Chandrasekhar extensively analyzed the time-like and null-geodesic features of the Schwarzschild spacetime in his book [1]. In addition, he has examined the orbital configurations of both the confined and unconfined trajectories using graphical representations. In addition, the authors in [2] have examined the geodesic structures of the Schwarzschild anti-de Sitter spacetime. The researchers assessed both radial and non-radial paths for time-like and null geodesics. Additionally, they have demonstrated that the geodesic structures of this black hole exhibit distinct forms of motion that are not permitted by the Schwarzschild spacetime. The geometric framework of the Schwarzschild spacetime is also examined in [3]. The Jacobi metric for time-like geodesics in static spacetimes has been examined in the reference [4]. They have demonstrated that the unrestricted movement of large particles in stationary spacetimes is determined by the geodesics of a Riemannian metric that depends on the particle’s energy. This metric is similar to Jacobi’s metric in classical dynamics. When the mass of an object approaches zero, Jacobi’s metric becomes identical to the Fermat or optical metric, which does not depend on energy. In addition, they have provided a detailed account of the characteristics of the Jacobi metric pertaining to the motion of heavy particles beyond the event horizon of a Schwarzschild black hole. The authors of [5] derived the Jacobi metric for different stationary metrics and developed the Jacobi-Maupertuis metric for time-dependent metrics by using the Eisenhart-Duval lift [6,7]. The authors in [8] have documented the remarkable characteristics of the time-like geodesic structure when dark energy is present in an emergent gravity framework, specifically for the Barriola-Vilenkin metric [9]. The K-essence emergent gravity metric is precisely correlated with the Barriola-Vilenkin (BV) metric for the Schwarzschild background, specifically for a certain form of K-essence scalar field [10]. The researchers have analyzed the various paths that time-like geodesics can take in the presence of dark energy in the Barriola-Vilenkin spacetime [8], which is equivalent to the Schwarzschild spacetime in terms of its fundamental structure. However, the permissible ranges for the maximum and minimum distances from the central object are significantly distinct. For a constant dark energy density, the orbits, both bound and unbound, are graphed.

In 1951, Vaidya proposed the first relativistic line element that properly represented the spacetime of a conceivable star [11]. It extended the specific solution of Schwarzschild by depicting the emission of radiation for a mass that is not in a static state. The Schwarzschild solution describes the geometry of spacetime around a spherically symmetric, non-rotating, black object with a constant mass. Therefore, it is clear that the model is incapable of accurately depicting spacetime outside the confines of a star. The solution proposed by Vaidya [11], known as the Vaidya spacetime or the radiating Schwarzschild metric, was introduced as a possible explanatory framework. The main distinction between the two metrics is that the Vaidya metric adds a time-dependent mass parameter, whereas the Schwarzschild metric uses a constant mass value. As a result, the spacetime in the Vaidya metric evolves with time. The Vaidya metric is primarily used to investigate gravitational collapse. The occurrence of gravitational collapse is widely acknowledged in the disciplines of general relativity and astrophysics, as demonstrated by the research conducted by Joshi et al. [12,13,14,15,16,17,18,21]. It plays a vital role in understanding several astrophysical aspects of our cosmos. The phenomenon of gravitational collapse provides useful insights into several elements of astronomy, including the evolution of structures, the features of stars, the genesis of black holes, and the construction of white dwarfs or neutron stars, among other events. Gravitational collapse refers to the phenomenon in which a star collapses as a result of its mass. The outcome of this collapse might vary depending on the exact beginning mass conditions, leading to distinct stages of collapse. Papapetrou [19] was the first to demonstrate that the solution of a null dust fluid with spherical symmetry in gravitational collapse can lead to the creation of naked singularities. This statement presents a counterexample of the cosmic censorship hypothesis (CCH) as proposed by Penrose [20]. The authors in [14,22] have provided a detailed account of the causal paths that connect the singularities in the continuing Vaidya scenario. Furthermore, a comprehensive classification of the non-spacelike geodesics that link the naked singularity in the past is presented, offering a rather thorough discussion of the restrictions involved. It is subsequently demonstrated to be a robust curvature singularity in a more significant manner.

The Vaidya solution, as a generalization, encompasses all the established solutions of Einstein’s field equations that include a mix of Type-I and Type-II matter fields [23,24,25,26,27,28]. The composition of this work is attributed to Husain [29] and Wang & Wu [30]. The extension of the Vaidya solution is sometimes referred to as the generalized Vaidya spacetime. The work performed in [31] examines the gravitational collapse of the generalized Vaidya spacetime within the framework of the cosmic censorship theory. They demonstrated that the categories of generalized Vaidya mass functions had emerged in the situation, suggesting the end of collapse with a locally visible central singularity. The authors computed the magnitude of these singularities. A comprehensive mathematical framework was created to examine the requirements for the mass function for non-spacelike geodesics going towards the future to end at the singularity in the past. Furthermore, they demonstrated that, when considering a certain generalized Vaidya mass function, the ultimate outcome of the collapse can be precisely defined as either a black hole or a naked singularity. The work by Patil [32] examines the phenomenon of gravitational collapse in higher dimensions within the context of the charged-Vaidya spacetime. It has been demonstrated that singularities occur in a charged null fluid in a higher dimension. These singularities consistently lack any form of covering, hence contradicting the strong CCH. This idea does not specifically pertain to weak cosmic censorship. The Vaidya metric has received significant attention in scholarly research, with several major contributions to our comprehension of this subject. The authors of the study [33] examined the geometric properties of Vaidya’s spacetime while considering a white hole that undergoes a decrease in mass. They found that the white hole can either stabilize and transform into a black hole within a limited or indefinite amount of time, or entirely evaporate. The researchers have focused specifically on the scenario of total evaporation over an indefinite period of time. They have successfully demonstrated the presence of an asymptotic light-like singularity in the conformal curvature, which connects both the past space-like singularity and the future time-like infinity. Vertogradov [34] conducted a study on the structure of the generalized Vaidya spacetime, specifically focusing on the case when the matter field of type-II follows the equation of state $P=\rho$. The findings of the study revealed the presence of an eternal naked singularity in this spacetime, which meets all energy conditions. Once formed, the singularity will remain perpetually uncovered by the apparent horizon. Nevertheless, the formation of the apparent horizon leads to the emergence of a white hole. Solanki et al. [35] have derived precise mathematical equations that describe the changes in the photon sphere and the angular radius of the shadow in a certain Vaidya spacetime. The mass function $m\left(v\right)$ has been seen as a function of time that either increases or decreases linearly. The initial scenario can function as a basic representation of a black hole that is accumulating matter, whereas the subsequent scenario can be seen as an illustration of a black hole that is emitting radiation, as theorized by Hawking.

In the realm of K-essence geometry, Manna et al. [36] were the first to establish a link between K-essence geometry and Vaidya spacetime. They achieved this by introducing a new definition of the generalized Vaidya mass function, which directly depends on the kinetic energy of the K-essence scalar field. Subsequently, Manna et al. [37] demonstrated that the K-essence emergent gravity metric bears a strong resemblance to the recently found generalized Vaidya metrics for the collapse of a null fluid. This similarity arises from the presence of a k-essence emergent mass function. Notably, Manna’s analysis exclusively considers the K-essence scalar field as a function of either the advanced or the retarded time. The recently developed K-essence model, known as the K-essence emergent Vaidya spacetime, has successfully met all the necessary energy conditions. The presence of the centrally exposed singularity and the intensity and stability of the singularities in the K-essence emergent Vaidya metric yield intriguing results in their research. The evaporation of the dynamical horizon with the Hawking temperature in the K-essence Vaidya Schwarzschild spacetime was investigated by Manna et al. in [38]. This study uses the dynamical horizon equation to quantify the reduction in mass caused by Hawking radiation. Additionally, the tunneling formalism, namely the Hamilton-Jacobi technique, is utilized to compute the Hawking temperature. In addition, Sawayama’s revised explanation of the dynamical horizon [39] is utilized to demonstrate that the results obtained differ from the conventional Vaidya spacetime geometry. The authors have established by analytical measures that the mass of the black hole, denoted as $m(v,r)$, in the K-essence emergent Schwarzschild-Vaidya spacetime, consistently decreases over time but does not fully evaporate.

The K-essence theory is a scalar field model that deviates from the canonical form. In this theory, the dominant energy component of the field is its kinetic energy, rather than its potential energy. This concept and related others have been extensively studied by several researchers [40,41,42,43,44,45,46,47,48,49,50,51]. The distinctions between the K-essence theory employing a non-canonical Lagrangian and the relativistic field theories utilizing a canonical Lagrangian are found in the sophisticated dynamical solutions of the K-essence equation of motion. These solutions not only spontaneously violate Lorentz invariance but also alter the metric for the perturbations around them. The disturbances propagate in the emergent or analogous curved spacetime, characterized by a metric distinct from the gravitational metric. The non-canonical Lagrangian may be expressed as $\mathcal{L}\left(X\right)=-V\left(\varphi \right)F\left(X\right)$, where, $X=\frac{1}{2}{g}_{\mu \nu }{\nabla }^{\mu }\varphi {\nabla }^{\nu }\varphi$, $\varphi$ is the K-essence scalar field, $V\left(\varphi \right)$ is the potential term. An alternate form of the Lagrangian, as described by Tian [52], may be represented as $\mathcal{L}=[1+f\left(y\right)]X+[1+g\left(y\right)]V_{exp}$, where $V_{exp}=V_{0}exp(-\lambda \varphi )$, $V_0$ and $\lambda$ are constants, $y=X/V_{exp}$, and $f\left(y\right)$ and $g\left(y\right)$ are arbitrary functions. The functions $f\left(y\right)$ and $g\left(y\right)$ are unrestricted and can have any form. Furthermore, it is important to mention that there exist examples of K-essence theories that are not minimally linked, as mentioned in the Refs. [53,54,55]. Nevertheless, this article only addresses the minimally coupled K-essence theory, as investigated by the Refs. [40,41,42,43,44,45,46,47,48,51]. In a general sense, the Lagrangian has the capacity to depend on any functions of $\varphi$ and X. The K-essence theory offers the benefit of circumventing both the fine-tuning problem and the coincidence problem [56] of the current universe. Additionally, it generates the negative pressure required for the universe’s acceleration only through the kinetic energy of the field. The kinetic term of the field dominates over the potential term. The article [45] presents attractor solutions where the dynamics of the cosmos are governed by the scalar field of the models. During the radiation-dominated phase, the K-essence field mimicked the equation of state of radiation and had a constant ratio to the radiation density. The K-essence field was unable to replicate the dust-like equation of state (EOS) due to dynamical limits during the time dominated by dust. However, it rapidly reduced its energy value by many orders of magnitude and eventually reached a constant value. Subsequently, over a period approximately equivalent to the current age of the universe, the density of matter was diminished by the K-essence field, leading to the commencement of cosmic acceleration. The equation of state (EOS) of the K-essence theory ultimately converges to a value within the range of 0 to -1. Although in theory, it has the potential to extend beyond $-1$. Another intriguing aspect of the K-essence idea is its potential to generate a type of dark energy where the speed of sound is consistently slower than that of light. This feature may mitigate the cosmic microwave background (CMB) disruptions on large angular scales [57,58,59]. In this specific situation, Manna et al. [8,10,36,37,38,63,64,65,66,67,68] have developed a fascinating emergent gravity metric referred to as ${\overline{G}}_{\mu \nu }$. This metric possesses distinct attributes in contrast to the standard gravitational metric $g_{\mu \nu }$ and is derived from the notions of the Dirac-Born-Infeld (DBI) type action, as outlined in the works [71,72,73,74]. Dirac et al. [74] proposed a non-canonical Lagrangian in order to eliminate the infinite self-energy of the electron, as described in their work. The specific reasons and objectives for selecting the non-canonical theory, such as the K-essence theory, may be found in the Refs.[69,70]. The Planck collaborations’ findings, as shown in Refs. [60,61,62], have examined the empirical evidence supporting the concept of K-essence with a DBI-type non-canonical Lagrangian, along with other modified theories. Furthermore, it has been noted that the K-essence theory may be applied in a model that combines dark energy and dark matter [8,10,46,63,64,65,66], as well as from a purely gravitational perspective [36,37,38,67,68].

This article is organized as follows: In Section 2, we provided a concise explanation of the K-essence geometry and its connection to the conventional generalized Vaidya spacetime, which leads to the construction of a new generalized K-essence Vaidya spacetime. Section 3 offers a comprehensive analysis of the geodesic structures observed in the generalized K-essence Vaidya spacetime. This analysis considers two forms of mass function while ensuring that the condition on the kinetic energy of the K-essence scalar field is maintained. This section also provides a detailed analysis of the radial and non-radial geodesics used to examine the structure of time-like and null geodesics in the given spacetime. This is achieved by solving the Euler-Lagrange equations. The graphical and numerical analysis is also done in this section. In Section 4, we have wrapped up both the discussion and conclusion.

This section offers a short introduction to the geometry of K-essence and the generalized Vaidya spacetime. Initially, we present a brief summary of the geometric aspects related to the K-essence, as extensively explored in many scholarly references [40,41,42,43,44,45,46,47,48,51]. The action performed by this geometry is where the expression $X=\frac{1}{2}{g}^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi$ represents the canonical kinetic term, whereas $\mathcal{L}(X,\varphi )$ denotes the non-canonical Lagrangian. In this scenario, the conventional gravitational metric $g_{\mu \nu }$ has formed a minimum coupling with the K-essence scalar field ($\varphi$).

The energy-momentum tensor that corresponds solely to the K-essence scalar field is: where ${\mathcal{L}}_{\mathrm{X}}=\frac{d\mathcal{L}}{dX},{\mathcal{L}}_{\mathrm{X}X}=\frac{d^2\mathcal{L}}{d{X}^2},{\mathcal{L}}_{\varphi }=\frac{d\mathcal{L}}{d\varphi }$ and ${\nabla }_{\mu }$ is the covariant derivative defined with respect to the gravitational metric $g_{\mu \nu }$.

The equation of motion (EOM) for the K-essence scalar field is where with $1+\frac{2X{\mathcal{L}}_{XX}}{{\mathcal{L}}_X}>0$ and $c_s^2(X,\varphi )\equiv {(1+2X\frac{{\mathcal{L}}_{XX}}{{\mathcal{L}}_X})}^{-1}$.

Following [8,10,36,37,63], the inverse metric can be written as

The Eqs. (4), (5) have physical relevance when ${\mathcal{L}}_X$ is nonzero, assuming a positive definite $c_s^2$. Eq. (5) states that the emergent metric, represented as ${\overline{G}}_{\mu \nu }$, differs in its conformal properties from the metric $g_{\mu \nu }$ when considering non-trivial configurations of the scalar field $\varphi$. Like canonical scalar fields, the variable $\varphi$ exhibits diverse local causal structural properties. It also differs from those that are defined using $g_{\mu \nu }$. The EOM, as stated in Eq. (3), is valid even when taking into account the implicit relationship between L and $\varphi$. Then the EOM Eq. (3) is:

This study addresses the Dirac-Born-Infeld (DBI) type non-canonical Lagrangian, which is represented as $\mathcal{L}(X,\varphi )\equiv \mathcal{L}\left(X\right)$ [8,10,36,69,70,71,72,73，74]:

The K-essence paradigm posits that the prevalence of kinetic energy over potential energy results in the exclusion of the potential term in the Lagrangian equation (7) [69,70,71]. The squared speed of sound, represented as $c_s^2$, is determined by the expression $(1-2X)$. Therefore, the Eq. (5) for the effective emergent metric is expressed as: since $\varphi$ is a scalar.

The Christoffel symbol corresponding to the emergent gravity metric given by Eq. (8) can be written as [8,10,69,70]: where ${\Gamma }_{\mu \nu }^{\alpha }$ is the usual Christoffel symbol associated with the gravitational metric $g_{\mu \nu }$.

Hence, the geodesic equation governing the K-essence geometry may be expressed as: where $\lambda$ is an affine parameter.

The covariant derivative $D_{\mu }$ [41,69,70] linked with the emergent metric ${\overline{G}}_{\mu \nu }$$(D_{\alpha }{\overline{G}}^{\alpha \beta }=0)$ gives and the inverse emergent metric is ${\overline{G}}^{\mu \nu }$ such as ${\overline{G}}_{\mu \lambda }{\overline{G}}^{\lambda \nu }={\delta }_{\mu }^{\nu }$.

Therefore, considering the extensive behavior that defines the dynamics of K-essence and general relativity [42,69,70], the Emergent Einstein’s Equation (EEE) may be formulated as: where $\kappa =8\pi G$ is constant, ${\overline{R}}_{\mu \nu }$ is Ricci tensor and $\overline{R}(={\overline{R}}_{\mu \nu }{\overline{G}}^{\mu \nu })$ is the Ricci scalar. Moreover, the energy-momentum tensor ${\overline{T}}_{\mu \nu }$ is linked to this emergent spacetime.

Now, we will provide a concise overview of the K-essence emergent generalized Vaidya spacetime. In the cited work [37], the author has introduced the concept of K-essence emergent generalized Vaidya spacetime. This framework considers the background gravitational metric to be the typical generalized Vaidya metric [29,30], while also satisfying the necessary energy requirements. The line element for the emergent generalized Vaidya metric in K-essence theory is as follows: with $d{\Omega }^2=d{\theta }^2+d{\Phi }^2$.

They have defined the K-essence emergent Vaidya mass function where $m(t,r)$ is the usual generalized Vaidya mass function and ${\varphi }_t^2$ (${\varphi }_t=\frac{\partial \varphi }{\partial t}$) is the non-zero kinetic energy of the K-essence scalar field.

The above-mentioned mass function pertains to the gravitational energy associated with the K-essence emergent gravity within a specified radius r. Here, we substitute the Eddington advanced time coordinate with the conventional time coordinate, without any loss of generality, denoted as $v\to t$. In this study [37], the author has examined the effective K-essence emergent metric, as denoted by Eq. (8). Additionally, the author has calculated all the components of the EEE (Eq. (12)) and the necessary energy conditions. It is important to mention that the assumption about $\varphi$ contradicts local Lorentz invariance since, in general, spherical symmetry only requires $\varphi \left(x\right)=\varphi (t,r)$. The inclusion of the assumption of the independence of $\varphi$, denoted as $\varphi (t,r)=\varphi \left(t\right)$, suggests that beyond this specific frame selection, a spherically symmetric $\varphi$ is indeed a function of both t and r. The K-essence theory permits the occurrence of Lorentz violation due to the fact that the dynamic solutions of the K-essence equation of motion spontaneously break Lorentz invariance and alter the metric for the perturbations around these solutions.

Furthermore, the authors in [36] have successfully established a connection between the geometry of K-essence and the Vaidya spacetime. The researchers have developed a model of the Vaidya spacetime with generalized K-essence, which takes into account any spherically symmetric static black hole as the underlying spacetime. The line element of the new geometry is (using Eq. (8): gives the mass function

In this article [36], the authors also have calculated all the components of EEE and required energy conditions. If we consider $f\left(r\right)=(1-2M/r)$, i.e., the background physical spacetime is Schwarzschild spacetime, the mass function may be expressed as:

Again, if we select the function $f\left(r\right)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$, Q represents the charge of the Reissner-Nordström (RN) black hole in the physical spacetime. In this case, the related mass function [36] is modified as

It is important to mention that the values of ${\varphi }_t^2$ must be between 0 and 1. Otherwise, the metric (13) and (15) cannot be specified properly, and the presence of a dynamical horizon is also questionable [36,37]. In order to maintain the energy conditions, it is evident that the ${\varphi }_t^2$ must be a monotonically increasing function of t, with the condition ${\varphi }_t^2<1$. The admissible configurations for the K-essence scalar field in the generalized Vaidya solution, in order to have a dynamical horizon, are subject to a highly restrictive constraint. It is important to note that the metrics mentioned above represent dynamical horizons rather than isolated or event horizons, as explained extensively in Ref. [36,37].

It is also noted that the time dependence in the given mass functions (Eqs. (17), (18)) arises from the kinetic energy of the K-essence scalar field. However, in the mass function Eq. (14), the time dependence comes from both the usual generalized Vaidya mass and the K-essence scalar fields. Thus, considering the above situations of the K-essence generalized Vaidya spacetime, we may conclude that the K-essence Vaidya mass function adheres to the general form specified in Eq. (14). Therefore, we may conclude that the background metric can be chosen from any standard gravitational metric, with the only alteration being the replacement of their masses with a background mass, which likewise satisfies the EEE equation.

This section focuses on analyzing the geodesic structure of the generalized K-essence Vaidya spacetime. In this context, we define our investigative metric as Eq. (13), where the K-essence emergent Vaidya mass function is represented by Eq. (14). For the metric (13), we can write the Lagrangian as [1,8,34,35] where $\dot{t}=\frac{dt}{d\tau },\dot{r}=\frac{dr}{d\tau },\dot{\theta }=\frac{d\theta }{d\tau },\dot{\Phi }=\frac{d\Phi }{d\tau }$, $\tau$ is to be identified with the proper time.

Now, using the Euler-Lagrange equation we have where ${\mathcal{M}}_t=\frac{\partial \mathcal{M}(t,r)}{\partial t}$ and we write $\mathcal{M}(t,r)$ as $\mathcal{M}$.

Because our object and metric are spherically symmetric, we can simplify everything by examining just motion on the equatorial plane $\theta =\frac{\pi }{2}$ and therefore, $\dot{\theta }=0$. For the above choice of equatorial plane, the Eq. (22) becomes

Thus, by employing Eq. (24) on the equatorial plane, we may write from Eq. (19) that the Lagrangian is

Due to the inclusion of the generalized K-essence Vaidya mass function ($\mathcal{M}(t,r)$) in the Lagrangian formulation provided above, further analysis is not possible as it can have varying values based on the gravitational mass. Within this particular situation, we have the option to select the mass function. For our subsequent analysis, we have selected two distinct mass functions. Moreover, it is mentioned that the K-essence Vaidya mass function (14) depends on ${\varphi }_t^2$, which has values between 0 and 1. Therefore, we can select ${\varphi }_t^2$ as an explicit function of time in order to keep the values of ${\varphi }_t^2$ throughout the article as [36] where $t_0$ is a positive constant.