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

Rh first calculating the $$\Gamma_\sigma^{\mu\nu}$$ from (107) and (108a). We have {{MathForm2|(108b)| $$ \left. \begin{align} \Gamma_{\alpha\beta}^\sigma & = \frac{1}{2}\frac{x_\sigma}{r}\cdot\frac{\lambda'x_\alpha x_\beta + 2\lambda r\delta_{\alpha\beta}}{1+\lambda r^2}\text{ (for }\alpha,\beta,\sigma = 1, 2, 3\text{)} \\ \Gamma_{44}^4 & = \Gamma_{4\beta}^\alpha = \Gamma_{\alpha\beta}^4 = 0\text{ (for }\alpha,\beta = 1, 2, 3\text{)} \\ \Gamma_{4\alpha}^4 & = \frac{1}{2}f^{-2}\frac{\delta f^2}{\delta x_\alpha}, \Gamma_{44}^\alpha = -\frac{1}{2}f^{-2}\frac{\delta f^2}{\delta x_\alpha} \end{align} \right\}. $$}}

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

in which we have put {{MathForm2|(109a)| $$ \left. \begin{align} x_4 & = l \\ x_1 & = r\sin\theta\sin\phi \\ x_2 & = r\sin\theta\cos\phi \\ x_3 & = r\cos\theta \\ A & = \frac{\kappa M}{4\pi} \end{align} \right\}. $$}}

$$M$$ 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