The interior Schwarzschild metric is framed in a
spherical coordinate system with the body's centre located at the origin, plus the time coordinate. Its
line element is : \begin{align} c^2\, d\tau^2 =& -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_s}{r_g}} - \sqrt{1 - \frac{r^2 r_s}{r_g^3}}\right)^2 c^2 \,dt^2 \\ &{} + \left(1 - \frac{r^2 r_s}{r_g^3}\right)^{-1} \,dr^2 + r^2 (d\theta^2 + \sin^2\theta \,d\varphi^2), \end{align} where : \tau is the
proper time (time measured by a clock moving along the same
world line with the
test particle), : c is the
speed of light, : t is the time coordinate (measured by a stationary clock located infinitely far from the spherical body), : r is the Schwarzschild radial coordinate; each surface of constant t and r has the geometry of a sphere with measurable (proper) circumference 2\pi r and area 4\pi r^2 (as by the usual formulas), but the warping of space means that the proper distance from each shell to the center of the body is greater than r, : \theta is the
colatitude (angle from north, in units of
radians), : \varphi is the
longitude (also in radians), : r_s is the
Schwarzschild radius of the body, which is related to its mass M by r_s = 2GM/c^2, where G is the
gravitational constant (for ordinary stars and planets, this is much less than their proper radius), : r_g is the value of the r coordinate at the body's surface; this is less than its proper (measurable interior) radius, although for the Earth the difference is only about 1.4 millimetres. This solution is valid for r \leq r_g. For a complete metric of the sphere's gravitational field, the interior Schwarzschild metric has to be matched with the exterior one, : -c^2 \,d\tau^2 = \left(1 - \frac{r_s}{r}\right) c^2 \,dt^2 - \left(1 - \frac{r_s}{r}\right)^{-1} \,dr^2 - r^2 (d\theta^2 + \sin^2\theta \,d\varphi^2), at the surface. It can easily be seen that the two have the same value at the surface, i.e., at r = r_g.
Other formulations Defining a parameter \mathcal{R}^2 = r_g^3 / r_s, we get : \begin{align} -c^2 d\tau^2 =& -\frac{1}{4} \left(3 \sqrt{1 - \frac {r^2_g}{\mathcal{R}^2}} - \sqrt{1 - \frac{r^2}{\mathcal{R}^2}}\right)^2 c^2 \,dt^2 \\ &{} + \left(1 - \frac{r^2}{\mathcal{R}^2}\right)^{-1} \,dr^2 + r^2 (d\theta^2 + \sin^2\theta \,d\varphi^2). \end{align} We can also define an alternative radial coordinate \eta = \arcsin \frac{r}{\mathcal{R}} and a corresponding parameter \eta_g = \arcsin \frac{r_g}{\mathcal{R}} = \arcsin \sqrt{\frac{r_s}{r_g}}, yielding : c^2 d\tau^2 = \left(\frac{3 \cos \eta_g - \cos \eta}{2}\right)^2 c^2 \,dt^2 - \frac{dr^2}{\cos^2 \eta} - r^2 (d\theta^2 + \sin^2\theta \,d\varphi^2). ==Properties==