Gravitational collapse in generalized K-essence emergent Vaidya spacetime via f(R,T) gravity

Gravitational collapse, a phenomenon central to general relativity and astrophysics, offers insights into black hole formation, star structure, and the creation of white dwarfs or neutron stars. Oppenheimer and Snyder's model, using a spherically symmetric dust cloud, demonstrated black hole formation from stellar collapse exceeding 20 solar masses [3]. Penrose's Cosmic Censorship Hypothesis suggests singularities are always hidden behind event horizons [5, 6], but the possibility of ‘naked singularities’ where singularities are not concealed has also been explored [7-17]. Vaidya spacetime, generalizing the Schwarzschild metric to include a time-dependent mass parameter, is relevant here [18-23]. Observations from type IA Supernovae, BAO, WMAP7, and Planck indicate accelerated universe expansion attributed to dark energy [24-29], leading to modifications of general relativity such as f(R) gravity [30], and its extensions f(G), f(R,Lm), f(R,G) [31-44]. f(R,T) gravity, proposed by Harko et al. [45, 46], couples the Ricci scalar to the trace of the energy-momentum tensor. This theory has been further developed and studied extensively, including thermodynamic aspects of cosmological models [47-49]. Typically, Lagrangian is L = T - V, however non-canonical theories like K-essence theory, which include a non-canonical Lagrangian [81-84], are necessary to explain cosmological phenomena not accounted for by standard models such as dark matter, dark energy, inflation etc. This study uses K-essence [85-95], characterized by a non-canonical Lagrangian L(X) = -V(φ)F(X) where X is the kinetic term and φ is the K-essence scalar field, and with the potential term potentially suppressed by the kinetic term of the field, thus providing a solution to the Cosmic Coincidence problem [100] . K-essence theories (minimally coupled to gravity as in this paper [85-95]) has the potential to act both as a dark energy framework, avoiding the fine tuning problem and mimicking the negative pressure required for universe acceleration, and as a purely gravitational or geometrical theory [109-113], considering that the existence of dark energy is still debated [128]. Manna et al. [104-113] have introduced an emergent gravity metric (Ḡµν) derived from a Dirac-Born-Infeld (DBI) type action [114-127], conformally different from the usual gravitational metric (gµν). This research examines the fate of singularities in f(R,T) gravity within generalized emergent Vaidya spacetime using the K-essence approach.

The literature review extensively covers existing works on gravitational collapse within the framework of general relativity and modified gravity theories. It highlights the significance of Vaidya spacetime as a model for radiating stars and its various generalizations. The review also details the development of f(R) gravity and its extensions, including f(R,T) gravity, as attempts to explain the accelerating expansion of the universe without resorting to dark energy. Numerous studies cited detail explorations into the thermodynamics and cosmological implications of these modified gravity theories. Finally, the review discusses the K-essence theory as a non-canonical approach to cosmology, focusing on its ability to explain dark energy through the kinetic energy of the scalar field. The previous work on emergent geometry and its implications for singularity analysis is also discussed, providing context for the current investigation.

This research investigates the fate of singularities within the framework of f(R,T) gravity in a generalized emergent Vaidya spacetime, employing the non-canonical K-essence approach. The starting point is the action of K-essence geometry, S_k[φ,g^µν] = ∫d⁴x√-gL(X,φ), where X is the canonical kinetic term and L(X,φ) is the non-canonical Lagrangian [85-95]. The energy-momentum tensor for the K-essence scalar field is derived, and the K-essence scalar field equation of motion is given. A conformal transformation leads to the emergent metric Ḡ^µν, which is not conformally equal to the gravitational metric g^µν [104-106]. The Dirac-Born-Infeld (DBI) type non-canonical Lagrangian L(X) = 1 - √(1 - 2X) is used [104-106, 114-117, 127]. The emergent metric is derived, which is then used with the action of f(R,T) gravity to obtain the modified field equations. The emergent Vaidya metric [110] is adopted as the primary metric, and the corresponding energy-momentum tensor is calculated, considering it as a combination of null radiation and a perfect fluid. Different forms of the function f(R,T) are considered: f(R,T) = g₁R^β1 + g₂T^β2, f(R,T) = g₁R^β1 + g₂e^β2T, f(R,T) = g₁e^β1R + g₂e^β2T, and f(R,T) = g₁e^β1R + g₂T^β2 [22]. The modified field equations are solved for each case to determine the mass function M(v,r). The analysis then focuses on determining the nature of the singularity (naked or black hole) formed through gravitational collapse by examining the radial null geodesics using the obtained mass functions. Graphical analysis is used to visualize and interpret the results.

The study yields several key findings. First, for specific choices of f(R,T) (Cases 1 and 2), the analysis of radial null geodesics indicates the formation of a global naked singularity during gravitational collapse. This is supported by graphical analysis showing that the slope of the tangent to the null geodesics at the singularity is always positive, regardless of the values of certain arbitrary constants. Second, in Cases 3 and 4, while the mass function does not explicitly depend on r, preventing a direct analysis of collapse, the analysis reveals a significant dependence on φ²_v, representing the kinetic energy of the K-essence scalar field. At φ²_v ≈ 0.75, which is close to the observed dark energy density, the mass function becomes zero, implying a transition to Minkowski spacetime where the mass is completely converted into energy. This indicates dark energy domination. Furthermore, plots of the mass function versus φ²_v and normalized time (v/v₀) reveal that the mass function can be positive or negative depending on the equation of state (EoS) parameter (ω). The presence of both positive and negative mass parameters suggests a possible gravitational dipole [143,144], explaining early inflation and later-time acceleration. The negative mass parameter values indicate dark energy domination, correlating to accelerated expansion. The study shows a potential concordance between the K-essence model used and observations, including the value of the dark energy density and the late-time acceleration of the universe.

The findings address the research question regarding the fate of singularities in f(R,T) gravity within emergent Vaidya spacetime. The demonstration of global naked singularity formation in certain scenarios challenges the cosmic censorship hypothesis. The results highlight the complex interplay between modified gravity (f(R,T)), non-canonical scalar field dynamics (K-essence), and the emergent geometric framework. The observation of both positive and negative masses suggests the presence of a gravitational dipole across different cosmological epochs, providing a potential mechanism for the observed accelerated expansion and the early inflation of the universe. The fact that the mass function vanishes at a value of φ²_v close to the observed dark energy density supports the K-essence theory as a viable dark energy model. This study suggests a potentially unified framework for understanding both gravitational collapse and dark energy phenomena within the context of emergent gravity.

This research successfully investigated gravitational collapse within a generalized emergent Vaidya spacetime using f(R,T) gravity and the K-essence theory. The study revealed conditions under which global naked singularities can form, contradicting the cosmic censorship hypothesis. It further demonstrated the potential of the K-essence theory to explain both dark energy and purely gravitational aspects of cosmology simultaneously. Future work could explore more complex f(R,T) functions, analyze different K-essence Lagrangians, and investigate the implications of the gravitational dipole phenomenon in greater detail.

The study primarily focused on specific functional forms of f(R,T) and a particular K-essence Lagrangian. A more exhaustive exploration of these functional forms might reveal additional insights. The analysis of Cases 3 and 4, due to the lack of explicit r dependence in the mass function, did not directly address the gravitational collapse itself but offered clues about dark energy and universe expansion. The assumption of a perfect fluid in the energy-momentum tensor could also be relaxed in future work.