When mass/energy is falling inward as radiation (lightlike “null dust”), the nicest coordinates follow the ingoing light fronts themselves. Use advanced (ingoing) Eddington–Finkelstein time \(v\): each constant-\(v\) surface is an inward-moving light wavefront. In these coordinates the collapsing/accreting spacetime is
\[ ds^2 \;=\; -\Big(1-\frac{2\,m(v)}{r}\Big)\,dv^2 \;+\; 2\,dv\,dr \;+\; r^2 d\Omega^2, \]
so the redshift factor is \(A(v,r)=1-\dfrac{2m(v)}{r}\). Einstein’s equations reduce the dynamics to a mass-balance law:
\[ \boxed{\;T_{vv}(v,r)=\frac{1}{4\pi r^2}\,\frac{dm}{dv}\;} \]
i.e., the energy flux density carried by the ingoing radiation is the luminosity \(dm/dv\) spread over the sphere’s area \(4\pi r^2\). Positive flux (\(T_{vv}\ge 0\)) makes the mass parameter \(m(v)\) non-decreasing. The object gains mass as radiation falls in. These coordinates are also horizon-friendly: they stay regular where Schwarzschild \(t\) would misbehave, so they naturally describe ongoing accretion/collapse.
Next step we’ll relate this EF picture back to our time-first flux law by translating \(T_{vv}\) into \(T_{tr}\)
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(v\) | advanced (ingoing) EF time | labels inward-moving light wavefronts | time | “Timestamp stamped on each raindrop sheet” |
| \(m(v)\) | Vaidya mass function | mass enclosed as of lightfront \(v\) | mass (or length with \(G=c=1\)) | “Reservoir level rising with the storm” |
| \(T_{vv}\) | ingoing null-flux density | energy per area per \(dv\) carried by radiation | energy/(area·time) | “Power per square meter of downpour” |
| \(A(v,r)\) | redshift factor | \(A=1-\dfrac{2m(v)}{r}\) in Vaidya | dimensionless | “Light-cone tilt knob during rain” |
| \(r\) | areal radius | sphere area \(4\pi r^2\) | length | “Radius painted on the umbrella” |
| \(dm/dv\) | accretion luminosity | rate mass grows per EF time | mass/time | “How fast the reservoir fills” |