Vaidya and Null-Fluid Spacetimes

Vaidya spacetime is the spherical geometry sourced by a directed stream of null energy. A single mass function replaces the Schwarzschild constant, but its derivative is physical: in advanced coordinates an increasing mass represents ingoing positive-energy radiation, while in retarded coordinates a decreasing mass represents outgoing positive-energy radiation. Confusing those two charts reverses the energy-condition statement.

Required background. Hilbert Stress–Energy from Metric Variation supplies the stress tensor and null energy condition.

Helpful background. The Einstein Equation fixes the field equations; The Riemann Curvature Tensor supplies invariant tests; Exact Solutions of Einstein’s Equations locates the family; GR Formulations develops null and spacelike formulations; and Continuum Kinematics and Balance Laws supplies continuum conservation.

This page uses geometrized units $G=c=1$. The function $m$ is the length $m_{\mathrm{geom}}=G M_{\mathrm{phys}}$; with dimensions restored, the ingoing flux below is $T_{vv}=(4\pi G r^2)^{-1}dm_{\mathrm{geom}}/dv$.

Ingoing and outgoing forms

In an advanced null coordinate $v$, the ingoing Vaidya metric is

$$ds^2=-\left(1-\frac{2m(v)}{r}\right)dv^2 +2\,dv\,dr+r^2d\Omega^2.$$

Direct substitution into Einstein’s equation gives

$$T_{ab}=\frac{1}{4\pi r^2}\frac{dm}{dv} (dv)_a(dv)_b.$$

The one-form $k_a=-(dv)_a$ is null, so this has the null-dust form $T_{ab}=\mu k_ak_b$ with $\mu=m_{,v}/(4\pi r^2)$. The null energy condition therefore requires

$$\frac{dm}{dv}\ge0.$$

The cross term $+2\,dv\,dr$ is part of this conclusion. It identifies $v$ as advanced time: radial ingoing null curves are $v=\text{constant}$.

For outgoing radiation use retarded time $u$:

$$ds^2=-\left(1-\frac{2m(u)}{r}\right)du^2 -2\,du\,dr+r^2d\Omega^2,$$ $$T_{ab}=-\frac{1}{4\pi r^2}\frac{dm}{du} (du)_a(du)_b.$$

Positive outgoing null energy thus requires $dm/du\le0$. The original radiating-star solution and its null-fluid interpretation are given in Vaidya 1951, pp. 264–276.

When the mass function is constant, either form is simply Schwarzschild written in ingoing or outgoing Eddington–Finkelstein coordinates. This limit simultaneously checks the sign of the cross term, the stress normalization, and the identification of the mass parameter.

Geometry carried by the mass function

The areal-radius identity gives

$$g^{ab}(\nabla_a r)(\nabla_b r)=1-\frac{2m}{r},$$

so $m$ is the spherical Misner–Sharp mass. The hypersurface $r=2m$ is marginal with respect to one radial null expansion in the natural spherical foliation. It is generally not the event horizon: the latter depends on the entire future mass history. Horizon mechanics and global causal boundaries are treated elsewhere; here $r=2m$ is a local geometric diagnostic.

Null dust is trace-free, hence

$$T^a{}_a=0, \qquad R=0.$$

Vanishing Ricci scalar does not mean vacuum. The tensor $R_{ab}=8\pi T_{ab}$ is nonzero whenever $m'$ is nonzero, and an observer or null frame measures a finite directed flux. Indeed $R_{ab}$ has the rank-one null form. Scalar polynomial invariants can be unusually insensitive to such a null Ricci contribution: the Kretschmann scalar is

$$R_{abcd}R^{abcd}=\frac{48m^2}{r^6},$$

with no explicit first derivative of $m$. Tensor components, Cartan invariants, or differentiated curvature still distinguish radiating Vaidya from an instantaneous Schwarzschild metric. This is a useful warning against classifying spacetimes from a short list of scalar invariants alone.

The central locus $r=0$ is singular whenever $m\ne0$, as the Kretschmann scalar shows. At $r=2m$ that scalar is finite for finite nonzero $m$, so the zero of $g^{ab}\nabla_ar\nabla_br$ is not itself a scalar-curvature singularity.

Flux regions and matching regimes

A useful model chooses $m$ piecewise smoothly. An initially constant mass can be followed by a finite interval with $m_{,v}>0$ and then another constant value. The two constant regions are Schwarzschild domains, and the middle region is an ingoing radiation pulse. If $m$ is continuous at the null interfaces, there is no delta-function shell there; a jump in $m$ produces an impulsive null shell and must be included in the distributional stress tensor.

Vaidya describes pure radial null dust, not arbitrary radiation from a star. It has no pressure transverse to the rays, no frequency distribution, and no backscattering sector. An exterior matched to a material interior must satisfy the appropriate timelike or null junction conditions, including the surface energy flux. Nor is outgoing classical Vaidya by itself a self-consistent semiclassical evaporation solution: choosing $m(u)$ prescribes the flux rather than deriving it.

Common pitfalls

Reading the sign of $m'$ without the cross term. The advanced metric has $+2\,dv\,dr$ and positive flux for $m_{,v}\ge0$; the retarded metric has $-2\,du\,dr$ and positive outgoing flux for $m_{,u}\le0$.

Equating $R=0$ with vacuum. Null dust is trace-free. Inspect $R_{ab}$ or a frame component, not only the Ricci scalar.

Exercises

Show that the displayed stress tensor satisfies the null energy condition precisely when its scalar coefficient is nonnegative.

Solution

For $T_{ab}=\mu k_ak_b$ and any null vector $\ell^a$,

$$T_{ab}\ell^a\ell^b =\mu(k_a\ell^a)^2.$$

The square is nonnegative. If $\mu\ge0$, the null energy condition holds for every $\ell^a$. Conversely, choose a null direction not orthogonal to $k_a$; if $\mu<0$, the contraction is negative. Thus ingoing Vaidya requires $m_{,v}\ge0$, while outgoing Vaidya requires $-m_{,u}\ge0$.

References

• P. C. Vaidya, “The Gravitational Field of a Radiating Star,” Proceedings of the Indian Academy of Sciences, Section A 33, 264–276 (1951). DOI.