalities play an essential role. Inequalities have become important tools for expressing profound understanding. On the other hand, relying solely on equalities, while capable of describing individual phenomena in detail, can lead to overlooking the overall picture or universal properties behind them. In other words, one might fall into the state of "not seeing the forest for the trees."

For example, in classical mechanics of many-body systems, one can describe individual phenomena deterministically by using the "equality" known as Newton's equation. However, simply understanding each phenomenon through equalities does not reveal the grand worldview obtained by integrating numerous phenomena, as seen in the second law of thermodynamics. There is a world of physics that is only revealed by inequalities—laws that possess universality beyond individual phenomena.

In general relativity, which describes curved spacetime, inequalities are actually the key to capturing the essence of physics. However, some might think, "The Einstein field equation, which is the fundamental equation, is an equality, isn't it?"

The Einstein field equation is an "equality" that equates the tensor representing spacetime curvature on the left side with the tensor representing the energy and momentum of matter on the right side. Interestingly, however, it can be said that this equality itself is not the essence of physics.

The Einstein field equation is mathematically very flexible, and it is possible to create solutions for any spacetime. For example, if you substitute an arbitrary metric tensor into the left side of the equation and define the matter distribution based on the resulting right side, it becomes a valid solution. You can freely set the metric tensor, including strange spacetimes like time machines, and the form of the matter distribution is determined as a result. However, many of these solutions have no physical meaning.

One reason for this is that there is a lower limit to the energy density of matter. In classical physics, energy density cannot take negative values. On the other hand, in quantum mechanics, negative energy density can appear due to zero-point fluctuations, but there are strict constraints on its value. For example, in flat spacetime, the total energy must be non-negative. Such conditions are called "energy conditions" of matter and are expressed in the form of inequalities. In other words, even if one can construct a solution to the Einstein field equation, a spacetime where the absolute value of negative energy density is extremely large cannot be physically realized.

This situation is different from Maxwell's equations of electromagnetism, which allow for both positive and negative charges. The non-negativity of energy has significant physical meaning. Since energy is closely related to time, the non-negativity of energy is also deeply connected to the "reason why time flows in one direction," or the problem of the arrow of time.

Even in undergraduate-level thermodynamics, attempts are made to understand the arrow of time through the law of entropy increase (the second law of thermodynamics). However, the reason for entropy increase may be related to the constraint that matter energy can take positive values but not negative values. This is also being discussed at the forefront of research.

Furthermore, in quantum mechanics, the nature of quantum entanglement is also related to the arrow of time. For example, quantum entanglement has a "monotonicity" where it is destroyed by local operations, because local quantum entanglement grows into more global entanglement through interactions. It can be said that this monotonicity specifies the arrow of time. Also, if time reversal is performed on only one of two quantumly entangled subsystems, the resulting density matrix will have negative eigenvalues and no longer describe a valid quantum state. Thus, quantum entanglement is closely linked to the arrow of time.

Furthermore, recent research has revealed that the non-negativity of relative entropy in quantum information theory is related to the non-negativity of matter energy appearing on the right side of the Einstein field equation. The concept of "information" is imposing constraints on the energy of "matter."

Looking at it this way, we can see that the essence of the Einstein field equation lies not in the "=" equality, but in the "≥" inequality that the matter distribution on the right side must satisfy. It is this inequality that generates the "arrow of time" even in classical general relativity. For example, when two black holes collide to form one large black hole, this inequality based on energy conditions also plays an important role in the behavior of the black hole's surface area.

This energy condition inequality also plays an essential role in proving that the area of a newly formed black hole is larger than the sum of the areas of the two colliding black holes. It is also known that the area of a black hole is proportional to its thermal entropy. Therefore, within the framework of classical general relativity, black hole entropy always increases.

However, the situation changes slightly when quantum effects are considered. Black holes evaporate due to Hawking radiation, and their area decreases. But even in this case, the sum of the entropy of the radiation from Hawking radiation and the thermodynamic entropy of the black hole itself increases monotonically over time. This suggests that the second law of thermodynamics is valid even at the quantum level.

Thus, comparing Maxwell's equations of electromagnetism with the Einstein field equation of general relativity clarifies the difference in the roles each equation plays in physics. While Maxwell's equations can be viewed as the "physics of equalities," the Einstein field equation is based on the "physics of inequalities." Understanding this distinction is extremely important for gaining a deeper knowledge of the essence of physics.