# Completeness of radial null geodesics in spacetime of Schwarzschild interior solution

The metric of static spherically symmetric spacetime can be written as $$ds^2=e^{2\nu}~c^2 dt^2-e^{2\lambda}~dr^2-r^2~(d\theta^2+\sin^2{\theta}~d \phi^2) \tag{1},$$ where $$r$$ is curvature radius of a 2-sphere ($$r\ge0$$).

Defining dimensionless variable $$r$$ and parameter $$\alpha$$ $$r\hat{=}\frac{r}{R},~~~r_S\equiv\frac{2 G M}{c^2},~~~\alpha\equiv\frac{r_S}{R},$$ the metric components of spacetime in question are

$$$${\rm e}^{-\lambda} =\left\{ \begin{array}{rcl} \sqrt{1-\alpha~r^2}~ & \mbox{,} & 0\leq r \leq 1 \\ \\\sqrt{1-\alpha~ r^{-1}}~ & \mbox{,} & 1 < r < \infty~, \end{array}\right.\tag{2}$$$$ and $$$${\rm e}^{\nu} =\left\{ \begin{array}{rcl} 3/2~\sqrt{1-\alpha}-1/2~\sqrt{1-\alpha~r^2})~ & \mbox{,} & 0\leq r \leq 1 \\ \\\sqrt{1-\alpha~ r^{-1}}~ & \mbox{,} & 1 < r < \infty~.\tag{3} \end{array}\right.$$$$ The corresponding affine parameter for radial null geodesics in this spacetime, expressed implicitly, reads
$$$$\label{affine lambda} \lambda(r,\theta,\phi) =\pm~ A~\left\{ \begin{array}{rcl} 3/2~\sqrt{1/\alpha-1}~ \arcsin{(\sqrt{\alpha}~r)}-1/2~r & \mbox{,} & 0 \leq r \leq 1 \\ \\r-1+3/2~\sqrt{1/\alpha-1}~ \arcsin{(\sqrt{\alpha}~1)}-1/2 & \mbox{,} & 1 < r < \infty~. \tag{4}\end{array}\right.$$$$

The constant $$A$$ in the equation is a scale factor, the sign plus denotes outgoing, the minus ingoing geodesics.

Below the Buchdahl limit ($$\alpha<8/9$$) the geodesics should be complete, i.e. their affine parameter $$\lambda$$ should extend from $$-\infty$$ to $$+\infty$$, see Valter Moretti.

However, due to equation (4) $$\lambda$$ extends only from $$0$$ to $$+\infty$$. Hence, every radial null geodesics is a half line. What am I missing here?

To resolve that problem I have thought of idea of antipodal identification that would equate the points of two half lines in combination with the sign switch of the affine parameter $$r(\lambda,\theta,\phi)\leftrightarrow r(-\lambda,\theta-\pi,\phi+\pi). \tag{5}$$ and thus representing full geodesics with affine parameter $$\lambda \in (-\infty,+\infty)$$

Would it be a right procedure?

What I was missing is the notion of analytical continuation. While the coordinate $$r$$ is always positive the geodesics should be continued to the antipodal direction $$\theta \rightarrow \pi-\theta,~~~\phi \rightarrow \phi+ \pi$$. Metrics that are solution of Einstein field equations for static spherically symmetric perfect fluid sphere are invariant to $$r~\leftrightarrow -r$$ transformation and therefore the antipodal continuation of geodesics there is always analytic.