Conformal Vaidya Metric & Extensions

• The conformal Vaidya metric is a linear-mass Vaidya spacetime recast as conformal to a static metric that admits a conformal Killing vector.
• It enables analytic study of photon spheres, black-hole shadows, and horizon thermodynamics through self-similar scaling and conformal Killing horizons.
• Generalized extensions incorporate charge and rotation, preserving separability and providing insights into dynamical spacetime properties and accretion effects.

Searching arXiv for papers on conformal Vaidya metrics, conformal Killing symmetry in Vaidya, and related rotating/generalized constructions. The conformal Vaidya metric denotes, in its most standard usage, the Vaidya spacetime with a linear mass function written in a form that is conformal to a static or stationary metric, so that the spacetime admits a conformal Killing vector rather than an exact Killing vector. In the spherically symmetric case, the starting point is the ingoing Vaidya line element with $m(v)$ linear in the null coordinate, a choice that yields self-similar scaling, a conformal Killing horizon, and analytic control over null geodesics, photon spheres, and black-hole shadows (Nielsen et al., 2017). Closely related constructions extend this idea to generalized Vaidya geometries with homothetic symmetry, to charged Vaidya metrics with linear mass and charge, and to Kerr–Vaidya-like rotating spacetimes obtained by combining a conformal factor with a Kerr-type seed metric (Solanki et al., 2022). The term is also used more broadly for Vaidya-type spacetimes whose conformal or Weyl geometry is the main object of study, but the physically most developed usage is the conformally static linear-mass Vaidya family (Tan, 2023).

1. Spherically symmetric seed metric and conformally static form

The standard ingoing Vaidya metric in Eddington–Finkelstein–like coordinates $(v,w,\theta,\phi)$ is

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$

with stress tensor

$$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$

For $m'(v)>0$ the spacetime describes accretion, while $m'(v)<0$ corresponds to radiation. A special role is played by the linear choice

$m(v)=\mu v,\qquad \mu>0,$

because with this choice Vaidya admits a conformal Killing symmetry (Tan, 2023).

A convenient coordinate transformation is

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$

which brings the metric to

$$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$

This displays the spacetime as conformal to a stationary metric, and after a further redefinition $T=t+\Upsilon(r)$ it becomes explicitly conformal to a static Schwarzschild-like line element with

$(v,w,\theta,\phi)$0

In this chart the conformal Killing vector is

$(v,w,\theta,\phi)$1

and the spacetime is “conformally stationary” in the sense that time translations preserve null cones but not lengths (Tan, 2023).

An alternative conformal representation of linear Vaidya uses

$(v,w,\theta,\phi)$2

for the linear mass profile

$(v,w,\theta,\phi)$3

In that description the dynamical Vaidya metric is mapped to a static but not asymptotically flat spacetime, and the Vaidya conformal Killing vector becomes a true Killing vector of the conformal image (Nielsen et al., 2017).

2. Conformal Killing vectors, homothety, and generalized Vaidya symmetry

For linear-mass Vaidya, the spacetime admits the conformal Killing vector

$(v,w,\theta,\phi)$4

which satisfies the conformal Killing equation with respect to the physical Vaidya metric. In the static limit $(v,w,\theta,\phi)$5, this vector reduces to the usual Schwarzschild Killing vector normalized to unit norm at infinity (Nielsen et al., 2017).

In generalized Vaidya spacetimes, the conformal Killing equation in the $(v,w,\theta,\phi)$6 sector leads to the most general proper conformal Killing vector

$(v,w,\theta,\phi)$7

The corresponding mass function is constrained by

$(v,w,\theta,\phi)$8

This result explains why pure Vaidya and charged Vaidya admit only homothetic Killing vectors but no proper conformal Killing vectors with non-constant conformal factors in the $(v,w,\theta,\phi)$9 plane (Ojako et al., 2019).

A particularly important homothetic Killing vector in generalized Vaidya is

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$0

For this to be a homothetic Killing vector, the generalized mass must satisfy

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$1

with solution

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$2

In coordinates

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$3

the homothetic vector becomes $$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$4, and the spacetime is manifestly conformally static (Vertogradov et al., 2022).

A recent extension of this viewpoint shows that generic Vaidya-like spacetimes admit a unique class of conformal Killing vectors that are homothetic when mass, charge, or rotation parameters are linear functions of the advanced null-time. For Schwarzschild–Vaidya, charged Vaidya, Husain-type Vaidya, and Kerr–Vaidya, the existence of the homothetic symmetry forces linear profiles in the dynamical parameters and yields a distinguished homothetic Killing horizon (Ghoshal et al., 10 Apr 2026).

3. Horizons: conformal Killing horizons, apparent horizons, and global structure

For a conformal Killing vector $$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$5, the conformal Killing horizon is defined by

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$6

In linear Vaidya with

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$7

this gives

$$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$8

so real conformal Killing horizons exist only when $$ds^2 = - \left( 1 - \frac{2m(v)}{w} \right) dv^2 + 2\, dv\, dw + w^2 \left( d\theta^2 + \sin^2 \theta\, d\phi^2 \right),$$9. The inner branch approaches $$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$0 in the static limit and is the physically relevant black-hole horizon, whereas the outer branch goes to infinity as $$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$1 (Nielsen et al., 2017).

The trapping horizon of Vaidya is instead at

$$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$2

where the outgoing null expansion vanishes. For $$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$3, the trapping horizon lies inside the inner conformal Killing horizon. The trapping horizon is typically spacelike when the area is increasing, while the conformal Killing horizon is null by construction (Nielsen et al., 2017).

In the conformally static chart built from the spherical seed with

$$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$4

the norm of the conformal Killing vector changes sign at

$$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$5

These are conformal Killing horizons of the spherical conformal Vaidya spacetime, with $$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$6 identified in the Vaidya literature with the event horizon of the accreting black hole and $$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$7 an outer “cosmological” conformal Killing horizon (Tan, 2023).

A different but complementary global perspective is provided by Israel coordinates for Vaidya,

$$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$8

with

$$T^{\mu\nu} = m'(v)\,K^\mu K^\nu,\qquad K^\mu \partial_\mu = \partial_w.$$9

These coordinates are regular at horizons and can be used to construct maximal extensions that are null geodesically complete up to the genuine curvature singularity at $m'(v)>0$0. In the cases studied, the Penrose diagrams display spacelike singularities, null infinities $m'(v)>0$1, event horizons traced by special radial null geodesics, and apparent horizons $m'(v)>0$2 that do not generally coincide with the event horizon (Nasereldin et al., 2023).

This suggests a useful distinction. The conformal Killing horizon is the natural horizon singled out by conformal symmetry, whereas the apparent horizon is the trapped-surface boundary defined by null expansions. In dynamical Vaidya spacetimes these surfaces generally differ, even though both are quasi-local (Nielsen et al., 2017).

4. Null geodesics, separability, photon spheres, and shadow geometry

A basic fact used repeatedly is that null geodesics are invariant under conformal transformations up to a reparametrization of the affine parameter. If

$m'(v)>0$3

then null trajectories in $m'(v)>0$4 coincide with those in $m'(v)>0$5 as unparametrized curves. This makes conformally static Vaidya metrics analytically tractable for lensing and shadow calculations (Tan, 2023).

For the spherically symmetric linear-mass Vaidya metric in conformally static coordinates, exact formulas can be derived for photon spheres and shadow radii. In the accreting case

$m'(v)>0$6

the metric admits the conformal Killing vector $m'(v)>0$7, and the unstable photon sphere in conformally static coordinates is located at

$m'(v)>0$8

Its physical areal radius grows with the conformal factor, and in the original Eddington–Finkelstein-like coordinates the corresponding null orbits form an outward logarithmic spiral. For a conformally static observer, the shadow is circular and its angular radius is time-independent, even though the physical photon-sphere area grows exponentially in the observer time (Solanki et al., 2022).

In the radiating case

$m'(v)>0$9

the conformally static metric has a single horizon and a photon sphere at

$m'(v)<0$0

The physical areal radius shrinks with time, and in the original coordinates the null orbits form an inward logarithmic spiral. Again, conformally static observers measure a time-independent angular shadow (Solanki et al., 2022).

The rotating generalization constructed by applying the Newman–Janis algorithm to the static seed

$m'(v)<0$1

with

$m'(v)<0$2

yields a Kerr-like metric with

$m'(v)<0$3

and the full conformal Kerr–Vaidya-like metric

$m'(v)<0$4

Because the null geodesics of the conformal metric coincide with those of the Kerr-like seed, the Hamilton–Jacobi equation separates exactly as in Kerr and a Carter-like constant survives. This permits analytic formulas for spherical photon orbits, shadow boundary curves, mean shadow radius, and asymmetry factor (Tan, 2023).

A central shadow result is the empirical scaling law

$m'(v)<0$5

$m'(v)<0$6

Thus, in this conformal Kerr–Vaidya-like model both the mean shadow radius and the asymmetry factor scale approximately with the square root of the remaining accretion margin to the bound $m'(v)<0$7, while the ratio $m'(v)<0$8 is approximately independent of $m'(v)<0$9 (Tan, 2023).

5. Rotating, charged, and generalized extensions

The charged Vaidya metric

$m(v)=\mu v,\qquad \mu>0,$0

admits a conformal Killing vector only if mass and charge are proportional and both are linear in advanced time: $m(v)=\mu v,\qquad \mu>0,$1 The resulting conformal Killing vector is

$m(v)=\mu v,\qquad \mu>0,$2

with constant conformal factor $m(v)=\mu v,\qquad \mu>0,$3, so it is actually a homothetic Killing vector (Koh et al., 2023).

For this linear charged Vaidya spacetime, solving $m(v)=\mu v,\qquad \mu>0,$4 gives three conformal Killing horizons, interpreted as a cosmological conformal Killing horizon, an inner conformal Killing horizon, and an outer conformal Killing horizon. Under the conformal transformation

$m(v)=\mu v,\qquad \mu>0,$5

these horizons map to Killing horizons of a static spacetime. For the simple choice $m(v)=\mu v,\qquad \mu>0,$6, the conformal image is

$m(v)=\mu v,\qquad \mu>0,$7

which contains a cosmological horizon but is not asymptotically de Sitter (Koh et al., 2023).

The fully rotating Kerr–Vaidya case extends the same idea. The conformal Killing equation has a solution if and only if both mass and rotation parameter become dynamic: $m(v)=\mu v,\qquad \mu>0,$8 with

$m(v)=\mu v,\qquad \mu>0,$9

The resulting conformal Killing vector is homothetic, and the null surface where it becomes null defines a homothetic Killing horizon. This provides the rotating counterpart of the conformal Vaidya construction and a route to horizon thermodynamics through conformal mapping to a stationary spacetime (Ghoshal et al., 10 Apr 2026).

In generalized Vaidya spacetimes with matter content

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$0

the homothetic Killing vector

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$1

exists only for a special power-law class

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$2

For this class, one can pass to conformally static coordinates, diagonalize the metric, define a new constant of motion along null and timelike geodesics generated by the homothety, and compute the surface gravity in the dust and stiff-fluid cases (Vertogradov et al., 2022).

6. Geometric interpretations, alternative usages, and limitations

The conformal Vaidya construction is valuable because it preserves enough symmetry to retain analytic control while incorporating time dependence. In particular, moving time dependence into a conformal factor can preserve a conformal Killing vector, and in rotating cases can preserve separability of null geodesics and a Carter-like constant. This is precisely why the conformal Kerr–Vaidya-like metric differs from the standard Kerr–Vaidya metric obtained by simply replacing $v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$3 with $v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$4 in Kerr form: the latter does not admit the conformal Killing structure used for separability and does not yield fully analytic shadow calculations (Tan, 2023).

At the same time, the conformally related static spacetimes are generally not asymptotically flat. In the linear-mass Vaidya thermodynamic construction, the conformal image is static but not asymptotically flat, and in the charged case the static image contains a cosmological horizon but is not asymptotically de Sitter (Nielsen et al., 2017). The conformal Kerr–Vaidya-like geometry is likewise not asymptotically flat and has a more complicated energy–momentum tensor (Tan, 2023).

The term “conformal Vaidya metric” is also used in other, distinct senses. In studies of conformal symmetries of generalized Vaidya, the emphasis is on conformal Killing vectors in the $v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$5 plane and the mass functions compatible with them, rather than on a single canonical metric (Ojako et al., 2019). In geometric studies of Vaidya–Bonner–de Sitter and related metrics, the phrase refers to the Weyl or conformal curvature structure: such spacetimes are not conformally flat but satisfy pseudosymmetry conditions such as

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$6

and their conformal 2-forms are recurrent (Shaikh et al., 2024).

There are also contexts in which the relevant Vaidya-type geometry is explicitly not conformally equivalent to the background metric. In generalized K-essence Vaidya spacetime, the emergent metric is

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$7

and the analysis emphasizes that this metric does not possess conformal equivalence to the conventional gravitational metric. In that setting, a “conformal Vaidya” description is not appropriate (Majumder et al., 2023).

A further limitation appears in geometric-flow applications. For the Vaidya metric

$v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$8

the conformal Ricci–Bourguignon soliton equations force the mass function to vanish, so the only admissible conformal Ricci–Bourguignon solitons occur in the flat Minkowski limit $v = r_0 e^{T/r_0},\qquad w = r\, e^{T/r_0},$9, not on genuinely radiating Vaidya backgrounds (Rehman et al., 13 Aug 2025).

These variations suggest a careful editorial distinction. In the narrow and most physically developed sense, the conformal Vaidya metric is the linear-mass Vaidya spacetime recast as conformal to a static metric, with its associated conformal Killing vector and conformal Killing horizon. In broader usage, the phrase may refer to Vaidya-type metrics studied through conformal symmetry, conformal curvature, or conformally related extensions, but those usages are not equivalent (Nielsen et al., 2017).

7. Astrophysical relevance and current directions

The conformal Kerr–Vaidya-like model has been applied to shadow calculations for M87$$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$0 and Sgr A$$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$1. For M87$$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$2, using

$$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$3

and accretion rates corresponding to

$$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$4

the model predicts shadow angular diameters $$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$5 and maximal $$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$6, with accretion-induced changes in $$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$7 and $$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$8 of order $$ds^2 = e^{2T/r_0}\left[ -\left(1-\frac{2\mu r_0}{r}-\frac{2r}{r_0}\right) dT^2 +2\, dT\, dr +r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \right].$$9. For Sgr A$T=t+\Upsilon(r)$0, with

$T=t+\Upsilon(r)$1

and

$T=t+\Upsilon(r)$2

the predicted shadow angular diameter is $T=t+\Upsilon(r)$3, while the fractional changes in shadow observables due to accretion backreaction are of order $T=t+\Upsilon(r)$4. In both cases the effect of $T=t+\Upsilon(r)$5 is negligible at current observational precision (Tan, 2023).

This supports a restrained interpretation. The conformal Vaidya metric is a phenomenological toy model intended to capture possible backreaction of accretion on the metric in an analytically tractable way, not a realistic GRMHD disk-plus-jet spacetime (Tan, 2023). Its importance lies in the combination of exact solvability, explicit conformal symmetry, and access to black-hole thermodynamics through conformal Killing horizons (Nielsen et al., 2017).

Current directions include extending homothetic Killing horizon methods to generic Vaidya-like metrics and rotating spacetimes, formulating flux-balance laws on these horizons, and using maximal extensions of charged Vaidya metrics to study particle creation in such backgrounds (Ghoshal et al., 10 Apr 2026). A plausible implication is that conformal symmetry provides one of the few systematic routes by which genuinely time-dependent black-hole metrics remain analytically tractable without collapsing back to the static case.