Vaidya used ingoing null time \(v\) (great for following light). Our time-first work is in the diagonal \((t,r)\) chart with \(ds^2=-A\,dt^2+A^{-1}dr^2+r^2d\Omega^2\). To compare the two, relate coordinates: ingoing EF time satisfies
\[ v \;=\; t + r_\ast, \qquad \frac{dr_\ast}{dr}=\frac{1}{A}. \]
So the Jacobian pieces are \(\partial v/\partial t=1\) and \(\partial v/\partial r=1/A\). Stress–energy is a covariant tensor, so
\[ T_{tr} \;=\; T_{\alpha\beta}\,\frac{\partial x^\alpha}{\partial t}\,\frac{\partial x^\beta}{\partial r}. \]
For null dust, only \(T_{vv}\neq 0\), hence
\[ \boxed{\,T_{tr} \;=\; T_{vv}\,\frac{\partial v}{\partial t}\,\frac{\partial v}{\partial r} \;=\; \frac{T_{vv}}{A}\, .} \]
Plug this into the flux law (Step 8), \(\partial_t\Phi=-4\pi r\,T_{tr}\), to get
\[ \partial_t\Phi \;=\; -\,\frac{4\pi r}{A}\,T_{vv}. \]
Using Vaidya’s mass-balance \(T_{vv}=\frac{1}{4\pi r^2}\frac{dm}{dv}\) (Step 13) gives
\[ \partial_t\Phi \;=\; -\,\frac{1}{A\,r}\,\frac{dm}{dv}, \]
making it explicit that the same physical inflow that raises \(m(v)\) in EF time is exactly what drives the time potential \(\Phi\) to evolve in our diagonal gauge. The “dictionary factor” \(1/A\) is just the coordinate tilt between \(v\) and \((t,r)\).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(v\) | advanced EF time | labels ingoing light fronts | time | “Timestamp on each rain sheet” |
| \(t\) | diagonal time | our time-first chart’s time label | time | “Flipbook page in our gauge” |
| \(r_\ast\) | tortoise radius | stretches \(r\) so light moves at 45° | \(dr_\ast/dr=1/A\) | “Log-stretched radial ruler” |
| \(A\) | redshift factor | packs lapse & radial stretch | \(A=e^{2\Phi}\) | “Light-cone tilt knob” |
| \(T_{vv}\) | null-flux density | energy carried by ingoing radiation | \(=\frac{1}{4\pi r^2}\frac{dm}{dv}\) | “Power per area of the downpour” |
| \(T_{tr}\) | radial flux (diag.) | flux seen in \((t,r)\) gauge | \(T_{tr}=T_{vv}/A\) | “Same rain, viewed through our window” |
| \(\Phi\) | time potential | drives lapse \(N=e^{\Phi}\) and \(A\) | \(\partial_t\Phi=-4\pi r\,T_{tr}\) | “Altitude of time responding to flow” |