In flat space-time the interval, referred to spherical polar coordinates and time, is \[ ds^{2} = -dr^{2} - r^{2}\, d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} + dt^{2}. \Tag{(38.11)} \] If we consider what modifications of this can be made without destroying the spherical symmetry in space, the symmetry as regards past and future time, or the static condition, the most general possible form appears to be \[ ds^{2} = -U(r)\, dr^{2} - V(r)\, (r^{2}\, d\theta^{2} + r^{2} \sin^{2}\theta\, d\phi^{2}) + W(r)\, dt^{2}, \Tag{(38.12)} \] where $U$, $V$, $W$ are arbitrary functions of~$r$. Let \[ r_{1}^{2} = r^{2} V(r). \] Then \Eq{(38.12)} becomes of the form \[ ds^{2} = - U_{1}(r_{1})\, dr_{1}^{2} - r_{1}^{2}\, d\theta^{2} - r_{1}^{2} \sin^{2}\theta\, d\phi^{2} + W_{1}(r_{1})\, dt^{2}, \Tag{(38.13)} \] where $U_{1}$ and $W_{1}$ are arbitrary functions of~$r_{1}$. There is no reason to regard $r$ in~\Eq{(38.12)} as more immediately the counterpart of~$r$ in~\Eq{(38.11)} than $r_{1}$~is. If the functions $U$, $V$, $W$ differ only slightly from unity, both $r$ and $r_{1}$ will have approximately the properties of the radius-vector in Euclidean geometry; but no length in non-Euclidean space can have exactly the properties of a Euclidean radius-vector, and it is arbitrary whether we choose $r$ or~$r_{1}$ as its closest representative. We shall here choose~$r_{1}$, and accordingly drop the suffix, writing \Eq{(38.13)} in the form \[ ds^{2} = -e^{\lambda}\, dr^{2} - r^{2}\, d\theta^{2} - r^{2} \sin^{2}\theta\, d\phi^{2} + e^{\nu}\, dt^{2}, \Tag{(38.2)} \] where $\lambda$ and~$\nu$ are functions of $r$~only. Moreover since the gravitational field (or disturbance of flat space-time) due to a particle diminishes indefinitely as we go to an infinite distance, we must have $\lambda$ and~$\nu$ tend to zero as $r$~tends to infinity. Formula~\Eq{(38.2)} will then reduce to~\Eq{(38.11)} at an infinite distance from the particle. Our coordinates are \[ x_{1} = r,\quad x_{2} = \theta,\quad x_{3} = \phi,\quad x_{4} = t, \] and the fundamental tensor is by~\Eq{(38.2)} \[ g_{11} = -e^{\lambda},\ g_{22} = -r^{2},\ g_{33} = -r^{2} \sin^{2}\theta,\ g_{44} = e^{\nu}, \Tag{(38.31)} \] and \[ g_{\mu\nu} = 0\quad\text{if $\mu \neq \nu$.} \] The determinant~$g$ reduces to its leading diagonal $g_{11} g_{22} g_{33} g_{44}$. Hence \[ -g = e^{\lambda + \nu} r^{4} \sin^{2}\theta, \Tag{(38.32)} \] and $g^{11} = 1/g_{11}$, etc., so that \[ g^{11} = -e^{-\lambda},\ \ g^{22} = -1/r^{2},\ \ g^{33} = -1/r^{2} \sin^{2}\theta,\ \ g^{44} = e^{-\nu}. \Tag{(38.33)} \] Since all the $g^{\mu\nu}$ vanish except when the two suffixes are the same, the summation disappears in the formula for the $3$-index symbols~\Eq{(27.2)}, and \[ \{\mu\nu, \sigma\} = \tfrac{1}{2} g^{\tau\sigma}\left( \frac{\dd g_{\mu\sigma}}{\dd x_{\nu}} + \frac{\dd g_{\nu\sigma}}{\dd x_{\mu}} + \frac{\dd g_{\mu\nu}}{\dd x_{\sigma}} \right)\quad\text{not summed.} \] \PageSep{84}