Hawking radiation, proposed by physicist Stephen Hawking, is a phenomenon predicted by quantum mechanics that describes black holes’ gradual loss of mass and energy. In the past, it was very difficult to accept that black holes had a temperature because they would then have to emit radiation, which went against the definition of a black hole. In 1974, S. Hawking theoretically discovered that a static black hole located in a vacuum must emit from its horizon in all directions a type of thermal radiation known as Hawking radiation [1]. According to Hawking’s mechanism, radiation occurs near the event horizon of a black hole and it arises from the steady conversion of quantum vacuum fluctuations into pairs of particles, one of which escapes at infinity while the other is trapped inside the black hole horizon. In Hawking’s most famous book, A Brief History of Time [2,3], he makes the analogy that space is filled with particle-antiparticle pairs and that one member can escape, carrying positive energy, while the other falls in, with negative energy. Since these pairs of particles were present outside the black hole, the flow of positive energy particles appears as radiation while the flow of negative energy particles reduces its mass leading to black hole decay. There are, however, two main issues to address: one of a pedagogical nature (i) and the other of a fundamental nature (ii).

(i) Hawking’s original derivation of black hole radiation involves complex mathematics due to the need to handle quantum fields in curved spacetime and solve intricate equations. Simplifying and intuitively explaining these concepts helps in making the ideas more accessible and comprehensible. Such approaches can provide a clearer understanding of the fundamental physics behind Hawking radiation and black hole thermodynamics without requiring advanced mathematical tools. Additionally, an intuitive approach can highlight the physical principles behind Hawking radiation - such as vacuum fluctuations and particle creation near the event horizon - without getting lost in technical details. This helps in grasping why and how black holes emit radiation.

(ii) At first glance, the Hawking radiation process seems to contradict both Prigogine’s second law of thermodynamics for open systems [4,7] and the second law of black hole dynamics [8]. Prigogine’s second law for open systems states that "During the evolution, the entropy production of the system is always positive". The second law of black hole dynamics states that "In any classical process involving black holes, the total area of the event horizon of the black hole cannot decrease". So, in the standard picture, since Hawking radiation causes a black hole to lose mass over time, Prigogine’s law and the second law of black holes are violated. The generalized second law (GSL) of black holes incorporates the entropy of both the black hole and the surrounding radiation, ensuring the total entropy (black hole entropy plus radiation entropy) never decreases (see for instance [9]). However, this all seems like a "gimmick, since a black hole that emits energy is an open system and must still satisfy Prigogine’s law.

This work deals with the above issues i) and ii). More specifically, the standard Hawking radiation mechanism will be revisited through the Generalized Uncertainty Principle (GUP) [10,11,12] or the Extended Uncertainty Principe (EUP) [13,14]. The GUP is a theoretical framework in quantum mechanics that extends the Heisenberg Uncertainty Principle by incorporating the effects of quantum gravity. The GUP modifies this principle by including terms that account for gravitational effects, particularly those that become significant at extremely small (Planck-scale) distances. These modifications suggest that at such scales, the uncertainties in position and momentum are influenced not just by quantum mechanical factors, but also by gravitational effects. The GUP suggests the existence of a minimal measurable length, typically on the order of the Planck length. This contrasts with the standard quantum mechanics view, where smaller and smaller distances could, in theory, be probed with higher momentum. The GUP has implications for black hole physics, particularly for the end stages of black hole evaporation. It suggests modifications to the Hawking radiation process and could imply that black holes do not evaporate completely but leave behind a Planck-sized remnant. The EUP addresses the possibility of both positive and negative values of the so-called EUP deformation parameter while still maintaining a minimum length scale. This flexibility in the EUP deformation parameter is significant because it broadens the range of possible physical interpretations and applications of the uncertainty principle in quantum gravity scenarios. We shall see that contrary to GUP, the possibility of a negative EUP parameter opens up the opportunity to set the Hawking temperature to zero. This adjustment can address the conflicts between Prigogine’s second law of thermodynamics and the second law of black hole mechanics. Indeed, by choosing a negative EUP parameter value such that the Hawking temperature becomes zero, the black hole ceases to radiate. However. even when the Hawking temperature is set to zero by adjusting the EUP parameter the black hole’s entropy does not vanish because entropy is primarily tied to the geometry of the black hole (its horizon area), not directly to the temperature. The entropy represents the number of hidden microstates or the information content of the black hole, which remains finite as long as the black hole has a non-zero mass and horizon area. This stabilization implies that the black hole’s mass and entropy would no longer decrease, thereby aligning with both the area theorem (second law of black hole mechanics) and Prigogine’s law of non-negative entropy production in open systems. This leads also to the situation where black hole entropy continues to increase despite no radiation. The increase in entropy without radiation suggests that the black hole is still evolving in some way, even if it is not losing mass through radiation.

The manuscript is organized as follows. Before starting our analysis, in Section 2 we shall revisit the steps leading to the determination of the Hawking temperature. This will allow us to focus on the main assumptions adopted by Hawking for getting the expression of the temperature of a black hole. In Section 3 we shall derive the exact expressions of the Hawking Temperature and Entropy for a Schwarzschild black hole. To do this task, we shall follow J. Pinochet’s arguments [15]. However, it should be stressed that we shall follow some of his mathematical steps but we shall not adopt his physical interpretation of the black hole radiation mechanism, remaining faithful to the original Hawking’s picture of the mechanism responsible for the thermal emission. For easy reference, Prigogine’s second law for open systems and the second law of black holes are recalled in Section 4. The connection between these two laws is shown in subSection 4.3. The method suggested by Pinochet will allow us to deal with more complex situations without resorting to complex mathematical methods. In particular, it will allow us to deal with, and resolve, the conflict between Hawking radiation and the second law of black holes. This will be the subject of Section 5. In Section 6 we determine the GUP and EUP deformation parameters that can reconcile Hawking radiation with Prigogine’s law and Bekenstein thermodynamic analogy. Concluding remarks can be found in Section 7.

In 1974 [1] Stephen W. Hawking published his celebrated result stating that if one takes quantum theory into account, black holes are not quite black, but they emit radiation consisting of photons, neutrinos, and to a lesser extent all sorts of massive particles. The Hawking radiation mechanism describes hypothetical particles and antiparticles formed by a black hole’s boundary, and it is based on the assumption that the horizon is the radiating surface. This radiation implies black holes have temperatures that are inversely proportional to their mass. Below, we resume in a few bullets the main assumptions made by Hawking to explain the mechanism of black hole radiation:

i) The universe is filled with particle-antiparticle pairs popping in and out of existence;

ii) These pairs of particle-antiparticle exist even in empty space, as a consequence of quantum field theory and the Heisenberg uncertainty relations;

iii) These pairs always find one another and re-annihilate after a very small time interval;

iv) Hawking radiation arises from quantum effects near the event horizon of a black hole. The strong gravitational field near the event horizon causes quantum vacuum fluctuations to become real particles;

v) This mechanism leads to a situation where one member of the pair falls in while a real particle escapes and is emitted with positive mass/energy from just outside the horizon itself;

vi) The paired member that falls into the event horizon must have negative energy that subtracts from the black hole’s total mass. The flow of particles of negative energy into the black hole reduces its mass until it disappears completely in a final burst of radiation;

Figure 1 illustrates the Hawking radiation process.

The Hawking radiation mechanism is a purely quantum effect. Hawking’s findings are practically irrelevant for classical black holes.

Stephen Hawking’s original work on Hawking radiation is mathematically complex for several reasons. Hawking’s derivation involves applying quantum field theory in the curved spacetime of a black hole. This requires complex calculations involving the behavior of quantum fields in a non-flat (curved) spacetime, which is inherently complicated. Additionally, Hawking used the semi-classical approximation where the spacetime is treated classically (using general relativity), but the fields are treated quantum mechanically. Combining these two frameworks is non-trivial and requires sophisticated mathematical tools. Hawking’s method involves decomposing the quantum fields into modes that interact with the black hole’s event horizon. Calculating the contributions of these modes to the radiation is mathematically intensive. The derivation involves evaluating complex integrals and solving differential equations that describe the behavior of fields near the event horizon. This requires advanced techniques from mathematical physics. Finally, showing that the radiation emitted by the black hole is thermal, and finding the exact temperature involves intricate calculations related to black body radiation and quantum statistics. So, deriving the exact expressions for Hawking temperature and entropy more simply and intuitively is necessary as simplifying and intuitively explaining these concepts helps in making the ideas more accessible and comprehensible. In our derivation of Hawking’s temperature, we shall follow the arguments of J. Pinochet [15]. However, this work differs conceptually from that of J. Pinochet. As the six points above state, we utilize the quantum vacuum fluctuation. In our derivation, according to Hawking’s assumption, the radiation does not come directly from within the black hole but near the horizon event. If particles are subject to the uncertainty principle, then we cannot know both the time and energy of a particle with perfect accuracy. Hawking radiation represents a situation where the energy-time uncertainty principle plays a crucial role in determining the characteristics of the emitted radiation. Denoting with $\Delta t$ the uncertainty in the time of the particle, the process effectively "chooses" the emission of particles with energy $\Delta E$ corresponding to the minimum value that the product of these quantities can take [16,17] with ℏ denoting the reduced Planck constant. In our energy situation, $\Delta E=c\Delta p$, with c denoting the speed of light and $\Delta p$ the momentum uncertainty, respectively. So, the Heisenberg Uncertainty Principle relating the uncertainties in position $\Delta l$ and momentum $\Delta p$ implies $c\Delta t=\Delta l$. In quantum field theory, the quantum vacuum state is the quantum state with the lowest possible energy $\Delta E$. Since $\Delta l$ and $\Delta E$ are inversely proportional, the minimum value $\Delta E$, compatible with Eq. (1), corresponds to the maximal uncertainty in the position of the particle ${\Delta l|}_{Max}$. Taking into account this observation, Eq. (1) leads to Assuming that the black hole event horizon is spherically symmetric with a radius equal to Schwarzschild’s radius $R_S$, it is easily checked that ${\Delta l|}_{Max}$ corresponds to the maximal uncertainty in the coordinate x (or y) of the particle on the event horizon. Hence, by taking into account that: with $\theta$ (with $0\le \theta \le \pi$) and $\varphi$ (with $0\le \varphi \le 2\pi$) denoting the spherical coordinates, we get: as the maximum indetermination is obtained for $\Delta \theta =\pi$, $\Delta \varphi =2\pi$, and at the coordinate values $(\theta ,\varphi )=(\pi /2,3/2\pi )$. So, By plugging in Eq. (5) the expression for the Schwarzschild’s radius $R_S$ [8] with M denoting the mass of the black hole and G is the universal gravitational constant, respectively, we get the Hawking expression for the black hole emitted energy: As said, Hawking radiation is a theoretical prediction that black holes can emit radiation due to quantum effects near the event horizon. This radiation has a thermal spectrum, and its associated temperature is known as the Hawking temperature. The energy of the particles (usually photons, but also other particles) emitted in Hawking radiation is directly related to the Hawking temperature $T_H$ by the relation [18,19,20]: with $K_B$ denoting the Boltzmann constant. This equation makes sense in this scenario because the emitted radiation is thermal, similar to black body radiation, but with Hawking temperature. So, the particles emitted by the black hole have an average energy that is proportional to the Hawking temperature $T_H$ through Boltzmann’s constant $K_B$. We finally get We note the subtlety implicit in Eq. (9). The temperature of a classical black hole is $0K$. According to Hawking’s radiation mechanism, due to the quantum effect, the black hole emits energy $\Delta E$ in the form of escaping matter showing a very weak temperature $T_H$ given by Eq. (9). The energy that the black hole loses $\Delta E_{B{H}_{loss}}$ is equal to the negative energy of the antimatter it has captured i.e., $\Delta E_{B{H}_{loss}}=-\Delta E$. In the absence of work, the first law of thermodynamics reads with $S_{BH}$ denoting the black hole entropy. Hawking radiation propagates away from the event horizon, and since real radiation carries energy, the only place where that energy $\Delta E$ can be taken from is from the mass of the black hole itself, via the classic Einstein’s equation, ${\Delta E|}_m=c^2\Delta M$. In this case, the mass lost by the black hole has to balance the energy of the emitted radiation. So, where Eq. (9) has been taken into account. The integration yields where $S_0$ is an arbitrary constant of integration. In physical contexts like black hole thermodynamics, $S_0$ typically corresponds to the minimal entropy or entropy at a particular reference point.

Expression (9) shows that a black hole loses mass through evaporation. This seems in disagreement with the law of thermodynamics expressed as The total entropy of an isolated system always increases. We soon realize that this disagreement is only apparent. Indeed, after simple calculations, we have that the total entropy change of the isolated system, composed of the black hole plus reservoir (i.e., the the rest of Universe) reads [9] with $M_i$ and $M_f$ denoting the original mass and the final mass of the black hole, respectively. Since the initial mass $M_i$ is greater than the final mass $M_f$, the result is a positive quantity showing an increase in entropy.

According to Hawking in [3] since the average temperature of the universe is about $2.7K$, most black holes absorb more energy than they emit and will not begin to evaporate for some time until the universe has expanded and cooled below their temperature. So, the expression for the total entropy (13) is valid only when the black hole is at thermodynamic equilibrium with the Universe. It is not applicable during the transition phase in which the black hole and the Universe tend to reach thermodynamic equilibrium. However, we know that a body with a temperature above absolute zero will emit radiation by Planck’s law. As said, during the transitional phase Eq. (13) does not apply and this is a big limitation of the demonstrations shown above. Briefly, what needs to be demonstrated is that a radiating black hole in non-equilibrium conditions satisfies Prigogine’s second law of thermodynamics. This is the subject of the next section.

Eqs (43) reconciles Hawking Radiation with Prigogine’s law but not with the thermodynamic analogy proposed by Bekenstein and Wheeler (see Eq. (21) in Section 4.3). For this to happen, the Hawking temperature must be zero (to satisfy Prigogine’s law). At the same time, the entropy of the black hole must be identical to that calculated by Bekenstein and Wheeler. It is easily checked that these two conditions are satisfied simultaneously by appropriately choosing the expressions for the GUP and EUP parameters. For this, let us consider the complete uncertainty principle (34) written in the following form: where $\xi$ is set to zero and the GUP and EUP parameters (i.e., $\beta$ and ${\beta }_0$, respectively) of positive value. We consider black holes with small mass near Planck’s mass. So, we may approximate forms of $\beta$ and ${\beta }_0$ that are valid for small deviations from $M_p$. $\beta$ and ${\beta }_0$ are therefore assumed to be a constant, or at least weakly dependent on the black hole mass. After simple algebra, we get: where we have taken into account that $\Delta x=2\pi R_S$ and $\Delta p=T_H{K}_B/c$. At the first order in $\beta$ and ${\beta }_0$, Eq. (47) reads: and Hence, Prigogine’s second law of thermodynamics and the Bekenstein-Wheeler thermodynamic interpretation of the horizon black hole surface are satisfied simultaneously by setting the EUP and GUP parameters respectively as with final result

Even the most extreme objects in the universe, like black holes, are bound by certain rules. G. Gibbons and S. Hawking demonstrated that black hole thermodynamics extends beyond black holes themselves, revealing that cosmological event horizons also possess entropy and temperature [34]. In 1974, Hawking further showed that when quantum mechanics is considered, black holes can emit light and particles through a process known as Hawking radiation. Since quantum black holes emit energy and light, they must have a temperature, in addition to mass, charge, and rotation. This makes them subject to the laws of thermodynamics. Hawking’s mechanism hinges on quantum vacuum fluctuations, suggesting that the radiation originates near the event horizon rather than from within the black hole itself. However, Hawking’s original derivation of this radiation is mathematically intricate, involving quantum fields in curved spacetime and complex equations. From a pedagogical standpoint, it’s beneficial to explain these concepts more intuitively, making them accessible without sacrificing the core physical principles underlying Hawking radiation. This work aims to offer a clearer understanding of the fundamental physics of Hawking radiation and black hole thermodynamics, without relying on advanced mathematical techniques. To achieve this goal, we have rederived the exact expressions of the Hawking temperature and entropy for a Schwarzschild black hole following the indications of J. Pinochet in [15] with the addition, however, of new elements and physical considerations that have allowed us to overcome the vulnerable aspects present in the original work of J. Pinochet. The way suggested by J. Pinochet allowed us to deal with more complex situations. In particular, we have shown that it is possible to ensure consistency with both Prigogine’s second law and black hole thermodynamics. The second law of black holes states that even if black holes merge or matter falls into a black hole, the total event horizon area of the resulting black hole(s) will be greater than or equal to the sum of the event horizon areas of the original black holes. The second law of black hole dynamics applies classically. In quantum mechanics, Hawking radiation introduces the possibility of black holes losing mass and thereby shrinking in area, which would appear to violate this law. With Hawking radiation effectively turned off by the negative EUP parameter, the entropy of the black hole stabilizes, eliminating the decrease that would otherwise violate Prigogine’s second law. So, when gravitational quantum effects are considered, black holes still obey the non-decreasing law, preserving the analogy with Prigogine’s second law of thermodynamics. While the classical interpretation of $dE=T_{H}d{S}_{BH}$ suggests that $d{S}_{BH}=0$ when $T_H=0^{\circ }\mathrm{K}$, the inclusion of EUP and quantum gravitational effects may lead to a scenario where entropy can still increase. This reflects a more complex interaction between the microstates of the black hole and quantum gravity, beyond what is described by classical thermodynamics alone. So, the key idea is that the traditional relation $dE=T_{H}d{S}_{BH}$ might not fully capture the behavior of entropy under the influence of the EUP, particularly when $T_H=0^{\circ }\mathrm{K}$,. In non-equilibrium thermodynamics, especially when considering open systems or systems influenced by quantum gravity, entropy can increase due to other processes, such as quantum fluctuations or interactions with external fields, even if the black hole is not radiating. We finally derived the modified entropy for a Schwarzschild black hole, which includes a logarithmic correction term, common in quantum gravity scenarios. In passing, we have shown that it is possible to determine the GUP and EUP deformation parameters such that Hawking radiation is turned off and the black hole entropy is the one proposed by Bekenstein and Wheeler in their thermodynamic analogy.