In the Schwarzschild spacetime, the tortoise coordinate $r_\ast$ is defined by the property that


Now, we cam integrate this. Multiply by $r$ on the numerator and denominator to get

$$r_\ast = \int \dfrac{1}{1-\frac{2M}{r}}dr=\int\dfrac{r}{r-2M}dr$$

Now integrate by parts with $u =r$ and $dv = dr/(r-2M)$ we get $du = dr$ and $v = \ln(r-2M)$. Then

$$r_\ast =r\ln(r-2M)-\int \ln(r-2M)dr$$

We integrate the last term to get

$$r_\ast = r\ln(r-2M)-(r-2M)\ln(r-2M)+(r-2M)+C$$

Reorganizing yields

$$r_\ast = r+2M\ln(r-2M)-2M+C$$

We can obviously choose $C$ to cancel the $2M$.

But anyway, virtually all references, shows a different $r_\ast$. The canonical one is

$$r_\ast = r+2M\ln \dfrac{r-2M}{2M}.$$

So one has one additional $2M$ on the denominator. This is equivalent of picking

$$C=2M-2M\ln 2M$$

Now why is that? It is clear to me that one can do that, after all the initial differential equation is still satisfied, but why everyone does it? What's the point with that $2M$ on the denominator inside the $\ln$?


Physicists really don't like to put dimensional variables inside a function like $\ln$, $\sin$, or $\exp$. By choosing $C = 2M - 2M \ln (2M)$, you can combine the logarithmic terms into the logarithm of the dimensionless ratio $(r-2M)/2M$.

Further details on why having "naked" dimensional variables inside a function like $\ln$ is a bad idea:


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