Accretion with back-reaction onto a cylindrically symmetric black hole with energy condition analysis

Cylindrical black holes (BHs) or black strings (BSs) are the cylindrically symmetric static solutions of the Einstein-Maxwell field equations (EMFEs) together with a negative cosmological constant. The study of BSs has a long history. The first exact solution for BSs was discovered by Lemos in [1]. Following this initial work, Cai and Zhang calculated a charged version [2]. Shortly thereafter, solutions for the rotating and rotating charged versions of BSs were also found [3].

Around a BS, there is an accretion disk. This accreting matter falls into the BS, known as the back-reaction of accreting matter onto the BS. In Newtonian gravity, the problem of matter accretion onto compact objects was first formulated in a self similar manner by Bondi [4]. The concept of matter accretion onto BHs traces back to the 1970s [5], when pioneering work by Shakura and Sunyaev (1973) introduced the ''disk model" for accretion flows around compact objects, including BHs [6]. This disk model emphasized how gas spirals into BHs, releasing energy and potentially affecting the BH's growth rate and observable properties. Shakura and Sunyaev's [6] approach set a foundational framework for understanding how accretion disks contribute to the observable luminosity of BHs and led to the recognition of accretion as a fundamental mechanism in BH evolution and active galactic nuclei (AGN) theory [7, 8].

Following these early models, the idea of back-reaction, or the impact of accreting matter on the BH's own properties, was explored [9]. In the early 1980s, seminal work by Bardeen and Wagoner [10] examined the effects of angular momentum and energy transfer during accretion, proposing that the BH spin and mass could be incrementally modified through this interaction [11]. This line of inquiry laid the groundwork for understanding back-reaction as not merely a passive process but as one where the accreting matter might influence the spacetime geometry near the BH, setting up a feedback mechanism that depends on the nature of accretion flow [12].

In recent years, with advancements in perturbative methods and numerical relativity, the study of BH accretion has evolved to include back-reaction effects [13] in more complex spacetimes. Modern treatments often apply perturbative methods [14] to investigate how small amounts of infalling matter influence the metric around BHs, allowing for a more detailed analysis of back-reaction effects on rotating and non-rotating BHs. Contemporary studies by Babichev et al. (2018) [15] and others highlight how research on accretion with a focus on energy conditions has extended to cosmological contexts, opening up new applications and insights in high-energy astrophysics [16].

In this study, we investigate back-reaction effects using a perturbative scheme onto a BS solution. Initially, we analyze a static BS case, approximating the mass at the zeroth order. This results in a expression for mass as a function of time t and radial coordinate r, which we refer to as the running mass. We then examine the self-consistency of the solution and energy condition analysis using the mass function. Furthermore, we explore three different accretion models, providing graphical insights based on the running mass. By considering the perturbed metric, we derive an expression for the corrected apparent horizon and investigate how accreting matter affects the horizon over time, as well as the energy density and pressure in these models. We also discuss the thermodynamics of the corrected horizon by calculating temperature and entropy expressions for each case. Additionally, we show that for a charged BS, the mass function expression resembles that of the static case, without the charge contributing to the mass in the perturbative approach. Hence, this approach demonstrates that the accretion back-reactions onto both charged and uncharged cylindrically symmetric BS solutions are identical.

Babichev et al. [17] applied a similar approach to study spherically symmetric static BHs. In our study, we extend this method to the case of BSs. Specifically, we employ this framework to analyze both charged and uncharged BSs and compare the results to explore how back-reaction affects their properties. Additionally, we provide a detailed analysis of the thermodynamics of the corrected BS metric for both cases.

Notably, we compare the corrected mass with the ADM mass and derive the relationship between them, highlighting their interdependence. We believe this constitutes a valuable contribution to the available literature on the study of BSs.

The structure of this paper is as follows. In Sec. II, we present the mathematical formalism necessary to investigate the back-reaction effects. In Sec. III, we apply the mathematical formalism to a static BS and derive the corrected mass term. Sec. IV addresses the self-consistency of the solution and energy condition analysis, providing a physical interpretation of the back-reaction phenomena. Sec. V discusses the analysis for different accreating models. In Sec. VI, we examine the back-reaction effects for a charged BS, and the paper concludes with a discussion and conclusion in Sec. VII.

In the study of BH accretion, back-reaction effects are commonly neglected owing to the typically negligible mass of the accreting matter compared to the BH mass [18]. This approximation allows for simpler analytic models but overlooks the complex interplay between the gravitational field of BH and the dynamics of infalling matter. The challenge of solving the equations governing accreting matter with back-reaction arises from the intricate coupling between matter and spacetime geometry [19], leading to non-linear effects that are difficult to analyze. Consequently, few analytic solutions have been derived that fully account for these back-reaction phenomena [20]. In this section, we investigate the implications of back-reaction effects due to matter accretion to enhance our understanding of their role in BS dynamics.

The complete solution of the Einstein field equations (EFEs) in the general case of BH accretion is still unknown; however, there are several special cases of the energy-momentum tensor for which the exact solution has been found, i.e., the Vaidya and Tolman solutions [21, 22]. In this section, we review the perturbative approach to find the solution. This approach was also used by Babichev et al. [17]. Here, we correct the metric because accreting matter has an energy-momentum tensor in the zeroth-order approximation (i.e., back-reaction is neglected) and find the first order correction of the metric with back-reaction. Mathematically, we begin with the EFEs with metric tensor $ g_{\mu v} $ , in which φ represents the degree of freedom associated with accreting matter,

and the equations of motion for the matter field are $ E\left[g_{\mu v}, \varphi\right]=0 $, which can also be derived from the Bianchi identities. In Eq. (1), if we neglect back-reaction, we obtain a zeroth-order approximation. Specifically, in this approximation, the solution for the metric is the vacuum solution, i.e., $ g_{\mu v}^{(0)}=g_{\mu v}^{v a c} $ , so that $ G_{\mu v}\left[g_{\mu v}^{(0)}\right]=0 $. Moreover, in the same zeroth order approximation, the solution for the equation for matter field(s), $ \varphi^{(0)} $, is computed as

Now, to proceed with the first-order approximation, we substitute $ g_{\mu\nu}^{(0)} $ and $ g_{\mu\nu}^{(1)} $ as follows. On the right-hand side of Eq. (1), we substitute $ g_{\mu\nu}^{(0)} $, and on the left-hand side, we substitute $ g_{\mu\nu}^{(0)} + g_{\mu\nu}^{(1)} $. This leads to the EEFs taking the form

We also assume that $ g_{\mu\nu}^{(0)} $ is greater than $ g_{\mu\nu}^{(1)} $. Next, we apply this on the static BS metric.

The static uncharged BS metric with a negative cosmological constant $\alpha^2 = -\dfrac{\Lambda}{3} > 0$ in anti-de Sitter [23] spacetime is given by

This metric represents the solution in the zeroth-order approximation. To include back-reaction effects, we introduce a perturbed EFE, as defined in Eq. (3), where the function $m(t, r)$ replaces the constant $m_0$. At zeroth order, $m(t, r)$ reduces to $m_0$, where $m_0$ represents the mass without the back-reaction effect and thus remains constant. Simplifying Eq. (4),

The components of the Einstein tensor are expressed as follows:

We denote derivatives with respect to t by a dot and those with respect to r by a prime. By substituting the components of the Einstein tensor into the EFEs, we obtain

To analyze the problem effectively, we begin by solving Eq. (9) to express $ m^{\prime}(t, r) $ in terms of $ T_{0}^{0} $ and $ T_{1}^{1} $. Subsequently, we solve Eq. (10), which provides an expression for $ \dot{m}(t, r) $ in terms of $ T_{1}^{0} $ and $ T_{1}^{1} $. However, note that not all components of this system of equations are independent. Using the Bianchi identities, we demonstrate that Eq. (11) is essentially a linear combination of Eqs. (9) and (10). Therefore, Eq. (11) does not contribute additional information and is not further considered. As a result, the expression for the mass is obtained through the solutions of Eqs. (9) and (10),

Eq. (12) represents the central finding of this study. In this context, $ m(t,r) $ denotes the corrected mass, which accounts for the back-reaction of the accreting matter, whereas $ m_{0} $ refers to the zeroth-order mass, representing the case without back-reaction. The components of the energy-momentum tensor are slowly varying functions of the radial coordinate. Using Eq. (12), we can express the mass $ m(t, r) $ in the vicinity of the BS horizon as r approaches $ r_0 $, indicating that near the horizon, the mass can be expressed as

Additionally, for our scheme, Eq. (13) is only applicable when the correction is small. Therefore, we require

In the context of general relativity (GR), self-consistency in the solutions of the EFEs is fundamental for ensuring the physical validity of any model, especially those involving back-reaction effects and accretion onto BHs [24]. A self-consistent solution requires that the energy-momentum tensor $ T^{\mu}_ {\nu} $ satisfies the EFEs in a way that aligns with the background metric while remaining dynamically consistent with the boundary conditions or any approximations used [25]. This approach is commonly found in studies of accreting systems and BHs, where the metric is perturbed by incoming matter [26], and any deviations are captured by the resulting back-reaction on spacetime geometry.

In accretion studies, the test-fluid approximation is frequently used to model the dynamics of accreting matter within a fixed background, generally under the assumption that the accreted mass has minimal impact on the BH mass and metric. This approach is valid under conditions of small energy densities ρ and slow accretion rates $ \dot{m} $. The back-reaction can be handled perturbatively. For instance, in works by Michel (1972) [27] and Babichev et al. (2004) [16], the self-consistency of accretion models was maintained by ensuring that the accreting matter's influence remained a small perturbation in comparison to the BH mass [28].

In our solution, this self-consistency is ensured by establishing small parameters associated with the energy density and accretion rate. By expressing the Einstein tensor components and solving the resulting field equations (Eqs. (6)−(11)), we derive the corrected mass function $ m(t, r) $, which incorporates these small parameters in a manner that respects the background symmetry. Our solution ensures consistency by adhering to the condition $ \rho_{\infty} m^2 \ll 1 $, meaning that the energy density of the accreting matter at infinity is sufficiently small to allow for the test-fluid treatment, where the BS spacetime is only minimally perturbed by the incoming matter.

Furthermore, our framework allows us to relate the mass function $ m(t, r) $ to the components of the energy-momentum tensor $ T^{\mu \nu} $ through a perturbative approach, where corrections appear linearly in terms of the accretion rate $ \dot{m} $. By introducing the back-reaction effect at the level of the EFEs, we maintain a first-order approximation that adequately captures the influence of accreting matter without violating the stability or consistency of the background metric.

In this sense, our solution is self-consistent, as the derived mass function reflects only slight modifications in the BS horizon owing to accretion, which is a scenario supported by the smallness of the perturbation terms $ m'(t, r) $ and $ \dot{m}(t, r) $. This is evident from Eq. $(12)$. Therefore, our solution is not only self-consistent but also corroborates the assumptions widely used in accretion theory, particularly in cylindrically symmetric spacetimes, where the energy-momentum tensor components remain compatible with the slowly varying behavior required by the underlying geometry.

The energy conditions serve as fundamental constraints that encapsulate the general properties of most forms of matter, helping to eliminate many non-physical solutions to the EFEs. Among these conditions, we find the null, weak, dominant, and strong energy conditions [29], all of which can be understood as limitations on the eigenvalues and eigenvectors associated with the energy-momentum tensor.

Null Energy Condition (NEC): This condition states that for any future-pointing null vector field $ \vec{k} $, the energy density must satisfy $\rho = T_{ab} k^a k^b \geq 0$.

Weak Energy Condition (WEC): The WEC asserts that for every timelike vector field $ \vec{X} $, the energy density also meets the requirement $\rho = T_{ab} X^a X^b \geq 0$.

Dominant Energy Condition (DEC): In addition to fulfilling the WEC for all future-pointing causal vector fields $ \vec{Y} $ (which can be either timelike or null), the DEC requires that the vector $ -T^b_a Y^b $ remains a future-pointing causal vector.

Strong Energy Condition (SEC): The SEC is defined such that for any timelike vector field $ \vec{X} $, the expression

holds true. These formulations do not depend on the specific matter source. However, if we consider an anisotropic fluid as the source, the energy-momentum tensor takes the form

where ρ denotes the energy density, $ u_q $ represents the four-velocity, $ n_q $ is the spacelike unit vector, and $ p_{\|} $ and $ p_{\perp} $ correspond to the pressures parallel and perpendicular to $ n_q $, respectively. For a static observer in this spacetime, the four-velocity primarily points in the t-direction (time coordinate) and has the following form:

The spacelike unit vector $ n_q $ is orthogonal to $ u_q $, and we may choose for it to point in the radial (in r) or angular directions θ. For simplicity, we consider $ n_q $ to be along the r-direction, yielding

The relations $ u_q u^q = -1 $, $ n_q n^q = 1 $, and $ u_q n^q = 0 $ are essential conditions that must be fulfilled.

To check each of the energy conditions rigorously, we use the components of the stress-energy tensor provided by Eqs. (9)−(11). We must verify that these components satisfy the NEC, WEC, SEC, and DEC. The NEC requires that for any null vector $ k^\mu $,

Consider a null vector $ k^\mu = (k^0, k^1, 0, 0) $ with the condition $ g_{\mu \nu} k^\mu k^\nu = 0 $. Using Eqs. (9)−(11),

For the NEC to hold, $ 3\alpha^{2} r^{2} - m' \geq 0 $. Hence, the NEC is satisfied if

The WEC requires $ T_{0}^{0} \geq 0 $. This implies $ \dfrac{1}{r^{2}} (3\alpha^{2} r^{2} - m') \geq 0 $, which leads to $ m' \leq 3\alpha^{2} r^{2} $, as in the NEC. For the timelike condition, for any timelike vector $ u^\mu $, the requirement is $ T_{\mu \nu} u^\mu u^\nu \geq 0 $. Using a vector along the t-direction, $ u^\mu = (u^0, 0, 0, 0) $, with $ g_{\mu \nu} u^\mu u^\nu = -1 $, we find

Therefore, the WEC holds if $ m' \leq 3\alpha^{2} r^{2} $, which is consistent with the NEC. The SEC is satisfied if

where $ T = g^{\mu \nu} T_{\mu \nu} $ denotes the trace of the stress-energy tensor. Because $ T_{0}^{0} = T_{1}^{1} $ and the other terms depend on the mass function and its derivatives, these terms consistently reduce. Given the previous results, if $ m' \leq 3\alpha^{2} r^{2} $, the SEC is also satisfied. The DEC requires $ T_{0}^{0} \geq 0 $, which is already ensured by the WEC. The vector $ T^\mu_{\nu} u^\nu $ must be causal, and because the $ T_{0}^{1} $ term, representing flux, is compatible with the causal structure, the DEC holds under the same condition $ m' \leq 3\alpha^{2} r^{2} $. All energy conditions (NEC, WEC, SEC, and DEC) are satisfied for this metric if

This inequality provides the constraint under which the energy conditions hold for our solution.

In this section, we analyze the accretion of dust onto a BS, represented by the energy-momentum tensor of dust given by

where ρ is the rest mass density of the dust, and $V^i$ is the four-velocity of the dust particles. To ensure that the four-velocity is normalized, we impose the condition

indicating that $V^i$ is a timelike vector. We define the four-velocity as

where u represents the spatial component of the velocity, and $f_0$ is a function characterizing the gravitational field of the BS. Using the normalization condition given by Eq. (26) and the expression for the four-velocity given by Eq. (27), we derive the components of the energy-momentum tensor as follows:

which represents the energy density of the dust, showing that it is affected by both the rest mass density ρ and the kinetic contribution from the dust's motion. For the momentum density, we have

Eq. (29) represents the flow of momentum owing to the dust particles, indicating how their motion contributes to the overall momentum density in the spacetime. Substituting the expressions for $T_{0}^{0}$ and $T_{0}^{1}$ into Eq. (13) gives

The above equation can be interpreted as describing how the BS mass evolves with accreting dust. The first term represents the mass contribution from dust at a reference radius $ r_0 $ near the apparent horizon, influenced by its energy density. The initial mass is denoted by $ m_{0} $, whereas the last term accounts for the dynamic effect of the dust's momentum flow over time on the mass.

We present graphs of Eq. (30), illustrating the evolution of mass with time and as a function of radial coordinates. The mass shows exponential growth over time while increasing linearly with the radial coordinate r. This indicates that the central object is steadily accreting mass, with the rate of accretion intensifying as time progresses. The mass distribution remains relatively uniform along the radius, suggesting an isotropic or homogeneously distributed accretion flow, particularly along the radial axis. These results are depicted in Figs. 1(a) and (b).

Figure 1.  (color online) Corrected mass function $ m(t, r) $ is plotted against time t for various fixed radial coordinates, illustrating different accretion behaviors across models. In Fig. 1(a), $ m(t, r) $ is shown for different radial values, where mass accretion increases with r. Fig. 1(b) demonstrates a nearly linear increase in $ m(t, r) $ over time, characteristic of a dust model, with higher accretion rates at larger radii. In Fig. 1(c), a non-linear growth in $ m(t, r) $ suggests additional pressure effects within a perfect fluid model, impacting the accretion rate. Fig. 1(d) presents $ m(t, r) $ at small radial values, where all models exhibit similar behaviors, likely due to dominant gravitational effects near the center. Finally, Fig. 1(e) shows the mass function over time for larger radii, where the radiative fluid model leads to accelerated accretion in outer regions owing to radiation pressure.

The energy-momentum tensor for a perfect fluid, which characterizes the distribution of energy and momentum in a fluid at rest or in motion, is expressed as

where ρ is the energy density, P is the pressure, $V^{i}$ is the fluid's 4-velocity, and $\delta_{j}^{i}$ is the Kronecker delta. Using Eqs. (27) and (31), we obtain the following specific components of the energy momentum tensor:

These components reveal how the energy density and pressure of the fluid contribute to the overall energy-momentum distribution in a dynamic setting, particularly under the influence of velocity u and the gravitational field characterized by $f_{0}$. Substituting Eqs. (32) and (33) into the mass function given by Eq. (13), we derive the corrected mass for the perfect fluid case as follows:

This equation for $m(t, r)$ shows the relationship between the fluid's dynamic properties and the gravitational field. The terms involving ρ and P indicate that the mass perceived by an observer can be significantly influenced by the fluid's velocity and pressure, revealing the rich structure of gravitational interactions in a perfect fluid context. Furthermore, the dependence on r suggests that mass accumulation is affected not just by the local energy density, but also by the spatial distribution of the fluid and its motion relative to the observer.

Figures 1(c) and 1(d) illustrate the variation in the corrected mass as a function of both coordinates t and r. The graphs show that the corrected mass increases with respect to t and r. Initially, as t increases, the mass increases rapidly, indicating a strong dependency on the temporal coordinate. For values of t close to zero, the mass still increases, although at a slower rate. Similarly, with respect to r, the corrected mass initially increases linearly, reflecting a direct proportionality. As r increases further, the various mass profiles begin to converge toward a single line, indicating that the radial dependency becomes less significant at larger radii. At smaller r values, the mass increase is more noticeable, with the slope of each line changing accordingly, suggesting a dynamic relationship between mass accumulation and radial distance in this model.

In the case of a radiative fluid, the equation of state is given by $\rho = 3P$, which characterizes it as a perfect fluid, where the energy density ρ is proportional to three times the pressure P. For such a fluid, the energy-momentum tensor takes the form

Using a similar procedure, we find the components of the energy-momentum tensor,

where u represents the radial velocity component, and $f_0$ is a function associated with the metric components, which typically represents gravitational effects on the fluid flow. Substituting these components into Eq. (13) yields the corrected mass function, which incorporates the effects of both radial motion and gravitational interactions:

In this case, the corrected mass exhibits a similar behavior to that in the previous cases. The mass increases with both r and t; initially, it increases linearly with r up to $r=1$, after which it increases exponentially for $r>1$. Along the time coordinate, the mass exhibits a linear increase for all values of t. This behavior can be observed in Figs. 1(e) and (f).

The position of the apparent horizon depends on the choice of the coordinate system. For metric (5), the location of the apparent horizon, $ r_{h} $, can be found as

Using the value of the corrected mass, we have

and further simplifying this equation using the binomial approximation produces the expression for the corrected apparent horizon for the line element (13),

The relationship between the apparent and event horizons, as depicted in Fig. 2, reveals their distinct but interrelated roles in the spacetime structure of BSs. The event horizon, shown in blue, serves as the ultimate causal boundary, beyond which no signals or matter can escape to infinity. It marks the point of no return, where causal connections with the external universe are severed. As time progresses, the event horizon moves inward, reflecting the increasing collapse of the BS spacetime. The apparent horizons (green dashed, orange dotted-dashed, and red dotted lines for dust, radiative fluid, and perfect fluid, respectively) represent the boundaries where light cones collapse, signaling the regions within the BS where the spacetime curvature becomes extreme. These apparent horizons are not static; they evolve over time, moving closer to the event horizon. The movement of these horizons reflects how different types of matter (dust, radiative fluid, and perfect fluid) interact with the gravitational field of the BS and influence the spacetime structure.

Figure 2.  (color online) Penrose diagram showing the evolution of the event and apparent horizons for different types of matter: dust (green dashed line), radiative fluid (orange dotted-dashed line), and perfect fluid (red dotted line). The event horizon is depicted as a blue curve, marking the boundary beyond which nothing can escape the gravitational pull of the BS. The apparent horizons for each type of matter move inward over time, with the perfect fluid causing the most rapid contraction. The singularity, marked by a black line at $r = 0$, represents the final collapse point at the center of the BS. The diagram provides the dynamic relationship between the event horizon, apparent horizons, and singularity in the context of the BS spacetime.

From a physical perspective, the dust case, represented by the green dashed line, reflects a more gradual and weak gravitational collapse owing to the absence of pressure in dust matter. The apparent horizon in this case moves slowly inward, as the dust contributes minimally to the curvature of spacetime. In contrast, the radiative fluid (orange dotted-dashed line) exhibits stronger gravitational effects owing to the pressure and energy flux associated with radiation, causing the apparent horizon to contract more rapidly. The perfect fluid (red dotted line), which has both energy density and pressure, produces the most intense gravitational effect, leading to the most rapid contraction of the apparent horizon. In all three cases, the apparent horizons approach the event horizon over time; however, they do not coincide, suggesting that the event horizon remains the definitive boundary of the BS, whereas the apparent horizons provide a time-dependent, matter-specific description of the BS internal structure. This demonstrates that the intricate relationship between matter distribution, spacetime curvature, and the dynamic evolution of BS horizons highlights the effect of matter accretion on the trapped region, with the apparent horizon adapting to local energy conditions, whereas the event horizon remains a global, coordinate-independent feature of the BS geometry.

We construct 3D plots in which the corrected apparent horizon, $ r_h $, is depicted in terms of time t and either the energy density ρ or pressure P for three distinct fluid models relevant to accretion. Figure 3(a) specifically illustrates the relationship between $ r_h $, t, and ρ for a dust model, where dust is characterized by a pressureless medium ($ P = 0 $). In this scenario, the horizon gradually shifts over time, indicating a steady, unresisted accretion process owing to the absence of internal pressure in the dust. As commonly discussed in the literature [16, 24], dust models serve as simplified representations of accretion, where the lack of opposing pressure allows the BH to gain mass and energy smoothly, leading to consistent linear expansion of the corrected horizon as accretion continues over time.

Figure 3.  (color online) Corrected apparent horizon $ r_h $ plotted against time t, energy density ρ, and pressure P for different accretion models, each providing distinct insights into BS growth. Fig. 3(a) depicts the dust model, representing the simplest case with a steady accretion rate owing to the absence of pressure. Figs. 3(b) and 3(c) depict the perfect fluid model, where pressure moderates accretion, leading to back-reaction effects on the system. Finally, Fig. 3(d) shows the radiative fluid model, where high radiation pressure induces complex interactions, significantly affecting accretion rates and stability through energy transfer and radiation dynamics.

Figures 3(b) and 3(c) show the corrected horizon $ r_h $ with respect to time t and pressure P or ρ in a perfect fluid model, which has both energy density and pressure (non-zero P). The presence of pressure introduces resistance to the accretion process, affecting the evolution of the event horizon. In GR, the presence of pressure leads to a more complex interaction with the BH's gravitational field. Consequently, as shown in the graphs, $ r_h $ exhibits a steeper growth with time when compared to that of the dust model, owing to the additional energy-momentum components associated with pressure. As a source of dynamic back-reaction, the pressure of the fluid influences the corrected apparent horizon of the BS. For instance, for higher pressures, the back-reaction effect can slow the rate of accretion owing to repulsive forces that counteract the BS gravitational pull.

Figure 3(d) shows the case for the radiative fluid, where $ r_h $ is plotted with respect to t and P. Radiative fluids, which are often represented by radiation pressure, involve particles moving at nearly the speed of light. This high-speed inflow creates a different profile for the growth of the corrected apparent horizon. The accretion rate can vary significantly because radiation exerts a strong pressure, potentially heating the surrounding medium and affecting the accretion dynamics. Intuitively, radiation pressure can significantly limit or even reverse the infall of matter under certain conditions. In this model, $ r_h $ exhibits a steep slope, suggesting that the horizon size could either increase rapidly or stabilize depending on the balance between gravitational attraction and radiation pressure.

To find the expression for entropy S, we use the Bekenstein-Hawking entropy formula [30], which is given by

where A is the area of the BS horizon, $k_B$ is the Boltzmann constant, and $l_p$ is the Planck length. For simplicity, we often use units of $k_B = 1$ and $l_p = 1$, such that the entropy is directly proportional to the horizon area,

and the area of the event horizon can be found as

If we assume that the BS extends infinitely along the z-direction, integration over z would contribute a factor proportional to the length of the z-direction, which is infinite. This length is denoted as $L_z$. Thus, the infinite extent of the z-direction naturally introduces $L_z$ as a scaling factor in the calculations. Specifically, the line element $5$ describes a BS with a cylindrical horizon, assuming that the coordinate z spans the entire real line, i.e., $ -\infty < z < \infty $. However, if the z-coordinate is restricted to the interval $0 \leq z < 2\pi$, the configuration corresponds to a closed BS with a toroidal horizon, where the z-coordinate is periodic, and the horizon takes the shape of a torus [31]. Therefore,

Substituting Eqs. (43) and (44) into (42), we have

$L_z$ is the length in the z direction, which can be treated as a constant depending on the physical scenario.

In Fig. 4(a), entropy is plotted as a function of time t and density ρ for dust. The surface illustrates an increase in entropy with both time and density. This suggests that, over time, accumulating dust leads to higher density, which contributes to gravitational potential and, consequently, raises entropy. This outcome aligns with the second law of thermodynamics, which states that entropy in an isolated system generally increases over time owing to the increasing matter density.

Figure 4.  (color online) Corrected entropy S as a function of time t, energy density ρ, and pressure P for various accretion models, highlighting thermodynamic principles in BS growth. Fig. 4(a) illustrates the dust model, where entropy increases primarily owing to increasing density, with minimal internal interactions. Figs. 4(b) and 4(c) display the perfect fluid model, where pressure contributes to entropy production by enhancing internal energy exchanges, resulting in greater entropy as pressure and density increase. Fig. 4(d) depicts the radiative fluid model, showing rapid entropy growth owing to radiative energy dissipation, which intensifies entropy with increasing pressure. These trends reveal how different matter types and energy interactions influence entropy evolution, providing insights into accretion dynamics and back-reaction effects in astrophysical contexts.

Figures 4(b) and 4(c) show the corrected entropy for the perfect fluid model, depicted as a function of in Fig. 4(b) and as a function of pressure P and time t in Fig. 4(c). Figure 4(b) exhibits a similar trend to the case of the dust model, with entropy increasing over time and with density, reflecting the added effects of pressure within the fluid. This pressure contributes to the gravitational potential and energy density, thereby enhancing entropy generation.

In Fig. 4(c), the relationship between entropy and pressure P and time t reveals that higher pressures correspond to greater entropy increases. This pattern highlights the role of pressure in energy exchanges within the fluid, enhancing entropy through compressibility and fluid interactions, thus amplifying entropy production in perfect fluids. Figure 4(d) presents the corrected entropy for the radiative fluid model, shown as a function of pressure P and time t. Here, entropy increases sharply with both pressure and time, underscoring the significant influence of radiation. Radiative fluids dissipate energy through radiation, leading to a rapid entropy increase. The steep gradient indicates that entropy in radiative fluids is highly responsive to changes in pressure and time, showing a more pronounced increase compared to that in the dust and perfect fluid models.

To calculate the temperature, we use the concept of Hawking radiation, which is particularly relevant for BHs. The temperature can be derived from the metric's geometry, specifically focusing on the surface gravity of the event horizon. The surface gravity κ at the event horizon is given by

Evaluating at the horizon $ r = r_h $, where $ g_{tt} = 0 $, we obtain

The corresponding temperature T is then

Thus, the temperature T is basically related to the location of the correct apparent horizon $ r_h $,

The temperature T of the corrected apparent horizon $ r_h $ for three different accretion models – dust, perfect fluid, and radiative fluid – is discussed graphically. In BH thermodynamics, the temperature of the horizon is typically inversely related to the horizon radius; as $ r_h $ increases, T tends to decrease.

Figure 5(a) shows the temperature for the dust fluid model, where the temperature T of the corrected apparent horizon decreases sharply as $ r_h $ increases, eventually stabilizing at a low value. With dust lacking pressure, the accretion process is straightforward, leading to a rapid expansion of $ r_h $ and a corresponding decrease in T. This trend reflects the cooling effect owing to the growing horizon, with temperature inversely scaling with horizon radius. In Fig. 5(b), for the perfect fluid model, a similar pattern is observed as T decreases with increasing $ r_h $, although the gradient is slightly different from that of the dust model. Here, the presence of pressure moderates the rate of accretion, causing a more gradual increase in $ r_h $. This moderation results in a smoother cooling effect, where the temperature decrease is less abrupt. The pressure in a perfect fluid acts as a resistive force, influencing the thermal behavior of the BS by limiting rapid horizon expansion.

Figure 5.  (color online) Corrected temperature T as a function of the apparent horizon $ r_h $ for different accretion models, showing how each model impacts temperature evolution. Fig. 5(a) illustrates the dust model, where rapid horizon growth leads to a quick temperature drop. Fig. 5(b) presents the perfect fluid model, in which pressure moderates horizon expansion, resulting in a more gradual temperature decrease. Fig. 5(c) shows the radiative fluid model, where radiation pressure causes slower horizon expansion, maintaining a relatively higher temperature at larger $ r_h $ values. These variations reveal how different physical factors influence temperature dynamics in BS growth.

Finally, Fig. 5(c) shows the case of the radiative fluid model. The temperature T also decreases with $ r_h $, but the curve differs significantly from that of the previous models, showing a slower decline at larger $ r_h $ values. The high-energy particles and radiation pressure in radiative fluids create a unique thermodynamic balance, affecting the BH properties in a distinct way. This slower temperature decrease aligns with findings in the literature [4, 24], where radiative fluids are often associated with steady-state horizon expansion owing to radiation’s outward pressure, which slows horizon growth and maintains a higher temperature for an extended period compared to those of other models.

In this section, we examine accretion with back-reaction effects for a charged BS. Cai and Zhang [15] derived cylindrically symmetric charged solutions to the EMFEs in the presence of a negative cosmological constant. The general form of the static charged BS metric in anti-de-Sitter space, where $\alpha^{2} = -\dfrac{\Lambda}{3} > 0$, is given as

Note that metric (50) represents the solution in the zeroth-order approximation. To calculate the back-reaction, we use Eq. (3). We find the perturbed EMFEs for Eq. (50) using the same procedure as in the case of the static uncharged BS. The corrected line element for the charged BS is then given by

The components of the energy-momentum tensor for the charged BS are obtained by replacing the following:

Using the CAS Maple, we compute the components of the Einstein tensor, which are provided in Appendix A. The field equations for the charged BS are given in Appendix B. By solving them through successive integration and substitution, we obtain the correct mass, or running mass, for the charged BS, which is given by

Note that Eq. (53) aligns with Eq. (12) in the first-order approximation, where charge-related terms involving q cancel out on both sides. This outcome is significant, as it indicates that, under the perturbative framework, the corrected mass $m(t,r)$ remains consistent across both scenarios, unaffected by charge contributions.

Assuming that the energy-momentum tensor varies slowly with the radial coordinate, Eq. (53) in the vicinity of the BS horizon is

The above equation closely resembles Eq. (13), further supporting the conclusion that charge does not influence the running mass in this perturbative scheme. The fact that the charge terms cancel out and do not affect the mass $m(t, r)$ underscores the robustness of the perturbative approach. This implies that the mass evolution near the BS horizon primarily depends on the matter content, as represented by the energy-momentum tensor, rather than on the presence of charge. This result shows that for slowly varying energy-momentum tensors, the impact of charge on mass dynamics is minimal, suggesting that gravitational effects dominate the system's evolution over electromagnetic influences in this approximation.

To discuss the thermodynamics in this case, we must first find the corrected apparent horizon for the charged BS. From Eq. (51), we have

The approximate solution is

This expression shows the leading term and a correction term due to the $ q^2 $-charge term. The first term, $\left( {4m(t, r)}/{\alpha^2} \right)^{1/3}$, represents the dominant radius, whereas the second term is a small correction.

The charge q primarily contributes a stabilizing outward-pushing correction to the apparent horizon, which is most significant at smaller radii (r). This effect counteracts gravitational collapse, slows the evolution of the apparent horizon, and delays its convergence to the event horizon. In all three cases, the charge acts to reduce the effective gravitational pull, leading to an apparent horizon that is slightly larger and less dynamic than in uncharged cases. However, at larger radii or as $t \to \infty$, the influence of charge diminishes, and the accretion dynamics (mass, pressure, or flux) dominate the evolution of the horizons.

We plot the apparent horizon $ r_h $ of a charged BS over time t for varying charge q values. The charge affects the gravitational pull, influencing the horizon's evolution. In Fig. 6(a), $ r_h $ increases linearly with time, with a slight increase in growth rate for higher q. The charge enhances gravitational attraction, accelerating accretion and horizon expansion. Unlike typical charged BSs, in which electrostatic repulsion counteracts gravity, the dust model produces a steady, near-uniform growth in the horizon.

Figure 6.  (color online) Corrected horizon $ r_h $ plotted against time t for various charge values q across different accretion models, highlighting the influence of charge on horizon growth. Fig. 6(a) illustrates the dust model, where the horizon grows linearly, and increasing charge causes moderate acceleration in growth. Fig. 6(b) displays the perfect fluid model, where the presence of both charge and pressure leads to non-linear horizon growth with significant acceleration. Fig. 6(c) represents the radiative fluid model, showing explosive horizon growth as q increases, driven by radiation pressure and charge effects. These results demonstrate that charge q has a progressively stronger impact on horizon growth across the models, with radiative fluids being most sensitive. This reflects the combined effects of charge and fluid type on accretion dynamics and horizon size in a charged BS scenario.

Figure 6(b) shows the perfect fluid model, where $ r_h $ increases non-linearly over time, especially for higher q. The fluid’s pressure adds dynamics to the accretion rate, with charge significantly accelerating horizon growth as both electromagnetic and fluid pressures drive expansion.

In Fig. 6(c), the radiative fluid model depicts a sharply increasing $ r_h $ beyond a certain threshold, with high q yielding near-exponential horizon growth. Radiative fluids, with their high-energy particles, exhibit strong charge sensitivity, where larger charges promote rapid expansion after an initial steady phase owing to increased radiation pressure alongside gravitational attraction.

To discuss the thermodynamics, we must first identify the form of the relevant parameters. The line element (51) resembles that of a charged BH, which is likely a generalization of the Reissner-Nordström solution in cylindrical symmetry. Using Eq. (46), the surface gravity κ at the event horizon $r = r_h$ is given by

Consequently, the expression for corrected temperature is given as

This shows how mass and charge influence the thermal behavior of charged BSs. The mass term decreases the temperature as the corrected apparent horizon grows, whereas the charge introduces further cooling, stabilizing the BS. The scaling parameter α modifies the temperature linearly with the horizon radius, revealing back-reaction effects on its thermal properties.

To find the expression for entropy, use Eqs. (42) and (43):

This expression is similar to Eq. (45) and shows the entropy dependence on the horizon radius $r_h$.

The study examines the back-reaction effects of an accreting fluid on a BS using a perturbative approach. Two primary methods are typically considered in accretion analysis. The first neglects back-reaction, which is valid when the mass of the accreting matter is negligible compared to that of the BH. The second approach includes the back-reaction, leading to complete solutions to the EFEs and matter equations. In this study, we consider a BS and apply a perturbative scheme to approximate the EFEs and derive an expression for the corrected mass of the BS near the horizon, as shown in Eq. (13).

We further analyze the energy conditions for our solution, demonstrating that it satisfies all energy conditions provided that the running mass m fulfills the condition $ m' \leq 3\alpha^{2} r^{2} $, where the prime denotes differentiation with respect to r. This indicates that the energy conditions and EFEs are satisfied, confirming the self-consistency of our solution.

Accretion with back-reaction is explored for three matter models: dust, perfect fluid, and radiative fluid. In each case, we graphically show that the running or corrected mass $m(t, r)$ increases over time t and with radial distance r, emphasizing the impact of back-reaction in each model. The dust model shows steady, linear mass growth, especially in outer regions. The perfect fluid model exhibits nonlinear accretion owing to pressure effects, whereas the radiative fluid model accelerates mass accumulation at larger radii owing to radiation pressure. In central regions, differences among models are less pronounced, as gravitational effects dominate uniformly. Overall, outer regions exhibit a faster mass increase, driven by fluid-specific dynamics and accretion properties, as depicted in Figs. 1(a)−(f).

We use the corrected metrics given by Eqs. (5) and (13) to calculate the corrected apparent horizon owing to the back-reaction effect. We demonstrate that the corrected apparent horizon $ r_h $ varies with time t and energy density ρ or pressure P for the dust, perfect fluid, and radiative fluid models in accretion. In the dust model, which has no internal pressure, accretion occurs steadily. In contrast, the perfect fluid model shows a steeper growth in the horizon owing to pressure resistance. For the radiative fluid case, radiation pressure can either accelerate or stabilize horizon expansion, depending on the balance with gravitational forces. These are illustrated graphically in Figs. 3(a)−(d). We present a Carter-Penrose diagram to illustrate the causal structure and relationship between the corrected apparent horizon and the event horizon for all three distinct cases. Our analysis shows that the accretion of matter plays a crucial role in modifying the spacetime geometry. Specifically, corrections arising from dust, radiative fluid, and the energy density and pressure of a perfect fluid alter the location of the apparent horizon. Furthermore, we discuss the thermodynamics of the corrected line element. Using the corrected apparent horizon, we calculate and plot the expressions for entropy and temperature for each model. The increase in entropy across all models is consistent with thermodynamic principles, whereas rapid horizon growth leads to a sharp decrease in temperature. The graphs of entropy and temperature for all three cases are shown in Figs. 4 and 5, respectively.

We also extend our analysis to a charged BS and apply the perturbation technique to examine its properties. The expression for the running mass $m(t, r)$ of the charged BS is identical to that of the uncharged BS, suggesting that charge does not influence the mass in this perturbative technique. Furthermore, we derive a corrected expression for the apparent horizon in the charged case and explore the impact of different charge values q on this correction. Additionally, we investigate the thermodynamic properties of the charged BS. Although the entropy remains the same for both the charged and uncharged cases, we observe a slight variation in the temperature owing to the different surface gravities in each case.

The running mass $ m(t, r) $ is associated with the asymptotically flat Arnowitt-Deser-Misner (ADM) mass, which measures the total mass-energy content of a spacetime. The ADM mass incorporates both gravitational energy and matter fields. For a spacetime to be asymptotically flat, the metric must approach the Minkowski metric at spatial infinity. In general, the ADM mass can be expressed in terms of the stress-energy tensor $ T^{\mu}_{\nu} $. For a spherically symmetric distribution of matter, the ADM mass is given by

The asymptotic behavior of the mass function relates to the ADM mass as $ r \to \infty $. If the energy density $ T_0^0 $ and pressure $ T_0^1 $ vanish or become negligible at infinity, the mass function approaches a constant value. At large r, the dominant contributions typically arise from the $ m_0 $ and $ \alpha^2 r^3 $ terms, assuming that the integral converges. For $ m(t, r) $ to represent the ADM mass, the spacetime must satisfy the condition of asymptotic flatness,

It is also worth considering that the perturbative technique can be extended to rotating BSs and rotating charged BSs. The perturbed solutions, along with the effects of various energy-momentum tensors on the corrected masses, are of significant interest.

The Einstein tensors for the considered spacetime are given by

In this appendix, we present the components of the energy-momentum tensor $ T^{\mu}_{\nu} $. The following relations are derived from the EFEs. The energy-momentum tensor components are given by