Single Copy of Vaidya: Kerr–Schild Map

• Single Copy of Vaidya is a Kerr–Schild mapping technique that converts a collapsing null-shell gravitational metric into an exact Maxwell configuration using a step-function mass profile.
• The method preserves key features such as the 1/r dependence, null direction, and step-function profile, ensuring that the electromagnetic potential faithfully replicates the gravitational collapse characteristics.
• The framework connects Vaidya metrics to self-similar, homothetic structures and elucidates eikonal scattering and Bogoliubov coefficient analyses, while clarifying the absence of traditional Hawking thermality.

“Single Copy of Vaidya” usually denotes the electromagnetic background obtained from a Vaidya collapse geometry by the Kerr–Schild classical double copy. In the explicit construction presently available, the gravitational spacetime is a collapsing-shell Vaidya metric written as

$$g_{\mu\nu}=\eta_{\mu\nu}-\frac{2GM}{r}\Theta(t+r)\,k_\mu k_\nu, \qquad k\!\cdot\! dx=d(t+r),$$

so that the shell lies on $t+r=0$, the region $t+r<0$ is flat, and the region $t+r>0$ is Schwarzschild in Kerr–Schild form. The single copy is obtained by replacing one factor of the Kerr–Schild vector, yielding an electromagnetic potential

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$

which the authors call the “$\sqrt{\textrm{Vaidya}}$” background (Aoude et al., 29 Oct 2025).

1. Classical definition and collapsing-shell realization

The direct realization of the single copy begins from a thin-shell collapsing Vaidya spacetime rather than from the full family of arbitrary mass profiles. The mass profile is

$M(t+r)=M\,\Theta(t+r),$

with preferred null coordinate

$v\equiv t+r.$

The null vector is written explicitly as

$$k_\mu(x)=\left(1,\frac{\mathbf{x}}{r}\right), \qquad k_\mu k^\mu=0,$$

so the gravitational metric is already in Kerr–Schild form $g_{\mu\nu}=\eta_{\mu\nu}+\phi\,k_\mu k_\nu$ with

$t+r=0$0

The single-copy step is then the replacement

$t+r=0$1

which gives the gauge potential

$t+r=0$2

and hence the gauge-theory analogue of Vaidya collapse (Aoude et al., 29 Oct 2025).

In this formulation, the term “single copy” has a precise and restricted meaning. It does not refer to an arbitrary preferred vector field on Vaidya, nor to a general null-fluid/gauge-field analogy. It refers to a Kerr–Schild map in which the same null shell profile $t+r=0$3, the same $t+r=0$4 dependence, and the same null direction $t+r=0$5 are inherited by the gauge field. This construction is exact in the sense emphasized in the source paper: the gravitational background is “exact although it is linear in Newton’s constant $t+r=0$6” (Aoude et al., 29 Oct 2025).

2. Electromagnetic background and source interpretation

The field strength of the single copy is

$t+r=0$7

Its interpretation is particularly simple. For $t+r=0$8, one has $t+r=0$9. For $t+r<0$0, $t+r<0$1 is the Coulomb field of a static point charge $t+r<0$2 at the origin. The shell itself therefore carries the infalling charge needed to switch on the Coulomb field, and the corresponding current on the null shell is

$t+r<0$3

The appendix further gives the point-charge contribution

$t+r<0$4

so the total current

$t+r<0$5

is conserved, $t+r<0$6 (Aoude et al., 29 Oct 2025).

This source picture mirrors the gravitational collapse only partially. Gravity describes a null shell that forms a black hole; the single copy describes a null shell of massless charge that accumulates at the origin and produces a Coulomb field. The source paper stresses an important caveat: unlike gravitational collapse, this electromagnetic process is less natural physically because assembling like charge at a point generally requires external forces. The background is nevertheless a consistent Maxwell configuration with a clear current interpretation (Aoude et al., 29 Oct 2025).

A useful technical feature is the “Kerr–Schild gauge,”

$t+r<0$7

The immediate consequence is that the scalar-background interaction is purely cubic, with no contact terms. That simplification is central to the later eikonal analysis (Aoude et al., 29 Oct 2025).

3. Eikonal scattering, Bogoliubov coefficients, and Hawking-like features

The single copy of Vaidya is not introduced only as a formal Kerr–Schild image; it is used as a dynamical background for scattering of a charged massless scalar. The probe charge $t+r<0$8 is packaged into

$t+r<0$9

which plays the role of a fine-structure constant for the problem. In the geometric-optics regime

$t+r>0$0

the ladder diagrams exponentiate, and the outgoing state acquires the phase

$t+r>0$1

Equivalently,

$t+r>0$2

The same logarithmic phase is reproduced semiclassically by ray tracing: the rays remain straight null lines in flat spacetime, but on the outgoing branch they accumulate the interaction phase

$t+r>0$3

after entering the switched-on Coulomb field (Aoude et al., 29 Oct 2025).

The Bogoliubov coefficients follow from the same structure. For the crossed amplitude one obtains

$t+r>0$4

while the corresponding number distribution contains the factor

$t+r>0$5

The source paper is explicit that this spectrum is not thermal in the naive Hawking sense. The exponential depends on $t+r>0$6, not on the emitted energy $t+r>0$7, so there is no Planck factor of the form $t+r>0$8. The construction therefore preserves the collapse profile, null-shell structure, and eikonal exponentiation, but not the horizon formation or the energy-dependent thermality of gravity (Aoude et al., 29 Oct 2025).

4. Relation to the broader Vaidya family

The collapsing-shell example sits inside a much larger Vaidya literature, much of which supplies geometric input rather than a direct single copy. For outgoing Vaidya, one standard form is

$t+r>0$9

with smooth decreasing $A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$0 and exact Schwarzschild regions outside a finite retarded-time interval (Coudray, 2024). For generalized ingoing Vaidya, a standard form is

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$1

with matter variables

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$2

so that pure Vaidya $A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$3 is a pure Type II null fluid (Ojako et al., 2019). A complementary double-null formulation writes charged Vaidya as

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$4

with source

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$5

making the radial null directions and time-dependent mass and charge explicit (Chirenti et al., 2012).

These geometries are closely aligned with the null-coordinate structure used in the direct single-copy construction. In particular, the outgoing Vaidya metric can be inferred to have the formal Kerr–Schild shape

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$6

when flat spacetime is written in retarded null coordinates (Coudray, 2024). This suggests that the collapsing-shell example is not an isolated curiosity but a particularly tractable member of a broader Kerr–Schild-compatible class. The cited papers, however, do not themselves construct a gauge field, a biadjoint scalar, or a double-copy map.

5. Symmetry-selected structures and obstructions

Several symmetry analyses identify distinguished sectors of Vaidya that appear especially relevant to any extension of the single-copy program. In generalized Vaidya, the most general conformal Killing vector in the $A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$7 subspace is

$A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$8

while the homothetic case is recovered for $A_\mu=\frac{Q_{\rm B}}{4\pi r}\Theta(t+r)\,k_\mu,$9,

$\sqrt{\textrm{Vaidya}}$0

For pure Vaidya $\sqrt{\textrm{Vaidya}}$1, consistency forces $\sqrt{\textrm{Vaidya}}$2, so pure Vaidya admits only the homothetic Killing vector and no proper conformal Killing vector with nonconstant conformal factor (Ojako et al., 2019). This suggests that the self-similar sector is the natural symmetry-reduced arena for single-copy constructions.

That suggestion is sharpened by later work on homothetic Killing horizons in Vaidya-like spacetimes. For the standard spherical Vaidya metric

$\sqrt{\textrm{Vaidya}}$3

a conformal or homothetic Killing vector exists for linear mass profiles such as

$\sqrt{\textrm{Vaidya}}$4

with corresponding vectors

$\sqrt{\textrm{Vaidya}}$5

The same analysis extends to charged and rotating Vaidya-like metrics, again singling out linear advanced-time profiles (Ghoshal et al., 10 Apr 2026). A plausible implication is that self-similar, linear-in-$\sqrt{\textrm{Vaidya}}$6 sectors may provide the cleanest nontrivial backgrounds beyond the step-function shell.

An important obstruction comes from a different direction. The conformal Ricci–Bourguignon soliton analysis on outgoing Vaidya spacetime shows that the distinguished soliton-compatible vector field and scalar potential exist if and only if the mass function vanishes,

$\sqrt{\textrm{Vaidya}}$7

In the coordinates used there, this means the geometry reduces to flat Minkowski spacetime in outgoing Eddington–Finkelstein-like null coordinates (Rehman et al., 13 Aug 2025). This does not rule out the Kerr–Schild single copy, but it does exclude one particular attempt to extract a preferred vector/scalar pair from genuinely radiating Vaidya.

6. Status, misconceptions, and scope of the term

The phrase “Single Copy of Vaidya” has a narrower meaning than much of the surrounding Vaidya literature might suggest. The direct construction presently available is the Maxwell background

$\sqrt{\textrm{Vaidya}}$8

obtained from the collapsing-shell Vaidya metric by the Kerr–Schild classical double copy (Aoude et al., 29 Oct 2025). It is not an electromagnetic black hole, it has no horizon, and it does not reproduce Hawking thermality in the ordinary Planckian sense. What it does preserve is the null-shell profile, the $\sqrt{\textrm{Vaidya}}$9 structure, the eikonal exponentiation pattern, and a Bogoliubov-coefficient construction parallel to the gravitational problem.

A second misconception is to equate any preferred geometric structure on Vaidya with a single copy. The conformal Ricci–Bourguignon soliton paper is explicitly not a double-copy paper, and its main relevance is negative: the extra soliton structure survives only for $M(t+r)=M\,\Theta(t+r),$0, not for genuinely radiating Vaidya (Rehman et al., 13 Aug 2025). Likewise, the broader EF and double-null Vaidya literature gives indispensable geometric data—null coordinates, stress tensors, matter decompositions, conformal symmetries, and homothetic sectors—but not a gauge-theory single copy in the usual sense [(Ojako et al., 2019); (Chirenti et al., 2012); (Coudray, 2024)].

Accordingly, the encyclopedic status of the subject is twofold. In the strict sense, the single copy of Vaidya is the explicit Kerr–Schild electromagnetic background associated with a collapsing null shell. In the broader research sense, it is a developing program centered on which nonstationary Vaidya sectors retain enough null, Kerr–Schild, or homothetic structure to admit equally explicit gauge-theory counterparts.