# Can an observer sitting at rest at infinity, simultaneously measure the proper time and proper distance of a particle travelling in any geodesic?

In the book General Relativity by M.P Hoson, the author tries to calculate the velocity of a particle radially infalling in Schwarzschild spacetime. The velocity is measured by the observer sitting at rest at infinity. For the same, he takes the ratio of proper distance and proper time of the particle, both measured by the observer. To calculate the proper distance, the path traveled by the particle should be spacelike, whereas, to calculate the proper time, the path traveled should be timelike. Since the particle can travel with only one of the paths, how does the observer simultaneously calculate the proper time and proper distance along the particle geodesic

In most texts, for an observer at infinity, and a particle falling from inifinity, that would be $$\frac{dr}{dt} = \left(1 - \frac{r_s}{r}\right)\left(\frac{r_s}{r}\right)^{1/2}$$ in the usual Schwarzschild coordinates. Whilst $$dt$$ is also an increment in proper time for the distant observer, $$dr$$ is not a proper distance. The proper distance increment between two radial coordinates could be written as $$ds = \frac{dr}{(1 - r_s/r)^{1/2}}$$ and you could then say $$\frac{ds}{dt} = \left(1-\frac{r_s}{r}\right)^{1/2}\left(\frac{r_s}{r}\right)^{1/2}\ .$$ Is that what Hoson means? This would be the equivalent of estimating a velocity by the observer receiving successive pictures of the falling particle passing "milestones" in $$r$$, applying a correction to the distance travelled between the milepstones to account for the curvature of space, and then dividing this distance by the time between successive pictures.