What unit of measurement are used for  and  in the Schwarzschild metric? So I understand that  in any metric solution stands for mass. But what unit of measurement do you use to define ?
 A: You can use whatever unit you want as long as they are consistent. The mass and radius appear in the factor:
$$ 1 - \frac{2GM}{c^2r} $$
and to be dimensionally consistent $2GM/c^2r$ has to be dimensionless. So for example if you use kilograms for mass, metres for $r$ and metres per second for $c$ then you need to give the gravitational constant $G$ in units of $\mathrm m^3 \mathrm {kg}^{-1} \mathrm s^{-2}$. Alternatively you could choose to express the mass in Solar masses then you would need to change the value of $G$ accordingly.
To avoid having to keep track of all the constants we tend to use geometric units in which we set both $G$ and $c$ to unity. However this can be a bit confusing for the beginner and if you're just starting out and trying to do calculations I suggest you stick to the SI units.
A: Physics does not depend on measurement units. Every consistent system of units may be used to write the GR metric. For instance, the Schwarzschild metric can be written as
$$
ds^2=\left( 1-\frac{2GM}{c^2r}  \right)c^2 dt^2 - \left( 1-\frac{2GM}{c^2r}  \right)^{-1}dr^2 - r^2 d\theta^2-r^2 \sin^2(\theta)d\phi^2
$$
where the constant $G$, $c$, and the dimensional physical quantities $M$ and $r$ can be expressed in any system of units.
Sometimes, it may be convenient to introduce a special system of units making the value of $G$ and $c$ equal to $1$. However, this choice does not modify the physical dimensions of $r$ and $M$. Therefore, dimensional analysis allows restoration of the (apparently missing) $G$'s and $c$'s in a formula like
$$
ds^2=\left( 1-\frac{2M}{r}  \right) dt^2 - \left( 1-\frac{2M}{r}  \right)^{-1}dr^2 - r^2 d\theta^2-r^2 \sin^2(\theta)d\phi^2
$$
