The third law of black hole thermodynamics: Quantum versus classical considerations

For over a century, physicists have debated a fundamental question about the nature of heat and cold: is it possible to reach absolute zero, the coldest temperature theoretically possible, in a finite amount of time? The third law of thermodynamics, a cornerstone of physics, says no. It asserts that no physical process can cool a system to absolute zero in a finite number of steps. This rule is not just about thermometers; it applies to the most extreme objects in the universe, black holes. In the 1970s, scientists discovered that black holes behave like hot objects, radiating energy and possessing a temperature linked to their surface gravity. When a black hole becomes "extremal," meaning it has the maximum possible electric charge for its mass, its temperature drops to absolute zero. For decades, the third law was thought to guarantee that nature would never allow a black hole to reach this zero-temperature state dynamically.

However, a recent challenge to this idea emerged from the work of Kehle and Unger, who demonstrated that within the strict rules of classical physics, a massless charged field could theoretically collapse to form an extremal black hole in a finite amount of time. If true, this would mean the universe could break its own thermodynamic laws, creating a zero-temperature object instantly. This paradox has sparked a new investigation by Shahar Hod and Tsvi Piran, who sought to determine if there is a physical mechanism that prevents this violation. They found that while classical physics might allow such a formation, the quantum nature of reality likely steps in to stop it, preserving the third law.

The researchers began by examining the specific conditions under which Kehle and Unger's model operates. In this scenario, a cloud of massless charged particles collapses under its own gravity. In the classical view, where quantum effects are ignored, this collapse could proceed smoothly to create a black hole with zero temperature. But Hod and Piran argued that the third law is intrinsically a quantum principle, meaning any self-consistent analysis must account for the behavior of the vacuum itself. In the quantum world, empty space is not truly empty; it is a seething sea of potential particles. When an electric field becomes strong enough, it can rip pairs of particles out of this vacuum, creating a positive and a negative charge from nothing.

The team calculated that if the electric charge of the collapsing system is sufficiently large, the electric field will spontaneously polarize the vacuum. This process creates pairs of massless charged particles. The negatively charged member of the pair is captured by the positively charged black hole, effectively neutralizing some of its charge, while the positive member escapes to infinity. This discharge mechanism acts as a natural safety valve. As long as the dimensionless charge parameter of the system is greater than one-half, this quantum discharge happens fast enough to prevent the black hole from ever reaching the extremal, zero-temperature state. In this regime, the third law is saved by the very quantum fluctuations that define the vacuum.

This leaves a narrow window of uncertainty: what happens if the charge is smaller than this critical threshold? In this lower range, the electric field is too weak to trigger the spontaneous creation of particle pairs, so the quantum safety valve does not open. Here, the researchers turned their attention back to classical physics. They noted that a recent mathematical theorem by Schneider had already shown that extremal black holes cannot form in the very smallest charge regimes, specifically below a value of one-third. However, the exact boundary between "possible" and "impossible" in the classical world remained unknown, sitting somewhere between one-third and one-half.

Hod and Piran propose a conjecture to close this gap. They suggest that the classical laws of physics themselves are sufficient to prevent the formation of extremal black holes in the entire range below the quantum threshold. Their argument is that the classical equations governing gravity and electromagnetism likely contain a hidden bound that stops the collapse before the black hole can become extremal, even when quantum effects are too weak to interfere. If this conjecture is correct, it means that the universe is doubly protected: quantum mechanics prevents the formation of zero-temperature black holes when the charge is high, and classical mechanics prevents it when the charge is low.

The paper does not claim to have proven this final bound mathematically; rather, it presents a compelling physical argument based on the consistency of thermodynamic laws. The authors suggest that the true limit for forming an extremal black hole is exactly at the point where quantum effects begin to take over. If this is true, the universe operates with a seamless transition where quantum discharge handles the strong charges, and classical dynamics handle the weak ones, ensuring that absolute zero is never reached in a finite time. This synthesis offers a resolution to the apparent violation of the third law, suggesting that the laws of thermodynamics are robust enough to withstand even the most extreme gravitational collapses.