Fuzzballs vs. black holes Now I am trying understand:
What are fuzzballs?
What is the difference between fuzzballs and black holes?
According to this presentation, one can construct fuzzball solution from ordinary BH solution in few steps:

*

*Start with standart metric, for example Schwarzschild metric:

$$
ds^2 = - \left(1-\frac{r_0}{r}\right) dt^2 + \frac{dr^2}{1-\frac{r_0}{r}} + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)
$$


*Make the analytic continuation $t \to -i \tau$
$$
ds^2 =  \left(1-\frac{r_0}{r}\right) d\tau^2 + \frac{dr^2}{1-\frac{r_0}{r}} + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)
$$


*Let the direction be a circle $0\le \tau \le 4\pi r_0$ and add time:

$$
ds^2 =  - dt^2 + \left(1-\frac{r_0}{r}\right) d\tau^2 + \frac{dr^2}{1-\frac{r_0}{r}} + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)
$$


*Dimencionally reduce on the circle $\tau$, and obtain 3+1 dimensional metric in $(t,r,\theta, \phi)$ coupled to  scalar field $\Phi$:

$$
g_{\tau\tau} = e^{\frac{2}{\sqrt{3}} \Phi}
\;\;\;\;\;
\Rightarrow
\;\;\;\;\;
\Phi = \frac{\sqrt{3}}{2} \ln (1-\frac{r_0}{r})
$$
$$
g^E_{\mu\nu} = e^{\frac{1}{\sqrt{3}}\Phi} g_{\mu\nu}
$$
$$
ds^2_E =  - \left(1-\frac{r_0}{r}\right)^{1/2} dt^2 + \frac{dr^2}{(1-\frac{r_0}{r})^{1/2}} + r^2 \left(1-\frac{r_0}{r}\right)^{1/2}(d\theta^2 + \sin^2 \theta d\phi^2)
$$
This is solution of Einstein equations with scalar field as source:
$$
T_{\mu\nu} = \partial_\mu \Phi \partial_\nu \Phi - \frac{1}{2} g^E_{\mu\nu} \partial^\mu \Phi \partial^\nu \Phi
$$
$$
T^\mu_{\;\nu} = diag (-\rho, p_r, p_\theta , p_\phi) =  diag (-f, f, -f , -f)
$$
$$
f = \frac{3r_0^2}{8r^4(1-\frac{r_0}{r})^{3/2}}
$$
This construction looks very unnatural. Could someone explain me motivation of such strange construction?
Is the fuzzball simply solution of Einstein eqyuation with matter or more complex object?
 A: Ramiro Hum-Sah comments answer well your question, but I will elaborate a bit more.
Fuzzballs are solutions of the equation of motion in string theory, that is specific configurations of strings and branes. Working in full non-perturbative string theory or M-theory is usually not feasible (yet) and anyway we may be interested in the 4D low energy description. That is why Fuzzballs are usually presented as solution of Supergravity (SUGRA), that is the low energy limit of string theory. Dimensionally reduced in 4D SUGRA appears as a gravity solution with additional vector and scalar fields, similarly to the example you mention.
The concrete difference between a fuzzball and a black hole in 4D is that while the black hole has no hairs, the fuzzballs have a non trivial multipole expansion, violating the no-hair theorem (and solving the information paradox). A fuzzball is expected to behave more like a star than a hole, meaning that it can exchange matter and information with the outside, like any other object.
