> for all $(r, \theta, \varphi, t) \in \mathbb{R}^4$ with $A,B: r \mapsto A(r),B(r)$.
??? note "*Proof*:"
Will be added later.
Reducing the determination of the metric to only two functions $A$ and $B$.
## Exterior solution
Outside of the mass distribution the energy-momentum tensor vanishes, so we can impose $\mathbf{W} = \mathbf{0}$. Then, by imposing the weak field limit we have the following.
> *Principle 1*: a metric outside a static, spherically symmetric mass distribution is described by the **Schwarzschild metric**
>
> $$
> \bm{g} = \Big(1 - \frac{2 G M}{c^2 r}\Big)^{-1} dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - c^2 \Big(1 - \frac{2 G M}{c^2 r} \Big) dt \otimes dt,
> $$
>
> for all $(r, \theta, \varphi, t) \in \mathbb{R}^4$ with $G$ the gravitational constant and $M$ the mass of the spherically symmetric mass distribution.
??? note "*Derivation*:"
Will be added later.
Notice that for $r_s = \frac{2 G M}{c^2}$ the metric with these coordinates is not defined. This radius is called the **Schwarzschild radius**.
> *Theorem 1 (Birkhoff's theorem)*: the Schwarzschild metric is the only spherically symmetric solution, outside a spherical mass distribution.
??? note "*Proof*:"
Will be added later.
Note that static is automatically implied by spherical symmetry. An important consequence of the theorem is that a purely radially pulsating star cannot emit gravitational radiation, because outside of this star such gravitational radiation would amount to a time-dependent spherically symmetric spacetime geometry in (approximate) vacuum, which, according to the Birkhoff’s theorem, cannot be consistent with Einstein’s field equations.
