Step 10. Second radial constraint: how energy density fixes the redshift profile

\[ \boxed{\;\frac{A}{r^{2}}\Big(-2\,r\,A\,\partial_{r}\Phi \;-\; A \;+\; 1\Big) \;=\; 8\pi\,T_{tt}\;, \qquad A=e^{2\Phi}\;} \]

This is the \(tt\)-Einstein equation in the same spherical, time-first gauge. It relates the energy density (in these coordinates) to a geometric combination of (i) the radial slope of the time potential \(\partial_r\Phi\) and (ii) how far the redshift factor \(A\) departs from flatness (\(A=1\)). Rearranged, it becomes a first-order ODE for \(A(r)\):

\[ \boxed{\;\partial_r A \;=\; -\,\frac{A-1}{r}\;-\;\frac{8\pi\,r}{A}\,T_{tt}\;} \]

Compare with Step 9’s ODE \(\partial_r A = 8\pi A r\,T_{rr} - (A-1)/r\): together they show how pressure (\(T_{rr}\)) and energy density (\(T_{tt}\)) pull on the same redshift dial \(A\) in complementary ways.

Intuition and checks:

• Vacuum (\(T_{tt}=0\)) gives \(\partial_r A = -(A-1)/r\) ⇒ \(A(r)=1+C/r\), and matching to asymptotic flatness makes \(C=-2M\), i.e. Schwarzschild.
• Positive energy density (\(T_{tt}>0\)) adds a negative contribution \(-8\pi r\,T_{tt}/A\): \(A\) drops faster with \(r\) (stronger redshift), exactly as you expect mass to deepen the potential well.
• Physical density seen by static observers with 4-velocity \(u^\mu=(1/\sqrt{A},0,0,0)\) is \(\rho=T_{\mu\nu}u^\mu u^\nu = T_{tt}/A\). So the same equation can be read as an \(A\)-ODE driven by \(\rho\).

Mini-Glossary