Undergrad GR: Attractors & Liouville's Theorem

[atyy]

TL;DR

Does GR have attractors?

In classical Hamiltonian mechanics, because of Liouville's theorem about the volume of phase space being preserved by time evolution, there are no attractors.

Naively, I think of the Raychaudhuri equation in GR as showing a shrinking volume. However, I guess Raychaudhri's equation does not deal with the phase space of GR.

Does GR have attractors?
Is there something like the Liouville theorem in GR, especially since GR has a Hamiltonian formulation?

 

[martinbn]

Just a guess, but given that very little is known about the global properties of solutions to the initial value problem, I would say that it is a very hard question.

 

[Demystifier]

Is there something like the Liouville theorem in GR, especially since GR has a Hamiltonian formulation?

Every Hamiltonian system obeys the Liouville theorem. GR is not an exception. Note, however, that the phase space in field theories is infinite dimensional. In particular, the 3-dimensional volume of a lump of matter (which may shrink due to gravitational collapse) has nothing to do with the infinite-dimensional phase-space volume.

 

[Demystifier]

Naively, I think of the Raychaudhuri equation in GR as showing a shrinking volume.

GR is invariant under the time reversal. Raychaudhuri equation can describe also the expansion. In the black-hole context, such an expanding solution is called a white hole. Such solutions are usually discarded because they require unrealistic initial conditions, or equivalently, because for such solutions entropy decreases with time.

 

[martinbn]

Every Hamiltonian system obeys the Liouville theorem. GR is not an exception. Note, however, that the phase space in field theories is infinite dimensional. In particular, the 3-dimensional volume of a lump of matter (which may shrink due to gravitational collapse) has nothing to do with the infinite-dimensional phase-space volume.

Are you sure the theorem holds in field theories?

 

[Demystifier]

Are you sure the theorem holds in field theories?

Formally yes, but I guess it's not fully rigorous due to the infinite number of degrees of freedom.