Page:The Meaning of Relativity - Albert Einstein (1922).djvu/117

first calculating the $\Gamma _{\sigma }^{\mu \nu }$ from (107) and (108a). We have

$\left.{\begin{aligned}\Gamma _{\alpha \beta }^{\sigma }&={\frac {1}{2}}{\frac {x_{\sigma }}{r}}\cdot {\frac {\lambda 'x_{\alpha }x_{\beta }+2\lambda r\delta _{\alpha \beta }}{1+\lambda r^{2}}}{\text{ (for }}\alpha ,\beta ,\sigma =1,2,3{\text{)}}\\\Gamma _{44}^{4}&=\Gamma _{4\beta }^{\alpha }=\Gamma _{\alpha \beta }^{4}=0{\text{ (for }}\alpha ,\beta =1,2,3{\text{)}}\\\Gamma _{4\alpha }^{4}&={\frac {1}{2}}f^{-2}{\frac {\delta f^{2}}{\delta x_{\alpha }}},\Gamma _{44}^{\alpha }=-{\frac {1}{2}}f^{-2}{\frac {\delta f^{2}}{\delta x_{\alpha }}}\end{aligned}}\right\}.$

(108b)

With the help of these results, the field equations furnish Schwarzschild's solution:

$ds^{2}=\left(1-{\frac {A}{r}}\right)dl^{2}-\left[{\frac {dr^{2}}{1-{\frac {A}{r}}}}+r^{2}(\sin ^{2}\theta d\phi ^{2}+d\theta ^{2})\right]$

(109)

in which we have put

$\left.{\begin{aligned}x_{4}&=l\\x_{1}&=r\sin \theta \sin \phi \\x_{2}&=r\sin \theta \cos \phi \\x_{3}&=r\cos \theta \\A&={\frac {\kappa M}{4\pi }}\end{aligned}}\right\}.$

(109a)

$M$ denotes the sun's mass, centrally symmetrically placed about the origin of co-ordinates; the solution (109) is valid only outside of this mass, where all the $T_{\mu \nu }$ vanish. If the motion of the planet takes place in the $x_{1}-x_{2}$ plane then we must replace (109) by

$ds^{2}=\left(1-{\frac {A}{r}}\right)dl^{2}-{\frac {dr^{2}}{1-{\frac {A}{r}}}}-r^{2}d\phi ^{2}.$

(109b)