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I started studying the Schwarzschild metric and have seen multiple versions of the metric online; specifically different versions of the Schwarzschild radius. In some cases it is $\frac{2GM}{c^2}$, other cases $2M$, and still other cases $2GM$. Even if they're interchangeable I'm curious how that could be since the units don't match.

To be fair I'm a bit out of my depth here so probably missing something simple.

I'll attach screenshots for reference, below:

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The Schwarzschild radius is $$R= \frac{2GM}{c^2}$$ where $M$ describes the specific black hole we are analyzing and $G$ and $c$ are universal constants.

All of them are dimensionful units, so their values depend on the units selected. Physicists often choose units to simplify the calculations. For instance, in units where $c=1$ we get $$R= 2GM$$ and in units where both $c=G=1$ then $$R= 2M$$ In fact, it is always possible to choose units such that $$R=1$$

Any physical scenario is described by the dimensionless ratio $$\frac{r}{R}$$ This does not change regardless of the change of units used to simplify $c$, $G$, or $R$.

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    $\begingroup$ Thanks for the quick reply Dale, that makes sense. $\endgroup$
    – Fulfilmaker
    Commented Aug 4 at 14:31

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