# Why falling body do not reach Schwarzschild radius in coordinate time but do in proper time?

I do still not really understand the solution for a radially falling body in the Schwarzschild metric:

One the one hand, in terms of proper time $$\tau$$ the equation of motion can be solved and gives a solution $$r(\tau)$$ for $$r\ge 0$$. The body crosses Schwarzschild radius $$r_s$$ without any problems and the solution is valid also for $$r\le r_s$$.

On the other hand, the same calculation, carried out in coordinate time t shows, that the particle is never reaching $$r_s$$ in finite time.

The body definitely crosses the Schwarzschild boundary (although is cannot be proven, because nobody can come back), but why is there no solution for $$r\le r_s$$? What is the deeper reason, that this region is not mapped by using t? I would expect, that a switch to a new coordinate system maps a point into another unique point. For points inside $$r_s$$ this seems to be not possible anymore, but, on the other hand, I can easily mark an inside point in the t-r-Diagram and the Schwarzschild Metric is, from its definition, not restricted to $$r>r_s$$.

I know, that its an artifact due to geometry, but how can it be, that the mapping of inside points between the two coordinate systems fails? Naively speaking, there should be a solution also for the region within Schwarzschild radius because moving astronauts COULD get there and make a meeting. This Meeting, taken as a real event is not visible in the t-r- coordinates. Thinking about that makes me crazy...

• youtube.com/watch?v=vNaEBbFbvcY Commented Aug 9, 2021 at 10:59
• In many coordinate systems, falling bodies reach the Schwarzschild radius in finite coordinate time. Conversely even in flat spacetime, you can choose a coordinate system (which, like traditional Schwarzschild coordinates, don't cover the whole spacetime) so that an object that does pass through an event $p$ doesn't reach that event in any finite coordinate time. Coordinate systems are arbitrary. Physically meaningful statements are independent of which coordinate system we use. Are you asking why a hovering observer never sees a falling body reach the Schwarzschild radius? Commented Aug 9, 2021 at 13:11
• Commented Sep 8, 2021 at 11:16

The causal structure, defined by light cones can be shown in t-r plane. The slope of the cones given by

$$$$\tag{1} \frac{dt}{dr} = \pm \frac{1}{(1-\frac{r_S}{r})}$$$$

increases to infinity for $$r\rightarrow r_S$$. (first picture below) Hence light rays asymptotically 'reaches' Schwarzschild radius in this coordinate system. The idea of tortoise coordinate is to make $$\frac{dt}{dr}$$ smaller. Just by integrating (1), we get $$r^* = r+r_S \text{ln}(\frac{r}{r_S}-1)+\text{const}$$. We can now map $$r < r_S$$ using tortoise coordinates ($$t,r^*$$) in which Schwarzschild metric beomes,

$$$$ds^2 = -(1-\frac{r_S}{r})(dt^2 + dr^{*2}) + r^2d\Omega^2)$$$$

(because $$dr^* = dr/(1-\frac{r_S}{r}))$$

Now $$dt/dr^*$$ is a constant hence we have light cones which are not asymptotic in $$t-r^*$$ plane (second picture).

The proper time and coordinate time can be related (from geodesic equation) as,

$$$$\tag{2} \frac{d\tau}{dt} = (1-\frac{r_S}{r})^{1/2}$$$$ Hence, distant observer (named B) will observe light coming from infalling observer (name him A), red shifted by (one over) this factor (and also we cannot define once $$r - physically A appears not only to be still but gets reddened and hence eventually dimmer to B).

A reaches $$r_S$$ in finite proper time but for B at rest, this would take infinite time. In other words, $$r_S$$ forms the Cauchy Horizon beyond which we have unique geodesics but within it, is a singularity (which does not belong to the Lorentzian manifold) with no future null-like geodesics.

So from (2) it is clear that, even though, A crosses $$r_S$$, B cannot see this event. Even though A can hold meeting inside the Horizon, B will never know of it. Is there a transformation $$t\rightarrow\tau$$? All such transformations are affine, in other words, proper time (atleast for null-like geodesics) are affine parameters which appear in the geodesic equations. But from the very definition of singularity, future null geodesics do not exists.