Rigid rotation in GR and a generalization of the virial theorem for gravitomagnetism

Abstract

In this work we study the properties of rigidly rotating neutral dust solutions in general relativity. This class of solutions gained relevance recently due to applications to the dynamics of spiral galaxies. We show that this class could be interpreted as a “rigid body” in general relativity and we analyze the different properties respect to the rigidly rotating disk in special relativity: for example, the general relativistic counterpart shows no Doppler effect for a light signal emitted and received from any two points at rest respect to the “rigid body”. This effect can be important to test the validity of the assumed model for our galaxy. In the second part we approach the problem from a low energy expansion perspective and we write down a generalization of the virial theorem for stationary spacetimes. The non-Newtonian contributions can lead to a re-weighting of dark matter in galaxies.

Can general relativity play a role in galactic dynamics?

“…The solution to the Einstein's equations is unique once that we have fixed these boundary conditions. In some sense, the grav-itational field is "carrying" energy and momentum on its own, and this is also confirmed by the generalization of the virial theorem found in Astesiano [28]:…”

“…4 In these works Ω 0 = 0, which is equivalent to a rigid rotation of the reference frame and ψ(r, z) := N (r, z) = v(r, z) r in their notation. The variable v(r, z), as discussed by Astesiano [28], is the velocity respect to the zero angular momentum observers (ZAMO) and, for this velocity, v/r is not constant.…”

Stationary rotating and axially symmetric dust systems as peculiar General Relativistic objects

“…This is not surprising: in fact, a particular case of this class is represented by the Balasin and Grumiller solution [23], which describes a rigidly rotating (i.e. Ω = const) dust [24]. A recent analysis by Costa et al [25] shows that this solution contains singularities along the axis, namely a pair of NUT rods and a cosmic string; we remember that a Newman-Unti-Tamburino (NUT) spacetime is a solution of Einstein's equations that generlises the Schwarzschild solution since, in addition to the mass parameter, it contains a second parameter, the so-called NUT charge, that can be interpreted as gravitomagnetic monopole (see e.g.…”

A tale of analogies: a review on gravitomagnetic effects, rotating sources, observers and all that

“…The off-diagonal terms responsible for the gravitomagnetic effects contribute to the modified virial theorem as shown in [226]. They can also give rise to a consistent definition of a 'gravitational mass' in this very specific set up [227].…”