Question: What theorem states only vacuum solution is spherically symmetric?
About the article "Black hole", section "Physical properties".

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In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static.
In fact, the static assumption is stronger than required, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; then one obtains the Schwarzschild solution.
Birkhoff's theorem has the consequence that any pulsating star which remains spherically symmetric cannot generate gravitational waves (as the region exterior to the star must remain static). 
In physics, in the context of electromagnetism, Birkhoff's theorem concerns spherically symmetric static solutions of Maxwell's field equations of electromagnetism.
The theorem is due to George D. Birkhoff.
It states that any spherically symmetric solution of the source-free Maxwell equations is necessarily static.
Pappas (1984) gives two proofs of this theorem.  
The conclusion that the exterior field must also be "stationary" is more surprising, and has an interesting consequence.
Suppose we have a spherically symmetric star of fixed mass which is experiencing spherical pulsations.
Then Birkhoff's theorem says that the exterior geometry must be Schwarzschild; the only effect of the pulsation is to change the location of the stellar surface.
This means that a spherically pulsating star cannot emit gravitational waves.
Another interesting consequence of Birkhoff's theorem is that for a spherically symmetric thin shell, the interior solution must be given by the Minkowski metric; in other words, the gravitational field must vanish inside a spherically symmetric shell.
This agrees with what happens in Newtonian gravitation. 
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of "nested round spheres".
In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is "adapted" to these nested round spheres.
The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere.
However, radial distances and angles are not accurately represented.
These charts have many applications in metric theories of gravitation such as general relativity.
They are most often used in static spherically symmetric spacetimes.
In the case of general relativity, Birkhoff's theorem states that every "isolated" spherically symmetric vacuum or electrovacuum solution of the Einstein field equation is static, but this is certainly not true for perfect fluids.
We should also note that the extension of the exterior region of the Schwarzschild vacuum solution inside the event horizon of a spherically symmetric black hole is not static inside the horizon, and the family of (spacelike) nested spheres cannot be extended inside the horizon, so the Schwarzschild chart for this solution necessarily breaks down at the horizon. 
A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the (rotation) group formula_1 and the orbits of this group are 2-dimensional spheres (2-spheres).
The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations".
The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere).
Spherical symmetry is a characteristic feature of many solutions of Einstein's field equations of general relativity, especially the Schwarzschild solution.
A spherically symmetric spacetime can be characterised in another way, namely, by using the notion of Killing vector fields, which, in a very precise sense, preserve the metric.
The isometries referred to above are actually local flow diffeomorphisms of Killing vector fields and thus generate these vector fields.
For a spherically symmetric spacetime formula_2, there are precisely 3 rotational Killing vector fields.
Stated in another way, the dimension of the Killing algebra is 3  formula_3.
It is known (see Birkhoff's theorem) that any spherically symmetric solution of the vacuum field equations is necessarily isometric to a subset of the maximally extended Schwarzschild solution.
This means that the exterior region around a spherically symmetric gravitating object must be static and asymptotically flat.