Scalar field solution

Lagrangian density for scalar field is taken to be

$$\mathcal{L}_M = \frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$$

and the equations to be solved are

As usual we simplify them a little bit by looking for a static spherically symmetrical solution. And the solution for this case turns out to be

Here it is $M$ and $q$ which should be interpreted as (total) mass and scalar charge of some matter configuration. The solution goes onto Schwarzschild upon setting $\alpha = 1$ (which corresponds to $q = 0$). But it turns out that this transition is not continuous, which is very important.

As you can see, this solution looks pretty simple, though its derivation is more complicated than in Reissner-Nordstrom case. But it took me less effort to derive it by solving the equations, than to find afterwards that it has already been discovered (by different people; at first, probably, by H. Buchdahl [8] in 1959; see also [6,7,5] and [9]). Not a single book (as far as I know) discusses it.

Discussion of any solution in GR starts with adopting some coordinate system (amongst infinite number of possible ones) which has the clearest physical interpretation. It turns out that only for static spherically symmetrical solutions there is a consensus on how to do so [1, p.120]. According to it, the metric above already has ``the right'' $t, \theta$and $\phi$. But $r$-coordinate there is not ``the true radial coordinate'', because the area of a ``symmetry orbit'' (i.e. of a two-dimensional surface invariant under rotations) is not $4{\pi}r^2$, but $4{\pi}r^2(1-\frac{2m}{r})^{1-\alpha}$.

So, we should change $r$ by $\bar{r} = r(1-\frac{2m}{r})^{\frac{1-\alpha}{2}}$. This, in principle, puts the metric in the form

$$ds^2 = -A(\bar{r})dt^2+\frac{d\bar{r}^2}{B(\bar{r})}+\bar{r}^2d\Omega^2,$$

but there is no way to write it explicitly, because $r$ and $\bar{r}$ are related via transcendental equation, which cannot be inverted. On the other hand, there is no need in doing this, because we can work with $r$-coordinate and then simply find the corresponding value of $\bar{r}$ when necessary.

One could have thought that this solution has a horizon at $r=2m$. Actually, this is not true, because $B(\bar{r}) = 0$ has no solution (note that this function gets contribution from the derivative $\frac{d\bar{r}}{dr}$). But there is an infinitely redshifted surface, which is where $dtdt$component of the metric vanishes:

$$A(\bar{r}) = 0 \quad \Leftrightarrow \quad r = 2m \quad \Leftrightarrow \quad \bar{r} = 0,$$

which means that it is a point-like surface (it has zero surface area). (In Schwarzschild solution the event horizon and the infinitely redshifted surface coincide, because $A(r)=B(r)=1-\frac{2M}{r}$; but when these metric functions are different one should be careful with what defines what).

This surface is singular, because all curvature invariants for this solution show $\frac{1}{(r-2m)^n}$ behavior (with different positive integer $n$'s). It is analogous to ``central singularity'' in Schwarzschild, but it is not covered by a horizon and can be ``seen'' by outside observers. Thus, this solution describes not a BH, but the so called ``naked singularity''.

What is interesting about this solution?

It goes very close to Schwarzschild for $\bar{r}>2M$ and the better the smaller is the scalar charge $q$! It has an infinitely redshifted surface (which is what one usually takes as a defining feature of a BH), but it is not a BH and does not have that fancy interior region. In fact, it is indistinguishable from Schwarzschild BH for the present day astrophysics and says that what is observed and interpreted as a BH may in reality be very different from it in the strong gravity region.

Why then is it neglected?

Well, because it contains a naked singularity, which people don't like, though no one really knows how harmful they are. And also this solution is proven to be unstable against small perturbations [9], but this does not immediately ``rule it out''. No one proved that being perturbed it goes to Schwarzschild, i.e. that the horizon forms and the BH interior develops in space-time (no one knows how to ``evolve singularities'').