When it comes to getting closer to the Schwarzschild radius, how is discrete a limit?

Question

From Keeton (2014) in Principles of Astrophysics: Using Gravity and Stellar Physics to Explore the Cosmos, Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows:

$$\frac{t_r}{t}=\sqrt{1-\frac{r_s}{r}}$$

where:

$t_r$ is the elapsed time for an observer at radial coordinate r within the gravitational field;

t is the elapsed time for an observer distant from the massive object (and therefore outside of the gravitational field);

r is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object);

$r_s$ is the Schwarzschild radius.

(source wiki).

When $r_1=r_s+n+1$ and $r_2=r_s+n+2$ and n is a positive integer, is it mathematically correct to study the ratio of $t_{r1}$ and $t_{r2}$?

Is it a valid approximation of the time dilation ratio for observers so close to each-other and so close to the Schwarzschild radius, when 1 is negligibly small compared to $r_s$?

Is this a simple manner to describe what some papers write about discrete mathematical hypothesis as “limits” to the infinity otherwise produced by formulas, when getting very close to the Schwarzschild radius?

Answer 1

You could use an additive factor as you suggest, but then the results will depend on the units in which you measure $r$, which does not seem very satisfactory.

It is more usual (and probably more useful) to use a multiplicative factor - say $r=r_s(1+\epsilon)$ - so that the ratio $\frac {r_s}{r}$ depends only on $\epsilon$ and not on the units used.

Answer 2

Note that the approximation is useful near a large body, which means you must have $r >> r_s$. The approximation will break down near the Schwarzschild radius.

Your question asks about infinities produced by formulas near the Schwarzschild radius, but in general such infinities are coordinate artifacts rather than anything inherent to the physics. The only physical singularity is inside the event horizon, not at it.