Page:The Meaning of Relativity - Albert Einstein (1922).djvu/127

$\gamma _{\mu \nu }$ will be such functions of $x_{1},x_{2},x_{3}$ as correspond to a three-dimensional continuum of constant positive curvature. We must now investigate whether such an assumption can satisfy the field equations of gravitation.

In order to be able to investigate this, we must first find what differential conditions the three-dimensional manifold of constant curvature satisfies. A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions,^[1] is given by the equations

${\begin{aligned}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}&=a^{2}\\dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}&=ds^{2}.\end{aligned}}$

By eliminating $x_{4}$, we get

$ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+{\frac {(x_{1}dx_{1}+x_{2}dx_{2}+x_{3}dx_{3})^{2}}{a^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}}}.$

As far as terms of the third and higher degrees in the $x_{\nu }$ we can put, in the neighbourhood of the origin of co-ordinates,

$ds^{2}=\left(\delta _{\mu \nu }+{\frac {x_{\mu }x_{\nu }}{a^{2}}}\right)dx_{\mu }dx_{\nu }.$

Inside the brackets are the $g_{\mu \nu }$ of the manifold in the neighbourhood of the origin. Since the first derivatives of the $g_{\mu \nu }$ and therefore also the $\Gamma _{\mu \nu }^{\sigma }$, vanish at the origin, the calculation of the $R_{\mu \nu }$ for this manifold, by (88), is very simple at the origin. We have

$R_{\mu \nu }={\frac {2}{a^{2}}}\delta _{\mu \nu }={\frac {2}{a^{2}}}g_{\mu \nu }.$

1. The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.