November 06, 2025
We present a general formalism linking modified entropy functions directly to a modified spacetime metric and, subsequently, to an effective matter sector of entropic origin. In particular, within the framework of general relativity, starting from the first law of black-hole thermodynamics we establish an explicit correspondence between the entropy derivative and the metric function, which naturally leads to an emergent stress-energy tensor representing an anisotropic effective fluid. This backreaction effect of horizon entropy may resolve possible inconsistencies recently identified in black hole physics with modified entropies. As specific examples, we apply this procedure to a wide class of modified entropies, such as Barrow, Tsallis-Cirto, Rényi, Kaniadakis, logarithmic, power-law, loop-quantum-gravity, and exponential modifications, and we derive the associated effective matter sectors, analyzing their physical properties and energy conditions.
The connection between gravity, thermodynamics, and quantum theory proves to be useful in studying the structure of spacetime. Since the pioneering works of Bekenstein and Hawking revealed that black holes behave as thermodynamic objects with
well-defined temperature and entropy [1], [2], it has become
evident that the laws of gravity are related to a thermodynamic structure. This feature suggests that spacetime itself may possess microscopic degrees of freedom, and that gravitational dynamics could emerge as a macroscopic manifestation of their
statistical behavior. The area law of black hole entropy and the holographic principle imply that information associated with a volume of space can be effectively stored on its boundary [3], [4]. Such ideas have inspired a broad paradigm in which gravity arises as an emergent, entropic phenomenon rather than a fundamental
force, linking the geometry of spacetime to the thermodynamic behavior of its underlying microscopic constituents [5], [6].
The modifications to the standard Bekenstein-Hawking entropy play a central role in exploring possible extensions of semiclassical gravity. The conventional area law, \(S = \tfrac{A}{4}\), represents the leading-order
contribution arising from quantum fields near the horizon, but various approaches to quantum gravity and non-equilibrium statistical mechanics suggest the presence of subleading corrections. Among these, logarithmic, power-law and exponential corrections
have received significant attention, as they naturally emerge in loop quantum gravity, quantum geometry, and generalized thermodynamic formalisms [7]–[17]. Power-law corrections to the Bekenstein-Hawking entropy naturally arise when entanglement entropy is computed for quantum fields in excited states rather than in the vacuum
configuration [18], and in such cases the leading area term receives subleading power-law suppressed contributions that depend on the excitation level
of the quantum state. Additionally, non-extensive entropy frameworks-such as those based on Tsallis and Rényi formalisms, quantum-gravitational corrections such as in Barrow entropy, relativistic corrections such as in Kaniadakis entropy, etc, provide a
generalized description of gravitational systems, leading to modifications of geometry and thermodynamics and thus to interesting black-hole and cosmological phenomenology [19]–[85]. Motivated by these developments, in this work, we investigate the implications of such generalized entropy functions,
incorporating logarithmic [7], [86], [87] and exponential corrections [88]–[91], on the structure of spherically symmetric black hole spacetimes.
On the other hand, in recent years, gravity has been increasingly viewed as an emergent phenomenon rather than a fundamental interaction, arising from the thermodynamic features of spacetime [5], [92]. In this context, the notion of an entropic force provides a useful conceptual framework. An entropic force is an effective macroscopic force that arises due to the statistical tendency of a system with many microscopic degrees of freedom to increase its entropy [6]. The force equation depends only on entropy differences and is independent of the details of the microscopic dynamics, with no fundamental field mediating it. Typical examples include colloidal interactions and osmotic pressure, both governed by entropy gradients rather than direct mechanical forces. A well-known example is the elasticity of a polymer molecule: when immersed in a heat bath, the polymer tends to adopt a randomly coiled configuration since such states maximize entropy. Stretching the polymer reduces the number of accessible microstates, and the system responds with a restoring force that drives it back toward the equilibrium configuration [93].
Similarly, in gravitational systems, entropy gradients associated with the microscopic degrees of freedom of spacetime can give rise to a macroscopic force that one can interpret as gravity. Within this picture, black hole thermodynamics provides a natural setting to explore the connection between entropy, temperature, and geometry [1], [2]. Hence, modifications to the entropy-area relation can lead to deviations from the standard Schwarzschild geometry and generate new classes of regular black hole metrics.
In the present work, we are interested in studying the effect of entropy modifications on the black-hole metric and consequently on the effective matter sector. We mention that our analysis differs from the one of [94], [95], since in those works the authors examined whether generalized non-extensive entropies are consistent with the Hawking temperature and the Arnowitt-Deser-Misner (ADM) mass, essentially testing the thermodynamic validity of such entropies within a fixed spacetime geometry, while our approach reverses the logic and we derive the modified spacetime metric itself from the chosen entropy function, establishing a direct entropy-geometry correspondence rather than treating geometry as given. Hence, by translating the entropy deformation into a modified effective fluid description, our framework provides an explicit physical realization of how entropy corrections manifest as new, effective matter sectors of entropic origin. Recently, a Lagrangian reconstruction of entropy deformations has been proposed in [96], where power-law modifications of the Bekenstein-Hawking entropy are studied from a complementary perspective. We mention here that the Lagrangian formulation follows a different route, since the entropy itself is elevated to a dynamical quantity derived from an underlying variational principle, allowing one to construct a consistent action whose extremization reproduces the desired non-extensive entropy structure.
The plan of the work is the following. In Section 2 we present the connection of modified entropy relations to a modified metric function and then to an effective matter sector of entropic origin. Then, in Section 3 we proceed to specific applications to the various modified entropy forms that exist in the literature, such as Barrow, Tsallis-Cirto, Rényi, Kaniadakis, logarithmic, power-law, loop-quantum-gravity, and exponential modifications, deriving the associated effective matter sectors, and analyzing their physical properties and energy conditions. Finally, Section 4 is devoted to the conclusions.
In this section we first show how a modified horizon entropy can lead to a modified spacetime geometry, and then we show how this modified geometry can be interpreted to arise from an effective matter sector of entropic origin.
In classical general relativity, the usual procedure is to specify an energy-momentum tensor, solve for the metric components, and subsequently compute the corresponding horizon entropy. On the other hand, ideas such as emergent gravity and the holographic principle indicate that gravity may arise from entropic or informational degrees of freedom, implying the role of entropy as a fundamental quantity. Guided by this viewpoint, we propose a simple approach in which the spacetime geometry is obtained directly from the horizon entropy.
Let us begin by assuming a static, spherically symmetric line element \[\label{Metric95ansatz} ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega^2 \;.\tag{1}\] The event horizon \(r_+\) satisfies \(f(r_+)=0\), and the Hawking temperature can be computed using \(T = f'(r_+)/4\pi\). On the other hand, the first law of black-hole thermodynamics is written as \[\label{First95Law} dM = T\,dS.\tag{2}\] Based on this expression, one can obtain the black-hole horizon entropy as \[S=\int \frac{1}{T}\frac{\partial M}{\partial r_+}dr_+ \;.\] Using this relation, it is well known that the entropy of a Schwarzschild black hole is proportional to the surface area of its horizon, as given by the Bekenstein-Hawking entropy formula \(S=A/4=\pi r_+^2\), where \(A=4 \pi r_+^2\). Conversely, one may follow the inverse procedure: starting from the Bekenstein-Hawking entropy, it is straightforward to verify that the corresponding metric function reproduces the Schwarzschild solution (see Appendix [Appn:Horizon Thermodynamics and Constraint on $f(r)$] for the details) and for that \(G_{\mu\nu}=0\). Thus, a natural question arises, namely what happens in the general case of non–Bekenstein–Hawking entropies, and if one can deduce the metric function directly from the modified horizon entropy.
We consider a general functional dependence of the black-hole entropy on the horizon radius, i.e. \[\label{General95Entropy} S = S(r_+) \, .\tag{3}\] To construct a metric consistent with this entropy we take the following ansatz for the metric function \[\label{tlozresx} f(r) = 1 - M\,g(r) \, ,\tag{4}\] where \(M\) denotes the Arnowitt-Deser-Misner (ADM) mass of the black hole and \(g(r)\) is an arbitrary function of the radial coordinate. The reason for choosing the form 1 is that in the limit where the entropy reduces to the Bekenstein-Hawking area law, one must recover the Schwarzschild metric. Therefore, the parametrization 1 is the minimal choice that ensures a smooth reduction to the Schwarzschild solution in the appropriate limit.
The location of the event horizon is determined from \(f(r_+)=0\), which gives \[M = \frac{1}{g(r_+)} \, .\] The Hawking temperature follows from the surface gravity and is given by \[\label{Temperature95expression} T = \frac{f'(r_+)}{4\pi} = -\,\frac{g'(r_+)}{4\pi\, g(r_+)} \,,\tag{5}\] where the prime denotes differentiation with respect to \(r\) and the expression is evaluated at the horizon. Now, using the first law of thermodynamics 2 and substituting relations 3 5 , we obtain \[\label{General95g40r41} g(r_+) = \frac{4\pi}{S'(r_+)} \, ,\tag{6}\] where we have considered the non-trivial case where \(g'(r_+)\neq0\) (ensuring a simple first-order zero of \(f(r)\) and hence a regular, non-extremal event horizon).
Relation 6 , which is the basic point of this work, encodes the backreaction effect of the entropy into the spacetime geometry. As one can see, it has been is derived locally at the event horizon. However, we can still
use it in order to determine the full spacetime geometry, provided that we make the following assumptions:
(i) Extending the horizon relation, namely promoting the horizon relation \(g(r_+)=4\pi/S'(r_+)\) to a global functional dependence, \(g(r) = 4\pi/S'(r)\), which uniquely
reconstructs the metric function from the chosen entropy functional form. This prescription ensures that the thermodynamic input \(S=S(r_+)\) is encoded directly in the geometry, and thus given \(S(r)\), the metric function is fixed everywhere.
(ii) Identifying the ADM mass with the parameter \(M\), which requires that the metric asymptotically reduces to the Schwarzschild form when \(S(r)=\pi r^2\). Under the above extension,
\(S' = 2\pi r\) implies \(g(r)=2/r\) and hence \(f(r)=1-2M/r\), confirming that the Schwarzschild solution is recovered when the entropy reduces to the
Bekenstein-Hawking area law. We emphasize that the present construction is limited to static, spherically symmetric spacetimes, and thus the proposed entropy-geometry correspondence should be viewed as a controlled realization within this symmetry class.
Generalizations to rotating or dynamical geometries (such as Kerr or time-dependent spacetimes) lie beyond the scope of the present work and will be explored in future investigations.
In summary, under the above assumptions, we can write \[f(r) = 1 - \frac{4\pi M}{S'(r)} \label{Generalfr},\tag{7}\] which shows that once the functional form of the entropy is specified, one can obtain effectively the corresponding metric function, and thereby the spacetime geometry. In the special case where \(S(r) = \pi r^2\) (Bekenstein-Hawking entropy), we recover the Schwarzschild solution, as expected. We would like to clarify the interpretation of the function \(S(r)\) in our construction. It is important to distinguish between Bekenstein-Hawking (thermodynamic) entropy, which is associated with a black hole horizon, and entanglement entropy of quantum fields across a spherical boundary. For radii \(r > r_+\), our \(S(r)\) should be understood as the entanglement entropy of vacuum fields across the spherical surface at radius \(r\), rather than as a thermodynamic entropy of matter contained in the sphere. Entanglement entropy counts quantum correlations between degrees of freedom inside and outside the surface and is known to satisfy an area law scaling \(S_{\rm ent}(r) \sim r^2.\) The Bekenstein bound, \(S \le 2 \pi E r,\) applies only to physical systems where the entropy is realized by energy \(E\) (or mass \(M\)) contained inside a radius \(r\). Since entanglement entropy does not correspond directly to the energy content, it is not constrained by this bound. Thus, the quadratic growth of \(S(r)\) with \(r\) does not violate any fundamental physical principle. In our metric construction, \(S(r)\) encodes the cumulative entanglement entropy of vacuum fields. Its role is not thermodynamic, and therefore the extension of \(S(r)\) to \(r > r_+\) is physically meaningful and consistent with semiclassical expectations. Our \(S(r)\) in the region \(r_+>r\) is the entanglement entropy and does not scale with energy, and the Bekenstein bound cannot be apply to such regions. This also suggest that one can modify the Bekenstein bound in order to include the total \(S(r)=S_{BH}(r_+)+S_{\rm ent}(r_+,r)\) and Bekenstein bound applies only the first term, while the second term is not constrained by the Bekenstein bound; it is constrained by the area (holographic/covariant) bound that can be written for the arbitrary surface \[S_{BH}(r_+)+S_{\rm ent}(r_+,r) \leq \frac{A(r)}{4G}\] Although we solved \(f(r)\) explicitly at the horizon, the solution can be extended to arbitrary \(r\) by invoking the continuity and smoothness of the metric function, together with the boundary conditions imposed at the horizon. In this way, the resulting metric satisfies the Einstein field equations for all \(r\) outside the horizon. The metric construction in our paper is therefore physically consistent.
In addition, our approach is consistent with uniqueness, namely the global extension does not violate any physical constraint: Birkhoff’s theorem ensures that any spherically symmetric vacuum solution outside the horizon is unique. Therefore, the choice of \(g(r)\) is not arbitrary, but is in fact constrained by the requirement that the solution remain the unique static, spherically symmetric vacuum geometry compatible with the given boundary conditions at infinity. In other words, our “entropy–geometry correspondence” is compatible with the uniqueness of the Schwarzschild solution, and the functional form of \(g(r)\) is consistent with Birkhoff’s theorem. In the case of Bekenstein-Hawking entropy we do get vacuum solution, however deviations from Bekenstein-Hawking entropy can be interpreted as non-vacuum solution, or contributions from contributions effective matter sectors.
We proceed by showing how in our setup the gravitational force emerges from the horizon entropy. From the relation \[f(r) \equiv 1+2 \phi_G\, ,\] where \(\phi_G \equiv - \frac{2\pi M}{S'(r)}\) is the gravitational potential, we can obtain the gravity force acting on a test particle with mass \(m\) near \(M\) using \[\vec{F}_G=-m \nabla \phi_G = 2\pi m M \nabla \left(\frac{1}{S'(r)}\right) \hat{r},\] obtaining the universal law of gravity as \[\vec{F}_G= -2\pi m M \frac{S''(r)}{S'(r)^2}\hat{r}.\] In other words, the attractive force of gravity can be viewed as an emergent effect arising from the change of horizon entropy. This is in line to the entropic force scenario proposed by Verlinde [6].
Let us now assume a general entropy relation, which can be expressed as a correction to Bekenstein–Hawking expression, namely \[S = S_{BH}+\mathcal{S}(A).\] Then using \[\frac{\partial S}{\partial r}=\frac{\partial S}{\partial A}\frac{\partial A}{\partial r}=\left(\frac{1}{4}+\frac{\partial \mathcal{S}}{\partial A} \right)8 \pi r\] and \[\frac{\partial^2 S}{\partial r^2}=2\pi \left(1+4 \frac{\partial \mathcal{S}}{\partial A}\right)+8 \pi r \frac{\partial }{\partial r}\left(\frac{\partial \mathcal{S}}{\partial A}\right),\] for the universal law of gravity we finally acquire \[\label{New} \vec{F}_G= - \frac{m M}{r^2} \left[\frac{ \left(1+4 \frac{\partial \mathcal{S}}{\partial A}\right)+ 4 r\frac{\partial }{\partial r}\left(\frac{\partial \mathcal{S}}{\partial A}\right)}{\left(\frac{1}{4}+\frac{\partial \mathcal{S}}{\partial A} \right)^2 16 }\right] \hat{r}.\tag{8}\] Note that if the entropy is just the Bekenstein-Hawking entropy, we reproduce the Newton’s law force, i.e. \(\vec{F}_G= - \frac{m M}{r^2}\hat{r}\). Hence, from Eq. 8 we see that a deviation from the Bekenstein-Hawking entropy can be interpreted as a modified gravitational law of gravity.
In this subsection we show how a modified entropy relation can be interpreted to lead to an effective matter sector. Let us now use the metric (7 ) to calculate the components of the Einstein tensor \(G^{\mu}_{\;\nu}\). We find \[\label{einsteintt} G_t^t =G_r^r = \frac{4 \pi M \left(r S''(r)-S'(r)\right)}{r^2 S'(r)^2},\tag{9}\] and \[\label{einsteinthth} G_\theta^\theta = G_\phi^\phi = \frac{2 \pi M \left\{S'(r) [r S^{'''}(r)+2 S''(r)]-2 r S''(r)^2\right\}}{r S'(r)^3}\;,\tag{10}\] which are non-zero in general for an entropy relation different from Bekenstein-Hawking one. Hence, from the field equations of general relativity \[\begin{align} G^{\mu}_{\;\nu} = 8\pi\, T^{\mu}_{\;\nu}, \end{align}\] we conclude that we obtain a non-zero, effective stress-energy tensor of entropic origin. In particular, we can define \(T^{\mu}_{\nu}=\left(-\rho, p_{r}, p_{t}, p_{t} \right)\), with \[\begin{align} \tag{11} \rho(r) &= -\frac{M\left[r S''(r) - S'(r)\right]}{2\, r^{2} S'(r)^{2}},\\[4pt] p_{r}(r) &= -\rho(r), \tag{12} \\[4pt] p_{t}(r) &= \frac{M\left\{S'(r)[r S^{'''}(r) + 2 S''(r)] - 2 r S''(r)^{2}\right\}}{4\, r\, S'(r)^{3}}. \tag{13} \end{align}\] Note that the relation \(p_{r} = -\rho\) reflects a vacuum-like or dark-energy-type equation of state in the radial direction. However, since \(p_{t} \neq p_{r}\), this effective matter sector of entropic origin is anisotropic. Thus, the deviation of the entropy function \(S(r)\) from the standard area law acts as a geometric source generating an anisotropic stress-energy tensor without introducing any explicit matter fields. In this sense, the modified entropy behaves as an effective, emergent gravitational matter content associated with horizon microstructure. This is the main result of the present work. In the following we apply it for the known modified entropy relations of the literature.
It is important to emphasize that the inverse thermodynamic construction adopted here does not aim to generate arbitrary vacuum geometries. Rather, once a modified entropy functional is specified, the resulting metric is determined by thermodynamic consistency and generally corresponds to a non-vacuum configuration. The associated stress-energy tensor obtained from the Einstein equations should therefore be interpreted as an effective matter sector encoding the corrections implied by the chosen entropy. In this sense, the framework establishes a mapping between entropy modifications and emergent matter content.
Next, let us show the conservation of the energy-momentum tensor for the general case in our setup. Let us start from the relation \[\begin{align} \nabla_\mu T^{\mu \nu}=0, \end{align}\] from this relation one can obtain \[\begin{align} \frac{d p_r}{dr}=-\frac{1}{2 g_{tt}}\frac{dg_{tt}}{dr}(\rho+p_r)+\frac{2}{r}(p_t-p_r). \end{align}\] From the last equation and using the condition \(\rho=-p_r\), it follows that \[\begin{align} \label{p95trhodrho} p_t=-\rho-\frac{r}{2} \frac{d \rho}{dr}. \end{align}\tag{14}\] Using Eq. 11 and Eq. 14 we indeed get Eq. 13 as follows \[\begin{align} \notag p_t &=\frac{M\left[r S''(r) - S'(r)\right]}{2\, r^{2} S'(r)^{2}}-\frac{r}{2} \frac{d}{dr}\left[-\frac{M\left[r S''(r) - S'(r)\right]}{2\, r^{2} S'(r)^{2}}\right] \\ &=\frac{M\left\{S'(r)[r S^{'''}(r) + 2 S''(r)] - 2 r S''(r)^{2}\right\}}{4\, r\, S'(r)^{3}}. \end{align}\] This result shows that the conservation of energy-momentum tensor holds in our setup.
We mention here that the effective stress-energy tensor derived from the reconstructed geometry can be interpreted as an anisotropic fluid, characterized by distinct radial and tangential pressures. Such effective matter sectors commonly arise in semiclassical gravity and quantum-corrected spacetimes. Violations of classical energy conditions in certain parameter regimes are therefore not necessarily pathological, but rather indicative of underlying quantum or statistical corrections to classical gravitational dynamics.
In the previous sections we showed how a modified entropy expression leads to an effective matter sector. Hence, we can now proceed to application to the various specific entropy forms that exist in the literature.
Barrow argued that quantum gravitational corrections may change the classical smoothness of the event horizon, giving rise to a horizon geometry with fractal characteristics. Such a modification implies that the standard area law for black hole entropy does not hold exactly. To quantify the degree of this geometric irregularity, a parameter \(\Delta\) is introduced, representing the extent to which the horizon departs from a smooth two-dimensional surface. With this modification, the entropy associated with a black hole is expressed as [12] \[\label{SB} S_B = \left( S_{BH} \right)^{1+\frac{\Delta}{2}} \;,\tag{15}\] where \(0 \leq \Delta \leq 1\). The case \(\Delta = 0\) corresponds to an undeformed horizon and reproduces the standard Bekenstein-Hawking entropy, while nonzero values of \(\Delta\) encode the influence of quantum-gravity-induced fractal structure. Conceptually, the presence of a nonzero \(\Delta\) indicates that the microstructure of spacetime at the horizon deviates from classical smoothness, potentially reflecting underlying quantum gravitational degrees of freedom. Thus, Barrow entropy provides an effective way to model such corrections without specifying the detailed microscopic theory.
Using 7 , the Barrow corrected metric function is \[f_{B}(r) = 1-\frac{4 M}{\sqrt{\pi ^\Delta}(\Delta +2)r^{\Delta +1}} \;,\] the corresponding Einstein tensor components (9 )-(10 ) are \[\begin{align} G_t^t &=&G_r^r = \frac{4 \pi ^{-\frac{\Delta }{2}} \Delta M r^{-\Delta -3}}{\Delta +2},\\ G_\theta^\theta &=& G_\phi^\phi = -\frac{2 \pi ^{-\frac{\Delta }{2}} \Delta (\Delta +1) M r^{-\Delta -3}}{\Delta +2} \;, \end{align}\] and thus the effective anisotropic matter sector (11 )-(13 ) becomes \[T^{\mu}{}_{\nu} = \rho\,\mathrm{diag}\!\left(-1,\,-1,\,\tfrac{\Delta+1}{2},\,\tfrac{\Delta+1}{2} \right) \;,\] where \[\rho = -\frac{\pi^{-\frac{\Delta}{2}}\Delta M\,r^{-\Delta-3}}{2\pi(\Delta+2)} \;.\] In the limit \(\Delta \to 0\), the effective stress tensor vanishes and the spacetime reduces to the Schwarzschild vacuum. Finally, note the gravitational force 8 for this case reads as \[\vec{F}_G= -2\pi M \frac{S_B''(r)}{S_B'(r)^2}\hat{r}= -\frac{2M (\Delta +1) }{\sqrt{\pi ^{\Delta}}\;(\Delta+2)r^{\Delta+2}} \hat{r}.\]
Let us briefly examine the energy conditions. For \(M>0\) and \(\Delta>0\), the effective energy density is negative (\(\rho<0\)). Since \(\rho+p_r = 0\) along the radial null direction, the null energy condition (NEC) is saturated. Along the tangential direction we have \(\rho+p_t = \tfrac{\Delta+3}{2}\rho < 0\), hence the NEC is violated. Since \(\rho<0\), the weak energy condition (WEC) is violated. Additionally, since \(\rho+p_r+2p_t=(\Delta+1)\rho<0\), the strong energy condition (SEC) is violated. Lastly, since \(\rho\ge0\) and \(|p_i|\le\rho\), the dominant energy condition (DEC) is violated, too. Thus, only the radial component marginally satisfies the NEC, while all other standard energy conditions are violated for \(\Delta>0\).
In summary, as we observe the effective matter sector behaves as an anisotropic fluid with negative energy density equal to the radial pressure, while the tangential pressure differs by a factor \((\Delta+1)/2\). Such stress-energy forms cannot arise from ordinary classical matter, and they reflect the quantum-gravitational or fractal corrections encoded by the Barrow entropy modification. The violation of the standard energy conditions is therefore not pathological but signals the presence of an effective, non-classical source required to support the modified horizon geometry.
The Tsallis-Cirto entropy represents a non-additive extension of the standard Bekenstein-Hawking entropy, inspired by the formalism of non-extensive statistical mechanics. In this framework, the entropy-area relation is modified to accommodate possible correlations or long-range interactions among the microscopic degrees of freedom associated with the horizon. This generalized entropy has been applied in gravitational and cosmological contexts, particularly in approaches where gravitational dynamics emerge from underlying thermodynamic principles. Within such scenarios, the Tsallis-Cirto entropy leads to modified cosmological evolution equations, offering an alternative route to explaining late-time cosmic acceleration and the effective behavior attributed to dark energy. For a black-hole horizon, the Tsallis-Cirto entropy is expressed as [10] \[S_{TC} = \left(S_{BH}\right)^{\delta} \;,\] where \(\delta\) denotes the non-extensive deformation parameter. The classical Bekenstein-Hawking entropy is recovered in the limit \(\delta \to 1\), signifying the absence of non-extensive effects and a return to the standard area law. The parameter \(\delta\) quantifies the degree to which horizon degrees of freedom are correlated or interacting at long ranges.
The corrected metric function 7 becomes \[\begin{align} f_{TC}(r) = 1-\frac{2M \pi ^{1-\delta }}{\delta \, r^{2\delta-1 }} \;, \end{align}\] while the Einstein tensor components (9 )-(10 ) read \[\begin{align} G_t^t &=& G_r^r = \frac{4M \pi ^{1-\delta } (\delta -1) }{\delta \; r^{2\delta +1}} \\ G_\theta^\theta &=& G_\phi^\phi = -\frac{2 M\pi ^{1-\delta } (\delta -1) (2 \delta -1) }{\delta \; r^{2\delta +1}} \;. \end{align}\] Moreover, the gravitational force 8 for this case reads \[\vec{F}_G= -2\pi M \frac{S_{TC}''(r)}{S_{TC}'(r)^2}\hat{r}= -\frac{M\pi^{1-\delta}(2 \delta -1)}{r^{2\delta} \;\delta } \hat{r} \;.\] Lastly, the effective stress-energy tensor (11 )-(13 ) becomes \[T^{\mu}{}_{\nu} =\rho\,\mathrm{diag}\!\left(-1,\,-1,\,\tfrac{2\delta-1}{2},\,\tfrac{2\delta-1}{2 }\right),\] where \[T^{\mu}{}_{\nu} =\rho\,\mathrm{diag}\!\left(-1,\,-1,\,\tfrac{2\delta-1}{2},\,\tfrac{2\delta-1}{2 }\right),\] with \[\rho=-\frac{M\,\pi^{-\delta}(\delta-1)}{2\,\delta\,r^{2\delta+1}} \;.\]
As we observe the Tsallis exponent \(\delta\) quantifies also the behavior of the effective matter sector. For \(\delta=1\) we reproduce the vacuum limit (\(T^{\mu}{}_{\nu}=0\)), while \(\delta\neq1\) describes an anisotropic effective matter source. The sign of \(\rho\) depends on \((\delta-1)\), i.e. for \(\delta>1\), \(\rho<0\) the effective source has negative energy density, while for \(0<\delta<1\), \(\rho>0\) we obtain a positive anisotropic matter distribution.
Concerning the energy conditions, we can see that radial NEC is saturated, but the tangential NEC \(\rho+p_t=\tfrac{2\delta+1}{2}\rho\) is satisfied for \(\rho>0\) and violated for \(\rho<0\). Moreover, the WEC holds if \(\rho>0\) (i.e.\(\delta<1\)), otherwise it is violated. Concerning SEC we find that \(\rho+p_r+2p_t=(2\delta-1)\rho\), and thus it is satisfied for \(\rho>0\) and \(\delta>\tfrac{1}{2}\), while it is violated otherwise. DEC holds only if \(\rho>0\) and \(|p_i|\le\rho\), which restricts \(\tfrac{1}{2}\le\delta\le1\). Hence, the stress tensor satisfies all standard energy conditions for \(0<\delta<1\) but violates them when \(\delta>1\), where \(\rho\) becomes negative.
In summary, the parameter \(\delta\) controls the power-law behavior of the effective energy density, \(\rho\propto r^{-(2\delta+1)}\), producing an anisotropic fluid with radial tension \(p_r=-\rho\) and tangential pressure proportional to \((2\delta-1)\rho/2\). For \(\delta<1\) the matter distribution is physically reasonable and satisfies the energy conditions, whereas for \(\delta>1\) the energy density becomes negative, indicating an exotic effective source required to sustain a regularized or non-classical geometry. In the limit \(\delta\to1\), all stress components vanish and the spacetime smoothly reduces to the Schwarzschild vacuum.
Rényi entropy offers a generalized measure of entropy that extends beyond the additive structure of the Bekenstein-Hawking formulation. The key feature of this framework is the parameter \(\lambda\), which controls the degree to which the entropy departs from extensivity. Such a modification is useful in black-hole thermodynamics since the microscopic degrees of freedom associated with the horizon may interact in ways that are not accounted for by standard Boltzmann-Gibbs statistics.
Interpreting \(\lambda\) as an additional thermodynamic parameter enlarges the black-hole phase space and enables a modified form of the first law and Smarr relation. This approach also leads to changes in the thermodynamic behavior and stability properties of black holes when compared to the usual area law [13], [14]. The Rényi entropy is given by [11] \[S_R = \frac{\log\!\left(1 + \lambda \, S_{BH}\right)}{\lambda} \;,\] and reduces to the usual Bekenstein-Hawking entropy in the limit \(\lambda \to 0\), indicating that the classical area law is recovered when no non-extensive effects are present.
The metric function is \[\begin{align} f_R(r) = 1-\frac{2M \left(1+ \pi \lambda r^2\right)}{r} \;, \end{align}\] the Einstein tensor components are \[\begin{align} G_t^t &=&G_r^r = -\frac{4 \pi \lambda M}{r}, \\ G_\theta^\theta &=& G_\phi^\phi = -\frac{2 \pi \lambda M}{r}, \end{align}\] and the gravitational force becomes \[\vec{F}_G= -2\pi M \frac{S_R''(r)}{S_R'(r)^2}\hat{r}= M \left(\pi \lambda -\frac{1}{r^2}\right)\hat{r} \;.\] Additionally, the corresponding effective stress-energy tensor (11 )-(13 ) is \[T^{\mu}{}_{\nu} = \rho\,\mathrm{diag}\!\left(-1,\,1,\,\tfrac{1}{2},\,\tfrac{1}{2}\right), \quad \text{where}\quad \rho = \frac{\lambda M}{2r} \;.\] In the limit \(\lambda\to0\), one recovers \(T^{\mu}{}_{\nu} \to 0\), i.e. the Schwarzschild vacuum. Finally, note that for \(M>0\) and \(\lambda>0\), the effective energy density is positive (\(\rho>0\)), and all standard energy conditions are therefore satisfied for \(\lambda>0\).
In summary, the effective matter sector behaves as an anisotropic fluid with positive energy density \(\rho\propto1/r\), a radial pressure equal to the energy density (\(p_r=\rho\)), and a smaller tangential pressure (\(p_t=\rho/2\)). This represents a non-vacuum configuration sustained by an extended, inhomogeneous distribution rather than a delta-function source. The parameter \(\lambda\) controls the strength of the deviation from the Schwarzschild vacuum, thus \(\lambda\to0\) restores vacuum geometry, whereas finite \(\lambda\) introduces a mild, physically reasonable anisotropy consistent with all energy conditions.
Kaniadakis proposed a generalized statistical framework that departs from the traditional Boltzmann-Gibbs formulation by introducing a deformation parameter \(\kappa\). This approach, often referred to as Kaniadakis statistics, is constructed to be compatible with relativistic dynamics while maintaining the foundational consistency of standard statistical mechanics [15], [16]. Within this framework, the usual Maxwell-Boltzmann distribution arises as a special limiting case, whereas nonzero values of \(\kappa\) encode deviations associated with generalized thermodynamic behavior.
When applied to gravitational systems, particularly black holes, this modified entropy provides a natural way to incorporate corrections to horizon thermodynamics. The corresponding Kaniadakis entropy for black holes takes the form \[S_K = \frac{\sinh\!\left(\kappa S_{BH}\right)}{\kappa} \;,\] where \(S_{BH}\) is the standard Bekenstein-Hawking entropy. In the limit \(\kappa \rightarrow 0\), the expression reduces smoothly to \(S_{BH}\), demonstrating that the Kaniadakis framework contains the conventional entropy law as a special case.
The parameter \(\kappa\) can be interpreted as a measure of deviations from the standard thermodynamic behavior encoded by the standard horizon geometry. A nonzero value of \(\kappa\) reflects the presence of additional microscopic correlations or fluctuations that are not captured by the ordinary Bekenstein-Hawking description. In this sense, the Kaniadakis entropy provides an effective macroscopic signature of statistical features near the event horizon, while still preserving continuity with conventional black-hole thermodynamics when \(\kappa \to 0\).
For this modified entropy the corrected metric function becomes \[f_\kappa(r) = 1-\frac{2 M \;\text{sech}\left(\pi \, \kappa \, r^2\right)}{r},\] the Einstein tensor components are \[\begin{align} G_t^t &=&G_r^r = \frac{4 \pi \kappa M \tanh \left(\pi \kappa r^2\right) \text{sech}\left(\pi \kappa r^2\right)}{r}\\ G_\theta^\theta &=& \frac{\pi \kappa M \left(\sinh \left(2 \pi \kappa r^2\right)-2 \pi \kappa r^2 \left(\cosh \left(2 \pi \kappa r^2\right)-3\right)\right)}{r\;\cosh^3\left(\pi \kappa r^2\right)} \;, \nonumber \end{align}\] and the gravitational force reads \[\vec{F}_G = -\frac{M \left(2 \pi \kappa r^2 \tanh \left(\pi \kappa r^2\right)+1\right) \text{sech}\left(\pi \kappa r^2\right)}{r^2}\, \hat{r} \;.\] Furthermore, the corresponding effective stress-energy tensor is \[\begin{align} \rho &=& -\frac{\kappa M}{2r}\, \tanh(\pi\kappa r^2)\,\text{sech}(\pi\kappa r^2), \\[3pt] p_r &=& -\rho, \\[3pt] p_t &=& \frac{\kappa M}{8r}\, \frac{\sinh(2\pi\kappa r^2)-2\pi\kappa r^2\bigl(\cosh(2\pi\kappa r^2)-3\bigr)}{\cosh^3(\pi\kappa r^2)} .\;\;\;\; \end{align}\] Hence, the effective matter distribution is anisotropic, with \(p_r=-\rho\), and the \(t\)-direction pressure \(p_t\) is determined by the hyperbolic functions of \(\pi\kappa r^2\). The parameter \(\kappa\) controls the strength of the deviation from the vacuum geometry, and for \(\kappa \to 0\) we find \(T^{\mu}{}_{\nu}\to0\) and the Schwarzschild limit is recovered.
Concerning the energy conditions, we can see that at small \(r\) we have \(\rho\simeq -\tfrac{\pi\kappa^2 M r}{2}<0\). Thus, near the origin \(\rho<0\) and the energy conditions are violated, while for intermediate \(r\) the sign of \(\rho\) may change depending on \(\kappa\). The radial NEC is saturated, but the tangential NEC can be negative near the core, indicating local NEC violation that softens the central singularity. Moreover, WEC and SEC are violated wherever \(\rho<0\), and they are satisfied only in outer regions where \(\rho>0\). DEC is generally violated near the center due to the negative \(\rho\). Hence, the effective source violates the classical energy conditions near the core but tends to restore them asymptotically.
In summary, the \(\kappa\)-dependent correction acts as an effective, anisotropic fluid that smoothly interpolates between a de Sitter-like core and an asymptotically vacuum exterior. Near \(r=0\) the negative energy density and the corresponding pressure \(p_r=-\rho\) regularize the central region, removing the curvature singularity, while for large \(r\) the stress-energy decays exponentially as \(\mathrm{sech}(\pi\kappa r^2)\). This behavior is consistent with a non-classical, quantum-gravity-induced core, where the violation of energy conditions is the price paid for achieving a regular black hole interior. In the standard limit \(\kappa\to0\), the effective stresses vanish and the standard Schwarzschild solution is recovered.
Logarithmic corrections to the Bekenstein-Hawking entropy naturally arise when quantum or statistical fluctuations of the horizon degrees of freedom are taken into account. While the leading-order entropy is proportional to the horizon area, subleading corrections appear once quantum fields, quantum geometry, or thermal fluctuations near the horizon are included. A general and widely encountered form of the corrected entropy is \[S_{\rm log} = S_{BH} + \lambda \, \ln S_{BH} \;,\] where \(\lambda\) depends on the underlying quantum gravity framework. These logarithmic corrections are remarkably universal: they have been derived in loop quantum gravity [7], in string theory [86], in the quantum geometry approach [87], and in treatments based on thermal fluctuations in canonical ensembles [8]. Physically, the logarithmic term reflects fluctuations around the classical equilibrium configuration of the horizon and becomes especially relevant for small black holes or near-extremal configurations. Such corrections can modify thermodynamic stability and phase behavior, providing an important probe into the microscopic origin of gravitational entropy.
For this entropy modification, the corrected metric function 7 becomes \[f_{\rm log}(r) = 1-\frac{2M \, \pi \, r}{\lambda +\pi r^2} \;,\] while the Einstein tensor components (11 )-(13 ) are \[\begin{align} G_t^t &=&G_r^r = -\frac{4 \pi \lambda M}{r \left(\lambda +\pi r^2\right)^2} \\ G_\theta^\theta &=& G_\phi^\phi = \frac{2 \pi \lambda M \left(3 \pi r^2-\lambda \right)}{r \left(\lambda +\pi r^2\right)^3} \;, \end{align}\] and the gravitational force 8 reads \[\vec{F}_G= -2\pi M \frac{S_{\rm log}''(r)}{S_{\rm log}'(r)^2}\hat{r}= \frac{\pi M \left(\lambda -\pi r^2\right)}{\left(\lambda +\pi r^2\right)^2}\hat{r} \;.\] The effective stress-energy tensor components (11 )-(13 ) become \[\begin{align} \rho = -p_r = \frac{\lambda M}{2 r \left(\lambda + \pi r^2\right)^2}, \end{align}\] and \[\begin{align} p_t = \frac{\lambda M \left(3 \pi r^2 - \lambda \right)}{4 r \left(\lambda + \pi r^2\right)^3} \;. \end{align}\] Thus, the stress tensor clearly shows an anisotropic pressure structure with \(p_r \neq p_t\), and a negative radial pressure similar to that found in de Sitter-like cores. NEC and WEC are satisfied. Concerning SEC, since we find that \[\rho + p_r + 2p_t = \frac{\lambda M \left(3 \pi r^2 - \lambda \right)}{2 r (\lambda + \pi r^2)^3},\] which can become negative for \(r^2 < \lambda / (3\pi)\), we conclude that SEC is violated near the core.
In summary, the effective matter source corresponds to an anisotropic fluid that mimics a regularized gravitational core. At small \(r\), the density approaches the finite \(\rho(r\to0) \sim \frac{M}{2 \pi \lambda r},\) while at large \(r\) it decreases as \(r^{-3}\). The negative radial pressure and SEC violation near the origin are signatures of a de Sitter-like vacuum behavior, ensuring a regular interior geometry. Thus, this stress-energy distribution effectively describes a smooth transition from a quantum-gravity-inspired core to an asymptotically Schwarzschild exterior.
In the context of Loop Quantum Gravity (LQG), non-extensive statistical mechanics gives the following modified entropy law [94], [97]: \[S_{LQG}(A) = \frac{1}{(1-q)} \exp \left[ \frac{(1-q)\Lambda(\gamma_0)A}{4} - 1 \right],\] where \(\Lambda(\gamma_0) = \ln 2 / (\sqrt{3}\pi \gamma_0)\), with \(\gamma_0\) the Barbero-Immirzi parameter which measures the size of area quanta in Planck units, determined by counting the number of spin-network states corresponding to an event horizon of area \(A\). The corrected metric is \[\begin{align} f_{\rm LQG} (r) = 1-\frac{2 M e^{\pi \Lambda (q-1) r^2+1}}{\Lambda r}, \end{align}\] the components of the Einstein tensor are \[G_t^t = G_r^r = -\frac{4 \pi M (q-1) e^{\pi \Lambda (q-1) r^2+1}}{r} \;\; \;\;\;\;\;\;\] and \[G_\theta^\theta= G_\phi^\phi = -\frac{2 \pi M (q\!-\!1) e^{\pi \Lambda (q\!-\!1) r^2+1} \left[2 \pi \Lambda (q\!-\!1) r^2+1\right]}{r},\] while the gravitational force for this case is written as \[\vec{F}_G= -2\pi M \frac{S_{\rm LQG}''(r)}{S_{\rm LQG}'(r)^2}\hat{r}= \frac{M \left[2 \pi \Lambda (q-1) r^2-1\right]}{\Lambda \,e^{-\left(\pi \Lambda (q-1) r^2+1\right)} r^2} \hat{r} \;.\] From the components of the effective stress-energy tensor we can easily compute the corresponding fluid parameters as \[\begin{align} \rho &=& -p_r = \frac{M (q-1) e^{\pi \Lambda (q-1) r^2+1}}{2 r} \;, \\ p_t &=& -\frac{M (q-1) \left[2 \pi \Lambda (q-1) r^2+1\right]}{4 e^{-\left[\pi \Lambda (q-1) r^2+1\right]} r}, \end{align}\] thus the matter source behaves as an anisotropic fluid. The radial WEC is satisfied, while for the tangential component we have \[\rho + p_t = \frac{M (q-1) \pi \Lambda (q-1) r^2}{2 e^{-\left(\pi \Lambda (q-1) r^2+1\right)} r} > 0,\] for \(\Lambda (q-1) > 0\). Hence, the WEC is satisfied in the physical region. Additionally, NEC coincides with WEC and it is also satisfied. Finally, concerning SEC we find \[\rho + p_r + 2p_t = -\frac{M (q-1) \left[2 \pi \Lambda (q-1) r^2+1\right]}{2e^{-\left[\pi \Lambda (q-1) r^2+1\right]} r},\] which becomes negative near the core (\(r\to0\)), indicating SEC violation.
In summary, the stress-energy distribution corresponds to an anisotropic matter source whose density increases exponentially with \(r^2\) for positive \(\Lambda(q-1)\). The negative radial pressure and the SEC violation near \(r=0\) imply the presence of a repulsive core that regularizes the central region, similar to a de-Sitter vacuum. For large \(r\), the exponential factor dominates and the energy density decays rapidly, leading to an asymptotically vacuum configuration. Thus, this effective source describes a smooth transition between a regular quantum-inspired core and an exterior Schwarzschild-like regime.
Exponential corrections to the Bekenstein-Hawking entropy offer an alternative approach to encoding possible quantum or statistical modifications to black hole thermodynamics. Unlike logarithmic corrections, which typically arise from quantum fluctuations, exponential corrections are motivated by non-perturbative or holographic effects that become significant near the Planck scale [88]–[91]. The corrected entropy is written as \[S_{\rm exp} = S_{BH} + \eta \, e^{- S_{BH}} \;, \label{exp95corr}\tag{16}\] where \(\eta\) is a model-dependent constant characterizing the strength and scale of the correction. Since the correction term is exponentially suppressed for large horizon area, classical black holes with large entropy are essentially unaffected, ensuring consistency with general relativity in the semiclassical regime. However, for small black holes (or near-extremal configurations), the exponential contribution can become non-negligible and affect the thermodynamic quantities such as heat capacity and free energy. These corrections have been studied in contexts including quantum tunneling methods, modified gravity theories, and non-perturbative quantum gravity models [88]–[91]. Their impact is particularly important in examining the late stages of black hole evaporation and the possible resolution of the final state problem.
In this case, the corrected metric is \[\begin{align} f_{\rm exp} (r) = 1-\frac{2 M}{r-\eta e^{-\pi r^2} r}, \end{align}\] the Einstein tensor components are \[G_t^t = G_r^r = \frac{4 \pi \eta M e^{\pi r^2}}{r \left(e^{\pi r^2}-\eta \right)^2},\] and \[G_\theta^\theta = G_\phi^\phi = \frac{-2 \pi \eta M \left[\eta +2 \pi \eta r^2+e^{\pi r^2} \left(2 \pi r^2-1\right)\right]}{e^{-\pi r^2}\,r \left(e^{\pi r^2}-\eta \right)^3} \;,\] and the gravitational force reads as \[\vec{F}_G= -2\pi M \frac{S_{\rm exp}''(r)}{S_{\rm exp}'(r)^2}\hat{r}= -\frac{M e^{\pi r^2} \left[\kappa \left(2 \pi r^2\!-\!1\right)+e^{\pi r^2}\right]}{r^2 \left(e^{\pi r^2}\!-\!\kappa \right)^2} \hat{r} \;.\] Furthermore, the effective stress-energy tensor components become \[\begin{align} \rho &=& -p_r = -\frac{\eta M e^{\pi r^2}}{2 r \left(e^{\pi r^2}-\eta \right)^2}, \\ p_t &=& -\frac{\eta M e^{\pi r^2}\left[\eta +2 \pi \eta r^2+e^{\pi r^2} \left(2 \pi r^2-1\right)\right]}{4 r \left(e^{\pi r^2}-\eta \right)^3}. \end{align}\] Therefore, the stress-energy tensor describes an anisotropic fluid with a de Sitter-like core, where the radial and tangential pressures differ (\(p_r \neq p_t\)). The WEC and NEC are satisfied for large \(r\), while for the SEC we find \[\rho + p_r + 2p_t = -\frac{\eta M e^{\pi r^2}\left[\eta + 2 \pi \eta r^2 - e^{\pi r^2}(1 - 2\pi r^2)\right]}{2 r \left(e^{\pi r^2}-\eta \right)^3},\] which can become negative near \(r=0\), showing SEC violation in the core region.
In summary, the effective matter source represents a regular anisotropic fluid distribution. For small \(r\) the density remains finite as \[\rho(r \to 0) \approx \frac{\eta M}{2 r (\eta - 1)^2} \;,\] and for large \(r\) it falls off exponentially due to the \(e^{\pi r^2}\) term in the denominator. The negative radial pressure and the SEC violation near the origin indicate a de-Sitter-like vacuum behavior, which prevents curvature singularities. This configuration thus provides a regular black-hole model interpolating between a finite-density quantum core and an asymptotically Schwarzschild regime.
There is well-known connection between gravity and thermodynamics, which can offer a useful perspective on the microscopic origin of spacetime geometry. Since the identification of black holes as thermodynamic systems possessing temperature and entropy, it has become clear that gravitational dynamics can be viewed as emergent phenomena arising from underlying statistical degrees of freedom. Within this framework, the gravitational field equations may be interpreted as thermodynamic relations between quantities defined on the horizon, and thus modifications in entropy can be expected to induce corresponding modifications in the geometry itself.
In the literature, various generalizations of the Bekenstein-Hawking entropy have been proposed, motivated by quantum gravitational, statistical, and non-extensive frameworks. In particular, Barrow, Tsallis-Cirto, Rényi, Kaniadakis, logarithmic, power-law, exponential and other entropy formulations introduce deviations from the standard area law, describing possible quantum or non-equilibrium effects near the horizon. Each of these modified entropies carries distinct implications for the thermodynamic stability, phase structure, and geometric regularization of black holes.
In this work, we have constructed a general framework linking a modified entropy function directly to a modified spacetime metric and, subsequently, to an effective matter sector of entropic origin. Starting from the first law of black-hole thermodynamics, we established an explicit correspondence between the entropy derivative and the metric function, which naturally leads to an emergent stress-energy tensor representing an anisotropic effective fluid. This procedure was then applied to a wide class of entropy models, allowing us to derive the associated effective matter sectors, and analyze their physical properties and energy conditions in a unified manner.
Although our analysis is performed within the framework of general relativity, assuming the Einstein field equations, deviations from Bekenstein-Hawking entropy can be interpreted in two complementary ways. They may be viewed either as effective matter sources when incorporated into the energy-momentum side of the Einstein field equations or, equivalently, as signatures of modified gravity when incorporated into the geometric side of the Einstein field equations. That being said, the definitive answer or the final interpretation remains open, and the correspondence between entropy and geometry allows for a dual viewpoint. However, the precise nature of this interpretation is not central to the present analysis. The essential result is that entropy itself governs the modification of spacetime geometry. Remarkably, the gravitational field equations need not be assumed a priori; rather, the geometry can be inferred directly from entropic considerations. This provides further evidence in favor of gravity as an emergent phenomenon.
Future work can extend this correspondence toward dynamical and cosmological spacetimes, studying how generalized entropy functions modify the Friedmann equations and the cosmological evolution. Furthermore, the framework can be used to investigate the thermodynamic origin of dark energy, to test consistency with quantum-gravity considerations, and to confront the results with astrophysical observations. Moreover, one could examine the connection of modified entropies to modified geometry and gravity. These interesting projects are currently under investigation.
Ankit Anand is financially supported by the Institute’s postdoctoral fellowship at IITK. S.D. acknowledges the financial support provided by DST vide Grant No. DST/INSPIRE Fellowship/2020/IF200537. A.A. and S.D. acknowledge fruitful discussions with Ayan Chatterjee and Anshul Mishra. K.J. thanks Sunny Vagnozzi for insightful and fruitful discussions. E.N.S. gratefully acknowledges the contribution of the LISA Cosmology Working Group (CosWG), as well as support from the COST Actions CA21136 - Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse) - CA23130, Bridging high and low energies in search of quantum gravity (BridgeQG) and CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and experiments (CosmicWISPers).
In this Appendix we show that starting from the Bekenstein-Hawking entropy, it is straightforward to verify that the corresponding metric function reproduces the Schwarzschild solution. We start with metric 1 , namely \[\label{Metric95ansatzA} f(r) = 1 - M\,g(r) \, ,\tag{17}\] whose event horizon \(r=r_{+}\) is defined by \(f(r_{+}) = 0\). We consider the Bekenstein-Hawking entropy \(S = \pi r_{+}^{2}\) and the Hawking temperature, determined by the surface gravity \(\kappa = f'(r_{+})/2\), namely \(T = \frac{f'(r_{+})}{4\pi}\). Using the first law of black hole thermodynamics \(dM = T\,dS\), we have \[\frac{dM}{dr_{+}} = T\,\frac{dS}{dr_{+}} = \frac{r_{+}}{2}\, f'(r_{+}) \;.\] Assuming that the metric function \(f(r)\) depends on the mass parameter \(M\), the horizon condition \(f(r_{+},M)=0\) determines \(M\) as a function of \(r_{+}\), i.e. \(M = M(r_{+})\). Differentiating the horizon condition implicitly gives \[\frac{dM}{dr_{+}} = -\frac{\partial_{r} f(r_{+},M)}{\partial_{M} f(r_{+},M)},\] and equating this with the thermodynamic relation above yields the general consistency condition \[-\frac{\partial_{r_+} f(r_{+},M)}{\partial_{M} f(r_{+},M)} = \frac{r_{+}}{2}\, f'(r_{+}) \;.\] This relation provides a constraint on the admissible metric functions \(f(r)\) for a black hole satisfying the first law with the Bekenstein-Hawking entropy. Starting with the simple power-law ansatz \(f(r)=1-c\,M\,r^{-p}\), imposing the above condition for arbitrary \(r_{+}\) uniquely selects \(p=1\) and \(c=2\), recovering the Schwarzschild form \[f(r)=1-\frac{2M}{r} \;.\] Thus, within this class of metrics, the Schwarzschild one is singled out solely by the requirement that the first law \(dM=T\,dS\) holds together with the Bekenstein entropy formula. Finally, we note that the uniqueness of the Schwarzschild solution obtained here follows from the imposed symmetry assumptions together with the thermodynamic constraint.