Suppose we have a cylinder of mass $m$, radius $R$ and height $h$ in rotation with speed $\omega$ around its symmetry axis with no friction (ideal situation).
I'd expect this cylinder to emit gravitational waves, is this the case?
If the cylinder is emitting waves (thus energy) should it stop rotating after some time?
There is no gravitational waves for a uniformly rotating axially symmetric body, because the metric doesn't depend on time. First of all, let me cite Landau, Lifshitz, The classical theory of fields, ยง88 The constant gravitational field:
However, for the field produced by a body to be a constant, it is not necessary for the body to be at rest. Thus the field of an axially symmetric body rotating uniformly about its axis still also be constant. However in this case the time directions are no longer equivalent by any means -- if the sign of time is changed, the sign of the angular velocity is changed. Therefore in such constant gravitational fields (we shell call them stationary fields) the components $g_{0i}$ of the metric tensor are in general different from zero.
The reason is very simple. For an axially symmetric body, the distribution of mass in the lab frame coincides with that in the rotating system, thus the solution of Einstein equation can be found in the rotating system where the body and metric are static and then in the lab frame by means of $r'=r$, $z' = z$, $\phi' = \phi + \Omega t$ coordinate transformation ($r$, $\phi$, $z$ are cylindrical coordinates). Therefore all derivatives $\partial x_{\alpha}/\partial x_{\beta}$ do not depend on time. Hence the metric of a uniformly rotating axially symmetric body is time-independent.
For example, the component $g_{0i}$ outside of the slow rotating body ($M\ll c m r_{g}$) has the form:
$$ g_{0i}=-\frac{2G}{c^{3}}\,M_{ij} \frac{n_{j}}{r^{2}}, $$
where $M_{ij}$ is the total angular momentum antisymmetric tensor.
UPD. Concerning the answer of David Bar Moshe. The conclusions presented above are valid only for axial symmetric body. The total power radiated by an non axial symmetric body is proportional to the third power of difference of inertia tensor eigen values for axes transverse to the rotation axis, i.e., $(I_{1}-I_{2})^{3}$.
The first-order approximation to the radiation power from the quadrupole term is given by
$$ P = - \frac{128}{5 c^5} G M^2 R^4 \Omega^6 $$
where $\Omega$ is the angular speed and $M$ and $R$ are the mass and radial separation of a binary system of masses. See this summary for a detailed derivation
Rotating bodies emit gravitational waves. The emission is of quadrupole radiation. The radiation power (which is proportional to the fifth power of the angular speed) causes gradual reduction of the angular speed. Please see the following review by: Alessandra Buonanno equations 7.14, 7.17.
