The famous 1939 paper by Oppenheimer (and Snyder) that proved that stars could collapse into Black Holes has me puzzled. I can’t get the signs to match in his first 2 field equations for the same metric, and those particular signs are crucial for his very first conclusion.
$$ds^2 = e^\nu dr^2 - e^\lambda dr^2 - r^2 (d\theta ^2 + \sin^2 \theta d\phi ^2).$$
Oppenheimer: (His spherical coordinates appear to be: 1,2,3,4 = radius,$\theta$,$\phi$ and time.)
$$-8\pi T_1^1 = e^{-\lambda} \left(\frac {\nu'}{r}+\frac{1}{r^2}\right)-\frac{1}{r^2}.$$
$$8\pi T_4^4 = e^{-\lambda} \left(\frac {\lambda'}{r}-\frac{1}{r^2}\right)+\frac{1}{r^2}.$$
Maxima program: (Maxima’s spherical coordinates are: 1,2,3,4 = time,radius,$\theta$,$\phi$)
$$E_2^2 = e^{-\lambda} \left(\frac {\nu'}{r}+\frac{1}{r^2}\right)+\frac{1}{r^2}.$$
$$E_1^1 = e^{-\lambda} \left(-\frac {\lambda'}{r}+\frac{1}{r^2}\right)+\frac{1}{r^2}.$$
The opposite signs on the one-over-radius-squared terms in each field equation are what he bases his very first conclusion on – “…that unless lambda vanishes at least as rapidly as radius-squared when $r$ goes to 0…” Can anyone explain what is going wrong? Maxima has never failed any of my hand-checks of its calculations.
I also cannot find Oppenheimer and Snyder’s arguments in any of the many physics textbooks I have bought from Amazon after retiring from engineering. If anyone knows where that might be in recent publications, please let me know.
