Generalized Vaidya Metric Overview

• Generalized Vaidya metric is a dynamic spacetime solution with variable mass functions driven by mixed null and timelike energy sources.
• It employs advanced Eddington–Finkelstein coordinates to model gravitational collapse and radiative processes in both classical GR and alternative gravity theories.
• Geometric corrections and integrability conditions in these metrics clarify horizon shifts, energy flux discontinuities, and constraints on cosmic censorship.

The generalized Vaidya metric is a class of nonstationary, spherically symmetric solutions to gravitational field equations, generalizing Vaidya's null-dust spacetime by incorporating arbitrary energy-momentum sources with both null and timelike components. Such metrics arise naturally in gravitational collapse, relativistic astrophysics, and extensions of general relativity, and have been systematically developed in frameworks ranging from classical GR to higher-curvature and third-rank gravity theories. Their unifying feature is the presence of a variable mass profile—generically a function of an advanced or retarded null coordinate and areal radius—which governs both the geometry and the matter content of the spacetime.

1. Metric Ansatz and Mass Functions

In advanced Eddington–Finkelstein-type coordinates $(u, r, \theta, \phi)$, the generalized Vaidya line element takes the form

$$ds^2 = -f(u, r)\, du^2 + 2\, du\, dr + r^2\, (d\theta^2 + \sin^2 \theta\, d\phi^2),$$

where

$f(u, r) = 1 - \frac{2\, M(u, r)}{r}$

and $M(u, r)$ is the generalized mass function ("Misner–Sharp mass" in GR parlance) whose radial and null dependence encode the distribution and flux of energy.

In the context of third-rank gauge theories such as Cotton gravity (CG) and Conformal Killing gravity (CKG), the mass functions admit additional geometric correction terms due to the higher-derivative structure of the field equations (Gürses et al., 3 Aug 2025):

• For Cotton gravity:

$$M_{\mathrm{CG}}(u, r) = C_0(u) + \frac{1}{2} C_1(u) r^2 + \frac{1}{3} C_2(u) r^3 + \frac{\kappa\, \rho_0(u)}{2 (1-2w)}\, r^{1 - 2w}$$

• For Conformal Killing gravity:

$$M_{\mathrm{CKG}}(u, r) = C_0(u) + \frac{1}{3} C_1 r^3 + \frac{1}{5} C_2 r^5 + \frac{\kappa\, \rho_0(u)}{2 (1-2w)}\, r^{1 - 2w}$$

Here, $C_0(u)$ is the standard radiating mass, $C_1, C_2$ geometric correction functions/parameters, and $\rho_0(u)$ the amplitude of a Type-I (timelike) fluid of barotropic index $w$, $p = w\, \rho$.

In the most general two-fluid GR-based models (e.g., (Vertogradov et al., 2022, Chakrabarti et al., 12 Nov 2025)), the mass function may depend arbitrarily on $u, r$, with special forms of $M(u, r)$ corresponding to specific physical sources (null dust alone, perfect fluid, combinations).

2. Field Equations and Matter Content

The generalized Vaidya metric supports a stress–energy tensor with both null and Type-I (timelike) fluid contributions: $$T_{\mu\nu} = \mu\, l_\mu l_\nu + (\rho + p)(l_\mu n_\nu + n_\mu l_\nu) + p\, g_{\mu\nu},$$ where $l_\mu$ is a future-directed null vector ($l_\mu dx^\mu = du$), $n_\mu$ the secondary null vector, $\mu$ the energy density of null dust, $\rho$ and $p$ the Type-I density and pressure.

The Einstein (or modified) field equations reduce to algebraic relations: $$\mu = \frac{2}{r^2} \frac{\partial M}{\partial u}, \quad \rho = \frac{2}{r^2} \frac{\partial M}{\partial r}, \quad p = -\frac{1}{r} \frac{\partial^2 M}{\partial r^2}$$ with modifications in higher-derivative theories due to additional tensorial terms (Gürses et al., 3 Aug 2025).

In CG and CKG, the geometric third-rank tensor $H_{\mu\nu}$ enters directly, leading to coupled equations:

• CG: $G_{\mu\nu} = \kappa\,T_{\mu\nu} + H_{\mu\nu}$ with Codazzi integrability $$\nabla_\alpha\tilde{H}_{\mu\nu} = \nabla_\mu\tilde{H}_{\alpha\nu}$$,
• CKG: $G_{\mu\nu} = \kappa\,T_{\mu\nu} + H_{\mu\nu}$ with cyclic symmetry.

Integration yields the mass and density profiles in terms of the freely specifiable functions $C_0(u)$, $C_1(u)$/$C_1$, $C_2(u)$/$C_2$, $\rho_0(u)$, and the barotropic index $w$.

3. Geometric Interpretation and Limits

Each term in the mass function admits a precise geometric and physical interpretation:

• $C_0(u)$: Standard Vaidya radiating mass. Its time derivative sources the null-dust energy density.
• $\rho_0(u)\, r^{-2(1+w)}$: Timelike fluid with equation of state $p = w \rho$, contributing to the mass via back-reaction.
• Geometric corrections ($C_1, C_2$): Arising from the non-Einsteinian, third-derivative (CG) or conformal (CKG) structure. For example, $C_1 r^2$ and $C_2 r^3$ in CG, $C_1 r^3$ and $C_2 r^5$ in CKG, giving asymptotically de Sitter-like contributions. In CG, $C_1, C_2$ can depend on null time, while in CKG, they are constants.

The pure Vaidya/GR limit is recovered by setting all correction functions and $\rho_0$ to zero, yielding: $M(u, r) = C_0(u), \quad f = 1 - \frac{2 C_0(u)}{r}$ with the classic null-dust stress tensor.

4. Integrability Conditions and Constraints

Several constraints are essential for physical and mathematical consistency:

• Barotropic index $w \neq 1/2$ is required to avoid logarithmic singularities unless compensated.
• The null-dust component must remain nonnegative: $$\mu(u, r) = (\dot C_0(u))/(\kappa r^2) + (\dot\rho_0(u))/(1-2w) r^{-(1+2w)} \geq 0$$.
• In the vacuum, CG allows time-dependent geometric corrections if $M$ depends on $r$, but no pure null-dust solution without matter; CKG vacuum solutions are static.
• In both theories, matching to an exterior Schwarzschild or Vaidya region is possible only if $\partial M / \partial r = 0$ at the boundary, otherwise a thin shell with surface energy–momentum is induced (Chakrabarti et al., 12 Nov 2025). Discontinuities occur in the extrinsic curvature, curvature invariants, and the Kodama quasi-local energy flux.

Violation of these matching conditions, or negative energy densities, signal physically inadmissible or unbounded configurations.

5. Horizons and Causal Structure

The apparent and event horizons are determined by the roots of $f(u, r) = 0$, i.e.,

$1 - \frac{2 M(u, r_{AH})}{r_{AH}} = 0.$

Corrections to the mass function shift horizon locations relative to Schwarzschild/Vaidya (Vertogradov et al., 2022). For matter with equation of state $P = \alpha \rho$, one obtains explicit horizon radii dependent on the integration constants and matter profiles.

Timelike and null geodesics acquire modified acceleration terms due to the radial derivatives of $M(u, r)$, producing corrections to precession and gravitational redshift outside the horizon (Vertogradov et al., 2022). New constants of motion, e.g., from homothetic or conformal Killing vectors, appear when the mass profile and metric admit self-similarity or conformal symmetry (Ojako et al., 2019).

6. Physical Applications and Generalizations

Generalized Vaidya metrics encompass:

• Gravitational collapse with both null and timelike matter fields (e.g., Bose–Einstein condensate dark matter (Rudra, 4 Nov 2024), anisotropic fluids (Culetu, 2016)).
• Extensions to higher dimensions and alternative theories (massive gravity, dRGT (Hu et al., 2016)).
• Inclusion of charge, cosmological constant, or geometric corrections, as in de Sitter, Reissner–Nordström–Vaidya, and conformally coupled solutions.
• Testing cosmic censorship via analysis of naked singularity formation in backgrounds with fluids beyond pure null dust (Ojako et al., 2019, Rudra, 4 Nov 2024).

The structure of conformal and homothetic Killing symmetries in the $(u, r)$ sector has been shown to strongly constrain the allowed mass profiles (Ojako et al., 2019). In particular, all known pure Vaidya and charged Vaidya metrics admit only homothetic (self-similar) symmetries, while proper conformal Killing vectors require genuine two-fluid Type I+II sources.

7. Limitations and Interpretation

A key result is that any generalized Vaidya spacetime with $\partial M /\partial r \neq 0$ cannot be consistently matched to an exterior static (Schwarzschild) or radiating (Vaidya) region across a finite boundary without introducing a surface layer (thin shell) and discontinuous invariants (Chakrabarti et al., 12 Nov 2025). This restricts the interpretation of $r$-dependent generalized Vaidya metrics to unbounded regions (not physically realistic stellar interiors).

Within their domain of validity, generalized Vaidya metrics with geometric corrections are of central relevance in the paper of nonstationary radiative geometries, the global structure of dynamical black holes, and the modeling of semiclassical processes such as evaporation and accretion in a broad class of metric and matter theories.