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

\begin{equation} {\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} \end{equation} and \begin{equation} {\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. \end{equation} The corresponding affine parameter for radial null geodesics in this spacetime, expressed implicitly, reads
\begin{equation} \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. \end{equation}

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?


1 Answer 1


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.

I have found that answer on the page 7 in the recent publication of Tian Zhou and Leonardo Modesto Geodesic incompleteness of some popular regular black holes (Phys. Rev. D 107, 044016 – Published 8 February 2023).


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