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

The paper investigates whether gravitational collapse in a generalized emergent Vaidya spacetime, when modeled within f(R,T) gravity and K-essence dynamics, results in black holes or naked singularities, and how different choices of f(R,T) influence cosmic evolution. Motivated by the long-standing issues of cosmic acceleration, dark energy modeling, and the Cosmic Censorship Hypothesis, the authors adopt the Vaidya framework (radiating spacetime with time-dependent mass) and a non-canonical scalar (K-essence) to explore collapse end-states and potential links to dark energy phenomenology. The study seeks to understand conditions under which naked singularities can form and whether K-essence can simultaneously describe dark energy and gravitational dynamics.

The work situates itself in extensive literature on gravitational collapse and the Cosmic Censorship Hypothesis (Oppenheimer-Snyder, Penrose, Joshi and others) and on Vaidya spacetimes as radiating generalizations of Schwarzschild. It reviews evidence for cosmic acceleration from SN Ia, BAO, WMAP and Planck, motivating modified gravity frameworks such as f(R), f(G), f(R,Lm), f(R,G), and specifically f(R,T) (Harko et al.). Prior studies have examined thermodynamics, cosmology, and constraints in f(R,T) models. K-essence, using non-canonical Lagrangians (often DBI-type), has been explored as a dark energy model capable of avoiding fine-tuning and addressing the coincidence problem, with possible attractor solutions and subluminal sound speed effects on CMB anisotropies. The emergent metric approach from K-essence (Manna et al.) yields an effective geometry Ḡµν distinct from gµν. Earlier works studied emergent gravity, singularity properties in generalized Vaidya metrics, and cosmology in emergent f(R,L(X)) and f(R,T) settings, showing links to observationally viable EoS parameters.

- Geometry and field content: Adopt K-essence with a DBI-type non-canonical kinetic Lagrangian L(X)=1−√(1−2X), X=1/2 gµν∇µφ∇νφ. The emergent metric is Ḡµν=gµν−∂µφ∂νφ, differing from the gravitational metric gµν. The sound speed satisfies c_s^2=1−2X. - Emergent gravity field equations: Using the emergent geometry, define Einstein-like equations Gµν(Ḡ)=κ Tµν with an emergent energy-momentum tensor derived either from the field equations or by variation with respect to Ḡµν. The modified gravity action in emergent geometry is S=∫ d^4x √(−Ḡ)[ f(R,T) + L(X) ] (κ=1), with R the Ricci scalar of Ḡ and T= Tµν Ḡµν. Variation yields the modified field equations: F Rµν − 1/2 f Ḡµν + (Ḡµν Ḋ−DµDν)F = 1/2 Tµν − f_T Tµν − f_T Θµν, where F=∂f/∂R, f_T=∂f/∂T, and Θµν depends on L(X). - Spacetime ansatz: Use the emergent generalized Vaidya metric dS^2= −(1−2M(v,r)/r) dv^2 + 2 dv dr + r^2 dΩ^2 with emergent mass M(v,r)= m(v,r) + r^2 φ_v^2, where φ_v=∂φ/∂v. The matter sector is modeled as a combination of null radiation and a perfect fluid: Tµν = Tµν^(n) + Tµν^(m), with Tµν^(n)=γ lµ lν (Vaidya null radiation), and Tµν^(m)=(ρ+p)(lµ nν + lν nµ) + Ḡµν p. EoS: p=ωρ, and density ρ = n M(v,r) (n>0). Null vectors lµ=(1,0,0,0), nµ=( (1−M/r)/2, −1, 0, 0). Energy conditions impose constraints on m(v,r) and φ_v. - Field equation components: Focus on the (01) and (22)/(33) components of the modified emergent field equations, as the (00) component is cumbersome (mixed r and v derivatives). Solve the system for specific additive forms f(R,T)=f1(R)+f2(T) across four cases: • Case 1: Power-law f1=g1 R^{β1}, f2=g2 T^{β2}. Solvable with β1=β2=1, reducing (22)/(33) to a second-order ODE in r with coefficients depending on ω, n, g1, g2, and φ_v. Solution for M(v,r) involves Airy functions and integration functions c1(v), c2(v). • Case 2: f1=g1 R^{β1}, f2=g2 e^{β2 T}. Solvable with β1=1, β2=0, yielding M(v,r) as a linear combination of Airy functions with time-dependent integration functions d1(v), d2(v). • Case 3: Double-exponential f1=g1 e^{β1 R}, f2=g2 e^{β2 T}. Due to exponential dependence on derivatives M′, M″, adopt approximations (β1=0 and Taylor expansion) leading to an r-independent M(v,r) depending on φ_v^2 and constants (g1,g2,β2,n,ω); no explicit r dependence remains. • Case 4: Mixed exponential-power f1=g1 e^{β1 R}, f2=g2 T^{β2}. Solvable with β1=0, β2=1, giving M(v,r) independent of r but time-dependent via φ_v^2. - Collapse analysis: Study outgoing radial null geodesics from the central singularity using dv/dr = 2/(1−2M/r) = 2/(1−2m/r + φ_v^2). Define Y=v/r and evaluate the limiting slope Y0=lim_{v,r→0} v/r. Positive real Y0 corresponds to naked singularity; negative to black hole. For Cases 1–2, insert the Airy-function solutions at the center, fix integration functions as c1(v)=ξ1 v, c2(v)=ξ2 v, and use known Airy values at 0 to obtain a quadratic for Y0 with positivity conditions determined by ξ1, ξ2, and φ_v^2. For Cases 3–4, r-independence of M prevents collapse analysis; instead, study M as a function of φ_v^2 and normalized time v/v0 (with φ_v^2 = e^{−v/v0} θ(v)).

- Collapse outcomes depend sensitively on the choice of f(R,T): • Cases 1 and 2 (effectively linear in R and T): The limiting equation for Y0 yields real, positive roots over broad ranges of parameters (ξ1, ξ2, 0<φ_v^2<1). Graphical analysis indicates Y0>0 generically, signaling globally naked singularities rather than black holes. • Cases 3 and 4 (exponential forms leading to r-independent M): No gravitational collapse analysis is possible (M lacks r dependence). Instead, M varies with φ_v^2 (and implicitly with v): for multiple ω, M(φ_v^2) decreases and crosses zero near φ_v^2≈0.75; similarly, as a function of v/v0, M crosses zero near v/v0≈0.287682. These behaviors point to epochs where spacetime becomes effectively Minkowski (M=0) and to accelerated expansion dominated by dark energy. - Dark energy interpretation: φ_v^2 (0<φ_v^2<1) can be associated with dark energy fraction; the crossing at φ_v^2≈0.75 aligns with the observed present dark energy density (~0.74), indicating a kinetic-energy-driven dark energy phase within K-essence. - Positive and negative mass regimes: Depending on ω and φ_v^2 (or v/v0), M can take positive or negative values, suggesting the possible presence of gravitational dipoles (coexistence of positive and negative mass components) consistent with discussions by Bondi, Bonnor, and Miller. - K-essence dual role: The framework can act both as a simple gravitational theory (emergent metric) producing global naked singularities in certain f(R,T) choices and as a dark-energy model driving acceleration in others.

The results demonstrate that in an emergent K-essence geometry, the additive f(R,T) structure decisively affects the end-state of gravitational collapse. Linear-like choices (Cases 1–2) tend to yield globally naked singularities, challenging strong forms of the Cosmic Censorship Hypothesis within this framework. In contrast, exponential choices that decouple r-dependence (Cases 3–4) preclude collapse analysis but naturally generate time-varying mass functions linked to the K-essence kinetic term, reproducing key dark-energy features such as late-time acceleration and a massless (M=0) Minkowski stage near the empirically inferred dark energy fraction. The emergence of both positive and negative mass sectors across epochs hints at gravitational dipoles, potentially relevant to high-energy astrophysical phenomena and early- or late-time acceleration mechanisms. Overall, the findings support K-essence as a unifying description that can encode both gravitational collapse phenomenology and dark energy behavior, with the emergent geometry providing a natural arena for such dual roles.

The study formulated emergent f(R,T) gravity within K-essence geometry using a DBI-type Lagrangian and analyzed a generalized emergent Vaidya spacetime. Solving selected components of the modified field equations for several additive f(R,T) forms yielded mass functions M(v,r) with distinct behaviors. For power-law or linear-effective cases (Cases 1–2), analysis of radial null geodesics indicated globally naked singularities. For exponential cases (Cases 3–4), M became r-independent and time-dependent via φ_v^2, showing zero crossings at φ_v^2≈0.75 and v/v0≈0.287682, consistent with dark-energy-dominated acceleration and transitional Minkowski behavior (M=0). The appearance of positive and negative mass intervals suggests gravitational dipoles. These results indicate that K-essence can simultaneously function as a dark energy model and a gravitational framework, enabling exploration of diverse cosmological phenomena including collapse outcomes, dark energy-driven acceleration, and potential dipole effects. Future work could address solving the full set of field equations (including the challenging (00) component), exploring broader functional classes of f(R,T), relaxing simplifying parameter choices, and confronting predictions with observational constraints.

- The full modified field system is not solved: the (00) component is avoided due to complexity (mixed r and v derivatives), and only the (01) and (22)/(33) components are used. - Solutions are obtained only for specific parameter choices (e.g., β1=β2=1 in Case 1; β1=1, β2=0 in Case 2; β1=0 approximations in Case 3; β1=0, β2=1 in Case 4), limiting generality. - Case 3 employs approximations (Taylor expansions) to handle exponentials in derivatives, yielding r-independent M and precluding direct collapse analysis. - Integration functions are chosen with specific time dependence (e.g., c1(v)=ξ1 v, c2(v)=ξ2 v) for tractability; results may depend on these choices. - Gravitational collapse conclusions rely on limiting/geodesic analysis and graphical exploration; no numerical simulations of full dynamics are presented. - No direct observational data is used; connections to observations (e.g., dark energy fraction) are qualitative.